College Algebra, 2015a
College Algebra, 2015a
College Algebra, 2015a
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<strong>College</strong> <strong>Algebra</strong><br />
SENIOR CONTRIBUTING AUTHOR<br />
JAY ABRAMSON, ARIZONA STATE UNIVERSITY
About Our Team<br />
Senior Contributing Author<br />
Jay Abramson has been teaching <strong>College</strong> <strong>Algebra</strong> for 33 years, the last 14 at Arizona State University, where he is a principal<br />
lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid<br />
and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing<br />
author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught<br />
at Texas State Technical <strong>College</strong> and Amarillo <strong>College</strong>. He received Teacher of the Year awards at both institutions.<br />
Contributing Authors<br />
Valeree Falduto, Palm Beach State <strong>College</strong><br />
Rachael Gross, Towson University<br />
David Lippman, Pierce <strong>College</strong><br />
Melonie Rasmussen, Pierce <strong>College</strong><br />
Rick Norwood, East Tennessee State University<br />
Nicholas Belloit, Florida State <strong>College</strong> Jacksonville<br />
Jean-Marie Magnier, Springfield Technical Community <strong>College</strong><br />
Harold Whipple<br />
Christina Fernandez<br />
The following faculty contributed to the development of OpenStax<br />
Precalculus, the text from which this product was updated and derived.<br />
Honorable Mention<br />
Nina Alketa, Cecil <strong>College</strong><br />
Kiran Bhutani, Catholic University of America<br />
Brandie Biddy, Cecil <strong>College</strong><br />
Lisa Blank, Lyme Central School<br />
Bryan Blount, Kentucky Wesleyan <strong>College</strong><br />
Jessica Bolz, The Bryn Mawr School<br />
Sheri Boyd, Rollins <strong>College</strong><br />
Sarah Brewer, Alabama School of Math and Science<br />
Charles Buckley, St. Gregory's University<br />
Kenneth Crane, Texarkana <strong>College</strong><br />
Rachel Cywinski, Alamo <strong>College</strong>s<br />
Nathan Czuba<br />
Srabasti Dutta, Ashford University<br />
Kristy Erickson, Cecil <strong>College</strong><br />
Nicole Fernandez, Georgetown University / Kent State University<br />
David French, Tidewater Community <strong>College</strong><br />
Douglas Furman, SUNY Ulster<br />
Erinn Izzo, Nicaragua Christian Academy<br />
John Jaffe<br />
Jerry Jared, Blue Ridge School<br />
Stan Kopec, Mount Wachusett Community <strong>College</strong><br />
Kathy Kovacs<br />
Sara Lenhart, Christopher Newport University<br />
Joanne Manville, Bunker Hill Community <strong>College</strong><br />
Karla McCavit, Albion <strong>College</strong><br />
Cynthia McGinnis, Northwest Florida State <strong>College</strong><br />
Lana Neal, University of Texas at Austin<br />
Steven Purtee, Valencia <strong>College</strong><br />
Alice Ramos, Bethel <strong>College</strong><br />
Nick Reynolds, Montgomery Community <strong>College</strong><br />
Amanda Ross, A. A. Ross Consulting and Research, LLC<br />
Erica Rutter, Arizona State University<br />
Sutandra Sarkar, Georgia State University<br />
Willy Schild, Wentworth Institute of Technology<br />
Todd Stephen, Cleveland State University<br />
Scott Sykes, University of West Georgia<br />
Linda Tansil, Southeast Missouri State University<br />
John Thomas, <strong>College</strong> of Lake County<br />
Diane Valade, Piedmont Virginia Community <strong>College</strong><br />
Reviewers<br />
Phil Clark, Scottsdale Community <strong>College</strong><br />
Michael Cohen, Hofstra University<br />
Matthew Goodell, SUNY Ulster<br />
Lance Hemlow, Raritan Valley Community <strong>College</strong><br />
Dongrin Kim, Arizona State University<br />
Cynthia Landrigan, Erie Community <strong>College</strong><br />
Wendy Lightheart, Lane Community <strong>College</strong><br />
Carl Penziul, Tompkins-Cortland Community <strong>College</strong><br />
Sandra Nite, Texas A&M University<br />
Eugenia Peterson, Richard J. Daley <strong>College</strong><br />
Rhonda Porter, Albany State University<br />
Michael Price, University of Oregon<br />
William Radulovich, Florida State <strong>College</strong> Jacksonville<br />
Camelia Salajean, City <strong>College</strong>s of Chicago<br />
Katy Shields, Oakland Community <strong>College</strong><br />
Nathan Schrenk, ECPI University<br />
Pablo Suarez, Delaware State University<br />
Allen Wolmer, Atlanta Jewish Academy<br />
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Brief Contents<br />
1 Prerequisites 1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
Equations and Inequalities 73<br />
Functions 159<br />
Linear Functions 279<br />
Polynomial and Rational Functions 343<br />
Exponential and Logarithmic Functions 463<br />
Systems of Equations and Inequalities 575<br />
Analytic Geometry 681<br />
Sequences, Probability and Counting Theory 755<br />
v
Contents<br />
Preface<br />
xi<br />
1 Prerequisites 1<br />
1.1 Real Numbers: <strong>Algebra</strong> Essentials 2<br />
1.2 Exponents and Scientific Notation 17<br />
1.3 Radicals and Rational Expressions 31<br />
1.4 Polynomials 41<br />
1.5 Factoring Polynomials 49<br />
1.6 Rational Expressions 58<br />
Chapter 1 Review 66<br />
Chapter 1 Review Exercises 70<br />
Chapter 1 Practice Test 72<br />
2<br />
3<br />
Equations and Inequalities 73<br />
2.1 The Rectangular Coordinate Systems and Graphs 74<br />
2.2 Linear Equations in One Variable 87<br />
2.3 Models and Applications 102<br />
2.4 Complex Numbers 111<br />
2.5 Quadratic Equations 119<br />
2.6 Other Types of Equations 131<br />
2.7 Linear Inequalities and Absolute Value Inequalities 142<br />
Chapter 2 Review 151<br />
Chapter 2 Review Exercises 155<br />
Chapter 2 Practice Test 158<br />
Functions 159<br />
3.1 Functions and Function Notation 160<br />
3.2 Domain and Range 180<br />
3.3 Rates of Change and Behavior of Graphs 196<br />
3.4 Composition of Functions 209<br />
3.5 Transformation of Functions 222<br />
3.6 Absolute Value Functions 247<br />
3.7 Inverse Functions 254<br />
Chapter 3 Review 267<br />
Chapter 3 Review Exercises 272<br />
Chapter 3 Practice Test 277<br />
vii
4<br />
Linear Functions 279<br />
4.1 Linear Functions 280<br />
4.2 Modeling with Linear Functions 309<br />
4.3 Fitting Linear Models to Data 322<br />
Chapter 4 Review 334<br />
Chapter 4 Review Exercises 336<br />
Chapter 4 Practice Test 340<br />
5<br />
Polynomial and Rational Functions 343<br />
5.1 Quadratic Functions 344<br />
5.2 Power Functions and Polynomial Functions 360<br />
5.3 Graphs of Polynomial Functions 375<br />
5.4 Dividing Polynomials 393<br />
5.5 Zeros of Polynomial Functions 402<br />
5.6 Rational Functions 414<br />
5.7 Inverses and Radical Functions 435<br />
5.8 Modeling Using Variation 446<br />
Chapter 5 Review 453<br />
Chapter 5 Review Exercises 458<br />
Chapter 5 Practice Test 461<br />
6<br />
Exponential and Logarithmic Functions 463<br />
6.1 Exponential Functions 464<br />
6.2 Graphs of Exponential Functions 479<br />
6.3 Logarithmic Functions 491<br />
6.4 Graphs of Logarithmic Functions 499<br />
6.5 Logarithmic Properties 516<br />
6.6 Exponential and Logarithmic Equations 526<br />
6.7 Exponential and Logarithmic Models 537<br />
6.8 Fitting Exponential Models to Data 552<br />
Chapter 6 Review 565<br />
Chapter 6 Review Exercises 570<br />
Chapter 6 Practice Test 573<br />
viii
7<br />
8<br />
9<br />
Systems of Equations and Inequalities 575<br />
7.1 Systems of Linear Equations: Two Variables 576<br />
7.2 Systems of Linear Equations: Three Variables 592<br />
7.3 Systems of Nonlinear Equations and Inequalities: Two Variables 603<br />
7.4 Partial Fractions 613<br />
7.5 Matrices and Matrix Operations 623<br />
7.6 Solving Systems with Gaussian Elimination 634<br />
7.7 Solving Systems with Inverses 647<br />
7.8 Solving Systems with Cramer's Rule 661<br />
Chapter 7 Review 672<br />
Chapter 7 Review Exercises 676<br />
Chapter 7 Practice Test 679<br />
Analytic Geometry 681<br />
8.1 The Ellipse 682<br />
8.2 The Hyperbola 697<br />
8.3 The Parabola 714<br />
8.4 Rotation of Axis 727<br />
8.5 Conic Sections in Polar Coordinates 740<br />
Chapter 8 Review 749<br />
Chapter 8 Review Exercises 752<br />
Chapter 8 Practice Test 754<br />
Sequences, Probability and Counting Theory 755<br />
9.1 Sequences and Their Notations 756<br />
9.2 Arithmetic Sequences 769<br />
9.3 Geometric Sequences 779<br />
9.4 Series and Their Notations 787<br />
9.5 Counting Principles 800<br />
9.6 Binomial Theorem 810<br />
9.7 Probability 817<br />
Chapter 9 Review 826<br />
Chapter 9 Review Exercises 830<br />
Chapter 9 Practice Test 833<br />
Try It Answer Section A-1<br />
Odd Answer Section B-1<br />
Index C-1<br />
ix
Preface<br />
<strong>College</strong> <strong>Algebra</strong>Th<br />
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<strong>College</strong> <strong>Algebra</strong><br />
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xi
About <strong>College</strong> <strong>Algebra</strong><br />
<strong>College</strong> <strong>Algebra</strong><br />
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xii
Pedagogical Foundations and Features<br />
Learning Objectives<br />
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Section Exercises<br />
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Verbal <br />
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Additional Resources<br />
Student and Instructor Resources<br />
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xiv
Prerequisites<br />
1<br />
Figure 1 Credit: Andreas Kambanls<br />
CHAPTER OUTLINE<br />
1.1 Real Numbers: <strong>Algebra</strong> Essentials<br />
1.2 Exponents and Scientific Notation<br />
1.3 Radicals and Rational Expressions<br />
1.4 Polynomials<br />
1.5 Factoring Polynomials<br />
1.6 Rational Expressions<br />
Introduction<br />
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2<br />
CHAPTER 1 PREREQUISITES<br />
LEARNING OBJECTIVES<br />
In this section students will:<br />
Classify a real number as a natural, whole, integer, rational, or irrational number.<br />
Perform calculations using order of operations.<br />
Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.<br />
Evaluate algebraic expressions.<br />
Simplify algebraic expressions.<br />
1.1 REAL NUMBERS: ALGEBRA ESSENTIALS<br />
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SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 3<br />
Example 1<br />
Writing Integers as Rational Numbers<br />
<br />
a. b. c. −<br />
Solution <br />
a. = <br />
b. = <br />
c. −=− <br />
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Try It #1<br />
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a. b. c. − <br />
Example 2<br />
Identifying Rational Numbers<br />
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a. − <br />
b.<br />
c.<br />
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Solution <br />
a. − <br />
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Irrational Numbers<br />
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irrational<br />
Thirrational numbers<br />
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Example 3<br />
Differentiating Rational and Irrational Numbers<br />
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a. √ — b. <br />
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Solution<br />
a. √ — √ — =5. Th√ —
4<br />
CHAPTER 1 PREREQUISITES<br />
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Example 4<br />
Classifying Real Numbers<br />
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Solution<br />
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a. − <br />
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b. √ — <br />
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d. −π <br />
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SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 5<br />
Try It #4<br />
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Sets of Numbers as Subsets<br />
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Example 5<br />
Differentiating the Sets of Numbers<br />
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Solution<br />
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a. √ — = × × × ×<br />
b.<br />
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=_ <br />
×<br />
c. √ — ×<br />
d. − × ×<br />
e. ×
6<br />
CHAPTER 1 PREREQUISITES<br />
Try It #5<br />
NWIQ<br />
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a. − <br />
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b. c. √ — d. √ — e. <br />
Performing Calculations Using the Order of Operations<br />
=∙ =<br />
exponential notationa n a<br />
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a n naabasenexponent<br />
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SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 7<br />
How To…<br />
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Example 6<br />
Using the Order of Operations<br />
<br />
a. (3 ∙ 2) −+ b. − <br />
− √— − c. −∣−∣+−<br />
−3 ∙ 2<br />
d.<br />
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2 ∙ 5− e. 7(5 ∙ 3)−−− +<br />
Solution<br />
a. 3 ∙ 2 −+= − <br />
=−<br />
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=−<br />
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=<br />
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b. − <br />
− √— −= − − √— <br />
= −<br />
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c. −−+−=−−+ <br />
=−+<br />
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=−+<br />
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=+<br />
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=<br />
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d. −3 ∙ 2 <br />
2 ∙ 5− =−3 ∙ 2 <br />
2 ∙ 5− <br />
= − <br />
−<br />
= <br />
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=<br />
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e 5 ∙ 3−−− +=−− + <br />
=−−+ <br />
=−−+ <br />
=++<br />
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=
8<br />
CHAPTER 1 PREREQUISITES<br />
Try It #6<br />
<br />
a. √ — − +− b. + <br />
7 ∙ 5−8 ∙ 4<br />
− <br />
c. −+√— +<br />
<br />
d. · − + · <br />
e. − −−−<br />
Using Properties of Real Numbers<br />
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Commutative Properties<br />
commutative property of additionff<br />
a+b =b +a<br />
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−+= +−=<br />
commutative property of multiplication<br />
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a ∙ b = b ∙ a<br />
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−∙ (−= −∙ (−=<br />
−<br />
−÷≠÷<br />
Associative Properties<br />
associative property of multiplication<br />
<br />
a(bc) = (ab)c<br />
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associative property of additionffff<br />
a + (b + c) = (a + b) + c<br />
Th<br />
+−+= +−+=<br />
<br />
−−−−<br />
÷÷÷÷<br />
−−−<br />
÷÷<br />
≠−<br />
≠<br />
<br />
Distributive Property<br />
distributive property<br />
<br />
a ∙ b + c = a ∙ b + a ∙ c
SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 9<br />
Th<br />
<br />
∙+−=4 ∙+4 ∙ (−<br />
<br />
=+−<br />
<br />
=<br />
−<br />
<br />
Th<br />
<br />
+3 ∙ 5+) ∙ (+<br />
+∙<br />
≠<br />
<br />
a − b = a + −b<br />
ff−+ff+<br />
+<br />
+−∙+<br />
−<br />
−+=+−∙+<br />
=+−∙ 5+−∙<br />
=+−<br />
=<br />
Th<br />
<br />
<br />
−+=+−−<br />
=+−<br />
=<br />
Identity Properties<br />
Thidentity property of addition<br />
<br />
a + = a<br />
Thidentity property of multiplication<br />
<br />
a ∙ = a<br />
−+=−∙=Th<br />
<br />
Inverse Properties<br />
inverse property of additiona<br />
−a<br />
a + −a = <br />
a =−−+=<br />
inverse property of multiplicationfi<br />
Tha<br />
a <br />
a ∙ a =<br />
a =− <br />
<br />
a −
10<br />
CHAPTER 1 PREREQUISITES<br />
a ∙ a = ( − <br />
properties of real numbers <br />
Thabc<br />
) ∙ ( − <br />
) =<br />
Addition<br />
Multiplication<br />
Commutative Property a + b = b + a a ∙ b = b ∙ a<br />
Associative Property a+b+c=a+b+c abc=abc<br />
Distributive Property<br />
a ∙ (b+c=a ∙ b + a ∙ c<br />
Identity Property<br />
Inverse Property<br />
Example 7<br />
Using Properties of Real Numbers<br />
Th<br />
<br />
a<br />
a + = a<br />
<br />
−a<br />
<br />
a+−a=<br />
Th<br />
<br />
a<br />
a ∙ = a<br />
a<br />
<br />
a <br />
a ∙ ( a )=<br />
<br />
a. ∙ 6+3 ∙ 4 b. ++− c. −+ d.<br />
<br />
∙ ( ∙ ) e. ∙ [0.75+−<br />
Solution<br />
a. ∙ 6+3 ∙ 4=3 ∙ (6+ <br />
= 3 ∙ 10 <br />
=<br />
<br />
b. ++−=++− <br />
=+<br />
<br />
=<br />
<br />
c. −+=+−+− <br />
=+−<br />
<br />
=−<br />
<br />
d.<br />
__ ∙ __<br />
( ∙ __ ) = __ ∙ __<br />
( ∙ __ )<br />
= __<br />
( ∙ __ ) ∙ __ <br />
= 1 ∙ __ <br />
<br />
<br />
<br />
= __ <br />
<br />
e. 100 ∙ [0.75+−=100 ∙ 0.75+100 ∙ (− <br />
=+−<br />
<br />
=−<br />
<br />
Try It #7<br />
<br />
a. ( − <br />
) ·[ · ( − ) ] b. ·+ c. −− d. <br />
+ [ <br />
+ ( − <br />
) ]<br />
e. ċ−+ċ
SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 11<br />
Evaluating <strong>Algebra</strong>ic Expressions<br />
<br />
x + <br />
πr √ — m n x +constant<br />
xvariable<br />
algebraic expression<br />
<br />
<br />
<br />
− =−) ∙ (−) ∙ (−) ∙ (−) ∙ (−<br />
x =x ∙ x ∙ x ∙ x ∙ x<br />
2 ∙ 7 =2 ∙ 7) ∙ (2 ∙ 7) ∙ (2 ∙ 7<br />
yz =yz) ∙ (yz) ∙ (yz<br />
<br />
<br />
ff<br />
<br />
<br />
<br />
<br />
Example 8<br />
Describing <strong>Algebra</strong>ic Expressions<br />
<br />
Solution<br />
a. x + b.<br />
πr c.<br />
<br />
√ — m n <br />
Constants<br />
Variables<br />
a. x + x<br />
b.<br />
πr <br />
<br />
<br />
π<br />
r<br />
c. √ — m n m, n<br />
Try It #8<br />
<br />
a. πrr+h b. L+W c. y +y<br />
Example 9<br />
Evaluating an <strong>Algebra</strong>ic Expression at Different Values<br />
x −x<br />
a. x = b. x = c. x = <br />
<br />
d. x =−<br />
Solution<br />
a. x x −=−<br />
=−<br />
=−<br />
b. x x −=−<br />
=−<br />
=−
12<br />
CHAPTER 1 PREREQUISITES<br />
Try It #9<br />
c. <br />
x<br />
x −= ( ) −<br />
=−<br />
=−<br />
d. −x x −=−−<br />
=−−<br />
=−<br />
−yy<br />
a. y = b. y = c. y = <br />
<br />
d. y =−<br />
Example 10<br />
Evaluating <strong>Algebra</strong>ic Expressions<br />
<br />
a. x +x =− b.<br />
t t = c.<br />
t−<br />
d. a +ab+ba =b =− e. √ — m n m =n =<br />
Solution<br />
a. −x x +=−+<br />
=<br />
b. t <br />
t <br />
t − = <br />
− <br />
<br />
=<br />
<br />
− <br />
= <br />
<br />
πr r =<br />
c. r<br />
<br />
πr = <br />
π<br />
= <br />
π<br />
= <br />
π<br />
d. a−b a + ab+ b =+−+−<br />
=−−<br />
=−<br />
e. mn √ — m n =√ — <br />
= √ — <br />
=√ — <br />
=<br />
Try It #10<br />
<br />
y + a. _ <br />
y − y = b. −tt =− c. <br />
πr r =<br />
d. p q p =−q = e. m−n−n−mm = <br />
n =
SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 13<br />
Formulas<br />
equationTh<br />
ThTh<br />
<br />
x +=x =x<br />
+=<br />
formulaft<br />
fift<br />
fiAr<br />
A =πr rAπr <br />
Example 11<br />
Using a Formula<br />
rhS <br />
S =πrr+hFigure 4<br />
π<br />
r<br />
h<br />
Figure Right circular cylinder<br />
Solution πrr + hr =h =<br />
S =πr+<br />
=π+<br />
=π<br />
=π<br />
Thπ<br />
Try It #11<br />
LWTh<br />
A =L+W+−L ∙ Figure 5<br />
<br />
<br />
<br />
<br />
<br />
Figure
14<br />
CHAPTER 1 PREREQUISITES<br />
Simplifying <strong>Algebra</strong>ic Expressions<br />
<br />
<br />
<br />
Example 12<br />
Simplifying <strong>Algebra</strong>ic Expressions<br />
<br />
a. x −y +x −y − b. r −−r+ c. ( t − <br />
) −( <br />
t +s ) <br />
d. mn−m +mn+n<br />
Solution<br />
a. x −y +x −y −=x +x −y −y − <br />
=x −y − <br />
b. r −−r+=r −+r + <br />
=r +r −+ <br />
=r − <br />
<br />
c. t −( t − <br />
) −( <br />
t + ) =t − <br />
s − t −s<br />
<br />
=t − <br />
t − s −s<br />
<br />
<br />
= <br />
t − <br />
s<br />
<br />
d. mn−m +mn+n =mn+mn−m +n <br />
=mn−m +n <br />
Try It #12<br />
<br />
a. <br />
y − ( <br />
y +z ) b. <br />
t −− + c. pq−+q−p d. r −s+r+−s<br />
t<br />
Example 13<br />
Simplifying a Formula<br />
LWPP =L +W +L +W<br />
Solution<br />
<br />
P =L +W +L +W<br />
P =L +L +W +W <br />
P =L +W <br />
P =L+W <br />
Try It #13<br />
PrtA<br />
A =P +Prt <br />
<br />
Simplify an Expression (http://openstaxcollege.org/l/simexpress)<br />
Evaluate an Expression1 (http://openstaxcollege.org/l/ordofoper1)<br />
Evaluate an Expression2 (http://openstaxcollege.org/l/ordofoper2)
SECTION 1.1 SECTION EXERCISES<br />
15<br />
1.1 SECTION EXERCISES<br />
VERBAL<br />
1. √ — <br />
t fi<br />
<br />
2. <br />
<br />
<br />
3. <br />
<br />
<br />
NUMERIC<br />
<br />
4. +·− 5. ÷−÷ 6. +− 7. −·÷− <br />
8. −+· 9. − 10. +−÷ 11. ÷÷+<br />
12. + ÷ 13. −·+ 14. +·÷ 15. ++−<br />
16. −÷ 17. ·÷− 18. −+· 19. +·−<br />
20. ÷+· 21. + 22. ÷· 23. ÷ −<br />
24. −·− 25. ·−− 26. −· <br />
<br />
27. −÷<br />
ALGEBRAIC<br />
<br />
28. x += 29. y +=y 30. a +−a =− 31. z −z+=<br />
32. y− =− 33. −x +=− 34. +−b =b 35. c −=<br />
36. −x = 37. <br />
w − =<br />
<br />
38. x +x− 39. y − y − 40. a <br />
−a ÷<br />
41. b −b+<br />
42. l ÷l ·− 43. z −+z · 44. ·+x ÷− 45. y +−<br />
46. ( <br />
t − ) 47. +b −·b 48. y −+y 49. ( <br />
) ·x<br />
50. −m+− 51. x +x+−x +x 52. −x
16<br />
CHAPTER 1 PREREQUISITES<br />
REAL-WORLD APPLICATIONS<br />
<br />
<br />
53. <br />
<br />
<br />
54. <br />
<br />
55. <br />
es. Th<br />
π<br />
<br />
<br />
56. <br />
<br />
<br />
<br />
<br />
<br />
ThTh<br />
<br />
<br />
57. 58. g<br />
<br />
59. <br />
<br />
<br />
−x =<br />
x <br />
TECHNOLOGY<br />
x<br />
60. −x = <br />
<br />
61. − x −=<br />
EXTENSIONS<br />
62. <br />
<br />
64. <br />
Th<br />
<br />
66. <br />
√ — −++<br />
63. <br />
Th<br />
<br />
65. <br />
√ — −−−<br />
67. Th<br />
<br />
68. <br />
+x −
SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 17<br />
LEARNING OBJECTIVES<br />
In this section students will:<br />
Use the product rule of exponents.<br />
Use the quotient rule of exponents.<br />
Use the power rule of exponents.<br />
Use the zero exponent rule of exponents.<br />
Use the negative rule of exponents.<br />
Find the power of a product and a quotient.<br />
Simplify exponential expressions.<br />
Use scientific notation.<br />
1.2 EXPONENTS AND SCIENTIFIC NOTATION<br />
<br />
fi<br />
<br />
<br />
Th<br />
o fil fi<br />
····ENTERTh1.304596316E13<br />
ThE13<br />
· fifi<br />
<br />
Using the Product Rule of Exponents<br />
x ∙ x xff<br />
<br />
<br />
<br />
<br />
x ·x =x· x · x · x · x · x · x<br />
<br />
<br />
<br />
= x · x · x · x · x · x · x<br />
<br />
=x <br />
Thx ∙ x =x + =x <br />
<br />
Th<br />
product rule of exponents<br />
<br />
a m ·a n =a m+ n<br />
<br />
<br />
· = + = <br />
fi <br />
Th∙ <br />
ff<br />
the product rule of exponents<br />
amn<br />
<br />
a m ·a n =a m+n
18<br />
CHAPTER 1 PREREQUISITES<br />
Example 1<br />
Using the Product Rule<br />
<br />
a. t ∙ t b. − ∙ (− c. x ∙ x ∙ x <br />
Solution <br />
a. t ∙ t =t + =t <br />
b. − ∙ (−=− ∙ (− =− + =− <br />
c. x ∙ x ∙ x <br />
fi<br />
e fi<br />
<br />
x ·x ·x =x ·x ·x =x + ·x =x ·x =x + =x <br />
<br />
Try It #1<br />
<br />
x ·x ·x =x ++ =x <br />
<br />
a. k ·k b. ( _ <br />
y ) ·( _ y ) c.t ·t ·t <br />
Using the Quotient Rule of Exponents<br />
quotient rule of exponents<br />
ff ym<br />
y n m >n<br />
y _ <br />
y <br />
<br />
y y · y · y · y · y · y · y ·y<br />
<br />
y=y· y· y · y · y <br />
·y<br />
<br />
<br />
<br />
<br />
= y· y· y· y· y· y · y · y ·y<br />
<br />
y· y· y· y· y <br />
<br />
y· y · y ·y<br />
= <br />
=y <br />
ff<br />
<br />
am<br />
<br />
a n=am−n<br />
<br />
<br />
<br />
y =y <br />
y =y−<br />
m >n m −n<br />
Th<br />
<br />
the quotient rule of exponents<br />
amnm > n<br />
<br />
am<br />
<br />
a n=am−n<br />
Example 2<br />
Using the Quotient Rule<br />
<br />
a. − <br />
− b. t <br />
t c. ( z√ — ) <br />
z√ —
SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 19<br />
Solution <br />
a. − =− <br />
− =−−<br />
b. t <br />
t =t − =t <br />
(<br />
c. z√ — ) <br />
z√ — = ( z√ — ) − =( z√ — ) <br />
Try It #2<br />
<br />
a. s <br />
s b. − <br />
− c. ef _ <br />
ef <br />
Using the Power Rule of Exponents<br />
<br />
x Th<br />
. Th<br />
c. − =− ∙ =− <br />
<br />
x =x ·x · x <br />
<br />
<br />
=( x·x )·( x·x )·( x·x )<br />
<br />
= x · x · x · x · x · x<br />
<br />
=x <br />
Thx =x ∙ =x <br />
<br />
<br />
a m n =a m∙ n<br />
ff<br />
<br />
<br />
Product Rule<br />
Power Rule<br />
∙ 5 = + = = 3 ∙ 4 = <br />
x ∙ x = x + = x x = x 5 ∙ 2 = x <br />
a ∙a = a + = a a = a 7 ∙ 1 = a <br />
the power rule of exponents<br />
amn<br />
<br />
a m n =a m∙n<br />
Example 3 Using the Power Rule<br />
<br />
a. x b. t c. − <br />
Solution <br />
a. x =x ∙ =x <br />
b. t =t ∙ =t
20<br />
CHAPTER 1 PREREQUISITES<br />
Try It #3<br />
<br />
a. y b. t c. −g <br />
Using the Zero Exponent Rule of Exponents<br />
m >nffm −n <br />
m =n<br />
<br />
<br />
t t<br />
<br />
t = t =<br />
<br />
<br />
t =t <br />
t<br />
=t− =Th<br />
<br />
<br />
a =<br />
Th Th<br />
fi<br />
the zero exponent rule of exponents<br />
a<br />
<br />
a =<br />
Example 4<br />
Using the Zero Exponent Rule<br />
<br />
a. c <br />
c b. j k <br />
−x x <br />
c. <br />
_ <br />
j k∙j k d. <br />
r _ <br />
r <br />
Solution <br />
a. c <br />
<br />
c =c−<br />
=c <br />
<br />
<br />
<br />
<br />
b. −x <br />
x =−3 ∙ x <br />
x <br />
=−∙x −<br />
=−∙x <br />
=−∙<br />
=−<br />
j k <br />
c. <br />
_ <br />
j k <br />
j k∙j k = <br />
j k +<br />
_ <br />
<br />
<br />
= j k <br />
j k <br />
=j k − <br />
=j k <br />
<br />
=<br />
d. r − <br />
r =r<br />
=r <br />
= 5 ∙ 1 <br />
=
SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 21<br />
Try It #4<br />
<br />
a. t <br />
t b. <br />
de _ <br />
de c. ·w <br />
w w d. t ·t <br />
t ·t <br />
Using the Negative Rule of Exponents<br />
m >n<br />
h <br />
hm
22<br />
CHAPTER 1 PREREQUISITES<br />
Try It #5<br />
<br />
a. −t <br />
−t b. f <br />
f ⋅f c. k <br />
k <br />
Example 6<br />
Using the Product and Quotient Rules<br />
<br />
a. b ∙ b − b. −x ∙ (−x − c. −z <br />
−z <br />
Solution<br />
a. b ∙ b − =b − =b − = <br />
b <br />
b. −x ∙ (−x − =−x − =−x =<br />
c. −z <br />
−z = −z =−z − = <br />
−z =−z− −z <br />
Try It #6<br />
<br />
a. t − ⋅t b. <br />
<br />
Finding the Power of a Product<br />
power of a product rule of exponents<br />
<br />
pq <br />
<br />
<br />
pq =pq·pq·pq<br />
<br />
= p · q · p · q · p · q<br />
<br />
<br />
=p· p · p · q · q ·q<br />
<br />
=p ·q <br />
(pq =p ·q <br />
the power of a product rule of exponents<br />
a n<br />
<br />
ab n =a n b n<br />
Example 7 Using the Power of a Product Rule<br />
<br />
<br />
<br />
a. ab b. t c. −w d. <br />
−z e. e− f <br />
Solution fi<br />
a. ab =a ∙ (b =a 1 ∙ 3 ∙ b 2 ∙ 3 =a b <br />
b. t = ∙ (t = t =t <br />
c. −w =− ∙ (w =−8 ∙ w 3 ∙ 3 =−w <br />
<br />
d. <br />
−z =<br />
<br />
− ∙ (z =<br />
<br />
z <br />
e. e − f =e − ∙ (f =e −2 ∙ 7 ∙ f 2 ∙ 7 =e − f = f <br />
e
SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 23<br />
Try It #7<br />
<br />
<br />
<br />
a. g h b. t c. −y d. <br />
a b e. r s − <br />
Finding the Power of a Quotient<br />
<br />
<br />
<br />
<br />
e − f = f <br />
e <br />
ff<br />
<br />
e − f =( f <br />
e )<br />
<br />
= f <br />
e <br />
<br />
<br />
e − f =( f <br />
e )<br />
<br />
<br />
<br />
<br />
<br />
= f <br />
e <br />
f·<br />
= <br />
e ·<br />
= f <br />
e <br />
the power of a quotient rule of exponents<br />
abn<br />
<br />
( a b ) n = an<br />
<br />
b n<br />
Example 8<br />
Using the Power of a Quotient Rule<br />
<br />
<br />
a. ( <br />
z ) b. (<br />
p <br />
q )<br />
c. ( − <br />
Solution<br />
a. ( <br />
z ) <br />
= <br />
z = <br />
z ∙= <br />
z <br />
b. (<br />
p <br />
q )<br />
= p <br />
<br />
<br />
q =p∙ q ∙=p q <br />
c. ( − <br />
t ) = <br />
− <br />
t =− <br />
t ∙=− <br />
t =− <br />
t <br />
t ) d. j k − e. m − n − <br />
d. j k − =( j <br />
k )= j <br />
<br />
k =j∙ k ∙=j <br />
k <br />
<br />
e. m − n − =( <br />
m n <br />
) =( <br />
m n ) = <br />
m n =<br />
<br />
m ∙ ∙ n ∙=<br />
<br />
m n
24<br />
CHAPTER 1 PREREQUISITES<br />
Try It #8<br />
<br />
<br />
<br />
c )<br />
a. ( b <br />
b. ( <br />
u ) c. ( − <br />
Simplifying Exponential Expressions<br />
w ) d. p − q e. c − d − <br />
<br />
. Th<br />
Example 9<br />
Simplifying Exponential Expressions<br />
<br />
a. m n − b. ∙ 17 − ∙ 17 − c. ( u− v<br />
<br />
v ) − d. −a b − a − b <br />
e. ( x √ — <br />
) ( x √ — −<br />
) <br />
f. w <br />
w − <br />
Solution<br />
a. m n − = m n − <br />
= m ∙ n −∙ <br />
=m n − <br />
= m n <br />
b. ∙ 17 − ∙ 17 − = −− Th<br />
= − <br />
<br />
= <br />
<br />
<br />
Th<br />
c. ( u− v <br />
<br />
v<br />
) − = u− v <br />
v − <br />
Th<br />
<br />
= u− v v − <br />
Th<br />
=u − v −− Th<br />
=u − v <br />
<br />
= v <br />
u <br />
Th<br />
d. −a b − a − b =−2 ∙ 5 ∙ a ∙ a − ∙ b − ∙ b <br />
=−0 ∙ a − ∙ b −+ Th<br />
=−ab <br />
e. ( x √ — ) ( x √ — ) − =( x √ — ) − Th<br />
=( x √ — ) <br />
= Th<br />
<br />
<br />
<br />
<br />
f. w ∙ (w <br />
w − = ∙ (w − <br />
<br />
= w ∙ <br />
w −∙<br />
= w <br />
w −<br />
Th<br />
Th<br />
<br />
= w −− Th<br />
= w
SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 25<br />
Try It #9<br />
<br />
a. uv − − b. x ·x − ·x c. ( e f − <br />
Using Scientific Notation<br />
<br />
f − ) <br />
d. r − s r s − e. ( <br />
<br />
) tw− − ( <br />
<br />
) tw− f. h k <br />
<br />
h − k <br />
× <br />
<br />
ff<br />
<br />
scientific notation<br />
fi<br />
fi<br />
n<br />
nn<br />
fi<br />
<br />
ft<br />
<br />
ftTh<br />
ft. Th<br />
× <br />
<br />
<br />
<br />
ThTh<br />
<br />
× −<br />
scientific notation<br />
fia × n ≤a
26<br />
CHAPTER 1 PREREQUISITES<br />
c. <br />
←<br />
× <br />
d. <br />
→<br />
× − m<br />
e. <br />
→<br />
× −<br />
Analysis Observe that, if the given number is greater than , as in examples a–c, the exponent of is positive; and if<br />
the number is less than , as in examples d–e, the exponent is negative.<br />
Try It #10<br />
fi<br />
a. <br />
b. <br />
c. <br />
d. <br />
e. <br />
Converting from Scientific to Standard Notation<br />
scientific notationn<br />
nn nn<br />
n<br />
Example 11<br />
Converting Scientific Notation to Standard Notation<br />
fi<br />
Solution<br />
a. × b. −× c × − d. −× −<br />
a. × <br />
<br />
→<br />
<br />
b. −× <br />
−<br />
→<br />
<br />
c. × −<br />
<br />
←<br />
<br />
d. −× −<br />
−<br />
←
SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 27<br />
Try It #11<br />
fi<br />
a. × b. −× c. −× − d. × −<br />
Using Scientific Notation in Applications<br />
fi<br />
1 <br />
Th<br />
× 1 × Th<br />
×× ×× ≈× <br />
fi<br />
fifi<br />
× ×× =× Thfi<br />
×. Th<br />
× =×× =×× =× <br />
Example 12<br />
Using Scientific Notation<br />
fi<br />
a. × − × <br />
b. × ÷−× <br />
c. × × <br />
d. × ÷× <br />
e. × −× × <br />
Solution<br />
a. × − × =× − × <br />
<br />
= <br />
=× fi <br />
b. × ÷−× =( <br />
− ) ( <br />
) <br />
<br />
<br />
≈− − <br />
=−× − fi <br />
c. × × =× × <br />
<br />
= <br />
=× fi <br />
d. × ÷× = ( ) ( <br />
) <br />
<br />
<br />
= <br />
=× fi <br />
e. × −× × =×−× × × <br />
<br />
≈− <br />
<br />
=−×
28<br />
CHAPTER 1 PREREQUISITES<br />
Try It #12<br />
fi<br />
a. −× × − <br />
b. × ÷× <br />
c. × × <br />
d. × ÷−× <br />
e. −× × −× <br />
Example 13<br />
Applying Scientific Notation to Solve Problems<br />
Th<br />
fififi<br />
fi<br />
Solution Th=× <br />
Th≈× <br />
o fi<br />
<br />
<br />
<br />
× ÷× = ( <br />
<br />
) <br />
) × ( <br />
≈× <br />
=× <br />
Th× <br />
Try It #13<br />
<br />
0.000008 <br />
<br />
fi<br />
Exponential Notation (http://openstaxcollege.org/l/exponnot)<br />
Properties of Exponents (http://openstaxcollege.org/l/exponprops)<br />
Zero Exponent (http://openstaxcollege.org/l/zeroexponent)<br />
Simplify Exponent Expressions (http://openstaxcollege.org/l/exponexpres)<br />
Quotient Rule for Exponents (http://openstaxcollege.org/l/quotofexpon)<br />
Scientific Notation (http://openstaxcollege.org/l/scientificnota)<br />
Converting to Decimal Notation (http://openstaxcollege.org/l/decimalnota)
SECTION 1.2 SECTION EXERCISES<br />
29<br />
1.2 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
3. 4. <br />
NUMERIC<br />
<br />
5. 6. − 7. · 8. ÷<br />
9. − 10. − 11. ÷ 12. · −<br />
13. 14. − ÷ <br />
<br />
<br />
15. · ÷ − 16. <br />
17. · 18. ÷ − <br />
19. − · − 20. ÷ <br />
fi<br />
21. 22. <br />
fi<br />
23. × 24. × −<br />
ALGEBRAIC<br />
<br />
25. a a <br />
a 26. mn <br />
m − 27. b c x 28. ( − y ) <br />
−<br />
29. ab ÷d − 30. w x − 31. m <br />
n <br />
32. y− y <br />
33. p− q <br />
p q − 34. l ×w 35. y ÷x 36. ( a <br />
) <br />
37. m ÷ m<br />
(<br />
38. √ — x ) y − 39. <br />
a −<br />
41. b − c 42. x y ÷y 43. z − y<br />
<br />
40. ma <br />
<br />
m a <br />
REAL-WORLD APPLICATIONS<br />
44. <br />
× <br />
<br />
46. Th<br />
<br />
<br />
45. . A<br />
× −3 <br />
<br />
47.
30<br />
CHAPTER 1 PREREQUISITES<br />
48. Th<br />
× <br />
<br />
49. × − <br />
<br />
50. Th<br />
<br />
<br />
TECHNOLOGY<br />
<br />
m<br />
51. ( <br />
− ) <br />
EXTENSIONS<br />
52. ÷ x <br />
<br />
− <br />
53. ( <br />
a ) ( a ) 55. m n <br />
a c −· a− n − <br />
m c <br />
56. (<br />
x y <br />
x y −· y − <br />
x ) −<br />
57. ( ab c − <br />
b − ) <br />
54. − ÷<br />
( x <br />
y ) − <br />
<br />
<br />
58. <br />
<br />
<br />
Th× <br />
<br />
59. <br />
<br />
. Th<br />
× −
SECTION 1.3 RADICALS AND RATIONAL EXPONENTS 31<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Evaluate square roots.<br />
Use the product rule to simplify square roots.<br />
Use the quotient rule to simplify square roots.<br />
Add and subtract square roots.<br />
Rationalize denominators.<br />
Use rational roots.<br />
1.3 RADICALS AND RATIONAL EXPONENTS<br />
<br />
fi<br />
Figure 1n Th<br />
<br />
c<br />
Figure 1<br />
a 2 +b 2 =c 2<br />
+ =c 2<br />
=c 2<br />
fi<br />
fifi<br />
Evaluating Square Roots<br />
=<br />
Th<br />
<br />
aa<br />
aTh<br />
Thprincipal square rootaTh<br />
<br />
Tha√ — a Th radical<br />
radicand radical expression<br />
<br />
√ — <br />
<br />
<br />
<br />
<br />
principal square root<br />
principal square rootaa<br />
radical expression radicalradicand√ — a
32<br />
CHAPTER 1 PREREQUISITES<br />
Q & A…<br />
Does √ — 25 = ±5?<br />
− <br />
Th√ — =<br />
Example 1 Evaluating Square Roots<br />
<br />
a. √ — b. √ — √ c. √ — + d. √ — −√ — <br />
Solution<br />
a. √ — = = <br />
b. √ — √ =√ — = = =<br />
c. √ — += √ — = = <br />
d. √ — −√ — =−=− = =<br />
Q & A…<br />
For √ — 25 + 144 , can we find the square roots before adding?<br />
√ — +√ — =+=Th√ — +=Th<br />
e fi<br />
Try It #1<br />
a. √ — b. √ — √ — c. √ — − d. √ — +√ — <br />
Using the Product Rule to Simplify Square Roots<br />
Th<br />
Thfi<br />
product rulefor simplifying square roots<br />
√ — √ — ∙ √ — <br />
<br />
the product rule for simplifying square roots<br />
ababab<br />
<br />
√ — ab =√ — a · √ — b<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
Example 2<br />
Using the Product Rule to Simplify Square Roots<br />
<br />
a. √ — b. √ — a b <br />
Solution<br />
a. √ — 100 ∙ 3 <br />
√ — ∙ √ — <br />
√ —
SECTION 1.3 RADICALS AND RATIONAL EXPONENTS 33<br />
b. √ — a b ∙ 2a <br />
√ — a b ∙ √ — a <br />
a b √ — a<br />
<br />
Try It #2<br />
√ — x y z <br />
How To…<br />
<br />
1. <br />
2. <br />
Example 3<br />
Using the Product Rule to Simplify the Product of Multiple Square Roots<br />
√ — ∙ √ — <br />
Solution<br />
√ — 2 ∙ 3 <br />
√ — <br />
<br />
<br />
Try It #3<br />
√ — x ·√ — x x ><br />
Using the Quotient Rule to Simplify Square Roots<br />
<br />
quotient rule for simplifying square roots<br />
<br />
<br />
√ <br />
<br />
√— <br />
√ — <br />
the quotient rule for simplifying square roots<br />
Th_<br />
a a b b ≠<br />
b<br />
<br />
<br />
√ __ a <br />
b = ____ a √—<br />
√ — b<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 4<br />
Using the Quotient Rule to Simplify Square Roots<br />
<br />
<br />
√ <br />
<br />
Solution<br />
<br />
√— √ — <br />
<br />
√—
34<br />
CHAPTER 1 PREREQUISITES<br />
Try It #4<br />
<br />
√ ____ x <br />
y .<br />
Example 5<br />
Using the Quotient Rule to Simplify an Expression with Two Square Roots<br />
√— x y _________<br />
√ — x y<br />
Solution<br />
√ x y<br />
x y <br />
<br />
√ — x <br />
x <br />
Try It #5<br />
________<br />
√— a b √ — a b . <br />
Adding and Subtracting Square Roots<br />
<br />
√ — √ <br />
— √ <br />
— ft<br />
Th√ — <br />
√ — √ <br />
— +√ <br />
— =√ <br />
— +√ <br />
— =√ <br />
— <br />
How To…<br />
<br />
1. <br />
2. <br />
Example 6<br />
Adding Square Roots<br />
√ — +√ — <br />
Solution<br />
√ — √ — √ — √ — √ — <br />
√ — √ — <br />
√ — +√ — =√ — <br />
Try It #6<br />
√ — +√ — <br />
Example 7<br />
Subtracting Square Roots<br />
√ — a b c−√ — a b c
SECTION 1.3 RADICALS AND RATIONAL EXPONENTS 35<br />
Solution<br />
<br />
<br />
√ — a b =√ — √ — √ — √ — a √ — a √ — b √ — c <br />
<br />
=∣a ∣b √ — ac <br />
<br />
=∣ a ∣b √ — ac <br />
<br />
√ — a b c =√ — √ — √ — a √ — a √ — b √ — c <br />
<br />
=∣ a ∣b √ — ac <br />
<br />
=∣ a ∣b √ — ac <br />
<br />
<br />
∣ a ∣b √ — ac −∣ a ∣b √ — ac =∣ a ∣b √ — ac <br />
Try It #7<br />
√ — x −√ — x .<br />
Rationalizing Denominators<br />
<br />
rationalizing the<br />
denominator.<br />
<br />
<br />
<br />
<br />
b √ — c ____<br />
√— c<br />
√ — c <br />
<br />
<br />
a +b √ — c a −b √ — c .<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 8<br />
Rationalizing a Denominator Containing a Single Term<br />
√— <br />
√ — <br />
Solution<br />
√ — √— <br />
√ — <br />
√— <br />
√ — · √— √ — <br />
√— <br />
√—
36<br />
CHAPTER 1 PREREQUISITES<br />
Try It #8<br />
√— <br />
√ — <br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 9<br />
Rationalizing a Denominator Containing Two Terms<br />
<br />
<br />
+√ — <br />
Solution<br />
<br />
+√ — −√ — −√— <br />
−√ — <br />
<br />
<br />
<br />
+√ — ∙ −√— −√ — <br />
<br />
−√— − <br />
√ — − <br />
Try It #9<br />
<br />
<br />
+√ — <br />
Using Rational Roots<br />
<br />
<br />
Th<br />
<br />
Understanding nth Roots<br />
a 3 =fi = <br />
<br />
na na−−<br />
− =−anprincipal nth roota<br />
ana<br />
Thna n √ — a n<br />
n index<br />
principal nth root<br />
a nprincipal nth roota n √ — a <br />
ana. Thindexn
SECTION 1.3 RADICALS AND RATIONAL EXPONENTS 37<br />
Example 10<br />
Simplifying n th Roots<br />
<br />
—<br />
a. √ <br />
— −<br />
b.√ <br />
— √ <br />
— c. − <br />
√<br />
x d. √ <br />
— − √ <br />
— <br />
Solution<br />
a. √ <br />
— −=−− =−<br />
b. √ <br />
— = =<br />
c. − √ <br />
— x <br />
√ <br />
— <br />
−x <br />
d. √ <br />
— − √ <br />
— <br />
√ <br />
— <br />
<br />
<br />
<br />
<br />
Try It #10<br />
<br />
a.√ <br />
− b. √ <br />
<br />
<br />
√ <br />
<br />
c. √ <br />
+ √ <br />
<br />
Using Rational Exponents<br />
<br />
Thna<br />
<br />
a n <br />
= √ n — a <br />
<br />
nTh<br />
<br />
<br />
a m n =( n √ — a ) m = n √ — a m <br />
<br />
rational exponents<br />
nTh<br />
<br />
<br />
a m n <br />
=( √ n — a ) m = √ n — a m <br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
Example 11<br />
Writing Rational Exponents as Radicals<br />
<br />
Solution<br />
<br />
<br />
=( √ <br />
— ) = √ <br />
—
38<br />
CHAPTER 1 PREREQUISITES<br />
√ <br />
— = =fifi<br />
o fit fi<br />
Try It #11<br />
<br />
<br />
=( √ <br />
— ) = =<br />
Example 12<br />
Writing Radicals as Rational Exponents<br />
<br />
<br />
<br />
√ <br />
— a <br />
Solution<br />
<br />
<br />
<br />
a <br />
<br />
<br />
√ <br />
— a =a − <br />
<br />
Try It #12<br />
x √ — y <br />
Example 13<br />
Simplifying Rational Exponents<br />
<br />
a. ( x <br />
Solution<br />
a. x <br />
x <br />
) ( x <br />
) b. ( <br />
<br />
x <br />
<br />
) − <br />
<br />
+ <br />
<br />
x <br />
<br />
<br />
b. ( ) <br />
<br />
√ <br />
<br />
√— √ — <br />
<br />
<br />
<br />
<br />
<br />
<br />
Try It #13<br />
x <br />
( x <br />
) <br />
<br />
Radicals (http://openstaxcollege.org/l/introradical)<br />
Rational Exponents (http://openstaxcollege.org/l/rationexpon)<br />
Simplify Radicals (http://openstaxcollege.org/l/simpradical)<br />
Rationalize Denominator (http://openstaxcollege.org/l/rationdenom)
SECTION 1.3 SECTION EXERCISES<br />
39<br />
1.3 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. <br />
<br />
2. <br />
<br />
4. <br />
<br />
NUMERIC<br />
<br />
5. √ — 6. √ — √ 7. √ — + 8. √ — −√ — <br />
9. √ — 10. √ — 11. √ — 12. <br />
<br />
√ <br />
<br />
13. <br />
√ <br />
14. √— 15. √ — +√ — <br />
<br />
16. √ <br />
<br />
17. <br />
√ — 18. √— 19. √ — −√ — 20. √ — +√ — <br />
21. √ — 22. <br />
<br />
√ 23. ( √ — )( √ — ) 24. √ — −√ — <br />
25. <br />
<br />
√ <br />
<br />
<br />
26. <br />
√ <br />
<br />
<br />
27. <br />
√ <br />
<br />
28. <br />
<br />
+√ — <br />
<br />
29. <br />
−√ — <br />
30. √ <br />
<br />
— <br />
31. √ <br />
— + √ <br />
— <br />
—<br />
√<br />
32. − <br />
<br />
33. √ <br />
— <br />
34. √ <br />
— −+ √ <br />
— <br />
√ <br />
— <br />
ALGEBRAIC<br />
<br />
35. √ — x 36. √ — y 37. √ — p 38. p q <br />
<br />
<br />
39. m<br />
√ — 40. √ — m +√ — 41. √ — ab −b√ — a 42. √— n <br />
√ — n <br />
43. √ x x 44. √— z +√ — z 45. √ — y 46. √ — bc <br />
47. <br />
<br />
√ d <br />
48. q √ — p <br />
√ —<br />
49. <br />
<br />
−√ — x <br />
50. √ d <br />
<br />
<br />
<br />
51. w<br />
√ — <br />
−w<br />
√ — 52. √ — x +√ — x 53. √— x +√ — <br />
54. √— k <br />
55. √ — n 56. √ q <br />
q <br />
57. √ m <br />
m <br />
58. √— c −√ — c
40<br />
CHAPTER 1 PREREQUISITES<br />
59. <br />
√ <br />
d <br />
60. √ <br />
<br />
— x + √ <br />
— x <br />
<br />
—<br />
61. x <br />
√<br />
x <br />
62. √ <br />
<br />
— y <br />
63. √ <br />
— z − √ <br />
— −z <br />
64. √ <br />
— c <br />
REAL-WORLD APPLICATIONS<br />
65. <br />
<br />
<br />
Th . Th<br />
<br />
<br />
Th<br />
<br />
√ — +<br />
<br />
66. − √— <br />
√ — t s t<br />
ft<br />
<br />
EXTENSIONS<br />
<br />
67. √— −√ — <br />
−√ — <br />
−<br />
<br />
<br />
68. − <br />
<br />
<br />
<br />
<br />
69. √— mn <br />
a √ — c− · a− n − <br />
√ — m c <br />
<br />
<br />
70. a<br />
a −√ — c <br />
71. x√— y +√ — y <br />
√ — y <br />
( 72. x √— √ — b )( √— b √ — x )<br />
<br />
<br />
73. √ √ <br />
— + √ <br />
— <br />
√ — +√ —
SECTION 1.4 POLYNOMIALS 41<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Identify the degree and leading coefficient of polynomials.<br />
Add and subtract polynomials.<br />
Multiply polynomials.<br />
Use FOIL to multiply binomials.<br />
Perform operations with polynomials of several variables.<br />
1.4 POLYNOMIALS<br />
Th<br />
fi<br />
Figure 1<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
t fi<br />
<br />
<br />
<br />
Figure 1<br />
A =s <br />
=x <br />
=x <br />
Thn fi<br />
<br />
<br />
<br />
A =_ bh <br />
=_ <br />
x( _ <br />
) <br />
=_ x <br />
t fi<br />
<br />
A =lw<br />
<br />
= x ·<br />
<br />
= x<br />
Th<br />
x +_ x −x ft x +_ x ft <br />
<br />
<br />
Identifying the Degree and Leading Coefficient of Polynomials<br />
Thpolynomialff<br />
<br />
πcoefficientffi<br />
a i<br />
x i πwterm of a polynomial<br />
constant
42<br />
CHAPTER 1 PREREQUISITES<br />
x monomial. <br />
x −binomial−x +x −trinomial<br />
fidegree<br />
Thleading termfiThffi<br />
leading coefficient<br />
<br />
<br />
<br />
a n<br />
x n ++a x +a x +a <br />
<br />
polynomials<br />
polynomial<br />
a n<br />
x n ++a x <br />
+a x +a <br />
a i<br />
coefficientTha <br />
constant<br />
a i<br />
x i term of a polynomialTh<br />
degreeThleading termffi<br />
leading coefficient<br />
How To…<br />
ffi<br />
1. x<br />
2. xo fi<br />
3. ffi<br />
Example 1<br />
Identifying the Degree and Leading Coefficient of a Polynomial<br />
ffi<br />
a. +x −x b. t −t +t c. p −p −<br />
Solution<br />
a. ThxTh−x <br />
Th −<br />
b. tt <br />
Th <br />
c. p−p <br />
Th −<br />
Try It #1<br />
<br />
x −x +x −<br />
Adding and Subtracting Polynomials<br />
<br />
x −x x xx
SECTION 1.4 POLYNOMIALS 43<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 2<br />
Adding Polynomials<br />
<br />
x +x −+x +x −x +<br />
Solution<br />
x +x +x +x −x+−+ <br />
x +x +x − <br />
Analysis We can check our answers to these types of problems usinggraphing calculator. To check, graph the problem<br />
as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to<br />
compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<br />
Try It #2<br />
<br />
x +x −x ++x −x −<br />
Example 3<br />
Subtracting Polynomials<br />
<br />
x −x +x +−x −x +x +<br />
Solution<br />
x −x +−x +x +x −x+− <br />
x −x +x +x − <br />
Analysis Note that finding the difference between two polynomials is the same as adding the opposite of the second<br />
polynomial to the first.<br />
Try It #3<br />
<br />
−x −x +x −−x −x −x +<br />
Multiplying Polynomials<br />
<br />
fi<br />
<br />
<br />
<br />
Multiplying Polynomials Using the Distributive Property<br />
Th<br />
x +x +<br />
fi
44<br />
CHAPTER 1 PREREQUISITES<br />
How To…<br />
<br />
1. e fi<br />
2. <br />
3. <br />
Example 4<br />
Multiplying Polynomials Using the Distributive Property<br />
<br />
x +x −x +<br />
Solution<br />
x x −x ++x x + <br />
x −x +x +x −x + <br />
x +−x +x +x −x+ <br />
x +x +x + <br />
Analysis We can usetable to keep track of our work, as shown in Table 1. Write one polynomial across the top and<br />
the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add<br />
all of the terms together, combine like terms, and simplify.<br />
x −x +<br />
x x −x x<br />
+ x −x <br />
Table 1<br />
Try It #4<br />
<br />
x + − +<br />
Using FOIL to Multiply Binomials<br />
o fi<br />
foil<br />
<br />
<br />
<br />
Thfi<br />
<br />
How To…<br />
<br />
1. e fi<br />
2. <br />
3. <br />
4. <br />
5. <br />
6.
SECTION 1.4 POLYNOMIALS 45<br />
Example 5<br />
Using FOIL to Multiply Binomials<br />
o fi<br />
x −x +<br />
Solution<br />
<br />
<br />
<br />
e fi<br />
x − x + xx = x <br />
<br />
<br />
<br />
<br />
x − x + x = x<br />
<br />
<br />
<br />
<br />
x − x + −x = −x<br />
<br />
<br />
<br />
<br />
x − x + − = −<br />
x +x −x − <br />
x +x −x− <br />
x −x − <br />
Try It #5<br />
o fi<br />
x+x −<br />
Perfect Square Trinomials<br />
perfect square<br />
trinomialfi<br />
<br />
<br />
<br />
x+ =x +x +<br />
<br />
x− =x −x +<br />
<br />
x − =x −x +<br />
fifi<br />
Th<br />
e fi<br />
perfect square trinomials<br />
fi<br />
<br />
<br />
x+a =x+ax+a=x +ax+a
46<br />
CHAPTER 1 PREREQUISITES<br />
How To…<br />
<br />
1. e fi<br />
2. <br />
3. <br />
4. <br />
Example 6<br />
x − <br />
Expanding Perfect Squares<br />
Solution fi<br />
<br />
x −x+− <br />
<br />
x −x +<br />
Try It #6<br />
x − <br />
Difference of Squares<br />
difference of squares<br />
x+x−<br />
<br />
<br />
x+x−=x −x +x −<br />
<br />
=x −<br />
Thff<br />
<br />
<br />
x+x−=x −<br />
<br />
x+x−=x −<br />
<br />
x +x −=x −<br />
<br />
e fi<br />
Q & A…<br />
Is therespecial form for the sum of squares?<br />
Thff<br />
Th<br />
difference of squares<br />
<br />
e fi<br />
<br />
a+ba−b=a −b <br />
How To…<br />
n, fiff<br />
1. e fi<br />
2. <br />
3. e fi
SECTION 1.4 POLYNOMIALS 47<br />
Example 7<br />
x +x −<br />
Multiplying Binomials Resulting in a Difference of Squares<br />
Solution fix =x =<br />
e fio fix −<br />
Try It #7<br />
x +x −<br />
Performing Operations with Polynomials of Several Variables<br />
<br />
<br />
<br />
a+ba −b −c<br />
aa −b −c+ba −b − c <br />
a −ab−ac+ab−b −bc <br />
a +−ab+ab−ac−b −bc <br />
a +ab−ac−bc−b <br />
Example 8 Multiplying Polynomials Containing Several Variables<br />
x+x −y +<br />
Solution <br />
xx −y ++x −y + <br />
x −xy+x +x −y + <br />
x −xy+x +x−y + <br />
x −xy+x −y + <br />
Try It #8<br />
x −x +y −<br />
<br />
Adding and Subtracting Polynomials (http://openstaxcollege.org/l/addsubpoly)<br />
Multiplying Polynomials (http://openstaxcollege.org/l/multiplpoly)<br />
Special Products of Polynomials (http://openstaxcollege.org/l/specialpolyprod)
48<br />
CHAPTER 1 PREREQUISITES<br />
1.4 SECTION EXERCISES<br />
VERBAL<br />
1. t: Th<br />
<br />
<br />
<br />
3. <br />
<br />
<br />
<br />
2. <br />
<br />
s a diff<br />
<br />
4. <br />
<br />
<br />
ALGEBRAIC<br />
<br />
5. x −x + 6. m +m −m + 7. −a +b 8. p −p m +m <br />
9. x +x + 10. y −y +y −<br />
, fiff<br />
11. x +x−x − 12. z +z −+−z +z +<br />
13. w +w +−w −w + 14. a +a −a −+−a −a +a +<br />
15. b −b +b −b +−b +b +b 16. p −+p −p +<br />
, fi<br />
17. x +x − 18. c +cc −c 19. b −b − 20. d −d +<br />
21. v −v − 22. t +t−t + 23. n −n +<br />
<br />
24. x + 25. y − 26. −x 27. p + <br />
28. m − 29. y − 30. b + <br />
<br />
31. c +c − 32. a −a + 33. n −n + 34. b +b −<br />
35. +m−m 36. p +p − 37. q −q +<br />
<br />
38. x +x +x − 39. t +t −t − 40. x −x −x +<br />
41. y −y −y − 42. k −k +k − 43. p +p −p −<br />
44. m −m −m + 45. a +ba −b 46. x −yx −y<br />
47. t −u 48. m +n −m + 49. t −xt −x +<br />
50. b −a +ab +b 51. r −dr +d 52. x +yx −xy+y <br />
REAL-WORLD APPLICATIONS<br />
53. <br />
. Th<br />
x +x − x <br />
<br />
<br />
EXTENSIONS<br />
<br />
54. <br />
. Th <br />
x + . Thx +<br />
x <br />
<br />
<br />
55. t − t +−t +t + 56. b +b −b −<br />
57. a +ac +c a −c
SECTION 1.5 FACTORING POLYNOMIALS 49<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Factor the greatest common factor of a polynomial.<br />
Factor a trinomial.<br />
Factor by grouping.<br />
Factor a perfect square trinomial.<br />
Factor a difference of squares.<br />
Factor the sum and difference of cubes.<br />
Factor expressions using fractional or negative exponents.<br />
1.5 FACTORING POLYNOMIALS<br />
fiTh<br />
Figure 1<br />
<br />
<br />
<br />
<br />
x<br />
<br />
Figure 1<br />
Th<br />
<br />
A =lw<br />
<br />
=x·x<br />
<br />
=x <br />
ThTh<br />
A = = = Th<br />
x −A =lw=x −=x − <br />
+x −=x <br />
Th −x Th<br />
xx − fi<br />
<br />
<br />
Factoring the Greatest Common Factor of a Polynomial<br />
x<br />
greatest common factor (GCF)<br />
<br />
Thxxx <br />
xx <br />
fi<br />
ffi
50<br />
CHAPTER 1 PREREQUISITES<br />
greatest common factor<br />
greatest common factor<br />
<br />
How To…<br />
<br />
1. ffi<br />
2. <br />
3. o fi<br />
4. <br />
5. <br />
Example 1<br />
x y +x y +xy<br />
Factoring the Greatest Common Factor<br />
Solution fiThThx x xx<br />
x n y y <br />
yyo fixy<br />
<br />
xyx y =x y xyxy=x y xy=xy<br />
<br />
xyx y +xy+<br />
Analysis After factoring, we can check our work by multiplying. Use the distributive property to confirm that<br />
xyx y + xy + = x y + x y + xy.<br />
Try It #1<br />
xb −a+b −a<br />
Factoring a Trinomial with Leading Coefficient 1<br />
<br />
Thx +x +<br />
x+x+<br />
x +bx+ficb <br />
x +x +<br />
. Thx+x+<br />
factoring a trinomial with leading coefficient 1<br />
x +bx+cx+px+qpq= cp +q =b<br />
Q & A…<br />
Can every trinomial be factored asproduct of binomials?<br />
. Th
SECTION 1.5 FACTORING POLYNOMIALS 51<br />
How To…<br />
x +bx+c<br />
1. c<br />
2. pqcb<br />
3. x+px+q<br />
Example 2 Factoring a Trinomial with Leading Coefficient 1<br />
x +x −<br />
Solution ffib =c =−fi<br />
−Table 1e fi<br />
Factors of −15<br />
−<br />
−<br />
−<br />
−<br />
Sum of Factors<br />
−<br />
<br />
−<br />
<br />
fipq−x−x+<br />
Analysis We can check our work by multiplying. Use FOIL to confirm that x − x + = x + x − .<br />
Table 1<br />
Q & A…<br />
Does the order of the factors matter?<br />
<br />
Try It #2<br />
x −x +<br />
Factoring by Grouping<br />
ffi<br />
factor by grouping <br />
Thx +x +x +x+<br />
x +x +x +<br />
xx++x+x+o fi<br />
factor by grouping<br />
ax +bx+c ac<br />
bx<br />
<br />
How To…<br />
ax +bx+c<br />
1. ac<br />
2. pqacb<br />
3. ax +px+qx+c<br />
4. ax +px<br />
5. qx+c<br />
6.
52<br />
CHAPTER 1 PREREQUISITES<br />
Example 3<br />
x +x −<br />
Factoring a Trinomial by Grouping<br />
Solution a =b =c =−ac=−fi<br />
−Table 2e fi<br />
p =−q =<br />
Factors of −30<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
Table 2<br />
Sum of Factors<br />
−<br />
<br />
−<br />
<br />
−<br />
<br />
x −x +x − ax +px+qx+c<br />
xx −+x − <br />
x −x+ <br />
Analysis We can check our work by multiplying. Use FOIL to confirm that x − x + = x + x − .<br />
Try It #3<br />
a. x +x + b. x +x −<br />
Factoring a Perfect Square Trinomial<br />
<br />
fi<br />
<br />
<br />
a +ab+b =a + b <br />
<br />
<br />
<br />
a −ab+b =a − b <br />
<br />
perfect square trinomials<br />
<br />
<br />
a +ab+b =a+b <br />
How To…<br />
<br />
1. fie fi<br />
2. fiab<br />
3. a + b
SECTION 1.5 FACTORING POLYNOMIALS 53<br />
Example 4<br />
x +x +<br />
Factoring a Perfect Square Trinomial<br />
Solution x x =x = Th<br />
x. Thx=x<br />
Thx + <br />
Try It #4<br />
x −x +<br />
Factoring a Difference of Squares<br />
ffff<br />
<br />
<br />
<br />
a −b =a + ba − b<br />
ff<br />
differences of squares<br />
A diff<br />
<br />
a −b =a + ba−b<br />
How To…<br />
ff<br />
1. fie fi<br />
2. a+ba−b<br />
Example 5<br />
x −<br />
Factoring a Difference of Squares<br />
Solution x x =x = Th<br />
ffx +x −<br />
Try It #5<br />
y −<br />
Q & A…<br />
Is thereaformula to factor the sum of squares?<br />
<br />
Factoring the Sum and Difference of Cubes<br />
ff<br />
<br />
<br />
a +b =a + ba −ab+b <br />
ff<br />
<br />
a −b =a − ba +ab+b
54<br />
CHAPTER 1 PREREQUISITES<br />
ffThfi<br />
SOAP<br />
<br />
x − =x−x +x +<br />
Thfisamex − Thxopposite<br />
x − always positive<br />
sum and difference of cubes<br />
<br />
<br />
e diff<br />
<br />
a +b =a + ba −ab+b <br />
a −b =a − ba +ab+b <br />
How To…<br />
ff<br />
1. fie fia +b a −b <br />
2. a+ba −ab+b ff<br />
a−ba +ab+b <br />
Example 6<br />
x +<br />
Factoring a Sum of Cubes<br />
Solution x =x+x −x +<br />
Analysis After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be<br />
factored further. However, the trinomial portion cannot be factored, so we do not need to check.<br />
Try It #6<br />
a +b <br />
Example 7<br />
x −<br />
Factoring a Difference of Cubes<br />
Solution x x =x = ff<br />
x −x +x +<br />
Analysis Just as with the sum of cubes, we will not be able to further factor the trinomial portion.<br />
Try It #7<br />
e diffx −
SECTION 1.5 FACTORING POLYNOMIALS 55<br />
Factoring Expressions with Fractional or Negative Exponents<br />
<br />
<br />
Thx +x <br />
x x ( +x ) <br />
Example 8<br />
Factoring an Expression with Fractional or Negative Exponents<br />
xx+ − +x+ <br />
<br />
Solution x+ − <br />
x+ − x +x+<br />
x+ − x +x +<br />
<br />
<br />
x+ − x +<br />
Try It #8<br />
a − +aa − − <br />
<br />
Identify GCF (http://openstaxcollege.org/l/findgcftofact)<br />
Factor Trinomials when a Equals 1 (http://openstaxcollege.org/l/facttrinom1)<br />
Factor Trinomials when a is not equal to 1 (http://openstaxcollege.org/l/facttrinom2)<br />
Factor Sum or Difference of Cubes (http://openstaxcollege.org/l/sumdifcube)
56<br />
CHAPTER 1 PREREQUISITES<br />
1.5 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
2. <br />
r a diff<br />
<br />
3. <br />
ALGEBRAIC<br />
, fi<br />
4. x +xy−xy 5. mb −m ba +ma <br />
6. x y −x y +xy 7. p m −p m +m <br />
8. j k −j k +j k 9. y −y +y −y<br />
<br />
10. x +x − 11. a +a − 12. c +c + 13. n −n −<br />
14. w −w + 15. p −p −<br />
<br />
16. x +x − 17. h −h − 18. b −b − 19. −d +<br />
20. v −v + 21. t +t − 22. n −n − 23. x −<br />
24. y − 25. p − 26. m − 27. d −<br />
28. x − 29. b −c 30. a −a + 31. n +n +<br />
32. x −x + 33. y +y + 34. m −m + 35. p −p +<br />
36. q +q +<br />
<br />
37. x + 38. y − 39. a + 40. b −d <br />
41. x − 42. q + 43. r +s <br />
44. xx − − +x − <br />
46. tt + +t + <br />
48. yy − −y − <br />
<br />
<br />
45. cc + − −c + <br />
47. xx + − +x + <br />
49. zz − − +z − − <br />
<br />
50. dd + − +d +
SECTION 1.5 SECTION EXERCISES<br />
57<br />
REAL-WORLD APPLICATIONS<br />
<br />
fiThfi<br />
Thx +x − fi<br />
Th<br />
l × w=x +x−<br />
51. <br />
<br />
53. <br />
in. Th<br />
x − <br />
<br />
52. k. Th<br />
x +x + <br />
<br />
<br />
<br />
flThfi<br />
. The flx −x +y <br />
x <br />
−x+<br />
<br />
<br />
54. e fl<br />
EXTENSIONS<br />
<br />
55. x −x + 56. y − 57. z −a <br />
58. xx + − +x + <br />
59. x +x −x − −
58<br />
CHAPTER 1 PREREQUISITES<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Simplify rational expressions.<br />
Multiply rational expressions.<br />
Divide rational expressions.<br />
Add and subtract rational expressions.<br />
Simplify complex rational expressions.<br />
1. 6 RATIONAL EXPRESSIONS<br />
fiTh<br />
x+x<br />
<br />
<br />
+x<br />
x<br />
<br />
<br />
<br />
Simplifying Rational Expressions<br />
Thrational expression<br />
<br />
fi<br />
<br />
x +x + <br />
<br />
x +x + <br />
<br />
x+<br />
<br />
<br />
x+x+ <br />
Thx+<br />
How To…<br />
<br />
1. <br />
2. <br />
x + x + <br />
Example 1<br />
Simplifying Rational Expressions<br />
x<br />
− <br />
x +x + <br />
x+x−<br />
Solution<br />
<br />
<br />
x+x+ <br />
<br />
<br />
x − x + <br />
x+<br />
Analysis We can cancel the common factor because any expression divided by itself is equal to .<br />
Q & A…<br />
Can the x 2 term be cancelled in Example 1?<br />
. Thx
SECTION 1.6 RATIONAL EXPRESSIONS 59<br />
Try It #1<br />
x − x − <br />
Multiplying Rational Expressions<br />
<br />
fifi<br />
<br />
ft<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 2<br />
Multiplying Rational Expressions<br />
<br />
Solution<br />
x +x − x + ∙ x − <br />
x + <br />
<br />
x+x− <br />
x+ ∙ <br />
x − <br />
x+ <br />
<br />
<br />
<br />
<br />
x+x−x <br />
− x+x+ <br />
x+x−x − x+ x+ <br />
<br />
<br />
x−x <br />
− x+ <br />
Try It #2<br />
<br />
x +x + <br />
<br />
x +x + · x +x + x +x + <br />
Dividing Rational Expressions<br />
<br />
fi<br />
x ÷x <br />
<br />
x ∙ <br />
x <br />
<br />
<br />
x · <br />
x = <br />
x
60<br />
CHAPTER 1 PREREQUISITES<br />
How To…<br />
<br />
1. e fi<br />
2. <br />
3. <br />
4. <br />
5. <br />
Example 3<br />
Dividing Rational Expressions<br />
<br />
Solution<br />
<br />
x +x − x − ÷<br />
x − <br />
x +x + <br />
x +x − <br />
x − +x + ∙x x − <br />
<br />
<br />
<br />
<br />
x −x+ <br />
<br />
x+x− ∙x+x+ <br />
<br />
x+x− <br />
x −x+x+x+ <br />
<br />
x+x−x+x− <br />
x −x+x+x+ <br />
<br />
x+x−x+x− <br />
x −x <br />
+ <br />
x − x − <br />
<br />
<br />
<br />
Try It #3<br />
<br />
x<br />
− −x − <br />
x +x − ÷x <br />
x +x − <br />
Adding and Subtracting Rational Expressions<br />
<br />
o fi<br />
<br />
<br />
<br />
+ <br />
= + <br />
<br />
= <br />
<br />
<br />
= <br />
<br />
<br />
<br />
Thleast common denominatorTh<br />
fi<br />
x+x+x+x+<br />
x+x+x+<br />
fi<br />
x+x+ x + <br />
x+x+ x + <br />
x + <br />
x +
SECTION 1.6 RATIONAL EXPRESSIONS 61<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
5. <br />
Example 4<br />
<br />
<br />
Adding Rational Expressions<br />
x + <br />
y <br />
Solution fixy<br />
xy<br />
<br />
<br />
x · y <br />
y + <br />
y · <br />
x x <br />
y <br />
xy +x <br />
xy <br />
o fi<br />
x +y<br />
<br />
xy<br />
<br />
Analysis Multiplying by y _<br />
y<br />
or<br />
_ x x<br />
does not change the value of the original expression because any number divided by<br />
itself is , and multiplying an expression by gives the original expression.<br />
Example 5 Subtracting Rational Expressions<br />
<br />
Solution<br />
<br />
<br />
<br />
x <br />
+x + − <br />
x − <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x+ − <br />
<br />
x+x− <br />
<br />
<br />
x+ ∙x <br />
− x − − <br />
<br />
x+x− ∙ x + x + <br />
x−<br />
−<br />
x+ x+<br />
<br />
x− x+ x− <br />
x −−x <br />
+ x+ x− <br />
x −<br />
<br />
x+ x− <br />
x−<br />
<br />
x+ <br />
x− <br />
<br />
<br />
<br />
<br />
<br />
<br />
Q & A…<br />
Do we have to use the LCD to add or subtract rational expressions?
62<br />
CHAPTER 1 PREREQUISITES<br />
Try It #4<br />
<br />
<br />
x + − <br />
x − <br />
<br />
Simplifying Complex Rational Expressions<br />
<br />
<br />
Th<br />
a <br />
fi<br />
<br />
b +c<br />
a + bc<br />
<br />
<br />
b<br />
a <br />
∙ b <br />
+bc <br />
<br />
ab <br />
+bc <br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
5. <br />
6. <br />
Example 6<br />
Simplifying Complex Rational Expressions<br />
y + <br />
x <br />
<br />
x y <br />
<br />
Solution <br />
<br />
<br />
<br />
y ∙ x x + <br />
x <br />
xy <br />
x + <br />
x <br />
xy+ x<br />
<br />
x <br />
x <br />
<br />
<br />
<br />
xy+ x<br />
__ x <br />
y <br />
<br />
<br />
xy+ <br />
x<br />
÷ x y <br />
xy+ <br />
x<br />
∙ y <br />
x <br />
<br />
<br />
yxy+ x
SECTION 1.6 RATIONAL EXPRESSIONS 63<br />
Try It #5<br />
__ x y −y __<br />
x y<br />
<br />
Q & A…<br />
Can a complex rational expression always be simplified?<br />
fi<br />
<br />
Simplify Rational Expressions (http://openstaxcollege.org/l/simpratexpress)<br />
Multiply and Divide Rational Expressions (http://openstaxcollege.org/l/multdivratex)<br />
Add and Subtract Rational Expressions (http://openstaxcollege.org/l/addsubratex)<br />
Simplify a Complex Fraction (http://openstaxcollege.org/l/complexfract)
64<br />
CHAPTER 1 PREREQUISITES<br />
1.6 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
<br />
2. <br />
<br />
ALGEBRAIC<br />
<br />
4.<br />
x − x −x + <br />
5. y +y + <br />
y +y + <br />
6. −a + a <br />
a − 7. +b + b <br />
b + <br />
8.<br />
m − m − <br />
9. +x − x x +x − <br />
10. x +x − <br />
<br />
x +x + <br />
11. a +a + a +a − <br />
12. c +c − <br />
c −c + <br />
13. −n − n <br />
<br />
n −n − <br />
<br />
14. x −x − x +x − · x +x − <br />
x − 15. c +c −<br />
<br />
c +c + · c −c + <br />
<br />
c −c + <br />
16. d +d − <br />
<br />
d +d + · d +d − <br />
d +d − <br />
18. b +b + <br />
b − · b +b − <br />
b −b − <br />
17. h −h − <br />
h −h + · h −h + <br />
h −h − <br />
19. +d + d <br />
d − · d −d + <br />
d − <br />
20. x −x − <br />
<br />
x −x − · x −x − <br />
x +x + 21.<br />
t − <br />
t +t + · t +t − <br />
t −t + <br />
22. n −n − <br />
n +n − · n −n + <br />
<br />
n −n + <br />
23. <br />
x −<br />
<br />
<br />
x +x + · x +x + <br />
x +x + <br />
<br />
24. y −y − +y −<br />
y −y − ÷y y +y − <br />
25. p +p − <br />
−p +<br />
p +p + ÷p <br />
p +p − <br />
q −<br />
26. −q −<br />
q +q + ÷q q +q − <br />
27. +d − d <br />
+d −<br />
d −d + ÷d <br />
<br />
d −d + <br />
28. x +x − <br />
+x +<br />
x −x − ÷x <br />
<br />
x +x + <br />
30. a −a + <br />
<br />
a +a − ÷<br />
a −<br />
<br />
a +a + <br />
29. <br />
b −<br />
<br />
−b +<br />
b −b − ÷b <br />
b −b − <br />
31. +y + y <br />
+y +<br />
y +y − ÷y <br />
<br />
y −y + <br />
32. x +x − <br />
+x −<br />
x −x + ÷x <br />
<br />
x −x +
SECTION 1.6 SECTION EXERCISES<br />
65<br />
<br />
33. <br />
x + y <br />
34. q − <br />
p<br />
<br />
35. <br />
a + + <br />
a − <br />
36. c + −c − <br />
39. z<br />
z + +z + <br />
z − <br />
37. y + y − +y − y + <br />
p<br />
40. p + −p + <br />
p<br />
38. x − x + −x + x + <br />
x<br />
41. x + + y<br />
y + <br />
<br />
42.<br />
y − <br />
x<br />
<br />
<br />
y<br />
<br />
43. _ + <br />
__<br />
a b<br />
b<br />
44. _ x<br />
− p <br />
p<br />
<br />
45. _ <br />
a<br />
+_ b <br />
<br />
b <br />
a<br />
46. <br />
<br />
x + + <br />
x − <br />
<br />
x − <br />
<br />
x + <br />
<br />
47. _ a<br />
b −b <br />
a <br />
<br />
a +b<br />
<br />
ab<br />
48.<br />
x <br />
+x <br />
49.<br />
c <br />
_ x c + +c − c + <br />
<br />
c + <br />
<br />
c + <br />
<br />
50. x _y − y _<br />
x<br />
<br />
x _<br />
y<br />
+ y _<br />
x<br />
REAL-WORLD APPLICATIONS<br />
51. Th<br />
x −x −7 ft Th<br />
x −x +1 ft <br />
x −x − <br />
<br />
x −x + <br />
=x −x −<br />
52. Thx −625 ft A p<br />
x −x +25 ft <br />
<br />
<br />
53. <br />
x +x +81 ft <br />
x −81 ft <br />
<br />
<br />
EXTENSIONS<br />
<br />
54. x +x − x −x − · x −x − +x +<br />
x −x − ÷x <br />
x +x + 55. y −y + <br />
<br />
y +y − ∙ −y − y <br />
<br />
y −y − <br />
<br />
y − <br />
<br />
56. a + a − +a − a + <br />
<br />
<br />
a + <br />
a<br />
57. x +x + +x +<br />
x +x − ÷x <br />
+x −<br />
x −x − ÷x x +x −
66<br />
CHAPTER 1 PREREQUISITES<br />
CHAPTER 1 REVIEW<br />
Key Terms<br />
algebraic expression <br />
associative property of addition <br />
a +b+c=a+b+c<br />
associative property of multiplication <br />
, a·b·c=a·b·c<br />
base <br />
binomial <br />
coefficient a i<br />
a n<br />
x n ++a <br />
x +a <br />
x +a <br />
commutative property of addition <br />
a +b =b +a<br />
commutative property of multiplication <br />
a·b =b ·a<br />
constant <br />
degree <br />
difference of squares <br />
<br />
distributive property <br />
a·b+c=a·b + a·c<br />
equation <br />
exponent <br />
<br />
exponential notation <br />
factor by grouping ax +bx+cx<br />
<br />
formula <br />
greatest common factor <br />
identity property of addition <br />
, a +=a<br />
identity property of multiplication <br />
, a·=a<br />
index n<br />
integers −−−<br />
inverse property of addition a<br />
−a, a +−a=<br />
inverse property of multiplication a<br />
a <br />
, a· a = <br />
irrational numbers <br />
<br />
leading coefficient ffi<br />
leading term <br />
least common denominator
CHAPTER 1 REVIEW 67<br />
monomial <br />
natural numbers <br />
order of operations <br />
perfect square trinomial <br />
polynomial <br />
principal nth root a tna<br />
principal square root f a naa<br />
radical <br />
radical expression <br />
radicand <br />
rational expression <br />
rational numbers __<br />
m n<br />
mnn ≠<br />
<br />
real number line <br />
<br />
real numbers <br />
scientific notation a × n ≤∣a ∣<<br />
n<br />
term of a polynomial a i<br />
x i a n<br />
x n ++a <br />
x +a <br />
x +a <br />
trinomial <br />
variable <br />
whole numbers <br />
Key Equations<br />
Rules of Exponents<br />
s a a<br />
<br />
a m ∙ a n =a m+n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
ff<br />
am<br />
=a<br />
a n m−n<br />
a m n =a m∙n<br />
a =<br />
a −n = <br />
<br />
a n<br />
a ∙ b n =a n ∙ b n<br />
( a __<br />
b ) n = an<br />
b n<br />
x+a =x+ax+a=x +ax+a <br />
a+ba−b=a −b <br />
a −b =a+ba − b<br />
a +ab+b =a+b <br />
a +b =a+ba −ab+b <br />
a −b =a−ba +ab+b
68<br />
CHAPTER 1 PREREQUISITES<br />
Key Concepts<br />
1.1 Real Numbers: <strong>Algebra</strong> Essentials<br />
Example 1Example 2<br />
Example 3<br />
ThExample 4<br />
Example 5<br />
ThExample 6<br />
Th<br />
Th<br />
Example 7<br />
<br />
Example 8Th<br />
Example 9Example 10Example 12.<br />
Th <br />
Example 11Example 13<br />
1.2 Exponents and Scientific Notation<br />
Example 1<br />
Example 2<br />
Example 3<br />
fiExample 4<br />
fiExample 5Example 6<br />
ThExample 7<br />
ThExample 8<br />
ThExample 9<br />
fiExample 10<br />
Example 11<br />
fiExample 12<br />
Example 13<br />
1.3 Radicals and Rational Expressions<br />
Thas a<br />
Example 1<br />
ababab<br />
Example 2Example 3<br />
ab __ a b ab<br />
Example 4 Example 5<br />
Example 6<br />
Example 7<br />
<br />
<br />
Example 8Example 9<br />
Thna ia tna. Th<br />
Example 10<br />
<br />
Example 11Example 12<br />
ThExample 13
CHAPTER 1 REVIEW 69<br />
1.4 Polynomials<br />
. Th<br />
. Th<br />
ffiffiExample 1<br />
Example 2 Example 3.<br />
e fi<br />
. ThExample 4<br />
Example 5<br />
Example 6 Example 7<br />
Example 8<br />
1.5 Factoring Polynomials<br />
The fi<br />
Example 1<br />
ffiy fi<br />
Example 2<br />
Example 3<br />
<br />
Example 4Example 5<br />
Th Example 6Example 7<br />
Example 8<br />
1.6 Rational Expressions<br />
<br />
Example 1<br />
<br />
Example 2<br />
Example 3<br />
s fiExample 4<br />
Example 5<br />
. Th<br />
Example 6
70<br />
CHAPTER 1 PREREQUISITES<br />
CHAPTER 1 REVIEW EXERCISES<br />
REAL NUMBERS: ALGEBRA ESSENTIALS<br />
<br />
1. −· − 2. ÷·+÷ 3. · +÷<br />
<br />
4. x +=− 5. y + =<br />
<br />
6. y +÷·+ 7. m+−m<br />
<br />
<br />
8. 9. 10. <br />
<br />
11. √ — <br />
EXPONENTS AND SCIENTIFIC NOTATION<br />
<br />
12. · 13. <br />
<br />
<br />
14. ( a <br />
b<br />
) <br />
15. a ·a <br />
a − <br />
16. xy <br />
y · <br />
x <br />
17. x y −<br />
− <br />
−<br />
y ) <br />
x 18. ( x <br />
19. ( a <br />
b<br />
) ab − −<br />
20. <br />
× −<br />
21. <br />
RADICALS AND RATIONAL EXPRESSIONS<br />
<br />
22. √ — 23. √ — 24. √ — 25. √ — <br />
26. √ — 27.<br />
√ <br />
<br />
28. √ <br />
<br />
29. √ <br />
<br />
<br />
30. <br />
+√ — 31. √— +√ — 32. √ — −√ — 33. √ <br />
— −<br />
34. √ — <br />
<br />
√ — −
CHAPTER 1 REVIEW 71<br />
POLYNOMIALS<br />
<br />
35. x +x −+x −x + 36. y +−y −y −<br />
37. x +x −+x −x + 38. a +a +−a −a +<br />
39. k +k − 40. h +h −<br />
41. x +x + 42. m −m +m −<br />
43. a +ba −b 44. x +yx −y<br />
FACTORING POLYNOMIALS<br />
<br />
45. p +pq −p q 46. x y +xy −xy 47. a b +a b −a <br />
<br />
48. x −x − 49. a +a − 50. d −d −<br />
51. x +x + 52. y −y + 53. h −hk +k <br />
54. x − 55. p + 56. x −<br />
57. q −p 58. xx − − +x − <br />
<br />
59. pp + −p + <br />
<br />
60. r r − − −r − <br />
<br />
RATIONAL EXPRESSIONS<br />
<br />
61. x −x − x −x + <br />
62. <br />
y −<br />
<br />
y −y + <br />
63. a −a − a −a − · a −a − <br />
a −a − <br />
64. d − <br />
d − · d − d − <br />
65. m +m + <br />
+m −<br />
m −m − ÷m <br />
m −m − <br />
66. d −d − <br />
+d +<br />
d −d + ÷d <br />
d +d − <br />
67. x + <br />
y<br />
<br />
<br />
68. <br />
a +a + − <br />
a − <br />
69.<br />
<br />
<br />
d + <br />
c <br />
<br />
c +d<br />
d <br />
70.<br />
x − <br />
y<br />
<br />
<br />
x
72<br />
CHAPTER 1 PREREQUISITES<br />
CHAPTER 1 PRACTICE TEST<br />
<br />
<br />
1. − 2. √ — <br />
<br />
3. x +−= 4. y+ −=<br />
5. × 6. <br />
<br />
<br />
7. −·+· + 8. x +−x + 9. · −<br />
10. ( <br />
) <br />
11. x <br />
x <br />
12. y y −<br />
13. √ — 14. √ — 15. <br />
√ x <br />
<br />
16. √— b <br />
+√ — b 17. √— +√ — −√ — 18. √ <br />
— − <br />
<br />
√ <br />
— <br />
19. q +q −−q +q − 20. p +p ++p − 21. n −n −n +<br />
22. a −ba +b<br />
<br />
23. x − 24. y +y + 25. c −<br />
<br />
26. xx − − +x − <br />
<br />
27. z +z + z − · z −z + <br />
z − 28. x <br />
y + <br />
x 29. a<br />
b −b <br />
a<br />
<br />
a −b<br />
<br />
a
Equations and Inequalities<br />
2<br />
Figure 1<br />
CHAPTER OUTLINE<br />
2.1 The Rectangular Coordinate Systems and Graphs<br />
2.2 Linear Equations in One Variable<br />
2.3 Models and Applications<br />
2.4 Complex Numbers<br />
2.5 Quadratic Equations<br />
2.6 Other Types of Equations<br />
2.7 Linear Inequalities and Absolute Value Inequalities<br />
Introduction<br />
<br />
<br />
<br />
A x <br />
y <br />
<br />
B x <br />
y <br />
<br />
73
74<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Plot ordered pairs in a Cartesian coordinate system.<br />
Graph equations by plotting points.<br />
Graph equations with a graphing utility.<br />
Find x-intercepts and y-intercepts.<br />
Use the distance formula.<br />
Use the midpoint formula.<br />
2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 1<br />
<br />
Figure 1<br />
<br />
<br />
Plotting Ordered Pairs in the Cartesian Coordinate System<br />
<br />
fl<br />
fl<br />
<br />
<br />
<br />
<br />
Cartesian coordinate system<br />
x-axis y-axis<br />
Th<br />
x- y-<br />
quadrantFigure 2.
SECTION 2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 75<br />
y<br />
<br />
<br />
x<br />
<br />
<br />
Figure 2<br />
Thorigin<br />
x-<br />
y- x- y-Th<br />
fiFigure 3<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 3<br />
x-coordinate<br />
y-coordinateordered pair<br />
x,y<br />
x- y-−<br />
Figure 4<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
–<br />
x<br />
Figure 4<br />
x-<br />
y- x- y-
76<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Cartesian coordinate system<br />
<br />
x<br />
y<br />
fix,y, x <br />
y <br />
Example 1<br />
Plotting Points in a Rectangular Coordinate System<br />
−−<br />
Solution −Th x-−ftTh<br />
y- y <br />
x-<br />
y- y <br />
−Th x-Th<br />
x- y- y Figure 5<br />
y<br />
–<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
–<br />
<br />
<br />
x<br />
Figure 5<br />
Analysis Note that when either coordinate is zero, the point must be on an axis. If the x-coordinate is zero, the point<br />
is on the y-axis. If the y-coordinate is zero, the point is on the x-axis.<br />
Graphing Equations by Plotting Points<br />
x y<br />
equation in two variablesgraph in two variables<br />
<br />
y =x − x <br />
y x- y-Table 1<br />
x −y<br />
x y = 2x − 1 (x, y)<br />
− y=−−=− −−<br />
− y=−−=− −−<br />
− y=−−=− −−<br />
y=−=− −<br />
y=−= <br />
y=−= <br />
y=−= <br />
Table 1
SECTION 2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 77<br />
Th<br />
Figure 6. Th<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–– –<br />
–<br />
–– –<br />
–<br />
–– –<br />
<br />
–<br />
Figure 6<br />
x-<br />
x <br />
<br />
Th<br />
<br />
<br />
<br />
<br />
<br />
x<br />
How To…<br />
<br />
1. x<br />
<br />
2. x-fi x-<br />
<br />
3. x- y-ff<br />
4. <br />
5. <br />
Example 2 Graphing an Equation in Two Variables by Plotting Points<br />
y =−x+<br />
Solution Table 2 x y<br />
x y = − x + 2 (x, y)<br />
− y=−−+= −<br />
− y=−−+= −<br />
− y=−−+= −<br />
y=−+= <br />
y=−+= <br />
y=−+=− −<br />
y=−+=− −<br />
Table 2
78<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Figure 7<br />
–<br />
<br />
<br />
<br />
–<br />
<br />
– <br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
–<br />
–<br />
x<br />
Try It #1<br />
Figure 7<br />
y= <br />
x+<br />
Graphing Equations with a Graphing Utility<br />
Th<br />
y = Th<br />
<br />
<br />
y =x −Figure 8aFigure 8b<br />
Th<br />
−≤ x ≤−≤ y ≤Figure 8c<br />
Plot1 Plot2 Plot3<br />
\Y 1<br />
= 2X–20<br />
\Y 2<br />
=<br />
\Y 3<br />
=<br />
\Y 4<br />
=<br />
\Y 5<br />
=<br />
\Y 6<br />
=<br />
\Y 7<br />
=<br />
WINDOW<br />
Xmin = −10<br />
Xmax = 10<br />
Xscl = 1<br />
Ymin = −10<br />
Ymax = 10<br />
Yscl = 1<br />
Xres = 1<br />
<br />
Figure 8 (a) Enter the equation. (b) This is the graph in the original window. (c) These are the original settings.<br />
x- y-<br />
x- y-Figure 9aFigure 9b<br />
WINDOW<br />
Xmin = −5<br />
Xmax = 15<br />
Xscl = 1<br />
Ymin = −30<br />
Ymax = 10<br />
Yscl = 1<br />
Xres = 1<br />
<br />
Figure 9 (a) This screen shows the new window settings. (b) We can clearly view the intercepts in the new window.<br />
y<br />
x<br />
<br />
x<br />
Example 3<br />
Using a Graphing Utility to Graph an Equation<br />
y =− <br />
x − <br />
<br />
Solution y = x-<br />
y-Figure 10<br />
– – – –<br />
–<br />
y<br />
<br />
<br />
<br />
–<br />
–<br />
Figure 10<br />
y=− x − <br />
<br />
<br />
<br />
<br />
x
SECTION 2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 79<br />
Finding x-intercepts and y-intercepts<br />
interceptsTh x-intercept<br />
x- y-Th y-intercept<br />
y- x-<br />
x- y x y- x<br />
ys fiy =x −<br />
o fi x-y =<br />
<br />
y =x −<br />
<br />
=x −<br />
<br />
=x<br />
<br />
<br />
=x<br />
( <br />
) <br />
<br />
x<br />
o fi y-x =<br />
<br />
y =x −<br />
<br />
y =−<br />
<br />
y =−<br />
− y<br />
fiFigure 11<br />
<br />
given an equation, find the intercepts.<br />
– – – –<br />
Figure 11<br />
xy =x<br />
yx =y<br />
–<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
y=x −<br />
<br />
<br />
<br />
<br />
x<br />
Example 4 Finding the Intercepts of the Given Equation<br />
y =−x −. Th<br />
Solution y =o fi x-<br />
<br />
y =−x −<br />
<br />
=−x −<br />
<br />
=−x<br />
<br />
− <br />
=x<br />
<br />
( − <br />
) <br />
x−<br />
x =o fi y-<br />
<br />
y =−x −<br />
<br />
y =−−<br />
<br />
y =−<br />
− y−
80<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Figure 12<br />
Try It #2<br />
Figure 12<br />
y=− <br />
x+<br />
Using the Distance Formula<br />
− <br />
<br />
– – – –<br />
–<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
Thdistance formulafi<br />
ThTha +b =c a b <br />
c Figure 13<br />
y<br />
<br />
<br />
<br />
–<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
x <br />
y <br />
<br />
<br />
x <br />
y <br />
<br />
d = c<br />
y <br />
− y <br />
= b<br />
x <br />
y <br />
<br />
x <br />
− x <br />
= a<br />
<br />
x<br />
Figure 13<br />
Th|x <br />
−x <br />
||y <br />
−y <br />
| d a b c<br />
<br />
|−|=Th|x <br />
−x <br />
||y <br />
−y <br />
|<br />
o ficn Th<br />
<br />
c =a +b →c =√ — a +b <br />
<br />
d =x <br />
−x <br />
+y <br />
−y <br />
→d =√ ——<br />
x <br />
−x <br />
+y <br />
−y <br />
<br />
fi<br />
the distance formula<br />
x <br />
y <br />
x <br />
y <br />
<br />
d=√ ——<br />
x <br />
−x <br />
+y <br />
−y <br />
<br />
Example 5 Finding the Distance between Two Points<br />
−−<br />
Solution fiFigure 14
SECTION 2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 81<br />
y<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
−− –<br />
–<br />
–<br />
<br />
<br />
<br />
−<br />
x<br />
Figure 14<br />
Th d <br />
d =√ ——<br />
x <br />
−x <br />
+y <br />
−y <br />
<br />
d =√ ——<br />
−− +−− <br />
<br />
=√ — + <br />
<br />
= √ <br />
— +<br />
<br />
=√ — <br />
Try It #3<br />
<br />
Example 6<br />
Finding the Distance Between Two Locations<br />
<br />
<br />
Figure 1<br />
d fi<br />
Solution Thfi<br />
<br />
fiTh<br />
ft<br />
Figure 15<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 15
82<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
<br />
<br />
<br />
<br />
Th<br />
<br />
<br />
<br />
Table 3<br />
From/To<br />
<br />
<br />
<br />
<br />
<br />
Number of Feet Driven<br />
<br />
<br />
<br />
<br />
<br />
Table 3<br />
ThTh<br />
fi<br />
<br />
<br />
<br />
<br />
d =√ — − +− <br />
=√ — +<br />
=√ — <br />
≈<br />
Th<br />
<br />
fl<br />
fl<br />
<br />
Using the Midpoint Formula<br />
fiTh<br />
midpoint formulax <br />
y <br />
<br />
x <br />
y <br />
o fiM<br />
M=( x +x y +y ) <br />
Figure 16<br />
<br />
y<br />
<br />
x +x y +y <br />
<br />
<br />
Figure 16<br />
x
SECTION 2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 83<br />
Example 7 Finding the Midpoint of the Line Segment<br />
−<br />
Solution o fi<br />
<br />
<br />
( x +x <br />
<br />
y +y <br />
<br />
−+ ) <br />
) = ( + <br />
= ( ) <br />
Try It #4<br />
−−−<br />
Example 8<br />
Finding the Center of a Circle<br />
Th−−−<br />
Solution ThTh<br />
<br />
<br />
<br />
( x +x <br />
<br />
y +y ) <br />
( −+ −− <br />
<br />
) = ( <br />
− ) =−<br />
<br />
Plotting Points on the Coordinate Plane (http://openstaxcollege.org/l/coordplotpnts)<br />
Find x- and y-intercepts Based on the Graph of a Line (http://openstaxcollege.org/l/xyintsgraph)
84 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
2.1 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. y-<br />
<br />
2. x-<br />
y-<br />
4. <br />
d=√ ——<br />
x <br />
−x <br />
+y <br />
−y <br />
<br />
<br />
<br />
ALGEBRAIC<br />
fi x- y-<br />
<br />
5. y =−x + 6. y =x − 7. x −y = 8. x −=y<br />
9. x +y = 10. x − <br />
= y +<br />
<br />
y x<br />
11. x +y = 12. x −y = 13. x =−y 14. x −y =<br />
15. y +=x 16. x +y =<br />
fi<br />
<br />
17. −− 18. − 19. 20. −<br />
21. <br />
<br />
fi<br />
22. −− 23. −− 24. −−−− 25. −<br />
26. −−<br />
GRAPHICAL<br />
<br />
27. 28. y-<br />
x-<br />
29. x-<br />
y-<br />
<br />
<br />
30. −− 31. −<br />
y<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x
SECTION 2.1 SECTION EXERCISES 85<br />
32. −−−− 33. <br />
y<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
A<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
B<br />
<br />
<br />
<br />
C<br />
<br />
x<br />
34. <br />
a. b. c. d. − e. −<br />
<br />
35. y = x + 36. y =−x + 37. y =x +<br />
<br />
NUMERIC<br />
fi x- y-<br />
<br />
38. x −y = 39. x −y = 40. y −=x 41. y =−x + 42. y = x − <br />
e fi<br />
y<br />
43. <br />
<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
44. <br />
<br />
45. −<br />
46. <br />
<br />
47. <br />
TECHNOLOGY<br />
Y=<br />
ft2 nd CALC1:valueENTER<br />
x= x y x <br />
x =s fi y-<br />
48. <br />
=−x + 49. <br />
= x− 50. =x + <br />
<br />
Y=<br />
ft2 nd CALC2:zeroENTER<br />
left bound? x-ENTER<br />
right bound? x-ENTERguess?<br />
x-ENTER<br />
x- y-fi x-
86 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
<br />
fi x-ft<br />
x-,<br />
fi <br />
51. <br />
=−x + 52. <br />
=x− 53. <br />
= x + <br />
EXTENSIONS<br />
54. <br />
<br />
<br />
<br />
<br />
56. AB−C<br />
D− <br />
_ AB <br />
_ CD<br />
58. <br />
<br />
<br />
55. <br />
<br />
<br />
57. ft <br />
<br />
<br />
59. <br />
<br />
–<br />
– – – – –<br />
––<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
x<br />
REAL-WORLD APPLICATIONS<br />
60. Th<br />
<br />
<br />
<br />
62. <br />
. Th<br />
<br />
<br />
<br />
<br />
<br />
61. −<br />
<br />
<br />
<br />
63. <br />
<br />
<br />
<br />
<br />
<br />
64. <br />
<br />
y x
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 87<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Solve equations in one variable algebraically.<br />
Solve a rational equation.<br />
Find a linear equation.<br />
Given the equations of two lines, determine whether their graphs are parallel or perpendicular.<br />
Write the equation of a line parallel or perpendicular to a given line.<br />
2.2 LINEAR EQUATIONS IN ONE VARIABLE<br />
<br />
<br />
<br />
<br />
<br />
Figure 1<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 1<br />
x<br />
Solving Linear Equations in One Variable<br />
linear equationTh<br />
ax+b =<br />
<br />
identity equation<br />
<br />
x =x +x<br />
solution set<br />
x <br />
conditional equation<br />
x +=x −<br />
<br />
x +=x −<br />
<br />
x =−<br />
<br />
x =−<br />
Th−<br />
<br />
inconsistent equationx −=x−<br />
<br />
<br />
x −=x −<br />
x −−x =x −−x x<br />
−≠−
88<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
−≠−. Th<br />
<br />
<br />
linear equation in one variable<br />
<br />
ax+b =<br />
a b a≠<br />
How To…<br />
<br />
Th<br />
x = x . Th<br />
1. <br />
<br />
2. ab+c=ab+ac<br />
3. <br />
4. ffifi<br />
ffi<br />
Example 1<br />
Solving an Equation in One Variable<br />
x +=<br />
Solution Thax+b =<br />
<br />
<br />
x +=<br />
x = <br />
x = <br />
<br />
Th<br />
Try It #1<br />
x+=−<br />
Example 2<br />
Solving an Equation <strong>Algebra</strong>ically When the Variable Appears on Both Sides<br />
x−+=−x+<br />
Solution <br />
<br />
x−+=−x+<br />
x −+=−x − <br />
x =−−x <br />
x =− x-<br />
<br />
x =− <br />
<br />
<br />
x =− <br />
<br />
Analysis This problem requires the distributive property to be applied twice, and then the properties of algebra are used<br />
to reach the final line, x = − _<br />
<br />
.
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 89<br />
Try It #2<br />
−x−+x=−x<br />
Solving a Rational Equation<br />
ft<br />
rational equation<br />
<br />
<br />
<br />
<br />
x + <br />
x − <br />
x − <br />
x +x − <br />
<br />
<br />
<br />
<br />
<br />
<br />
Example 3<br />
Solving a Rational Equation<br />
<br />
x − <br />
<br />
x = <br />
<br />
Solution xxThxxx<br />
<br />
x<br />
<br />
x( <br />
x − <br />
x ) = ( <br />
) x<br />
<br />
x( <br />
x ) −x( <br />
x ) =( <br />
) x<br />
<br />
<br />
x( <br />
x ) − x( <br />
x ) =( <br />
<br />
) x <br />
−=x <br />
<br />
−=x<br />
<br />
=x<br />
<br />
<br />
<br />
=x<br />
<br />
=x<br />
fi<br />
x+<br />
xx −<br />
x −xx−x−<br />
x−Th x <br />
fi x x−ff<br />
<br />
Thxx−Th<br />
<br />
<br />
xx−=xx−<br />
xx−
90<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
x x +x<br />
x +x =xx+ x xx+<br />
<br />
<br />
<br />
a <br />
b =c d <br />
fi<br />
<br />
<br />
a <br />
b =c <br />
d a <br />
b =c <br />
d <br />
adbc,ad=bc<br />
<br />
rational equations<br />
rational equation<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. ft<br />
5. <br />
6. <br />
Example 4<br />
Solving a Rational Equation without Factoring<br />
<br />
<br />
x − <br />
= <br />
x <br />
Solution xxThfi<br />
x<br />
x<br />
<br />
x ( <br />
x − <br />
) = ( <br />
x ) x<br />
<br />
x( <br />
x ) − x( <br />
)<br />
<br />
=( <br />
x )<br />
x x<br />
−x = <br />
<br />
−x =<br />
<br />
−x =<br />
<br />
x =−<br />
<br />
−<br />
Th−−−<br />
<br />
Try It #3<br />
<br />
x = <br />
− <br />
<br />
x
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 91<br />
Example 5<br />
Solving a Rational Equation by Factoring the Denominator<br />
x = <br />
<br />
− <br />
<br />
x <br />
Solution fiThx=ċ<br />
x =ċċxThxx =<br />
x<br />
<br />
<br />
<br />
<br />
Th <br />
<br />
<br />
− <br />
x ) x<br />
=x −<br />
=x<br />
x ( x ) = ( <br />
<br />
=x<br />
Try It #4<br />
− <br />
x + <br />
<br />
x =− <br />
<br />
Example 6 Solving Rational Equations with a Binomial in the Denominator<br />
<br />
<br />
a. <br />
x− = <br />
x b. x <br />
x− = <br />
x− − <br />
<br />
c. <br />
x <br />
x− = <br />
x− − <br />
<br />
Solution<br />
a. Thxx −n. Thxx−<br />
<br />
<br />
<br />
<br />
x − = <br />
x <br />
x =x− <br />
<br />
x =x −<br />
<br />
−x =−<br />
<br />
x =<br />
Th. Th<br />
b. Thx−x−<br />
<br />
x<br />
x−(<br />
x − ) = <br />
( <br />
x − − ) x −<br />
<br />
x−x<br />
x − <br />
x− = x − − x− <br />
<br />
<br />
x =−x−<br />
<br />
x =−x +<br />
<br />
x =−x<br />
<br />
x =<br />
<br />
Th . Th<br />
<br />
x =
92<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
<br />
<br />
<br />
<br />
<br />
c. Thx−xx−<br />
x−( <br />
x <br />
x − ) = <br />
( <br />
x − − <br />
) x −<br />
Th. Th<br />
x =−x−<br />
x =−x<br />
x =<br />
x =<br />
Try It #5<br />
−<br />
<br />
x+ = <br />
x+ <br />
Example 7 Solving a Rational Equation with Factored Denominators and Stating Excluded Values<br />
<br />
ft <br />
x + − <br />
x − = x<br />
<br />
x − <br />
Solution x −ff<br />
x−x+Thx−x+<br />
<br />
<br />
<br />
x−x+( <br />
x + − <br />
x − ) = ( x <br />
x−x+ )<br />
x −x+<br />
<br />
x−−x+=x<br />
x −−x −=x <br />
<br />
−−x =<br />
<br />
−=x<br />
Th−. Th−<br />
Try It #6<br />
<br />
<br />
x− + <br />
x+ = <br />
x −x− <br />
Finding a Linear Equation<br />
y =mx+b<br />
m =b =y<br />
The Slope of a Line<br />
slope y x <br />
<br />
<br />
<br />
m = y −y <br />
x −x <br />
<br />
ft<br />
Figure 2Thm =−<br />
m =m =
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 93<br />
y<br />
y = −x − <br />
<br />
<br />
<br />
<br />
<br />
<br />
––– – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
y = x + <br />
<br />
<br />
<br />
y = x + <br />
<br />
<br />
x<br />
Figure 2<br />
the slope of a line<br />
m y xx <br />
y <br />
x <br />
y <br />
<br />
<br />
<br />
m = y −y <br />
x −x <br />
<br />
Example 8<br />
Finding the Slope of a Line Given Two Points<br />
−−<br />
Solution y- x-<br />
<br />
<br />
<br />
Th− <br />
<br />
m = −− <br />
−− <br />
= <br />
− <br />
=− <br />
<br />
Analysis It does not matter which point is called x <br />
, y <br />
or x <br />
, y <br />
. As long as we are consistent with the order of the y<br />
terms and the order of the x terms in the numerator and denominator, the calculation will yield the same result.<br />
Try It #7<br />
−<br />
Example 9<br />
Identifying the Slope and y-intercept of a Line Given an Equation<br />
y- y =− x −<br />
<br />
Solution y =mx+b m =− y-b =−<br />
<br />
Analysis The y-intercept is the point at which the line crosses the y-axis. On the y-axis, x = . We can always identify<br />
the y-intercept when the line is in slope-intercept form, as it will always equal b. Or, just substitute x = and solve for y.<br />
The Point-Slope Formula<br />
n fi<br />
<br />
y −y <br />
=mx−x <br />
<br />
Thftfi<br />
ft
94<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
the point-slope formula<br />
<br />
<br />
y −y <br />
=mx−x <br />
<br />
Example 10<br />
Finding the Equation of a Line Given the Slope and One Point<br />
m =−fi<br />
<br />
Solution − m x <br />
y <br />
<br />
<br />
y −y <br />
=mx−x <br />
<br />
<br />
y −=−x−<br />
<br />
y −=−x +<br />
<br />
y =−x +<br />
Analysis Note that any point on the line can be used to find the equation. If done correctly, the same final equation will<br />
be obtained.<br />
Try It #8<br />
m= <br />
Example 11<br />
Finding the Equation of a Line Passing Through Two Given Points<br />
−fi<br />
<br />
Solution <br />
<br />
<br />
m = −− <br />
− <br />
= − <br />
− <br />
<br />
= <br />
<br />
<br />
x y <br />
<br />
<br />
y −= <br />
x−<br />
y −= <br />
x − <br />
<br />
<br />
y = x −<br />
<br />
y = x −<br />
<br />
Analysis To prove that either point can be used, let us use the second point , − and see if we get the same equation.<br />
<br />
<br />
y −−= <br />
x−<br />
y += <br />
x<br />
<br />
y = x −<br />
<br />
We see that the same line will be obtained using either point. This makes sense because we used both points to calculate<br />
the slope.
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 95<br />
Standard Form of a Line<br />
<br />
<br />
Ax+By=<br />
AB C Th x- y-<br />
<br />
Example 12<br />
Finding the Equation of a Line and Writing It in Standard Form<br />
m =−( <br />
− ) <br />
Solution <br />
<br />
y −−=−( x − <br />
) <br />
<br />
y +=−x + <br />
<br />
<br />
<br />
<br />
<br />
<br />
Th<br />
y+=( −x + <br />
) <br />
y +=−x +<br />
x +y =−<br />
Try It #9<br />
m =− <br />
( <br />
) <br />
Vertical and Horizontal Lines<br />
Th<br />
. Th<br />
<br />
x =c<br />
c Thfi y-<br />
x-c<br />
fi−−−−<br />
−l fi<br />
−<br />
<br />
m = <br />
−−− = <br />
fi<br />
x-fi<br />
x =−Figure 3<br />
Th<br />
<br />
y =c<br />
c Th x- y-<br />
c<br />
fi−−−−<br />
−e fi
96<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
<br />
m = −−− <br />
−− <br />
<br />
= <br />
<br />
<br />
=<br />
x <br />
y <br />
y-<br />
<br />
y −−=x−<br />
<br />
y +=<br />
<br />
y =−<br />
Thy =− y-Figure 3<br />
x= −<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
y= −<br />
x<br />
Example 13<br />
Figure 3 The line x = −3 is a vertical line. The line y = −2 is a horizontal line.<br />
Finding the Equation of a Line Passing Through the Given Points<br />
−<br />
Solution x-. Thx =<br />
Try It #10<br />
−<br />
Determining Whether Graphs of Lines are Parallel or Perpendicular<br />
ff y-<br />
Figure 4m =<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
y = x − <br />
y = x + <br />
y = x + <br />
x<br />
<br />
Figure 4 Parallel lines<br />
ff y-<br />
Th
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 97<br />
−m <br />
ċm <br />
=−<br />
Figure 5− <br />
<br />
m <br />
ċm <br />
=−<br />
<br />
ċ( − <br />
) =−<br />
y<br />
y = x − <br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
y = −<br />
<br />
x − <br />
<br />
Example 14<br />
Figure 5 Perpendicular lines<br />
Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither<br />
y =−x +<br />
x −y =<br />
Solution The fi<br />
<br />
<br />
y =−x +<br />
<br />
<br />
<br />
<br />
<br />
Figure 6<br />
y = −<br />
<br />
x + <br />
<br />
– – – –<br />
x −y =<br />
y =− x +<br />
<br />
−y =−x +<br />
–<br />
y = x −<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
y =<br />
<br />
x − <br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 6<br />
<br />
<br />
<br />
m <br />
=− <br />
<br />
m <br />
= <br />
<br />
<br />
m <br />
ċm <br />
=( − <br />
) ( <br />
) =−<br />
Thfi
98<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Try It #11<br />
y−x=y=x+<br />
Writing the Equations of Lines Parallel or Perpendicular to a Given Line<br />
fi<br />
fi<br />
ftr fi<br />
How To…<br />
<br />
1. . Th<br />
2. <br />
3. <br />
Example 15<br />
Writing the Equation of a Line Parallel to a Given Line Passing Through a Given Point<br />
x +y =<br />
Solution o fi<br />
<br />
x +y =<br />
<br />
y =x +<br />
<br />
y =− <br />
x + <br />
<br />
Th m =− <br />
y- <br />
<br />
Th y-<br />
. Th<br />
<br />
<br />
<br />
Th y =− <br />
y −=− <br />
x−<br />
y −=− x +<br />
<br />
y =− x +<br />
<br />
x +Figure 7 Figure 7<br />
y = −<br />
<br />
x +<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
y = −<br />
<br />
x + <br />
<br />
<br />
x<br />
Try It #12<br />
x=+y−−
SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 99<br />
Example 16<br />
Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point<br />
x −y +=−<br />
Solution The fi<br />
<br />
x −y +=<br />
<br />
−y =−x −<br />
<br />
m = <br />
y = <br />
x + <br />
<br />
Th<br />
− <br />
<br />
<br />
<br />
<br />
<br />
y −=− <br />
x−−<br />
y −=− <br />
x − <br />
<br />
y =− <br />
x − <br />
+ <br />
<br />
y =− <br />
x − <br />
<br />
<br />
<br />
<br />
<br />
findperpline)
100 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
2.2 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
y =x +<br />
<br />
<br />
5. <br />
<br />
x − = <br />
x + <br />
x =x =−<br />
<br />
<br />
2. <br />
<br />
<br />
4. <br />
<br />
ALGEBRAIC<br />
x<br />
6. x +=x − 7. x−= 8. x +−=x +<br />
9. −x +=x − 10. <br />
− <br />
x= <br />
<br />
11. x <br />
− <br />
=x + <br />
<br />
12. <br />
x + <br />
= <br />
13. x −+x =x + 14. x<br />
<br />
− <br />
15. x + <br />
−x − <br />
=<br />
=x <br />
+ <br />
<br />
x x-<br />
16. x − <br />
= <br />
<br />
x<br />
19. <br />
x − += <br />
x − <br />
<br />
17. − <br />
x+ =x+ <br />
x+ <br />
20. <br />
x+ + <br />
x− =<br />
− <br />
x −x− <br />
18. <br />
x − = <br />
x − + <br />
<br />
x −x − <br />
21. <br />
x = <br />
+ <br />
x <br />
fifi<br />
<br />
22. <br />
<br />
23. − <br />
<br />
24. x− 25. y−<br />
26. −− 27. <br />
28. y =x +<br />
<br />
29. y =x −<br />
−<br />
, fi<br />
30. −− 31. <br />
32. Th <br />
<br />
34. Th <br />
<br />
33. Th<br />
−<br />
35. −−
SECTION 2.2 SECTION EXERCISES 101<br />
GRAPHICAL<br />
<br />
<br />
x +<br />
38. y =<br />
<br />
36. y =x +<br />
37.<br />
y=− x −<br />
<br />
NUMERIC<br />
x −y =<br />
y −x =<br />
y =x +<br />
, fi<br />
39. x =<br />
y =−<br />
40. 41. −− 42. − 43. −−<br />
44. −−<br />
fi<br />
<br />
45. −<br />
−<br />
TECHNOLOGY<br />
46. <br />
−−<br />
<br />
Y1yminymax<br />
yyminymax<br />
47. x−y= 48. x −y = −y<br />
49. <br />
x =<br />
EXTENSIONS<br />
50. <br />
y −y <br />
=mx −x <br />
x <br />
x <br />
yy <br />
m<br />
52. <br />
<br />
x −y =<br />
54. <br />
<br />
51. <br />
Ax +By =C y <br />
ABCx. Th<br />
<br />
53. <br />
<br />
<br />
<br />
−<br />
REAL-WORLD APPLICATIONS<br />
55. Th<br />
<br />
<br />
s 2.5 ft, fin<br />
<br />
<br />
56. fi x<br />
y <br />
p =x +y y p =<br />
x =<br />
Th<br />
y =+x x <br />
57. 58. <br />
<br />
59.
102<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Set up a linear equation to solve a real-world application.<br />
Use a formula to solve a real-world application.<br />
2.3 MODELS AND APPLICATIONS<br />
Figure 1 Credit: Kevin Dooley<br />
fifi<br />
fi<br />
<br />
<br />
<br />
x <br />
Th<br />
<br />
Setting up a Linear Equation to Solve a Real-World Application<br />
fifi<br />
fiTh<br />
<br />
ThxTh<br />
<br />
<br />
fi<br />
o fiC<br />
<br />
C =x +<br />
<br />
Table 1<br />
Verbal<br />
a<br />
<br />
a<br />
a<br />
Thab<br />
Tha<br />
<br />
Th<br />
b c<br />
Table 1<br />
Translation to Math Operations<br />
xx +a<br />
x<br />
xx +a<br />
xx −a<br />
ax−b<br />
x<br />
<br />
x +a =x<br />
xx−b=c
SECTION 2.3 MODELS AND APPLICATIONS 103<br />
How To…<br />
o fi<br />
1. <br />
2. <br />
3. , fie fi<br />
4. <br />
5. <br />
Example 1<br />
Modeling a Linear Equation to Solve an Unknown Number Problem<br />
<br />
<br />
Solution x fiThfi<br />
x +. Th<br />
<br />
x +x+=<br />
x += <br />
<br />
x =<br />
<br />
x =<br />
<br />
x +=+<br />
<br />
=<br />
Th<br />
Try It #1<br />
<br />
<br />
Example 2 Setting Up a Linear Equation to Solve a Real-World Application<br />
Thffff A <br />
B <br />
a. ff<br />
b. ff<br />
c. ff<br />
d. <br />
Solution<br />
a. Th A A =x +Thx<br />
B<br />
xBB =x +<br />
b. <br />
<br />
A =+<br />
<br />
=+<br />
<br />
=<br />
<br />
B =+<br />
<br />
=+<br />
<br />
=<br />
B ffff<br />
A
104<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
c. <br />
<br />
A =+<br />
<br />
=+<br />
<br />
=<br />
<br />
B =+<br />
<br />
=+<br />
<br />
=<br />
A ff<br />
B<br />
d. <br />
xyxy<br />
x<br />
<br />
x +=x +<br />
<br />
x =<br />
<br />
x =<br />
x-<br />
<br />
+=<br />
<br />
+=<br />
ThFigure 2.<br />
<br />
y<br />
<br />
<br />
B=x+<br />
<br />
<br />
<br />
A=x+<br />
<br />
<br />
<br />
Figure 2<br />
x<br />
<br />
Try It #2<br />
<br />
s. Th<br />
<br />
Using a Formula to Solve a Real-World Application<br />
Thfi<br />
<br />
<br />
areaA =LWperimeter<br />
P =L +WV =LWHfi<br />
<br />
Example 3<br />
Solving an Application Using a Formula
SECTION 2.3 MODELS AND APPLICATIONS 105<br />
Solution Thd =rt<br />
<br />
<br />
<br />
<br />
r <br />
<br />
d. A tTable 2ft<br />
<br />
<br />
d r t<br />
To Work d r <br />
<br />
To Home d r− <br />
<br />
Table 2<br />
d =r ( <br />
) <br />
<br />
<br />
d =r−( <br />
) <br />
r<br />
<br />
r( <br />
) =r−( <br />
) <br />
<br />
<br />
r = <br />
r − <br />
<br />
<br />
<br />
r − <br />
r =− <br />
<br />
<br />
<br />
<br />
− <br />
r =− <br />
<br />
r =− <br />
−<br />
r =<br />
<br />
d<br />
<br />
<br />
Th<br />
d =( <br />
) <br />
=<br />
Analysis Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the<br />
LCD to solve for r.<br />
<br />
r( <br />
) =r−( <br />
) <br />
<br />
r( <br />
) =r−( <br />
) <br />
<br />
r =r−<br />
<br />
r =r −<br />
<br />
−r=−<br />
<br />
r =<br />
Try It #3<br />
er 3.6 h t<br />
er 4 h t
106<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 4<br />
Solving a Perimeter Problem<br />
ThftTh <br />
<br />
Solution ThP =L +W<br />
L =W +<br />
ftFigure 3<br />
W<br />
L = W +<br />
Figure 3<br />
<br />
<br />
P =L +W<br />
<br />
=W++W<br />
<br />
=W ++W<br />
<br />
=W +<br />
<br />
=W<br />
<br />
=W<br />
<br />
+=L<br />
<br />
=L<br />
ThL = W =ft<br />
Try It #4<br />
f a r<br />
Example 5<br />
Solving an Area Problem<br />
ThTh<br />
<br />
Solution ThA =LW<br />
Th<br />
<br />
ft<br />
L =W +<br />
d fi<br />
<br />
P =L +W<br />
<br />
=W++W<br />
<br />
=W ++W<br />
<br />
=W +<br />
<br />
=W<br />
<br />
=W<br />
<br />
+=L<br />
<br />
=L
SECTION 2.3 MODELS AND APPLICATIONS 107<br />
e fiL =W =<br />
<br />
A =LW<br />
<br />
A =<br />
<br />
= <br />
Th <br />
Try It #5<br />
<br />
<br />
s fiy ft <br />
Example 6 Solving a Volume Problem<br />
<br />
<br />
Solution ThV =LWH<br />
L =WH =. Th<br />
<br />
V =LWH<br />
<br />
=WW<br />
<br />
=W <br />
<br />
=W <br />
<br />
=W<br />
ThL =W =H =<br />
Analysis<br />
Note that the square root of W 2 would result in a positive and a negative value. However, because we are<br />
describing width, we can use only the positive result.
108 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
2.3 SECTION EXERCISES<br />
VERBAL<br />
1. o fi<br />
<br />
3. <br />
x <br />
<br />
2. <br />
n n +=n<br />
4. <br />
n <br />
n <br />
5. v <br />
<br />
REAL-WORLD APPLICATIONS<br />
fi<br />
<br />
6. <br />
<br />
<br />
<br />
<br />
8. y fi<br />
. Thy fi<br />
<br />
y fi<br />
7. <br />
<br />
<br />
ffff<br />
y y <br />
<br />
9. <br />
m <br />
11. <br />
<br />
10. <br />
m <br />
12. <br />
<br />
: ff<br />
Th<br />
Th<br />
P <br />
<br />
13. 14. <br />
<br />
15. <br />
<br />
<br />
16.
SECTION 2.3 SECTION EXERCISES 109<br />
: <br />
Th<br />
<br />
17. x<br />
<br />
<br />
19. es fl<br />
<br />
<br />
21. ft<br />
<br />
. Th<br />
<br />
18. <br />
<br />
<br />
20. <br />
ft<br />
<br />
<br />
22. <br />
<br />
<br />
<br />
23. <br />
<br />
<br />
<br />
<br />
ff<br />
<br />
24. <br />
<br />
26. <br />
<br />
25. <br />
<br />
27. <br />
<br />
fi<br />
<br />
28. A=P+rt P<br />
rt<br />
P<br />
A=<br />
30. F =maFm<br />
a<br />
f 12 N i<br />
29. ThF= mv <br />
R Fv<br />
mRRm=<br />
v=F=<br />
<br />
31. Sum = <br />
−r<br />
r<br />
ft<br />
<br />
32. WP=L+W 33. <br />
W<br />
<br />
34. f <br />
p + <br />
q = f <br />
35. f<br />
p =q =<br />
36. m <br />
y =mx +b<br />
37. <br />
m <br />
b =
110 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
38. ThA = <br />
hb +b <br />
<br />
<br />
h =b <br />
=b <br />
=<br />
39. hA = <br />
hb +b <br />
40. <br />
A =b <br />
=<br />
b <br />
=<br />
42. d =rt<br />
<br />
55 mi/h f<br />
44. <br />
3 h if o<br />
<br />
46. hA = <br />
bh<br />
48. ThV =πr h<br />
π <br />
r<br />
<br />
50. <br />
<br />
π<br />
52. <br />
<br />
π<br />
41. <br />
Th <br />
<br />
P =L +W<br />
43. <br />
<br />
<br />
45. A = bh <br />
<br />
<br />
<br />
47. <br />
<br />
<br />
49. hV =πr h<br />
51. rV =πr h<br />
53. Th<br />
C =πr<br />
r =r π<br />
<br />
54. π<br />
π
SECTION 2.4 COMPLEX NUMBERS 111<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Add and subtract complex numbers.<br />
Multiply and divide complex numbers.<br />
Simplify powers of i<br />
2.4 COMPLEX NUMBERS<br />
Figure 1<br />
Th<br />
<br />
fiTh<br />
<br />
<br />
fift<br />
Thfi Thfi<br />
<br />
t fid fi<br />
Expressing Square Roots of Negative Numbers as Multiples of i<br />
fifi<br />
Thff<br />
. Th i fi−<br />
<br />
√ — −=i<br />
<br />
<br />
i =( √ — −) =−<br />
i−<br />
<br />
√ — −=√ — ċ−<br />
<br />
=√ — √ — −<br />
<br />
=i<br />
i−i<br />
. <br />
a +bia b +i<br />
+i√ — <br />
+i<br />
<br />
<br />
ff
112<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
imaginary and complex numbers<br />
complex numbera +bi<br />
a<br />
b<br />
b =a +bia = b <br />
imaginary number<br />
How To…<br />
<br />
1. √ — −a √ — a √ — −<br />
2. √ — −i<br />
3. √ — a ċ i <br />
Example 1<br />
√ — −<br />
Solution<br />
<br />
+i<br />
Try It #1<br />
√ — −<br />
Expressing an Imaginary Number in Standard Form<br />
Plotting a Complex Number on the Complex Plane<br />
√ — −=√ — √ — −<br />
=i<br />
<br />
<br />
complex plane<br />
<br />
a,b,a b <br />
−+iTh−<br />
−−+iFigure 2<br />
y<br />
−+i<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 2<br />
complex plane<br />
<br />
Figure 3<br />
imaginary<br />
real<br />
Figure 3
SECTION 2.4 COMPLEX NUMBERS 113<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 2 Plotting a Complex Number on the Complex Plane<br />
−i<br />
Solution Th−−<br />
Figure 4<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 4<br />
Try It #2<br />
−−i<br />
Adding and Subtracting Complex Numbers<br />
<br />
<br />
complex numbers: addition and subtraction<br />
<br />
a +bi+c+di=a + c+b+di<br />
<br />
a+bi−c+di=a − c+b−di<br />
How To…<br />
, fiff<br />
1. <br />
2. <br />
3. <br />
Example 3 Adding and Subtracting Complex Numbers<br />
<br />
a. −i++i b. −+i−−+i
114<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Solution <br />
<br />
<br />
<br />
<br />
<br />
<br />
a. −i++i=−++i<br />
=++−i+i<br />
=++−+i<br />
=+<br />
b. −+i−−+i=−++−i<br />
=−++i −i<br />
=−++−i<br />
=+i<br />
Try It #3<br />
+i−i<br />
Multiplying Complex Numbers<br />
Thff<br />
<br />
Multiplying a Complex Number by a Real Number<br />
<br />
+i<br />
+i=·+·i <br />
=+i <br />
How To…<br />
o fi<br />
1. <br />
2. <br />
Example 4<br />
Multiplying a Complex Number by a Real Number<br />
+i<br />
Solution <br />
<br />
<br />
+i=ċ+ċi<br />
=+i<br />
Try It #4<br />
<br />
−i<br />
Multiplying Complex Numbers Together<br />
fi<br />
<br />
. Thffi −<br />
<br />
a+bic+di=ac+adi+bci+bdi <br />
=ac+adi+bci−bd i =−<br />
=ac−bd+ad+bci
SECTION 2.4 COMPLEX NUMBERS 115<br />
How To…<br />
o fi<br />
1. <br />
2. i =−<br />
3. <br />
Example 5<br />
+i−i<br />
Solution<br />
<br />
<br />
<br />
Try It #5<br />
−i+i<br />
Dividing Complex Numbers<br />
Multiplying a Complex Number by a Complex Number<br />
+i−i=−i+i−ii<br />
=−i +i −i <br />
=++−+i<br />
=−i<br />
<br />
<br />
a +bifi<br />
<br />
Th<br />
a +bia −bi<br />
a +bia −bi<br />
<br />
a+bia−bi=a −abi+abi−b i <br />
<br />
=a +b <br />
Th<br />
Tha +bia −bi<br />
a −bia +biffi<br />
<br />
c +dia +bia b fi<br />
n fi<br />
c +di<br />
a≠ b ≠<br />
a +bi<br />
<br />
<br />
<br />
<br />
i =−<br />
<br />
<br />
+di <br />
a+bi ċa−bi <br />
a−bi =+dia−bi <br />
a+bia−bi <br />
= ca−bi+adi−bdi <br />
<br />
a −abi+abi−b i <br />
= ca−cbi+adi−bd− <br />
<br />
a −abi+abi−b − <br />
= ca+bd+ad−cbi<br />
<br />
a +b
116<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
the complex conjugate<br />
complex conjugatea +bia −bi<br />
. Th <br />
<br />
<br />
Example 6<br />
Finding Complex Conjugates<br />
<br />
a. +i√ — b. − <br />
i<br />
Solution<br />
a. Tha +bi. Tha −bi−i√ — <br />
b. a +bi− <br />
i. Tha−bi+ <br />
i<br />
Th <br />
i<br />
Analysis Although we have seen that we can find the complex conjugate of an imaginary number, in practice we<br />
generally find the complex conjugates of only complex numbers with bothreal and an imaginary component. To obtain a<br />
real number from an imaginary number, we can simply multiply by i.<br />
Try It #6<br />
−+i<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 7<br />
+i−i<br />
Dividing Complex Numbers<br />
Solution <br />
+i <br />
−i <br />
Th<br />
+i <br />
−i ċ+i <br />
+i <br />
<br />
<br />
+i <br />
−i ċ+i <br />
+i =+i +i +i <br />
+i −i −i <br />
<br />
<br />
<br />
<br />
<br />
<br />
= +i +i <br />
+− <br />
+i −i −− i =−<br />
= +i<br />
<br />
<br />
= <br />
+ i
SECTION 2.4 COMPLEX NUMBERS 117<br />
Simplifying Powers of i<br />
Th i i <br />
<br />
i =<br />
<br />
i =−<br />
<br />
i =i ċi =−ċi =−i<br />
<br />
i =i ċi =−iċi =−i =−−=<br />
<br />
i =i ċi =ċi =<br />
fiftifi i <br />
i<br />
<br />
i =i ċi =i ċi =i =−<br />
<br />
i =i ċi =i ċi =i =−i<br />
<br />
i =i ċi =i ċi =i =<br />
<br />
i =i ċi =i ċi =i =<br />
Thi−−i<br />
Example 8<br />
i <br />
Simplifying Powers of i<br />
Solution i =i fi<br />
=ċ+<br />
i =i ċ+ =i ċ ċi =i ċi = ċi =i =−i<br />
Try It #7<br />
i <br />
Q & A…<br />
Can we write i 35 in other helpful ways?<br />
Example 8i i fi<br />
fii Table 1<br />
<br />
Factorization of i 35 i ċi i ċi i ċi i ċi <br />
Reduced form i ċi i ċ− i ċ i ċi <br />
Simplified form − ċi −i i i <br />
Table 1<br />
<br />
<br />
<br />
Adding and Subtracting Complex Numbers (http://openstaxcollege.org/l/addsubcomplex)<br />
Multiply Complex Numbers (http://openstaxcollege.org/l/multiplycomplex)<br />
Multiplying Complex Conjugates (http://openstaxcollege.org/l/multcompconj)<br />
Raising i to Powers (http://openstaxcollege.org/l/raisingi)
118 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
2.4 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
<br />
3. <br />
<br />
ALGEBRAIC<br />
<br />
4. <br />
<br />
5. y=x +x−yix=i 6. y=x −yix=i<br />
7. y=x +x+yix=+i 8. y=x +x−yix=−i<br />
9. y = x + y ix =i<br />
−x<br />
10. y =+x<br />
<br />
x + y ix =i<br />
GRAPHICAL<br />
<br />
11. −i 12. −+i 13. i 14. −−i<br />
NUMERIC<br />
fi<br />
15. +i+−i 16. −−i++i 17. −+i−−i 18. −i−+i<br />
19. −+i−−+i 20. +ii 21. −ii 22. −i<br />
23. −+i 24. +i−i 25. −+i−+i 26. −i+i<br />
27. +i−i 28. +i<br />
29. −i 30. −+i<br />
i<br />
31. +i <br />
i<br />
32. −i <br />
+i<br />
33. +i <br />
−i<br />
<br />
34. +i <br />
−i <br />
35. √ — −+√ — − 36. −√ — −−√ — − 37. +√— − 38. − +√— <br />
39. i 40. i 41. i <br />
TECHNOLOGY<br />
<br />
42. +i k k=<br />
k=<br />
44. l +i k −l −i k k =<br />
k =<br />
43. −i k k=<br />
k=<br />
45. x += √— <br />
+ <br />
i<br />
46. x −= √— <br />
+√— <br />
i<br />
EXTENSIONS<br />
fi<br />
47. <br />
i + <br />
i <br />
48. <br />
i − <br />
i 49. i +i 50. i − +i <br />
51. +i−i <br />
+i 52. +i−i <br />
+i 53. +i <br />
+i <br />
55. +i<br />
+ −i <br />
i −i<br />
56. +i<br />
+i −−i <br />
+i<br />
54. +i +i ++i
SECTION 2.5 QUADRATIC EQUATIONS 119<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Solve quadratic equations by factoring.<br />
Solve quadratic equations by the square root property.<br />
Solve quadratic equations by completing the square.<br />
Solve quadratic equations by using the quadratic formula.<br />
2.5 QUADRATIC EQUATIONS<br />
Figure 1<br />
Th Figure 1<br />
<br />
<br />
ff<br />
Solving Quadratic Equations by Factoring<br />
<br />
x +x −=x −=Thfi<br />
, fi<br />
ftfi<br />
<br />
<br />
aċb =a =b =a b <br />
<br />
<br />
<br />
<br />
x−x+<br />
<br />
x−x+=x +x −x −<br />
<br />
=x +x −<br />
Thx +x −=<br />
<br />
<br />
fifi<br />
ax +bx+c =ab c a≠. Thx +x −=<br />
fi<br />
ff
120<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
the zero-product property and quadratic equations<br />
zero-product property<br />
aċb =a =b =<br />
a b <br />
quadratic equation<br />
ax +bx+c =<br />
ab c a≠<br />
Solving Quadratics with a Leading Coefficient of 1<br />
x +x −=ffiffix <br />
<br />
How To…<br />
ffi<br />
1. c b<br />
2. x+kx−k, k <br />
−x+x−<br />
3. <br />
Example 1 Factoring and Solving a Quadratic with Leading Coefficient of 1<br />
x +x −=<br />
Solution x +x −=−<br />
−<br />
ċ−<br />
−ċ<br />
ċ−<br />
ċ−<br />
Thċ−Th<br />
<br />
<br />
x−x+=<br />
<br />
<br />
x−x+=<br />
<br />
x−=<br />
<br />
x =<br />
<br />
x+=<br />
<br />
x =−<br />
Th−Figure 2Th<br />
x- y =x +x −=<br />
y<br />
x +x− =<br />
−<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x<br />
Figure 2
SECTION 2.5 QUADRATIC EQUATIONS 121<br />
Try It #1<br />
x −x−=<br />
Example 2<br />
Solve the Quadratic Equation by Factoring<br />
x +x +=<br />
Solution <br />
ċ<br />
ċ<br />
−ċ−<br />
−ċ−<br />
Th. Th<br />
<br />
x+x+=<br />
<br />
x+=<br />
<br />
x =−<br />
<br />
x+=<br />
<br />
x =−<br />
Th−−<br />
Try It #2<br />
x −x−=<br />
Example 3<br />
Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares<br />
ffx −=<br />
Solution ff<br />
<br />
<br />
<br />
x −=<br />
<br />
x−x+=<br />
<br />
x−=<br />
<br />
x =<br />
<br />
x+=<br />
<br />
x =−<br />
Th−<br />
Try It #3<br />
x −=<br />
Factoring and Solving a Quadratic Equation of Higher Order<br />
ffi<br />
<br />
1. ax +bx+c =a ċc<br />
2. acb<br />
3. bx x<br />
4. Th<br />
<br />
5. <br />
6.
122<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 4<br />
Solving a Quadratic Equation Using Grouping<br />
x +x +=<br />
Solution ac=. Th<br />
<br />
ċ<br />
<br />
ċ<br />
<br />
ċ<br />
<br />
ċ<br />
<br />
ċ<br />
Th+ b x<br />
ffixe fi<br />
<br />
x +x +x +=<br />
<br />
xx ++x +=<br />
<br />
x +x+=<br />
<br />
<br />
x +x+=<br />
<br />
x +=<br />
<br />
<br />
<br />
Th− −Figure <br />
3<br />
<br />
x =− <br />
<br />
x+=<br />
x =−<br />
y<br />
x <br />
+x+ =<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 3<br />
Try It #4<br />
x +x+=<br />
Example 5<br />
Solving a Higher Degree Quadratic Equation by Factoring<br />
−x −x −x =<br />
Solution Thfi<br />
−x<br />
<br />
<br />
−x −x −x =<br />
<br />
−xx +x +=<br />
<br />
<br />
−xx +x +x +=<br />
<br />
−xxx++x+=<br />
<br />
−xx +x+=
SECTION 2.5 QUADRATIC EQUATIONS 123<br />
<br />
<br />
−x=<br />
<br />
x =<br />
<br />
x +=<br />
<br />
x =− <br />
<br />
<br />
x +=<br />
<br />
x =−<br />
Th− <br />
−<br />
Try It #5<br />
x +x +x=<br />
Using the Square Root Property<br />
square<br />
root propertyx <br />
x <br />
<br />
the square root property<br />
x <br />
x =kx =±√ — k <br />
k <br />
How To…<br />
x x <br />
1. x <br />
2. ±<br />
<br />
3. ±<br />
Example 6<br />
Solving a Simple Quadratic Equation Using the Square Root Property<br />
x =<br />
Solution ±<br />
<br />
<br />
x =<br />
<br />
x =±√ — <br />
<br />
=±√ — <br />
Th√ — −√ — <br />
Example 7 Solving a Quadratic Equation Using the Square Root Property<br />
x +=<br />
Solution x . Th<br />
<br />
x +=<br />
<br />
x =<br />
<br />
<br />
Th √— <br />
−√— <br />
<br />
x = <br />
<br />
x =± √—
124<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Try It #6<br />
x− =<br />
Completing the Square<br />
<br />
completing the square<br />
<br />
ffial <br />
aTh<br />
<br />
x +x +=<br />
1. a = <br />
<br />
<br />
x +x =−<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2. b <br />
<br />
<br />
=<br />
=<br />
3. ( <br />
b ) <br />
x +x +=−+<br />
x +x +=<br />
4. Th <br />
x +x +=<br />
x+ =<br />
5. <br />
√ — x+ =±√ — <br />
6. Th−+√ — −−√ — <br />
Example 8<br />
x +=±√ — <br />
Solving a Quadratic by Completing the Square<br />
x =−±√ — <br />
x −x −=<br />
Solution <br />
<br />
x −x =<br />
Th b <br />
<br />
<br />
<br />
−=− <br />
<br />
<br />
( − <br />
) = <br />
<br />
<br />
<br />
x −x +( − <br />
) =+( − <br />
) <br />
<br />
x −x + <br />
=+ <br />
<br />
<br />
<br />
( x − <br />
) =
SECTION 2.5 QUADRATIC EQUATIONS 125<br />
<br />
<br />
<br />
<br />
Th +√— <br />
−√— <br />
Try It #7<br />
x −x=<br />
Using the Quadratic Formula<br />
√ ( x − <br />
<br />
<br />
) =±<br />
√ <br />
( x − <br />
) =± √— <br />
<br />
x = <br />
±√— <br />
<br />
Th<br />
<br />
<br />
<br />
ffi<br />
−aax +bx+c =a≠<br />
<br />
1. <br />
<br />
ax +bx=−c<br />
2. a<br />
<br />
x + a b x =−c <br />
a <br />
3. Th <br />
( b __<br />
a ) = b <br />
a <br />
<br />
x + a b x +b <br />
<br />
<br />
<br />
a =b a −c <br />
a <br />
4. eft<br />
<br />
<br />
( x + b a ) = b −ac<br />
a <br />
<br />
5. <br />
<br />
x + b <br />
a =± √ b −ac<br />
a <br />
<br />
<br />
x + b <br />
a b −ac =±√— a<br />
<br />
6. − b . Th<br />
a<br />
x = −b±√— b −ac <br />
a<br />
<br />
the quadratic formula<br />
ax +bx+c =quadratic formula<br />
<br />
x = −b±√— b −ac <br />
<br />
a<br />
ab c a≠
126<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
How To…<br />
<br />
1. ax +bx+c =<br />
2. ffiabc<br />
3. <br />
<br />
4. <br />
Example 9 Solve the Quadratic Equation Using the Quadratic Formula<br />
x +x +=<br />
Solution ffi: a =b =c =. Th<br />
<br />
<br />
<br />
x = −±√— − <br />
<br />
<br />
= −±√— − <br />
<br />
= −±√— <br />
<br />
Example 10 Solving a Quadratic Equation with the Quadratic Formula<br />
x +x +=<br />
Solution ffi: a =b =c =<br />
<br />
<br />
<br />
<br />
<br />
<br />
b<br />
x = −b±√— −ac <br />
<br />
a<br />
<br />
= <br />
−±√— −ċċ <br />
<br />
ċ <br />
<br />
=<br />
− −±√— <br />
=<br />
− −±√— <br />
= −±i√— <br />
Th −+i√— <br />
−−i√— <br />
Try It #8<br />
x +x−=<br />
The Discriminant<br />
<br />
b −acTh<br />
Table 1
SECTION 2.5 QUADRATIC EQUATIONS 127<br />
Value of Discriminant<br />
b −ac=<br />
b −ac><br />
b −ac><br />
b −ac<<br />
Results<br />
<br />
<br />
<br />
<br />
Table 1<br />
the discriminant<br />
ax +bx+c =ab c discriminant<br />
b −ac<br />
<br />
Example 11<br />
Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation<br />
o fi<br />
a. x +x += b. x +x += c. x −x −= d. x −x +=<br />
Solution b −ac<br />
a. x +x +=<br />
b −ac= −=. Th<br />
b. x +x +=<br />
b −ac= −=<br />
c. x −x −=<br />
b −ac=− −−=<br />
d. x −x +=<br />
b −ac=− −=−. Th<br />
Using the Pythagorean Theorem<br />
Pythagorean Theorem<br />
a +b =c a b <br />
c <br />
<br />
Th<br />
<br />
<br />
<br />
Thn Th<br />
<br />
a +b =c <br />
a b c <br />
Figure 4<br />
b<br />
c<br />
a<br />
Figure 4
128<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 12<br />
Finding the Length of the Missing Side of a Right Triangle<br />
Figure 5<br />
<br />
<br />
Figure 5<br />
Solution b a<br />
a<br />
<br />
<br />
<br />
<br />
<br />
<br />
a +b =c <br />
a + = <br />
a +=<br />
a =<br />
a =√ — <br />
=√ — <br />
Try It #9<br />
n Thab<br />
<br />
<br />
Solving Quadratic Equations by Factoring (http://openstaxcollege.org/l/quadreqfactor)<br />
The Zero-Product Property (http://openstaxcollege.org/l/zeroprodprop)<br />
Completing the Square (http://openstaxcollege.org/l/complthesqr)<br />
Quadratic Formula with Two Rational Solutions (http://openstaxcollege.org/l/quadrformrat)<br />
Length of a Leg of a Right Triangle (http://openstaxcollege.org/l/leglengthtri)
SECTION 2.5 SECTION EXERCISES 129<br />
2.5 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
<br />
5. <br />
<br />
<br />
2. <br />
<br />
<br />
ax +bx +c =<br />
y =ax +bx +cx<br />
4. <br />
b −ac<br />
<br />
<br />
ALGEBRAIC<br />
<br />
6. x +x−= 7. x −x += 8. x +x −= 9. x +x +=<br />
10. x −x+= 11. x −= 12. x +x −= 13. x =<br />
14. x +x = 15. x =x + 16. x =x 17. x +x =<br />
18. x <br />
− <br />
x =<br />
<br />
19. x = 20. x = 21. x − = 22. x − =<br />
23. x + = 24. x − =<br />
<br />
25. x −x −= 26. x −x −= 27. x −x = 28. x + <br />
x − <br />
=<br />
29. +z =z 30. p +p −= 31. x −x −=<br />
<br />
<br />
32. x −x += 33. x +x += 34. x +x −= 35. x −x +=<br />
36. x −x −= 37. x −x −=<br />
<br />
No Real Solution<br />
38. x +x += 39. x +x = 40. x −x −= 41. x −x +=<br />
42. x +x += 43. + x − <br />
x =<br />
TECHNOLOGY<br />
fi<br />
x-2 nd CALC 2:zerofi <br />
,,<br />
<br />
44. <br />
=x +x − 45. <br />
=−x +x − 46. <br />
=x +x −
130 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
47. x +x −=<br />
<br />
<br />
=x +x − <br />
=<br />
2 nd CALC<br />
5:intersection <br />
<br />
EXTENSIONS<br />
49. <br />
ax +bx +c = x <br />
<br />
<br />
51. <br />
<br />
<br />
119 ft. <br />
<br />
53. <br />
p =−x +x − x <br />
p <br />
fi<br />
fi<br />
<br />
<br />
2 nd CALC maximum<br />
x y <br />
REAL-WORLD APPLICATIONS<br />
54. <br />
A<br />
P =A −A +<br />
<br />
<br />
56. <br />
d =t +t t <br />
<br />
<br />
48. x +x −=<br />
<br />
<br />
=x +x − <br />
=<br />
2 nd CALC<br />
5:intersection <br />
<br />
50. <br />
− a b <br />
52. <br />
P =t −t + t <br />
t =<br />
<br />
<br />
55. Th<br />
C =x +<br />
R =x −x fi<br />
<br />
x fi<br />
57. <br />
den. Th<br />
f 378 ft <br />
<br />
x<br />
<br />
x<br />
x<br />
<br />
58. <br />
infl<br />
P<br />
e fl t ft<br />
P =−t +t +≤ t ≤<br />
e fl<br />
t <br />
x
SECTION 2.6 OTHER TYPES OF EQUATIONS 131<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Solve equations involving rational exponents.<br />
Solve equations using factoring.<br />
Solve radical equations.<br />
Solve absolute value equations.<br />
Solve other types of equations.<br />
2.6 OTHER TYPES OF EQUATIONS<br />
<br />
<br />
<br />
<br />
<br />
Solving Equations Involving Rational Exponents<br />
<br />
<br />
<br />
√ — √ <br />
— Th<br />
<br />
<br />
Th<br />
<br />
<br />
( <br />
) =( <br />
) =<br />
rational exponents<br />
Th<br />
<br />
a m <br />
n =( a <br />
n ) <br />
m<br />
=a m <br />
n = √ n — a m =( √ n — a ) m <br />
Example 1<br />
Evaluating a Number Raised to a Rational Exponent<br />
<br />
<br />
Solution fififi<br />
<br />
( <br />
<br />
<br />
Try It #1<br />
− <br />
<br />
<br />
<br />
) <br />
<br />
( <br />
<br />
) <br />
= <br />
=
132<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 2<br />
Solve the Equation Including a Variable Raised to a Rational Exponent<br />
x <br />
=<br />
Solution Th x <br />
<br />
<br />
<br />
<br />
x <br />
=<br />
<br />
( x <br />
) <br />
<br />
= <br />
<br />
x = Th <br />
<br />
=<br />
Try It #2<br />
x <br />
=<br />
Example 3<br />
Solving an Equation Involving Rational Exponents and Factoring<br />
x <br />
=x <br />
<br />
Solution Th<br />
<br />
<br />
<br />
x <br />
−( x <br />
) =x <br />
−( x <br />
−x <br />
x <br />
=<br />
<br />
x <br />
x <br />
. Thx <br />
ft<br />
<br />
x <br />
−x <br />
=<br />
<br />
x <br />
( x <br />
−) =<br />
x <br />
<br />
Thx <br />
<br />
<br />
Th <br />
x <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x <br />
( x <br />
−) =<br />
Th <br />
<br />
Try It #3<br />
x+ <br />
<br />
=<br />
x <br />
=<br />
x =<br />
) <br />
x <br />
−=<br />
x <br />
=<br />
x <br />
= <br />
<br />
( x <br />
<br />
) =( <br />
) <br />
<br />
x =
SECTION 2.6 OTHER TYPES OF EQUATIONS 133<br />
Solving Equations Using Factoring<br />
<br />
ffi<br />
ft<br />
polynomial equations<br />
n <br />
a n<br />
x n +a n−<br />
x n − +ċċċ+a <br />
x +a <br />
x +a <br />
n a n<br />
a <br />
a n<br />
≠<br />
polynomial equationTh<br />
n<br />
Example 4<br />
Solving a Polynomial by Factoring<br />
x =x <br />
Solution . Th<br />
<br />
x −x =<br />
<br />
x x −=<br />
ffx −<br />
Thfix <br />
x = x −=<br />
x = x−x+=<br />
x −= x +=<br />
x = x =−<br />
Th−<br />
Analysis We can see the solutions on the graph in Figure 1. The x-coordinates of the points where the graph crosses the<br />
x-axis are the solutions—the x-intercepts. Notice on the graph that at , the graph touches the x-axis and bounces back.<br />
It does not cross the x-axis. This is typical of double solutions.<br />
y<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
– –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 1<br />
Try It #4<br />
x =x
134<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 5<br />
Solve a Polynomial by Grouping<br />
x +x −x −=<br />
Solution Th<br />
fi<br />
<br />
<br />
x +x −x −=<br />
<br />
x x+−x+=<br />
<br />
x −x+=<br />
Thx −ff<br />
<br />
x −x+=<br />
<br />
x−x+x+=<br />
x −= x += x +=<br />
x = x =− x =−<br />
Th−−<br />
x-Figure 2<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 2<br />
Analysis We looked at solving quadratic equations by factoring when the leading coefficient is . When the leading<br />
coefficient is not , we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term<br />
of quadratic equations. We can also use grouping for some polynomials of degree higher than , as we saw here, since<br />
there were already four terms.<br />
Solving Radical Equations<br />
Radical equations,<br />
<br />
<br />
<br />
√ — x +=x<br />
√ — x +=x −<br />
√ — x +−√ — x −=<br />
<br />
fiextraneous solutions<br />
Th<br />
<br />
fi
SECTION 2.6 OTHER TYPES OF EQUATIONS 135<br />
radical equations<br />
radical equation<br />
How To…<br />
<br />
1. <br />
2. <br />
nn<br />
<br />
3. <br />
4. <br />
5. fi<br />
Example 6<br />
Solving an Equation with One Radical<br />
√ — −x=x<br />
Solution Th <br />
<br />
√ — −x =x<br />
<br />
<br />
( √ — −x ) =x <br />
−x =x <br />
<br />
<br />
=x +x −<br />
<br />
=x+x−<br />
=x+ =x −<br />
−=x =x<br />
Th−−<br />
<br />
<br />
<br />
<br />
√ — −x =x<br />
√ — −−=−<br />
√ — =−<br />
≠−<br />
Th<br />
<br />
<br />
<br />
√ — −x=x<br />
<br />
√ — −=<br />
<br />
√ — =<br />
<br />
=<br />
Th<br />
Try It #5<br />
√ — x+=x−
136<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 7<br />
Solving a Radical Equation Containing Two Radicals<br />
√ — x ++√ — x −=<br />
Solution <br />
<br />
<br />
√ — x ++√ — x −=<br />
√ — x +=−√ — x − √ — x −<br />
( √ — x +) =( −√ — x −) <br />
a − b =a −ab+b <br />
<br />
x += −√ — x −+( √ — x −) <br />
<br />
x +=−√ — x −+x−<br />
x +=+x −√ — x − <br />
x −=−√ — x − <br />
x− =( −√ — x −) <br />
x −x +=x−<br />
<br />
<br />
x −x +=x −<br />
<br />
x −x +=<br />
x−x−= <br />
x −= x −=<br />
x = x =<br />
Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
√ — x ++√ — x −=<br />
√ — x +=−√ — x −<br />
√ — +=−√ — −<br />
√ — =−√ — <br />
=<br />
√ — x ++√ — x −=<br />
√ — x +=−√ — x −<br />
√ — +=−√ — −<br />
√ — =−√ — <br />
≠−<br />
Th<br />
Try It #6<br />
√ — x++√ — x+=
SECTION 2.6 OTHER TYPES OF EQUATIONS 137<br />
Solving an Absolute Value Equation<br />
|x −|=<br />
−Thff<br />
<br />
x −= x −=−<br />
x = x =−<br />
x = x =−<br />
<br />
fi<br />
absolute value equations<br />
Th x |x|<br />
<br />
x ≥|x|=x<br />
<br />
x
138<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
c. |x −|−=<br />
<br />
<br />
|x −|−=<br />
<br />
|x −|=<br />
x −= x −=−<br />
x = x =−<br />
x = x =− <br />
<br />
Th− <br />
<br />
d. |−x +|=<br />
Th<br />
<br />
−x +=<br />
<br />
−x =−<br />
<br />
x =<br />
Th<br />
Try It #7<br />
|−x|+=<br />
Solving Other Types of Equations<br />
Th<br />
<br />
<br />
Solving Equations in Quadratic Form<br />
Equations in quadratic formThfiTh<br />
Th<br />
x −x +=x +x −=<br />
x <br />
+x <br />
+=<br />
<br />
quadratic form<br />
equation in<br />
quadratic form<br />
<br />
How To…<br />
<br />
1. <br />
2. u<br />
3. <br />
4. <br />
5. <br />
6.
SECTION 2.6 OTHER TYPES OF EQUATIONS 139<br />
Example 9<br />
Solving a Fourth-degree Equation in Quadratic Form<br />
x −x −=<br />
Solution Thfi<br />
u =x u<br />
<br />
<br />
<br />
<br />
u −u −=<br />
u −u −=<br />
u +u−=<br />
u<br />
u += u −=<br />
<br />
u =−<br />
u =<br />
<br />
u =− x <br />
=<br />
<br />
x =±<br />
<br />
x =− <br />
<br />
<br />
x =±i <br />
√ <br />
<br />
Th±i<br />
√ <br />
±<br />
Try It #8<br />
x −x −=<br />
Example 10<br />
Solving an Equation in Quadratic Form Containing a Binomial<br />
x+ +x+−=<br />
Solution ThTh<br />
fiff<br />
u =x +. Thu<br />
<br />
u +u −=<br />
<br />
u+−=<br />
u <br />
<br />
u +=<br />
<br />
u =−<br />
<br />
x +=−<br />
<br />
x =−<br />
Th<br />
<br />
u −=<br />
<br />
u =<br />
<br />
x +=<br />
<br />
x =−<br />
−−<br />
Try It #9<br />
x− −x−−=
140<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Solving Rational Equations Resulting in a Quadratic<br />
<br />
<br />
<br />
Example 11 Solving a Rational Equation Leading to a Quadratic<br />
−x <br />
x − + <br />
x + = − x − <br />
Solution o fi<br />
x −=x+x−Thx+x−x<br />
<br />
<br />
x+x−( −x <br />
x − + <br />
x + ) = ( <br />
− <br />
x+x− ) x+x−<br />
<br />
−xx++x−=−<br />
<br />
−x −x +x −=−<br />
<br />
−x +=<br />
<br />
−x −=<br />
<br />
−x+x−=<br />
<br />
x =−x =<br />
. Th<br />
Try It #10<br />
x+ <br />
x− + <br />
x = − <br />
x −x <br />
ff<br />
Rational Equation with No Solution (http://openstaxcollege.org/l/rateqnosoln)<br />
Solving Equations with Rational Exponents Using Reciprocal Powers (http://openstaxcollege.org/l/ratexprecpexp)<br />
Solving Radical Equations part 1 of 2 (http://openstaxcollege.org/l/radeqsolvepart1)<br />
Solving Radical Equations part 2 of 2 (http://openstaxcollege.org/l/radeqsolvepart2)
SECTION 2.6 SECTION EXERCISES 141<br />
2.6 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. e − <br />
<br />
ERROR<br />
<br />
2. must<br />
<br />
4. |x +|=−<br />
5. <br />
<br />
ALGEBRAIC<br />
<br />
<br />
6. x =<br />
<br />
7. x =<br />
<br />
8. x <br />
<br />
−x <br />
=<br />
9. x − <br />
=<br />
10. x + <br />
=<br />
<br />
11. x <br />
−x <br />
+=<br />
<br />
12. x <br />
−x <br />
<br />
−x <br />
=<br />
<br />
13. x +x −x −= 14. x −x −x += 15. y −y =<br />
16. x +x −x −= 17. m +m −m −= 18. x −x =<br />
19. x +x =x +<br />
<br />
20. √ — x −−= 21. √ — x −= 22. √ — x −=x −<br />
23. √ — t += 24. √ — t ++= 25. √ — −x =x<br />
26. √ — x +−√ — x += 27. √ — x ++√ — x += 28. √ — x +−√ — x +=<br />
<br />
29. |x −|= 30. |x −|=− 31. |−x|−= 32. |x +|−=<br />
33. |x −|−=− 34. |x +|−=− 35. |x +|= 36. −|x +|=−<br />
fi<br />
<br />
37. x −x += 38. t − −t −=− 39. x − +x −−=<br />
40. x + −x +−= 41. x − −=<br />
EXTENSIONS<br />
<br />
42. x − −x − −= 43. √ — |x| =x 44. t −t += 45. |x +x −|=<br />
REAL-WORLD APPLICATIONS<br />
TT =π<br />
√ L <br />
g <br />
L g<br />
46. <br />
<br />
=<br />
47. s 1 s,<br />
=<br />
<br />
BSA=<br />
√ wh w =<br />
<br />
h =<br />
48. <br />
BSA =<br />
49. <br />
BSA =
142<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Use interval notation.<br />
Use properties of inequalities.<br />
Solve inequalities in one variable algebraically.<br />
Solve absolute value inequalities.<br />
2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES<br />
Figure 1<br />
<br />
<br />
ff<br />
<br />
Using Interval Notation<br />
x ≥<br />
Figure 2Thx =<br />
fi<br />
<br />
Figure 2<br />
x|x≥ x x <br />
<br />
Thinterval notationTh<br />
x ≥∞Th<br />
<br />
Th<br />
<br />
fififi<br />
interval<br />
−−−−<br />
−−∞Table 1<br />
<br />
Set Indicated Set-Builder Notation Interval Notation<br />
abab x|a< x a a,∞<br />
bb x| x
SECTION 2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 143<br />
Set Indicated Set-Builder Notation Interval Notation<br />
bb x| x ≤b −∞b<br />
aba x|a≤ x ba≥b<br />
a
144<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Solution Th<br />
<br />
a. x−<<br />
x −+<br />
x +−>− <br />
<br />
x ><br />
Try It #3<br />
x−<<br />
Example 4<br />
Demonstrating the Multiplication Property<br />
<br />
a. x < b. −x −≥ c. −x ><br />
Solution<br />
<br />
<br />
a. x<<br />
<br />
x< <br />
<br />
x <<br />
b. −x−≥<br />
<br />
−x ≥<br />
( − <br />
) −x≥( − <br />
) <br />
− <br />
<br />
x ≤− <br />
c. −x><br />
<br />
−x><br />
−−x>− −<br />
x
SECTION 2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 145<br />
Example 5<br />
Solving an Inequality <strong>Algebra</strong>ically<br />
−x ≥x −<br />
Solution <br />
<br />
−x ≥x −<br />
−x ≥− <br />
−x ≥− <br />
x ≤ −<br />
Th−∞<br />
Try It #5<br />
−x+< <br />
x+<br />
Example 6<br />
Solving an Inequality with Fractions<br />
− <br />
x ≥− <br />
+ <br />
x<br />
Solution <br />
<br />
<br />
<br />
<br />
<br />
− <br />
x ≥− <br />
+ <br />
x<br />
− <br />
x − <br />
x ≥− <br />
<br />
− <br />
x − <br />
<br />
x ≥− <br />
<br />
− <br />
x ≥− <br />
<br />
x ≤− <br />
( − <br />
) <br />
<br />
x ≤ <br />
<br />
Th( −∞ <br />
]<br />
Try It #6<br />
<br />
− <br />
x≤ <br />
+ <br />
x<br />
Understanding Compound Inequalities<br />
<br />
<br />
compound inequality< x ≤< x <br />
x ≤Th<br />
<br />
<br />
Example 7<br />
Solving a Compound Inequality<br />
≤x +<<br />
Solution ≤x +x +<<br />
<br />
≤x + x +<<br />
≤x x <<br />
<br />
<br />
≤x<br />
x
146<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Th<br />
<br />
≤ x <<br />
<br />
[ <br />
) <br />
Th<br />
<br />
<br />
≤x +<<br />
≤x < <br />
<br />
≤ x <<br />
<br />
[ <br />
) <br />
Try It #7<br />
x −>x −<br />
Solution e fi<br />
+x >x − x −>x −<br />
>x − x −>−<br />
>x x >−<br />
<br />
<br />
>x<br />
x >−<br />
<br />
x < <br />
<br />
−< x<br />
Th−< x < <br />
( − <br />
) <br />
fi <br />
Figure 3<br />
<br />
<br />
<br />
Figure 3<br />
Try It #8<br />
y
SECTION 2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 147<br />
ThTh<br />
Th<br />
ffi<br />
<br />
x- x <br />
x |x−||x−|≤<br />
<br />
−≤x −≤<br />
<br />
−+≤x −+≤+<br />
<br />
≤ x ≤<br />
Th<br />
<br />
<br />
absolute value inequalities<br />
X k >absolute value inequality<br />
<br />
|X|< k −k < X < k<br />
<br />
|X|> k X k<br />
Th|X|≤ k |X|≥k<br />
Example 9<br />
Determining a Number within a Prescribed Distance<br />
x <br />
Solution x <br />
Figure 4fi<br />
<br />
<br />
Figure 4<br />
Th x |x−| x <br />
<br />
|x−|≤<br />
<br />
x −≤ x −≥−<br />
x ≤ x ≥<br />
x ≤ x ≥<br />
<br />
|x−|≤<br />
<br />
Try It #9<br />
x<br />
Example 10<br />
|x−|≤<br />
Solution<br />
<br />
<br />
<br />
Solving an Absolute Value Inequality<br />
|x−|≤<br />
−≤x −≤<br />
−≤ x ≤<br />
−
148<br />
CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Example 11<br />
Using a Graphical Approach to Solve Absolute Value Inequalities<br />
<br />
y =− |x −|+ x- y-<br />
<br />
Solution y
SECTION 2.7 SECTION EXERCISES 149<br />
2.7 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
−x ><br />
x<br />
<br />
5. y =|x −|<br />
ALGEBRAIC<br />
r fi<br />
6. x −≤ 7. x +≥x − 8. −x +>x −<br />
9. x +≥x − 10. − <br />
x≤− <br />
+ <br />
x<br />
12. −x +>−x + 13. x+ <br />
−x+ <br />
≥ <br />
<br />
<br />
11. −x −+>x −−x<br />
14. x− <br />
+x+ <br />
≤ <br />
<br />
fi<br />
15. |x+|≥− 16. |x+|< 17. |x−|><br />
18. |x +|+≤ 19. |x−|+≥ 20. |−x+|≤<br />
21. |x−|− 23. | x− | <<br />
x-<br />
24. 25. <br />
26. 27. <br />
<br />
<br />
28. −x−>x−<br />
30. y
150 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
GRAPHICAL<br />
x-<br />
r fi<br />
33. |x−|> 34. |x+|≥ 35. |x+|≤ 36. |x −|< 37. |x−|<<br />
fty <br />
y <br />
<br />
y-<br />
<br />
38. x +x + 40. x +>x + 41. x +> x −<br />
<br />
42. x +< x +<br />
<br />
NUMERIC<br />
<br />
43. x|−< x < 44. x|x≥ 45. x|x< 46. x| x <br />
<br />
47. −∞ 48. ∞ 49. − 50. −∪∞<br />
<br />
51.<br />
−<br />
−<br />
52.<br />
−<br />
−<br />
53.<br />
<br />
TECHNOLOGY<br />
fts 1 <br />
Y2=MATHNum1:abs(<br />
2 nd CALC 5:intersection1st curve, enter, 2 nd curve, enter, guess, enter<br />
x-fi<br />
54. |x +|−< 55. − |x +|< 56. |x +|−> 57. |x −|<<br />
<br />
58. |x +|≥<br />
EXTENSIONS<br />
59. |x +|=|x +| 60. x −x><br />
61. x − ≤ x ≠−<br />
x + 62.<br />
REAL-WORLD APPLICATIONS<br />
63. <br />
V =T V T <br />
<br />
<br />
p=−x +x −fi<br />
x-<br />
fi<br />
64. <br />
<br />
. Th<br />
C =+x −
CHAPTER 2 REVIEW 151<br />
CHAPTER 2 REVIEW<br />
Key Terms<br />
absolute value equation <br />
<br />
area fi<br />
A =LW<br />
Cartesian coordinate system <br />
completing the square <br />
<br />
complex conjugate <br />
<br />
complex number a +bia<br />
b<br />
complex plane <br />
i<br />
compound inequality <br />
conditional equation <br />
discriminant <br />
<br />
distance formula o fi<br />
equation in two variables xy<br />
equations in quadratic form <br />
<br />
extraneous solutions <br />
graph in two variables <br />
<br />
identity equation <br />
imaginary number −i =√ — −<br />
inconsistent equation <br />
intercepts xy<br />
interval <br />
interval notation <br />
<br />
linear equation fi<br />
<br />
linear inequality <br />
midpoint formula o fi<br />
ordered pair <br />
xy<br />
origin <br />
perimeter fi<br />
P =L +W<br />
polynomial equation ffi
152 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
Pythagorean Theorem <br />
<br />
quadrant <br />
quadratic equation <br />
quadratic formula <br />
radical equation <br />
rational equation <br />
slope yx<br />
solution set <br />
square root property x <br />
x<br />
volume V =LWH<br />
x-axis <br />
x-coordinate <br />
x-intercept xy<br />
y-axis <br />
y-coordinate <br />
y-intercept yx<br />
zero-product property <br />
<br />
Key Equations<br />
<br />
x = −b±√— b −ac a <br />
<br />
Key Concepts<br />
2.1 The Rectangular Coordinate Systems and Graphs<br />
fi<br />
xyExample 1<br />
Example 2<br />
<br />
y = Example 3<br />
xyfiTh<br />
Example 4<br />
ThThfi<br />
Example 5Example 6<br />
Thfi<br />
xyExample 7Example 8<br />
2.2 Linear Equations in One Variable<br />
ax+b =<br />
e Example 1 Example 2<br />
<br />
Example 3Example 4<br />
fifi<br />
Example 5, Example 6, Example 7<br />
n fiExample 8
CHAPTER 2 REVIEW 153<br />
yExample 9<br />
n fiExample 10<br />
fi<br />
Example 11<br />
ThExample 12<br />
fiy =c<br />
fifix =c<br />
Example 13<br />
ffyExample 14Example 15<br />
<br />
Example 16<br />
2.3 Models and Applications<br />
Example 1<br />
<br />
Example 2<br />
Th<br />
d =rtExample 3<br />
P =L +WA =LW<br />
V =LWHExample 4Example 5Example 6<br />
2.4 Complex Numbers<br />
ThiExample 1<br />
Th<br />
Example 2<br />
<br />
Example 3<br />
<br />
◦Example 4Example 5<br />
◦<br />
Example 6 Example 7<br />
ThExample 8<br />
2.5 Quadratic Equations<br />
ffi<br />
ffThfiExample 1<br />
Example 2Example 3<br />
ffi<br />
Example 4Example 5<br />
Th<br />
Th<br />
Example 6Example 7<br />
<br />
Example 8<br />
ffi<br />
Example 9Example 10<br />
Th<br />
Example 11
154 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
ThTh<br />
fi<br />
Example 12<br />
2.6 Other Types of Equations<br />
<br />
<br />
Example 1Example 2Example 3<br />
<br />
Example 4Example 5<br />
<br />
Example 6Example 7<br />
<br />
Example 8<br />
fi<br />
<br />
Example 9Example 10<br />
Example 11<br />
2.7 Linear Inequalities and Absolute Value Inequalities<br />
<br />
<br />
Table 1 Example 1 Example 2<br />
Th<br />
Example 3Example 4Example 5Example 6<br />
ft<br />
<br />
Example 7 Example 8<br />
<br />
Example 9<br />
Example 10<br />
<br />
Example 11
CHAPTER 2 REVIEW 155<br />
CHAPTER 2 REVIEW EXERCISES<br />
THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS<br />
, fi x- y-<br />
1. x −y = 2. y −=x<br />
y x<br />
3. x =y − 4. x −y =<br />
, fi<br />
5. −− 6. −−−<br />
7. −<br />
<br />
, fi<br />
8. − 9. −<br />
<br />
10. y = x +<br />
<br />
11. x −y =<br />
LINEAR EQUATIONS IN ONE VARIABLE<br />
x<br />
12. x +=x − 13. x +−=x + 14. x −=<br />
15. −x +=x − 16. x <br />
− <br />
=x <br />
+ <br />
<br />
x x-<br />
17.<br />
x <br />
x − + <br />
x + = x ≠−<br />
x − 18. <br />
+ <br />
x = <br />
<br />
, fi<br />
19. − 20. −−<br />
<br />
<br />
21. − 22.<br />
y = x +<br />
<br />
MODELS AND APPLICATIONS<br />
<br />
<br />
23. Ths fi<br />
<br />
<br />
<br />
24.
156 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
25. <br />
<br />
COMPLEX NUMBERS<br />
<br />
26. x −x += 27. x +x +=<br />
<br />
28. −i 29. −−i<br />
<br />
30. −i−−i 31. +i−−−i 32. √ — −+√ — <br />
33. √ — −+√ — − 34. −ii − 35. −i <br />
36. √ — −√ — − 37. √ — −( √ — −−√ — ) <br />
<br />
38. <br />
−i <br />
39. +i<br />
i<br />
<br />
QUADRATIC EQUATIONS<br />
<br />
40. x −x −= 41. x +x += 42. x −=<br />
43. x −x =<br />
<br />
44. x = 45. x − =<br />
<br />
46. x +x −= 47. x +x −=<br />
<br />
No real solution<br />
48. x −x += 49. x −x −=<br />
<br />
50. x − = 51. x =x +
CHAPTER 2 REVIEW 157<br />
OTHER TYPES OF EQUATIONS<br />
<br />
<br />
52. x =<br />
53. x <br />
<br />
−x <br />
=<br />
54. x +x −x −=<br />
55. x −x = 56. √ — x +=x − 57. √ — x ++√ — x +=<br />
58. |x −|= 59. |x +|−=<br />
LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES<br />
r fi<br />
60. x −≤ 61. −x +>x − 62. x − +x + ≤ <br />
63. |x +|+≤ 64. |x −|> 65. |x −|
158 CHAPTER 2 EQUATIONS AND INEQUALITIES<br />
CHAPTER 2 PRACTICE TEST<br />
1. yx 2. x- y-<br />
x −y =<br />
3. x- y-<br />
<br />
x −y =<br />
4. −−<br />
<br />
<br />
5. <br />
x|x≤<br />
6. xx +=x −<br />
7. xx −−x −=x − 8. x x <br />
+= <br />
x <br />
<br />
9. x <br />
x + =+<br />
<br />
x − <br />
10. Ths 30 in. Th<br />
<br />
<br />
<br />
11. x 12. x −≤<br />
x <br />
−x =− <br />
<br />
13. |x +|< 14. |x −|≥<br />
, fi<br />
15. −− 16. <br />
<br />
17. <br />
18.<br />
y =− x +<br />
<br />
−+−i<br />
19. √ — −+√ — − 20. i−<br />
21. −i<br />
+i <br />
22. <br />
a +bix −x +=<br />
23. x − −= 24. x −x =<br />
25. x −x −= 26. √ — x −=x −<br />
27. +√ — −x =x 28. x − <br />
=<br />
, fi<br />
29. x −x −x += 30. x + −x +−=
3<br />
Functions<br />
P<br />
<br />
<br />
<br />
y<br />
<br />
Figure 1 Standard and Poor’s Index with dividends reinvested<br />
(credit "bull": modification of work by Prayitno Hadinata; credit "graph": modification of work by MeasuringWorth)<br />
CHAPTER OUTLINE<br />
3.1 Functions and Function Notation<br />
3.2 Domain and Range<br />
3.3 Rates of Change and Behavior of Graphs<br />
3.4 Composition of Functions<br />
3.5 Transformation of Functions<br />
3.6 Absolute Value Functions<br />
3.7 Inverse Functions<br />
Introduction<br />
<br />
Figure 1<br />
<br />
Thfi<br />
<br />
Th<br />
<br />
fi<br />
<br />
<br />
159
160<br />
CHAPTER 3 FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Determine whether a relation represents a function.<br />
Find the value of a function.<br />
Determine whether a function is one-to-one.<br />
Use the vertical line test to identify functions.<br />
Graph the functions listed in the library of functions.<br />
3.1 FUNCTIONS AND FUNCTION NOTATION<br />
flTh<br />
Th<br />
<br />
Determining Whether a Relation Represents a Function<br />
relationThfidomain<br />
range<br />
ThfififiTh<br />
e fi<br />
<br />
ThTh<br />
inputindependent variableft<br />
xoutputdependent variable<br />
fty<br />
f<br />
xfifi<br />
<br />
<br />
fifi<br />
<br />
<br />
not<br />
<br />
ThfiFigure<br />
1<br />
<br />
<br />
p<br />
p<br />
x<br />
x<br />
m<br />
p<br />
q<br />
q<br />
y<br />
y<br />
n<br />
q<br />
r<br />
r<br />
z<br />
z<br />
<br />
Figure 1 ( a ) This relationship is a function because each input is associated with a single output. Note that input q and r both give output n.<br />
( b ) This relationship is also a function. In this case, each input is associated with a single output.<br />
( c ) This relationship is not a function because input q is associated with two different outputs.
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 161<br />
function<br />
function<br />
<br />
inputdomainoutputrange<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
<br />
Example 1 Determining If Menu Price Lists Are Functions<br />
ThffFigure 2<br />
a. b.<br />
Menu<br />
Item<br />
Price<br />
Plain Donut ..............................1.49<br />
Jelly Donut ..............................1.99<br />
Chocolate Donut ..........................1.99<br />
Figure 2<br />
Solution<br />
a. ThFigure 2<br />
<br />
b. <br />
Figure 3<br />
Menu<br />
Item<br />
Price<br />
Plain Donut ..............................1.49<br />
Jelly Donut ..............................1.99<br />
Chocolate Donut ..........................<br />
Th<br />
Figure 3<br />
Example 2<br />
Determining If Class Grade Rules Are Functions<br />
<br />
Table 1
162<br />
CHAPTER 3 FUNCTIONS<br />
Percent grade <br />
Grade-point average <br />
Table 1<br />
Solution <br />
fi<br />
<br />
<br />
. Th<br />
Try It #1<br />
Table 2 e fi<br />
Player<br />
<br />
<br />
<br />
<br />
<br />
Rank<br />
<br />
<br />
<br />
<br />
<br />
a. <br />
b. <br />
Table 2<br />
Using Function Notation<br />
fi<br />
<br />
Th<br />
<br />
ha<br />
ThfghftxyzAB<br />
C<br />
hfa f <br />
h =f a <br />
f a<br />
f fa<br />
hahaTha<br />
hTh<br />
<br />
f a +bfiab<br />
fTh<br />
function notation<br />
Thy =f xfifThyxThx<br />
. Thyf x
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 163<br />
Example 3<br />
Using Function Notation for Days in a Month<br />
<br />
<br />
Solution Thf<br />
=f d =f mThfi<br />
<br />
output<br />
31f (January)<br />
rule<br />
Figure 4<br />
f =Thd =f m<br />
d m<br />
Analysis Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels<br />
of geometric objects, or any other element that determines some kind of output. However, most of the functions we will<br />
work with in this book will have numbers as inputs and outputs.<br />
Example 4<br />
Interpreting Function Notation<br />
N =f yffiNyf =<br />
Solution f =Th<br />
ffiNN =f yThf =<br />
ffi<br />
input<br />
Try It #2<br />
d<br />
Q & A…<br />
Instead of a notation such as y = f (x), could we use the same symbol for the output as for the function, such as<br />
y = y (x), meaning “y is a function of x?”<br />
<br />
f<br />
yfxTh<br />
y =f xP =Wd<br />
Representing Functions Using Tables<br />
<br />
<br />
<br />
Table 3==<br />
Th<br />
fifD =f mfi<br />
<br />
Month number, m (input) <br />
Days in month, D (output) <br />
Table 3
164<br />
CHAPTER 3 FUNCTIONS<br />
Table 4fiQ =g ng<br />
nQ<br />
n <br />
Q <br />
Table 4<br />
Table 5Th<br />
<br />
ff<br />
Age in years, a (input) <br />
Height in inches, h (output) <br />
Table 5<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 5<br />
Identifying Tables that Represent Functions<br />
Table 6Table 7Table 8<br />
Input<br />
<br />
<br />
<br />
Output<br />
<br />
<br />
<br />
Input<br />
−<br />
<br />
<br />
Output<br />
<br />
<br />
<br />
Input<br />
<br />
<br />
<br />
Table 6 Table 7 Table 8<br />
Output<br />
<br />
<br />
<br />
Solution Table 6Table 7fi<br />
Table 8fiff<br />
fi<br />
<br />
ThTable 6<br />
f =f =f =<br />
<br />
g −=g =g =<br />
Table 7.<br />
Table 8<br />
Try It #3<br />
Table 9<br />
Input<br />
<br />
<br />
<br />
Table 9<br />
Output
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 165<br />
Finding Input and Output Values of a Function<br />
<br />
<br />
<br />
<br />
<br />
ff<br />
Evaluation of Functions in <strong>Algebra</strong>ic Forms<br />
<br />
f x=−x <br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 6<br />
Evaluating Functions at Specific Values<br />
fx=x +x −<br />
a. b. a c. a +h d. f a +h−f a __<br />
<br />
<br />
h<br />
Solution xfi<br />
a. <br />
<br />
f = +−<br />
<br />
=+−<br />
<br />
=<br />
b. <br />
<br />
f a=a +a −<br />
c. a +h<br />
<br />
f a +h=a +h +a +h−<br />
<br />
=a +ah +h +a +h −<br />
d. <br />
<br />
<br />
f a +h=a +ah+h +a +h −<br />
<br />
<br />
fa=a +a −<br />
<br />
<br />
<br />
f a +h−fa __<br />
<br />
=____<br />
a +ah+h +a +h −−a +a − <br />
<br />
h<br />
h<br />
=___________<br />
ah+h +h<br />
<br />
h<br />
<br />
=___________<br />
ha +h <br />
+ h<br />
h<br />
=a +h +
166<br />
CHAPTER 3 FUNCTIONS<br />
Example 7 Evaluating Functions<br />
hp=p +ph<br />
Solution hp<br />
<br />
hp=p +p<br />
<br />
h= +<br />
<br />
=+<br />
<br />
=<br />
Th<br />
Try It #4<br />
gm=√ — m−g<br />
Example 8<br />
Solving Functions<br />
hp=p +php=<br />
Solution<br />
hp=<br />
p +p = hp=p +p<br />
p +p −= <br />
p+p−= 0 <br />
p+p−=p+=p−=<br />
p<br />
p+= p =−<br />
p−= p =<br />
ThThhp=p =p =−<br />
Figure 5. Thfih=h−=h=<br />
h(p)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
p<br />
Try It #5<br />
gm=√ — m−gm=<br />
p <br />
h(p) <br />
Figure 5<br />
Evaluating Functions Expressed in Formulas<br />
fi<br />
fi<br />
n +p =np<br />
pn
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 167<br />
How To…<br />
<br />
1. <br />
only<br />
2. <br />
<br />
Example 9<br />
Finding an Equation of a Function<br />
n +p =p =f n<br />
Solution p<br />
np =n<br />
<br />
n +p =<br />
p =−n n<br />
<br />
<br />
<br />
Thpn<br />
<br />
Example 10<br />
p = −n _______<br />
<br />
p =__<br />
<br />
−n ___<br />
<br />
p =− <br />
n<br />
p =f n=−<br />
__ n<br />
Expressing the Equation of a Circle as a Function<br />
x +y =xy<br />
y =f x<br />
Solution x <br />
y<br />
y =−x <br />
y =±√ — −x <br />
=+√ — −x −√ — −x <br />
<br />
y =f x<br />
Try It #6<br />
x−y =yx<br />
Q & A…<br />
Are there relationships expressed by an equation that do represent a function but that still cannot be represented<br />
by an algebraic formula?<br />
x =y + y yx<br />
xyxy<br />
y<br />
yx
168<br />
CHAPTER 3 FUNCTIONS<br />
Evaluating a Function Given in Tabular Form<br />
<br />
<br />
Thfifi<br />
fi<br />
Th<br />
<br />
Th<br />
Table 10 Pet Memory span in hours<br />
<br />
<br />
<br />
fi<br />
a fi<br />
<br />
<br />
<br />
<br />
<br />
Table 10<br />
P<br />
<br />
PfiPfi=<br />
<br />
ThP<br />
<br />
How To…<br />
fi<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 11<br />
Evaluating and Solving a Tabular Function<br />
Table 11<br />
a. g b. gn=<br />
n <br />
g (n) <br />
Solution<br />
Table 11<br />
a. g gn =Th<br />
n =g =<br />
b. g n=nTable 11<br />
gg
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 169<br />
Try It #7<br />
Table 11g<br />
Finding Function Values from a Graph<br />
fi<br />
fifi<br />
<br />
Example 12<br />
Reading Function Values from a Graph<br />
Figure 6<br />
a. f <br />
b. f x=<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–– – – – –<br />
<br />
<br />
<br />
<br />
<br />
– <br />
– <br />
Solution<br />
Figure 6<br />
a. f x =yTh<br />
f =Figure 7<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–– – – – –<br />
– <br />
– <br />
<br />
Figure 7<br />
<br />
<br />
<br />
f =<br />
b. f x= y =<br />
−Th<br />
f x=− f −=f =−Figure 8<br />
f x<br />
<br />
<br />
−<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–– – – – –<br />
– <br />
– <br />
<br />
Figure 8
170<br />
CHAPTER 3 FUNCTIONS<br />
Try It #8<br />
Figure 7f x=<br />
Determining Whether a Function is One-to-One<br />
<br />
Figure 1fiff<br />
e fiff<br />
<br />
<br />
Table 12<br />
Letter grade<br />
<br />
<br />
<br />
<br />
Grade-point average<br />
<br />
<br />
<br />
<br />
Table 12<br />
Th<br />
<br />
Figure 1(a)Figure 1(b)Th<br />
qr<br />
nTh<br />
<br />
one-to-one function<br />
one-to-one functionTh<br />
xy<br />
Example 13<br />
Determining Whether a Relationship Is a One-to-One Function<br />
<br />
Solution rA =πr r<br />
A. Thr<br />
<br />
AA =πr <br />
r =<br />
√ A <br />
π <br />
Try It #9<br />
a. <br />
b. <br />
c. <br />
Try It #10<br />
a. <br />
<br />
b.
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 171<br />
Using the Vertical Line Test<br />
<br />
Thft<br />
<br />
<br />
Thxyyxy =f x<br />
fThxyfi<br />
y =f xfi<br />
xy<br />
Figure 9f =f =<br />
xyy =f xTh<br />
<br />
y<br />
x<br />
Figure 9<br />
vertical line test<br />
fi<br />
Figure 10<br />
<br />
Figure 10<br />
How To…<br />
<br />
1. <br />
2.
172<br />
CHAPTER 3 FUNCTIONS<br />
Example 14<br />
Applying the Vertical Line Test<br />
Figure 11y =f x<br />
f x<br />
y<br />
f x<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
x<br />
x<br />
<br />
Figure 11<br />
Solution <br />
Figure<br />
11Th<br />
xFigure 12<br />
y<br />
x<br />
Figure 12<br />
Try It #11<br />
Figure 13<br />
y<br />
x<br />
Figure 13
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 173<br />
Using the Horizontal Line Test<br />
fi<br />
horizontal line test<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 15<br />
Applying the Horizontal Line Test<br />
Figure 11(a)Figure 11(b)<br />
Solution ThFigure 11(a)ThFigure 14<br />
n fi<br />
f x<br />
x<br />
Figure 14<br />
ThFigure 11(b)<br />
Try It #12<br />
<br />
y<br />
x<br />
Identifying Basic Toolkit Functions
174<br />
CHAPTER 3 FUNCTIONS<br />
<br />
fixy =f x<br />
<br />
<br />
Th<br />
Table 13<br />
Toolkit Functions<br />
Name Function Graph<br />
f x<br />
f x=cc x<br />
x<br />
<br />
<br />
<br />
f(x)<br />
<br />
<br />
<br />
f x<br />
f x=x x<br />
x<br />
<br />
<br />
<br />
f(x)<br />
<br />
<br />
<br />
f x<br />
f x=x x<br />
x<br />
<br />
<br />
<br />
f(x)<br />
<br />
<br />
<br />
f x=x x<br />
f x<br />
x<br />
<br />
<br />
<br />
<br />
<br />
f(x)<br />
<br />
<br />
<br />
<br />
<br />
f x=x x<br />
f x<br />
x<br />
<br />
<br />
<br />
<br />
<br />
f(x)
SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 175<br />
f x<br />
x<br />
f(x)<br />
<br />
<br />
<br />
<br />
<br />
f x=__<br />
x <br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f x<br />
x<br />
f(x)<br />
<br />
<br />
<br />
<br />
<br />
f x=_<br />
<br />
x <br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f x<br />
f x=√ — x x<br />
x<br />
<br />
<br />
<br />
f(x)<br />
<br />
<br />
<br />
f x<br />
x<br />
f(x)<br />
<br />
<br />
<br />
f x= √ <br />
— x <br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 13<br />
<br />
Determine if a Relation is a Function (http://openstaxcollege.org/l/relationfunction)<br />
Vertical Line Test (http://openstaxcollege.org/l/vertlinetest)<br />
Introduction to Functions (http://openstaxcollege.org/l/introtofunction)<br />
Vertical Line Test of Graph (http://openstaxcollege.org/l/vertlinegraph)<br />
One-to-one Functions (http://openstaxcollege.org/l/onetoone)<br />
Graphs as One-to-one Functions (http://openstaxcollege.org/l/graphonetoone)
176 CHAPTER 3 FUNCTIONS<br />
3.1 SECTION EXERCISES<br />
VERBAL<br />
1. e diff<br />
<br />
3. <br />
<br />
5. <br />
<br />
2. e diff<br />
<br />
4. <br />
<br />
ALGEBRAIC<br />
<br />
6. abcdac 7. abbcc<br />
yx<br />
8. x + y = 9. y=x 10. x=y <br />
11. x +y= 12. x+y = 13. y=−x +x<br />
14. y=<br />
__ <br />
x <br />
15. x=y+ <br />
y− <br />
16. x=√— −y <br />
17. y =<br />
______ x+ x− 18. x +y = 19. xy=<br />
20. x=y 21. y=x 22. y=√ — −x <br />
23. x=± √ — −y 24. y=± √ — −x 25. y =x <br />
26. y =x <br />
ff −f f −a−f af a +h<br />
27. f x=x− 28. fx=−x +x− 29. fx=√ — −x +<br />
30. fx=<br />
______ x− x+ <br />
31. fx=∣x−∣−∣x+∣<br />
32. gx=−x gx + h−gx __<br />
<br />
h h≠<br />
33. gx=x +x gx−ga _<br />
x−a<br />
x≠a<br />
34. kt=t−<br />
a. k<br />
b. kt=<br />
36. pc=c +c<br />
a. p−<br />
b. pc=<br />
38. f x= √ — x+<br />
a. f <br />
b. f x=<br />
35. f x=−x<br />
a. f−<br />
b. fx=−<br />
37. f x=x −x<br />
a. f <br />
b. f x=<br />
39. r+t=<br />
a. r=f t<br />
b. f −<br />
c. ft=
SECTION 3.1 SECTION EXERCISES 177<br />
GRAPHICAL<br />
<br />
40.<br />
y 41.<br />
y 42.<br />
y<br />
x<br />
x<br />
x<br />
43.<br />
y 44. y<br />
45.<br />
y<br />
x<br />
x<br />
x<br />
46.<br />
y 47.<br />
y 48.<br />
y<br />
x<br />
x<br />
x<br />
49.<br />
y 50.<br />
y 51.<br />
y<br />
x<br />
x<br />
x
178 CHAPTER 3 FUNCTIONS<br />
52. <br />
a. f −<br />
b. fx=<br />
y<br />
53. <br />
a. f <br />
b. fx=−<br />
y<br />
54. <br />
a. f <br />
b. fx=<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
55.<br />
y 56.<br />
y 57.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
58.<br />
y 59.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
π<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
π<br />
x<br />
NUMERIC<br />
<br />
60. −−−−−− 61. 62. <br />
yx<br />
63. x <br />
y <br />
64. x <br />
y <br />
65. x <br />
y <br />
fTable 14<br />
x <br />
f (x) <br />
66. f 67. fx=<br />
Table 14
SECTION 3.1 SECTION EXERCISES 179<br />
ff −f −f f f <br />
68. f x=−x 69. f x=−x 70. f x=x −x+<br />
71. f x=+√ — x+ 72. f x= x− <br />
x+ <br />
73. f x= x<br />
fgh<br />
f x=x − gx=−x hx=−x +x −<br />
74. f −g− 75. f ( <br />
) −h−<br />
TECHNOLOGY<br />
y =x <br />
<br />
76. − 77. − 78. −<br />
y =x <br />
<br />
79. − 80. − 81. −<br />
y =√ — x <br />
<br />
82. 83. 84. <br />
y = √ <br />
— x <br />
<br />
85. − 86. − 87. −<br />
REAL-WORLD APPLICATIONS<br />
88. ThG<br />
pG =f pG<br />
p<br />
<br />
a. <br />
<br />
f<br />
b. f =<br />
89. ThD<br />
a<br />
D =ga<br />
a. <br />
g<br />
b. g=<br />
90. f ttft<br />
<br />
a. f =<br />
b. f =<br />
91. ht<br />
tft<br />
<br />
a. h=<br />
b. h=<br />
92. f x=x − +
180<br />
CHAPTER 3 FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the domain of a function defined by an equation.<br />
Graph piecewise-defined functions.<br />
3.2 DOMAIN AND RANGE<br />
fi<br />
I am LegendHannibal The Ring The GrudgeThe ConjuringFigure 1<br />
<br />
<br />
ff<br />
<br />
<br />
Inflation-adjusted Gross,<br />
in Millions of Dollars<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Top-Five Grossing Horror Movies<br />
for years 2000–2013<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 1 Based on data compiled by www.the-numbers.com. <br />
Finding the Domain of a Function Defined by an Equation<br />
Market Share of Horror Moveis, by Year<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Functions and Function Notation<br />
fi<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2<br />
<br />
a<br />
b<br />
c<br />
<br />
<br />
Figure 2<br />
interval notation<br />
<br />
<br />
<br />
<br />
Thffi<br />
<br />
x<br />
y<br />
z
SECTION 3.2 DOMAIN AND RANGE 181<br />
fiftfi<br />
ff<br />
<br />
Th<br />
<br />
<br />
Th <br />
Th<br />
<br />
<br />
Figure 3<br />
Inequality Interval Notation Graph on Number Line Description<br />
x>a<br />
a∞<br />
a <br />
xa<br />
x
182<br />
CHAPTER 3 FUNCTIONS<br />
Example 2<br />
Finding the Domain of a Function<br />
f x=x −<br />
Solution Thx<br />
<br />
. Th<br />
f−∞∞<br />
Try It #2<br />
fx=−x+x <br />
How To…<br />
n, fi<br />
1. <br />
2. <br />
x<br />
<br />
3. <br />
Example 3<br />
Finding the Domain of a Function Involving a Denominator<br />
f x= x + −x <br />
Solution <br />
x<br />
<br />
−x =<br />
<br />
−x=−<br />
x =<br />
Thx Figure<br />
4∪<br />
−∞∪∞<br />
Try It #3<br />
f x= +x ______<br />
x− <br />
<br />
x<br />
↓ ↓<br />
−∞∪∞<br />
Figure 4<br />
How To…<br />
, fi<br />
1. <br />
2. <br />
x<br />
3. Th
SECTION 3.2 DOMAIN AND RANGE 183<br />
Example 4<br />
Finding the Domain of a Function with an Even Root<br />
f x=√ — −x <br />
Solution <br />
<br />
x<br />
<br />
−x ≥<br />
<br />
−x≥−<br />
x ≤<br />
Th<br />
−∞<br />
Try It #4<br />
f x=√ — +x <br />
Q & A…<br />
Can there be functions in which the domain and range do not intersect at all?<br />
<br />
f x=− _<br />
<br />
√ — x <br />
<br />
ff<br />
<br />
Using Notations to Specify Domain and Range<br />
<br />
fi<br />
set-builder notationx|≤x
184<br />
CHAPTER 3 FUNCTIONS<br />
<br />
∪<br />
or<br />
fi<br />
<br />
<br />
x||x |≥=−∞−∪∞<br />
set-builder notation interval notation <br />
Set-builder notation<br />
x|xxx<br />
<br />
x|<br />
x |≤x ≤x > <br />
∪∞
SECTION 3.2 DOMAIN AND RANGE 185<br />
Try It #5<br />
s fi <br />
a. <br />
b. <br />
c. <br />
<br />
Figure 7<br />
Finding Domain and Range from Graphs<br />
<br />
xTh<br />
y<br />
Figure 8<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– ––<br />
– – – –<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
– <br />
– <br />
Figure 8<br />
−−∞<br />
Th−∞<br />
<br />
<br />
Example 6 Finding Domain and Range from a Graph<br />
fFigure 9<br />
y<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
f<br />
–<br />
– <br />
– <br />
–<br />
Figure 9
186<br />
CHAPTER 3 FUNCTIONS<br />
Solution −f−<br />
Th−−Figure 10<br />
y<br />
–<br />
–<br />
–<br />
<br />
<br />
– –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
f<br />
–<br />
– <br />
– <br />
–<br />
Figure 10<br />
<br />
Example 7<br />
Finding Domain and Range from a Graph of Oil Production<br />
fFigure 11.<br />
Thousand barrels per day<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Alaska Crude Oil Production<br />
<br />
Figure 11 (credit: modification of work by the U.S. Energy Information Administration) <br />
Solution ThtTh<br />
b. Th<br />
<br />
≤t ≤≤b ≤<br />
<br />
<br />
Try It #6<br />
Figure 12<br />
Millions of people<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
World Population Increase<br />
<br />
Year<br />
Figure 12<br />
<br />
Q & A…<br />
Can a function’s domain and range be the same?<br />
<br />
===
SECTION 3.2 DOMAIN AND RANGE 187<br />
Finding Domains and Ranges of the Toolkit Functions<br />
<br />
f x<br />
f x=c<br />
−∞∞<br />
cc<br />
x<br />
constant functionfx=c<br />
Th<br />
cc<br />
<br />
c, cc.<br />
Figure 13<br />
f x<br />
−∞∞<br />
−∞∞<br />
x<br />
identity functionfx=x<br />
x<br />
Figure 14<br />
f x<br />
−∞∞<br />
∞<br />
x<br />
absolute value functionfx=∣ x ∣<br />
xfi<br />
<br />
<br />
Figure 15<br />
f x<br />
−∞∞<br />
∞<br />
x<br />
quadratic functionfx=x <br />
<br />
<br />
<br />
<br />
Figure 16<br />
f x<br />
−∞∞<br />
−∞∞<br />
x<br />
cubic functionfx=x <br />
<br />
Th<br />
<br />
<br />
Figure 17
188<br />
CHAPTER 3 FUNCTIONS<br />
f x<br />
−∞∪∞<br />
−∞∪∞<br />
x<br />
reciprocal functionfx= x <br />
<br />
<br />
x|x ≠<br />
<br />
Figure 18<br />
f x<br />
Figure 19<br />
f x<br />
Figure 20<br />
f x<br />
−∞∪∞<br />
∞<br />
x<br />
x<br />
∞<br />
∞<br />
−∞∞<br />
−∞∞<br />
x<br />
reciprocal squared functionfx= <br />
x <br />
Th<br />
x<br />
<br />
<br />
<br />
square root functionfx=√ — x <br />
<br />
Th<br />
xfi<br />
<br />
−√ — x x.<br />
cube root functionfx= √ <br />
— x <br />
<br />
<br />
<br />
<br />
Figure 21<br />
How To…<br />
<br />
1. <br />
2. fi<br />
3. <br />
4. fi<br />
Example 8 Finding the Domain and Range Using Toolkit Functions<br />
f x=x −x<br />
Th<br />
. Th−∞∞−∞∞
SECTION 3.2 DOMAIN AND RANGE 189<br />
Example 9 Finding the Domain and Range<br />
<br />
f x=____<br />
<br />
x + <br />
Solution −fiTh−∞−∪−∞<br />
. Th−∞∪∞<br />
Example 10<br />
Finding the Domain and Range<br />
f x=√ — x +<br />
Solution <br />
x+≥x ≥−<br />
Thf x−∞<br />
fif −=x<br />
f∞<br />
Analysis Figure 22 represents the function f.<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
f<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 22<br />
Try It #7<br />
f x=−√ — −x <br />
Graphing Piecewise-Defined Functions<br />
<br />
f x=|x|<br />
<br />
<br />
<br />
<br />
f x=xx ≥<br />
<br />
f x=−xx <<br />
ff<br />
piecewise functionfiff<br />
<br />
<br />
ft<br />
<br />
<br />
ThSSS ≤+S−<br />
S >
190<br />
CHAPTER 3 FUNCTIONS<br />
piecewise function<br />
piecewise functionfi<br />
<br />
{<br />
x<br />
f x= x<br />
x<br />
<br />
x x ≥<br />
|x|= −x x <<br />
How To…<br />
<br />
1. h diff<br />
2. <br />
3. <br />
Example 11<br />
Writing a Piecewise Function<br />
fi<br />
nC<br />
Solution<br />
C =<br />
ffnC =nn<br />
n
SECTION 3.2 DOMAIN AND RANGE 191<br />
Analysis The function is represented in Figure 24. We can see where the function changes from a constant to a shifted<br />
and stretched identity at g=. We plot the graphs for the different formulas on a common set of axes, making sure each<br />
formula is applied on its proper domain.<br />
Cg<br />
<br />
<br />
<br />
Cg<br />
<br />
<br />
– – – – <br />
– <br />
–<br />
<br />
<br />
<br />
<br />
<br />
g<br />
Figure 24<br />
How To…<br />
<br />
1. x <br />
2. <br />
<br />
Example 13<br />
<br />
Graphing a Piecewise Function<br />
<br />
x x≤<br />
f x= <br />
Solution <br />
<br />
<br />
<br />
<br />
Figure 25<br />
f x<br />
f x<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– <br />
– <br />
– <br />
<br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
– <br />
– <br />
– <br />
<br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
– <br />
– <br />
– <br />
<br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
Figure 25 ( a) f (x)=x 2 if x≤1; ( b) f (x)=3 if 12
192<br />
CHAPTER 3 FUNCTIONS<br />
Figure 26<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
– <br />
– <br />
– <br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
Figure 26<br />
Analysis Note that the graph does pass the vertical line test even at x= and x= because the points and<br />
are not part of the graph of the function, though and are.<br />
Try It #8<br />
<br />
<br />
x x
SECTION 3.2 SECTION EXERCISES 193<br />
3.2 SECTION EXERCISES<br />
VERBAL<br />
1. in diffr diff 2. <br />
<br />
3. fx= √ <br />
— x diff<br />
fx=√ — x <br />
4. <br />
<br />
<br />
5. <br />
ALGEBRAIC<br />
, fi<br />
6. f x=−xx − x − 7. f x=−x 8. f x=√ — x−<br />
9. f x=−√ — −x 10. f x=√ — −x 11. f x=√ — x +<br />
12. f x= √ <br />
— −x <br />
15. f x= ______<br />
x + x + <br />
<br />
18. f x= ________<br />
<br />
x −x − <br />
21. f x= _<br />
x+ √ — − x <br />
24. f x= x __<br />
x <br />
13. f x= √ <br />
— x−<br />
16. f x= <br />
_______ x + <br />
√— x − <br />
x<br />
19. f x= __________<br />
− x −x− <br />
22. f x= _<br />
√— x − √ — x − <br />
25. f x= x −x _<br />
x − <br />
<br />
14. f x= _____<br />
<br />
x − <br />
x − <br />
17. f x= __________<br />
<br />
x +x − <br />
<br />
20. f x= _<br />
<br />
√ — x − <br />
23. f x= _ x − √— √ — x − <br />
26. f x=√ — x −x <br />
a. <br />
b. x<br />
<br />
GRAPHICAL<br />
<br />
27.<br />
y<br />
28.<br />
y<br />
29.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – – –<br />
<br />
<br />
<br />
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<br />
x<br />
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<br />
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– – – – – –<br />
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<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–
194 CHAPTER 3 FUNCTIONS<br />
30.<br />
y<br />
31.<br />
y<br />
32.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
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x<br />
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– – – – – –<br />
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<br />
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x<br />
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– – – – – –<br />
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x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
33.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
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<br />
<br />
<br />
<br />
x<br />
34.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
35.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
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<br />
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x<br />
–<br />
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–<br />
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–<br />
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–<br />
–<br />
–<br />
36.<br />
y<br />
37.<br />
y<br />
<br />
<br />
<br />
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––<br />
– – – –<br />
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<br />
<br />
<br />
<br />
x<br />
<br />
<br />
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<br />
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––<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x + x <br />
41. f x=<br />
<br />
x <<br />
√ — x x ≥<br />
42. f x=<br />
<br />
x x <<br />
− x x ><br />
43. f x=<br />
<br />
x x <<br />
x + x ≥<br />
44. f x=<br />
<br />
x + x <<br />
x x ≥<br />
| x | x <<br />
45. f x= x ≥
SECTION 3.2 SECTION EXERCISES 195<br />
NUMERIC<br />
ff −f −f −f <br />
x + x −<br />
f x=<br />
<br />
<br />
<br />
−x + x ≤−<br />
x − x >−<br />
ff −f f f <br />
x + x <<br />
x<br />
49. f x=<br />
− x <<br />
50.<br />
x + x ≥<br />
f x=<br />
<br />
x x <<br />
51. f x= ≤x ≤<br />
+ | x − |x ≥<br />
x x ><br />
<br />
x + x <br />
f x= −x x ≥<br />
TECHNOLOGY<br />
55. y=<br />
__ <br />
x −−<br />
<br />
56. y=_<br />
x<br />
−−<br />
<br />
EXTENSION<br />
57. f−fx<br />
58. <br />
59. x><br />
REAL-WORLD APPLICATIONS<br />
60. h<br />
tt<br />
ht=−t +t<br />
<br />
<br />
61. x<br />
Cx=x+<br />
a. <br />
<br />
b. <br />
c. <br />
Cx
196<br />
CHAPTER 3 FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the average rate of change of a function.<br />
Use a graph to determine where a function is increasing, decreasing, or constant.<br />
Use a graph to locate local maxima and local minima.<br />
Use a graph to locate the absolute maximum and absolute minimum.<br />
3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS<br />
flTable 1 <br />
Th<br />
y <br />
C(y) <br />
Table 1<br />
<br />
<br />
per year<br />
Finding the Average Rate of Change of a Function<br />
Thrate of change<br />
Table 1<br />
fiaverage<br />
rate of changefifi<br />
<br />
<br />
= __<br />
<br />
<br />
= Δy _<br />
Δx <br />
= y _ −y x <br />
−x<br />
<br />
<br />
= f x −f x __<br />
<br />
x <br />
−x<br />
<br />
<br />
ThΔfiyx<br />
yxΔf Δy<br />
<br />
<br />
<br />
<br />
Δy _<br />
Δx = _ <br />
≈<br />
<br />
<br />
<br />
<br />
<br />
Th<br />
Th<br />
=
SECTION 3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 197<br />
rate of change<br />
Th<br />
<br />
Th<br />
<br />
Δy _<br />
Δx =f x −f x __<br />
<br />
x <br />
−x<br />
<br />
<br />
How To…<br />
ff<br />
x <br />
x <br />
<br />
1. e diffy <br />
−y <br />
=Δy<br />
2. ffx <br />
−x <br />
=Δx<br />
3. Δy _<br />
Δx <br />
Example 1<br />
Computing an Average Rate of Change<br />
Table 1, fi<br />
Solution . Th<br />
<br />
<br />
___ Δy<br />
Δx =y _ −y x <br />
−x<br />
<br />
<br />
<br />
=__<br />
− <br />
− <br />
<br />
=______<br />
− <br />
<br />
<br />
=−<br />
Analysis Note that a decrease is expressed by a negative change or “negative increase.” A rate of change is negative when<br />
the output decreases as the input increases or when the output increases as the input decreases.<br />
Try It #1<br />
Table 1 <br />
Example 2 Computing Average Rate of Change from a Graph<br />
g tFigure 1, fi−<br />
gt<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
t<br />
–<br />
–<br />
–<br />
–<br />
Figure 1
198<br />
CHAPTER 3 FUNCTIONS<br />
Solution t =−Figure 2g −=t=g =<br />
gt<br />
<br />
<br />
<br />
Δgt=–<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
Δt=<br />
t<br />
Figure 2<br />
ThΔ t =Δ gt=−<br />
ThTh–<br />
<br />
−<br />
_______<br />
<br />
−− =− ___<br />
<br />
=−<br />
Analysis Note that the order we choose is very important. If, for example, we use y − _ y<br />
x <br />
− x<br />
, we will not get the correct<br />
<br />
answer. Decide which point will be and which point will be , and keep the coordinates fixed as ( x <br />
, y <br />
) and ( x <br />
, y <br />
).<br />
Example 3<br />
Computing Average Rate of Change from a Table<br />
ft<br />
. ThTable 2e fi<br />
t (hours) <br />
D(t) (miles) <br />
Table 2<br />
Solution <br />
<br />
<br />
Th<br />
_______<br />
− <br />
− = ___<br />
<br />
<br />
=<br />
Analysis Because the speed is not constant, the average speed depends on the interval chosen. For the interval ,<br />
the average speed is miles per hour.<br />
Example 4<br />
Computing Average Rate of Change for a Function Expressed as a Formula<br />
f x=x − __<br />
<br />
x <br />
Solution <br />
<br />
f = −<br />
__ <br />
=−__<br />
<br />
<br />
=<br />
__ <br />
<br />
f= −<br />
__ <br />
=− __<br />
<br />
= __
SECTION 3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 199<br />
<br />
<br />
= f −f _<br />
− <br />
__<br />
<br />
− __<br />
=_<br />
− <br />
<br />
<br />
Try It #2<br />
__<br />
<br />
=_<br />
<br />
<br />
=__<br />
<br />
<br />
fxx−√ — x <br />
Example 5<br />
Finding the Average Rate of Change of a Force<br />
F<br />
dFd= _<br />
<br />
d<br />
<br />
<br />
Solution<br />
F d=<br />
__ <br />
d l [<br />
<br />
<br />
=__<br />
F−F − <br />
_<br />
<br />
<br />
=<br />
− _<br />
<br />
_<br />
−<br />
<br />
<br />
<br />
<br />
__ <br />
− __<br />
=<br />
_<br />
<br />
<br />
<br />
=− __<br />
<br />
Th− __<br />
<br />
Example 6<br />
Finding an Average Rate of Change as an Expression<br />
− __<br />
=_<br />
<br />
<br />
gt=t +t +aTh<br />
a <br />
Solution <br />
<br />
<br />
= ga−g _<br />
a − <br />
=___<br />
a +a +− ++ <br />
<br />
a −<br />
<br />
=_____________<br />
a +a +− <br />
<br />
a<br />
<br />
<br />
<br />
=<br />
_______ aa+ <br />
a<br />
=a +<br />
a<br />
That=t=a<br />
+=
200<br />
CHAPTER 3 FUNCTIONS<br />
Try It #3<br />
f x=x +x−aa<br />
Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<br />
fi<br />
<br />
<br />
Th<br />
Figure 3 <br />
<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
<br />
fb>fa<br />
b>a<br />
fba<br />
fb>fa<br />
b>a<br />
Figure 3 The function f(x) = x 3 − 12x is increasing on (−∞, −2) ∪ (2, ∞) and is decreasing on (−2, 2).<br />
<br />
<br />
local maximum<br />
local<br />
minimumThlocal extrema<br />
Thftlocal<br />
relativelocal<br />
<br />
local<br />
fi<br />
Figure 4x =−Th<br />
−x =<br />
=<br />
x=–<br />
fx<br />
<br />
–<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
=–<br />
x=<br />
Figure 4
SECTION 3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 201<br />
<br />
<br />
Th<br />
Figure 5<br />
fx<br />
fb<br />
<br />
<br />
<br />
<br />
<br />
a<br />
b<br />
c<br />
Figure 5 Definition of a local maximum<br />
x<br />
Thfi<br />
local minima and local maxima<br />
fincreasing functionf b>f aab<br />
b>a<br />
fdecreasing functionf ba<br />
fx=baca
202<br />
CHAPTER 3 FUNCTIONS<br />
Analysis Notice in this example that we used open intervals (intervals that do not include the endpoints), because the<br />
function is neither increasing nor decreasing at t=, t=, and t=. These points are the local extrema (two minima<br />
and a maximum).<br />
Example 8<br />
Finding Local Extrema from a Graph<br />
f x=<br />
__ <br />
x +x __<br />
Th<br />
<br />
Solution fiFigure 7<br />
x =x =<br />
x=−x=−<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 7<br />
Analysis Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 8<br />
provides screen images from two different technologies, showing the estimate for the local maximum and minimum.<br />
<br />
y<br />
<br />
<br />
<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
=–<br />
Figure 8<br />
<br />
=–<br />
Based on these estimates, the function is increasing on the interval (−∞, −) and (, ∞). Notice that, while we<br />
expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing<br />
approximation algorithms used by each. (The exact location of the extrema is at ± √ — , but determining this requires<br />
calculus.)<br />
Try It #4<br />
f x=x −x −x+
SECTION 3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 203<br />
Example 9 Finding Local Maxima and Minima from a Graph<br />
fFigure 9, fi<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
f<br />
Figure 9<br />
Solution fThx=<br />
x=. Thyx=<br />
Thx=−x=−<br />
Thyx=−−<br />
Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<br />
Figure 10Figure 11Figure 12<br />
Function Increasing/Decreasing Example<br />
y<br />
<br />
fx=c<br />
<br />
<br />
x<br />
y<br />
<br />
fx=x<br />
<br />
x<br />
y<br />
<br />
fx=x <br />
∞<br />
−∞<br />
x=<br />
x<br />
Figure 10
204<br />
CHAPTER 3 FUNCTIONS<br />
Function Increasing/Decreasing Example<br />
y<br />
<br />
fx=x <br />
<br />
x<br />
y<br />
<br />
fx=<br />
__ <br />
x <br />
<br />
−∞∪∞<br />
x<br />
y<br />
<br />
fx=<br />
_ <br />
x <br />
−∞<br />
∞<br />
x<br />
Figure 11<br />
Function Increasing/Decreasing Example<br />
y<br />
<br />
fx= √ <br />
— x <br />
<br />
x<br />
y<br />
<br />
fx=√ — x <br />
∞<br />
x<br />
y<br />
<br />
fx=∣ x ∣ <br />
∞<br />
−∞<br />
x<br />
Figure 12
SECTION 3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 205<br />
Use A Graph to Locate the Absolute Maximum and Absolute Minimum<br />
Thff<br />
Thy<br />
absolute maximumabsolute minimum<br />
<br />
Figure 13<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
f<br />
<br />
<br />
<br />
<br />
<br />
f =<br />
x<br />
–<br />
–<br />
–<br />
–<br />
f = −<br />
Figure 13<br />
Thf x=x <br />
absolute maxima and minima<br />
absolute maximumfx=cf cf c≥f xxf<br />
absolute minimumfx=df df d≤f xxf<br />
Example 10 Finding Absolute Maxima and Minima from a Graph<br />
fFigure 14, fi<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
f<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 14<br />
Solution fThx=−x=<br />
Thy<br />
x=−x=<br />
Thx=<br />
Thyx=−<br />
<br />
Average Rate of Change (http://openstaxcollege.org/l/aroc)
206<br />
CHAPTER 3 FUNCTIONS<br />
3.3 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
d diff<br />
2. fab<br />
bc<br />
fac<br />
4. <br />
<br />
y=x <br />
<br />
ALGEBRAIC<br />
fifi<br />
bh<br />
5. f x=x −b 6. g x=x −<br />
7. px=x++h 8. kx=x−+h<br />
9. f x=x +xx+h 10. gx=x −xx+h<br />
<br />
11. at=<br />
<br />
t+ +h<br />
<br />
12. bx=<br />
<br />
x+ +h<br />
13. jx=x +h 14. rt=t +h<br />
f x+h−fx 15. __<br />
f(x=x −xx, x+h<br />
h<br />
GRAPHICAL<br />
fFigure 15 <br />
16. x=x=<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
17. x=x=<br />
<br />
<br />
<br />
–<br />
<br />
<br />
<br />
Figure 15<br />
<br />
<br />
<br />
x<br />
<br />
<br />
18.<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
x<br />
19.<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
20.<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
x
SECTION 3.3 SECTION EXERCISES 207<br />
21.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Figure 16<br />
y<br />
22. <br />
23. f<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 17.<br />
Figure 16<br />
y<br />
24. <br />
<br />
25. <br />
<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 17<br />
NUMERIC<br />
26. Table 3<br />
ab<br />
Year <br />
Sales (millions of dollars) <br />
Table 3<br />
27. Table 4<br />
ab<br />
Year <br />
Population (thousands) <br />
Table 4
208 CHAPTER 3 FUNCTIONS<br />
, fifi<br />
28. f x=x 29. hx=−x −<br />
30. qx=x − 31. g x=x −−<br />
32. y= x <br />
34. k t=t +<br />
__ <br />
t −<br />
33. pt= t −t+ <br />
<br />
t + −<br />
TECHNOLOGY<br />
<br />
<br />
35. f x=x −x + 36. hx=x +x +x +x −<br />
37. gt=t √ — <br />
t+ 38. kt=t −t <br />
39. mx=x +x −x −x+ 40. nx=x −x +x −x+<br />
EXTENSION<br />
41. ThfFigure 18<br />
<br />
= <br />
Figure 18<br />
= <br />
42. f x= c <br />
x<br />
f c<br />
− __<br />
<br />
REAL-WORLD APPLICATIONS<br />
44. <br />
<br />
<br />
<br />
<br />
46. <br />
<br />
dt=t tdt<br />
<br />
<br />
t=t=<br />
<br />
<br />
a. <br />
b. <br />
c. <br />
d. <br />
43. f(x)= __<br />
<br />
x .b<br />
fb<br />
− <br />
__ <br />
<br />
45. o fi<br />
<br />
<br />
<br />
<br />
<br />
<br />
47. ThFigure 19<br />
t<br />
Amount (milligrams)<br />
<br />
<br />
<br />
<br />
<br />
<br />
A<br />
<br />
<br />
<br />
Time (days)<br />
Figure 19<br />
<br />
t=t=<br />
<br />
<br />
t
SECTION 3.4 COMPOSITION OF FUNCTIONS 209<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Combine functions using algebraic operations.<br />
Create a new function by composition of functions.<br />
Evaluate composite functions.<br />
Find the domain of a composite function.<br />
Decompose a composite function into its component functions.<br />
3.4 COMPOSITION OF FUNCTIONS<br />
Th<br />
<br />
fiTh<br />
<br />
ThCTC<br />
TThTd<br />
d=CTd<br />
ThTd<br />
Te Th<br />
CT<br />
<br />
CT<br />
<br />
<br />
<br />
Combining Functions Using <strong>Algebra</strong>ic Operations<br />
<br />
<br />
fi<br />
<br />
<br />
wyhy<br />
yTfi<br />
Ty=hy+wy<br />
<br />
<br />
T=h+w<br />
fiff
210<br />
CHAPTER 3 FUNCTIONS<br />
f xgxfif + gf − gfg f _<br />
g<br />
<br />
<br />
<br />
f + gx = fx + gx<br />
<br />
<br />
f − gx = fx − gx<br />
fgx = fxgx<br />
<br />
Example 1<br />
( f _<br />
g ) x=fx <br />
gx <br />
Performing <strong>Algebra</strong>ic Operations on Functions<br />
x≠<br />
g − f x<br />
( g _<br />
f ) xf x=x −gx=x −<br />
<br />
Solution <br />
<br />
<br />
<br />
<br />
g−f x=gx−f x<br />
g−f x=x −−x−<br />
g−f x=x −x<br />
g−f x=xx−<br />
<br />
<br />
( g _<br />
f ) x=gx <br />
f x <br />
<br />
( g _ −<br />
f ) x=x <br />
x − <br />
x ≠<br />
<br />
<br />
<br />
( g _<br />
f ) x=x+x− x ≠<br />
x −<br />
( g _<br />
x=x +<br />
f )<br />
x ≠<br />
<br />
<br />
( g _<br />
xx ≠x =<br />
f )<br />
fi<br />
Try It #1<br />
fgxf−gx<br />
fx=x −gx=x −<br />
<br />
Create a Function by Composition of Functions<br />
<br />
<br />
Th<br />
. Th<br />
composite function<br />
f∘gx=f gx
SECTION 3.4 COMPOSITION OF FUNCTIONS 211<br />
ft fgxfgxTh<br />
Th∘<br />
<br />
<br />
<br />
f gx≠f xgx<br />
<br />
fi<br />
gxfigxThfgx<br />
f gx<br />
gxg<br />
f<br />
f ° gx = fgx<br />
xg<br />
f∘gg∘ffff gx≠gfxx<br />
fi<br />
f x=x gx=x +<br />
f gx=f x+<br />
=x+ <br />
<br />
=x +x +<br />
gfx=gx <br />
=x +<br />
Thx<br />
x =− <br />
<br />
fi<br />
<br />
composition of functions<br />
<br />
xfgficomposite functionf∘g<br />
<br />
f∘gx=f gx<br />
Thf∘gxxggx<br />
ffgf gx<br />
f xgx≠f gx<br />
Example 2 Determining whether Composition of Functions is Commutative<br />
fif gxgfx<br />
<br />
fx=x + gx=−x
212<br />
CHAPTER 3 FUNCTIONS<br />
Solution gxf x<br />
<br />
<br />
f gx=−x+<br />
=−x +<br />
=−x<br />
f xgx<br />
gfx=−x +<br />
<br />
=−x −<br />
<br />
=−x +<br />
e figfx≠f gx<br />
Example 3<br />
Interpreting Composite Functions<br />
Thcssst<br />
tcs<br />
Solution Thsst =<br />
s<br />
scs<br />
<br />
Example 4<br />
Investigating the Order of Function Composition<br />
f xxgyy<br />
f gygfx<br />
Solution Thy =f x<br />
<br />
=f <br />
Thgy<br />
. Th<br />
=g <br />
ThgyThf x<br />
Thf gy<br />
Thf xThgy<br />
f xgy<br />
Thgfx<br />
gf xx<br />
Q & A…<br />
Are there any situations where f (g(y)) and g(f (x)) would both be meaningful or useful expressions ?<br />
<br />
ff<br />
<br />
Try It #2<br />
ThrGrTh<br />
FaF
SECTION 3.4 COMPOSITION OF FUNCTIONS 213<br />
Evaluating Composite Functions<br />
<br />
fi<br />
<br />
<br />
Evaluating Composite Functions Using Tables<br />
<br />
fi<br />
<br />
Example 5 Using a Table to Evaluate a Composite Function<br />
Table 1f ggf<br />
x f (x) g(x)<br />
<br />
<br />
<br />
<br />
Table 1<br />
Solution f g<br />
gfigg=f<br />
gf . Thfife fif =<br />
g=<br />
fg=f =<br />
gffif fif =Th<br />
g<br />
gf=g=<br />
Table 2f∘gg∘f<br />
x g(x) f (g(x)) f (x) g(f (x))<br />
<br />
Table 2<br />
Try It #3<br />
Table 1fggf<br />
Evaluating Composite Functions Using Graphs<br />
<br />
xy
214<br />
CHAPTER 3 FUNCTIONS<br />
How To…<br />
<br />
<br />
1. x<br />
2. y<br />
3. x<br />
4. y. Th<br />
Example 6<br />
Using a Graph to Evaluate a Composite Function<br />
Figure 1f g<br />
gx<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x<br />
Figure 1<br />
Solution f gFigure 2<br />
gx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
g=<br />
x<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
f =<br />
x<br />
Figure 2<br />
ggx x <br />
g=f<br />
fg=f <br />
f x x<br />
f =f g=
SECTION 3.4 COMPOSITION OF FUNCTIONS 215<br />
Analysis<br />
Figure 3 shows how we can mark the graphs with arrows to trace the path from the input value to the output value.<br />
gx<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Figure 3<br />
Try It #4<br />
Figure 1gf<br />
Evaluating Composite Functions Using Formulas<br />
<br />
Th<br />
<br />
fi<br />
f gx<br />
f t=t −t<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
Example 7<br />
Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input<br />
f t=t −thx=x +f h<br />
Solution hhx<br />
f h=f f t<br />
h=+<br />
h=<br />
f h=f <br />
f h= −<br />
f h=<br />
Analysis It makes no difference what the input variables t and x were called in this problem because we evaluated for<br />
specific numerical values.
216<br />
CHAPTER 3 FUNCTIONS<br />
Try It #5<br />
f t=t −thx=x +<br />
a. hf b. hf−<br />
Finding the Domain of a Composite Function<br />
f∘gg<br />
f<br />
f ∘gfg<br />
xf gxx<br />
g<br />
gxff gx<br />
fiThf ∘ g<br />
ggffg<br />
xxggxf<br />
domain of a composite function<br />
Thfgxxggx<br />
f<br />
How To…<br />
f gx<br />
1. g.<br />
2. f.<br />
3. xggxf.Thx<br />
ggxf. Thf∘g.<br />
Example 8<br />
<br />
Finding the Domain of a Composite Function<br />
<br />
f∘gxf x =____<br />
gx =_____<br />
<br />
x − x − <br />
Solution Thgxx =<br />
__ <br />
fgx<br />
xgx =<br />
<br />
<br />
_____<br />
<br />
x − =<br />
<br />
=x −<br />
<br />
=x<br />
x =<br />
f∘g__<br />
. Th<br />
<br />
x≠<br />
__ x ≠<br />
<br />
<br />
( −∞ __<br />
<br />
) ∪ ( __<br />
<br />
) ∪∞
SECTION 3.4 COMPOSITION OF FUNCTIONS 217<br />
Example 9<br />
<br />
Finding the Domain of a Composite Function Involving Radicals<br />
f∘gxf x=√ — x +gx =√ — −x <br />
Solution g−∞<br />
<br />
<br />
<br />
f∘gx=√√ — −x +<br />
f∘gx=√ √ — −x +√ <br />
— −x +≥<br />
√ — −x≥−x ≥−∞<br />
Analysis This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful<br />
in finding the domain of a composite function. It also shows that the domain of f∘g can contain values that are not in<br />
the domain of f, though they must be in the domain of g.<br />
Try It #6<br />
<br />
f∘gxfx=____<br />
<br />
x− gx=√— x+<br />
Decomposing a Composite Function into its Component Functions<br />
<br />
Th<br />
<br />
Example 10<br />
Decomposing a Function<br />
f x=√ — −x <br />
Solution ghf x=ghx<br />
f x−x <br />
<br />
<br />
Try It #7<br />
hx=−x gx=√ — x <br />
ghx=g−x =√ — −x <br />
<br />
fx=__<br />
<br />
−√ — +x <br />
<br />
Composite Functions (http://openstaxcollege.org/l/compfunction)<br />
Composite Function Notation Application (http://openstaxcollege.org/l/compfuncnot)<br />
Composite Functions Using Graphs (http://openstaxcollege.org/l/compfuncgraph)<br />
Decompose Functions (http://openstaxcollege.org/l/decompfunction)<br />
Composite Function Values (http://openstaxcollege.org/l/compfuncvalue)
218 CHAPTER 3 FUNCTIONS<br />
3.4 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
f <br />
g <br />
3. <br />
<br />
<br />
<br />
2. f∘g<br />
4. <br />
f∘g<br />
ALGEBRAIC<br />
<br />
5. fx=x +xgx=−x f+g<br />
f−gfg, f <br />
g <br />
6. fx=−x +xgx= f+g<br />
f−gfg, f <br />
g <br />
7. fx=x +xgx= f+g<br />
x<br />
f−gfg, f <br />
g <br />
<br />
8. fx= <br />
x− gx= <br />
−x ,fi<br />
f+gf−gfg, f <br />
g <br />
9. fx=x gx=√ — x− f+g<br />
f−gfg, f <br />
g <br />
10. fx=√ — x gx=x− g <br />
f <br />
11. fx=x +gx=x−<br />
a. fg b. fgx c gfx d. g∘gx e. f∘f −<br />
o fif gxgfx<br />
12. fx=x +gx=√ — x+ 13. fx=√ — x +gx=x +<br />
14. fx=xgx=x+ 15. fx= √ <br />
— <br />
x gx= x+ x <br />
<br />
16. fx= <br />
x − gx= <br />
x +<br />
<br />
17. fx=___<br />
<br />
x− gx= x +<br />
o fif ghx<br />
18. fx=x +gx=x−hx=√ — x 19. fx=x +gx= x hx=x+<br />
20. fx= x ,gx=x− <br />
<br />
a. f∘gx<br />
b. f∘gx<br />
c. g∘f x<br />
d. g∘f x<br />
e. ( f <br />
g ) x<br />
21. fx=√ — −x gx=− x <br />
<br />
a. g∘f x<br />
b. g∘f x
SECTION 3.4 SECTION EXERCISES 219<br />
22. fx= −x<br />
<br />
x gx= <br />
+x <br />
<br />
a. g∘f x<br />
b. g∘f<br />
<br />
<br />
23. px = <br />
√ — x mx=x −<br />
<br />
<br />
px<br />
a. <br />
mx <br />
b. pmx<br />
c. mpx<br />
24. qx= <br />
√ — x hx=x −<br />
<br />
<br />
25. fx= x gx=√— x−<br />
f∘gx<br />
a. qx <br />
hx <br />
b. qhx<br />
c. hqx<br />
, fif xgxhx=f x<br />
<br />
26. hx=x+ 27. hx=x− 28. hx= <br />
x − <br />
30. hx=+ √ <br />
— x <br />
_______<br />
31. hx= √ <br />
x− <br />
<br />
<br />
<br />
32. hx= <br />
x − −<br />
<br />
<br />
29. hx= <br />
x+ <br />
<br />
_______<br />
33. hx= <br />
√ x− x+ <br />
34. hx = ( +x <br />
<br />
−x )<br />
35. hx=√ — x+ 36. hx=x− 37. hx= √ <br />
— x−<br />
<br />
38. hx=x + 39. hx= <br />
x− <br />
<br />
<br />
40. hx=( <br />
x− ) <br />
41. hx=<br />
√ x− <br />
x+ <br />
GRAPHICAL<br />
fFigure 4g,Figure 5<br />
f(x)<br />
f(x)<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
<br />
<br />
g<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 4<br />
Figure 5<br />
42. fg 43. fg 44. gf 45. gf<br />
46. ff 47. ff 48. gg 49. gg
220 CHAPTER 3 FUNCTIONS<br />
f x,Figure 6gx,Figure 7hx,Figure 8<br />
<br />
f x<br />
f x<br />
f x<br />
<br />
<br />
<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
gx<br />
hx<br />
<br />
<br />
<br />
<br />
<br />
<br />
– <br />
– <br />
– <br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
– <br />
– <br />
– <br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
– <br />
– <br />
– <br />
– <br />
–<br />
<br />
<br />
<br />
<br />
x<br />
Figure 6<br />
Figure 7<br />
Figure 8<br />
50. gf 51. gf 52. fg 53. fg<br />
54. fh 55. hf 56. fgh 57. fgf−<br />
NUMERIC<br />
fgTable 3<br />
x <br />
fx <br />
gx <br />
Table 3<br />
58. fg 59. fg 60. gf 61. gf<br />
62. ff 63. ff 64. gg 65. gg<br />
fgTable 4<br />
x − − − <br />
f (x) −<br />
g(x) − − − −<br />
Table 4<br />
66. f∘g 67. f∘g 68. g∘f <br />
69. g∘f 70. g∘g 71. f∘f <br />
o fif ggf<br />
72. fx=x+gx=−x 73. fx=x+gx=−x <br />
74. fx=√ — x+gx=−x <br />
75. fx=_<br />
<br />
x + gx=x+<br />
f x=x +gx=x +fi<br />
<br />
76. fg 77. fgx 78. g(f− 79. g∘g x
SECTION 3.4 SECTION EXERCISES 221<br />
EXTENSIONS<br />
f x=x +x= √ <br />
— x −<br />
80. f∘gxg∘fx 81. f∘gg∘f<br />
82. g∘fx 83. f∘gx<br />
84. fx=<br />
__ <br />
x <br />
a. f∘fx<br />
b. f∘f xf<br />
F x=x+ f x=x gx=x +<br />
85. g∘fx=Fx 86. f∘gx=Fx<br />
, fif x=x +x ≥gx=√ — x −<br />
87. f∘gg∘f 88. g∘faf∘ga 89. f∘gg∘f<br />
REAL-WORLD APPLICATIONS<br />
90. ThDp<br />
p. Th<br />
Cxx<br />
<br />
<br />
a. DC<br />
b. CD<br />
c. DCx=<br />
d. CDp=<br />
92. ff<br />
xs. Th<br />
<br />
Px <br />
x<br />
<br />
94. <br />
<br />
<br />
rt=t+<br />
t<br />
96. Thr<br />
VrV= ___<br />
√ <br />
V <br />
π . <br />
ftt<br />
Vt=+t<br />
a. rVt<br />
b. exact<br />
10 in<br />
91. ThAd<br />
d mi<br />
em. Th<br />
ftt<br />
m(t)<br />
<br />
<br />
a. Am<br />
b. mA<br />
c. Amt=<br />
d. mAd=<br />
93. <br />
<br />
rt= √ <br />
— t+ <br />
<br />
t=<br />
95. <br />
ft<br />
97. Th<br />
<br />
NT=T −T+
222<br />
CHAPTER 3 FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Graph functions using vertical and horizontal shifts.<br />
Graph functions using reflections about the x-axis and the y-axis.<br />
Determine whether a function is even, odd, or neither from its graph.<br />
Graph functions using compressions and stretches.<br />
Combine transformations.<br />
3.5 TRANSFORMATION OF FUNCTIONS<br />
Figure 1 (credit: "Misko"/Flickr)<br />
fl<br />
fl<br />
<br />
<br />
<br />
Graphing Functions Using Vertical and Horizontal Shifts<br />
ft<br />
<br />
Th<br />
<br />
Identifying Vertical Shifts<br />
ftftTh<br />
vertical shift<br />
<br />
g x=f x+kf xftkFigure 2
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 223<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
fx+<br />
fx<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 2 Vertical shift by k = 1 of the cube root function f ( x) = 3 √ — x .<br />
fty =f xThf x+ky +k<br />
yy +kykTh<br />
<br />
vertical shift<br />
f xgx=f x+kkvertical shift<br />
f xkk k<br />
<br />
<br />
Example 1<br />
Adding a Constant to a Function<br />
flFigure<br />
3V ftt<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
V<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t<br />
Figure 3<br />
Solution <br />
ThffftFigure 4<br />
<br />
<br />
<br />
<br />
<br />
V<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
t<br />
Figure 4
224<br />
CHAPTER 3 FUNCTIONS<br />
Figure 4St<br />
<br />
St=Vt+<br />
ThtStV<br />
ThfiSV <br />
ftTable 1<br />
t <br />
V(t) <br />
S(t) <br />
Table 1<br />
How To…<br />
ft<br />
1. <br />
2. ft<br />
3. <br />
Example 2<br />
Shifting a Tabular Function Vertically<br />
f xTable 2gx=f x−<br />
x <br />
f (x) <br />
Table 2<br />
Solution Thg x=f x−fig<br />
f<br />
f =<br />
<br />
gx=f x− <br />
g=f −<br />
<br />
=−<br />
<br />
=−<br />
f xgxTable 3<br />
x <br />
f (x) <br />
g(x) − <br />
Table 3<br />
Analysis As with the earlier vertical shift, notice the input values stay the same and only the output values change.<br />
Try It #1<br />
Thht=–t +thftt<br />
btht<br />
bt
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 225<br />
Identifying Horizontal Shifts<br />
. <br />
<br />
horizontal shiftFigure 5<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
fx+<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 5 Horizontal shift of the function f (x) = 3 √ — x . Note that h = +1 shifts the graph to the left, that is, towards negative values of x.<br />
f x=x gx=x− <br />
Thft<br />
f<br />
horizontal shift<br />
fgx=f x−hhhorizontal shiftf<br />
h h ft<br />
Example 3<br />
Adding a Constant to an Input<br />
flFigure 3<br />
<br />
<br />
Solution VtF t<br />
Vt=<br />
<br />
Ft=<br />
flV2 <br />
Vfl8 Ffl<br />
ThV=FFigure 6fi<br />
ft 0 ft t 8 aV=F<br />
F th =−Th<br />
F t=Vt−−=Vt +<br />
<br />
<br />
<br />
<br />
<br />
<br />
V<br />
<br />
–<br />
–<br />
– <br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t<br />
Figure 6
226<br />
CHAPTER 3 FUNCTIONS<br />
Analysis Note that Vt + has the effect of shifting the graph to the ft.<br />
Horizontal changes or “inside changes” affect the domain of a function (the input) instead of the range and often seem<br />
counterintuitive. The new function F(t) uses the same outputs as V(t), but matches those outputs to inputs 2 hours<br />
earlier than those of V(t). Said another way, we must add 2 hours to the input of V to find the corresponding output<br />
for FFt=Vt+<br />
How To…<br />
ft<br />
1. <br />
2. ft<br />
3. <br />
Example 4<br />
Shifting a Tabular Function Horizontally<br />
f xTable 4g x=f x−<br />
x <br />
f (x) <br />
Table 4<br />
Solution Thgx=f x−gf<br />
f =<br />
glargerg x<br />
f<br />
g =f −<br />
<br />
=f <br />
<br />
=<br />
Table 5<br />
x <br />
x − 3 <br />
f (x) <br />
g(x) <br />
Table 5<br />
Thg xftg x<br />
f xxftfift<br />
ftftft<br />
Analysis Figure 7 represents both of the functions. We can see the horizontal shift in each point.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–––<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f x<br />
<br />
<br />
<br />
g x=fx – <br />
Figure 7<br />
<br />
x
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 227<br />
Example 5<br />
Identifying a Horizontal Shift of a Toolkit Function<br />
Figure 8f x=x g xf x<br />
fig x<br />
<br />
<br />
<br />
<br />
<br />
f (x)<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 8<br />
Solution f x=x xft<br />
ThTh<br />
ft<br />
g x=f x−<br />
x =y =x<br />
f x<br />
g xf x−<br />
f x=x <br />
g x=f x−<br />
g x=f x−=x− <br />
Analysis To determine whether the ft is + or −, consider a single reference point on the graph. For a quadratic,<br />
looking at the vertex point is convenient. In the original function, f = . In our shifted function, g = . To<br />
obtain the output value of from the function f, we need to decide whether a plus or a minus sign will work to satisfy<br />
g () = f x − = f = . For this to work, we will need to subtract units from our input values.<br />
Example 6<br />
Interpreting Horizontal versus Vertical Shifts<br />
ThG mmG m+G m+<br />
Solution G m+Thm<br />
. Thft<br />
Gm+<br />
m. Thft<br />
Try It #2<br />
fx=√ — x fxgx = fx+<br />
hift<br />
Combining Vertical and Horizontal Shifts<br />
ftff<br />
y ftffx<br />
ftftftft<br />
andft
228<br />
CHAPTER 3 FUNCTIONS<br />
How To…<br />
ft<br />
1. <br />
2. Th <br />
<br />
3. Thftft<br />
<br />
4. ft<br />
Example 7<br />
Graphing Combined Vertical and Horizontal Shifts<br />
f x= x h x=f x+−<br />
Solution Thf<br />
Thhff x+<br />
f x+−<br />
. ThFigure 9<br />
f x= x <br />
→−<br />
−ft−→−−<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
y = x+<br />
y = x|<br />
y = x+–<br />
<br />
x<br />
Figure 10h<br />
Figure 9<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y=x+–<br />
<br />
x<br />
Figure 10<br />
Try It #3<br />
fx= x hx=fx − +
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 229<br />
Example 8<br />
Identifying Combined Vertical and Horizontal Shifts<br />
Figure 11<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
h(x)<br />
<br />
<br />
<br />
Figure 11<br />
Solution Thft<br />
<br />
hx=f x−+<br />
<br />
hx=√ — x −+<br />
<br />
<br />
<br />
<br />
x<br />
Analysis Note that this transformation has changed the domain and range of the function. This new graph has domain<br />
∞ and range ∞.<br />
Try It #4<br />
fx=<br />
__ <br />
x<br />
<br />
Graphing Functions Using Reflections about the Axes<br />
flxyvertical reflection<br />
flxhorizontal reflectionfly<br />
ThflFigure 12<br />
y<br />
<br />
eflectio<br />
fx<br />
<br />
<br />
<br />
eflectio<br />
f–x<br />
x<br />
–fx<br />
Figure 12 Vertical and horizontal reflections of a function.<br />
fl<br />
x-Thfl<br />
y
230<br />
CHAPTER 3 FUNCTIONS<br />
reflections<br />
f xgx=−fxvertical reflectionf x<br />
flx<br />
f xgx=f −xhorizontal reflectionf tf x<br />
fly<br />
How To…<br />
fl<br />
1. – Th x<br />
2. –flThfl<br />
y<br />
Example 9<br />
flst=√ — t<br />
Solution<br />
Reflecting a Graph Horizontally and Vertically<br />
a.b.<br />
a. t-<br />
Figure 13<br />
st<br />
vt<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
Figure 13 Vertical reflection of the square root function<br />
<br />
Vt=−stVt=−√ — t <br />
ffst<br />
<br />
b. Figure 14<br />
st<br />
Ht<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
Figure 14 Horizontal reflection of the square root function
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 231<br />
<br />
Ht=s−tHt=√ — −t <br />
ff<br />
<br />
ff<br />
∞∞ Vt−∞<br />
Ht−∞<br />
Try It #5<br />
flf x=|x−|a.b.<br />
Example 10<br />
Reflecting a Tabular Function Horizontally and Vertically<br />
f xTable 6<br />
a. gx=−f x b. hx=f −x<br />
Solution<br />
x <br />
f (x) <br />
Table 6<br />
a. gx x<br />
Table 7<br />
x <br />
g (x) – – – –<br />
Table 7<br />
b. hx <br />
hxf xTable 8<br />
x – – – –<br />
h(x) <br />
Table 8<br />
Try It #6<br />
f xTable 9<br />
x − <br />
f(x) <br />
a. gx=−f x<br />
b. hx=f −x<br />
Table 9
232<br />
CHAPTER 3 FUNCTIONS<br />
Example 11<br />
Applying a Learning Model Equation<br />
kt=− −t +k<br />
fttThf t= t Figure 15<br />
kt<br />
f t<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
Figure 15<br />
Solution Th<br />
f −t= −t<br />
−f−t=− −t<br />
−f−t+=− −t +<br />
<br />
<br />
1. −<br />
2. Th −−<br />
3. <br />
Thft<br />
Figure 16fiflThflTh<br />
<br />
f t<br />
f t<br />
kt<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
t<br />
<br />
Figure 16<br />
Analysis As a model for learning, this function would be limited to a domain of t ≥ , with corresponding range .<br />
Try It #7<br />
fx=x gx=−fxhx=f−x
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 233<br />
Determining Even and Odd Functions<br />
flfl<br />
f x=x f x=⎪x⎪<br />
yyeven function<br />
f x=x f x=_<br />
x<br />
flboth<br />
Figure 17<br />
y<br />
f x=x <br />
f −x<br />
y<br />
y<br />
−f −x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function<br />
(c) Horizontal and vertical reflections reproduce the original cubic function.<br />
<br />
odd function<br />
f x= x <br />
f x=<br />
even and odd functions<br />
even functionx: f x=f −x<br />
Thy<br />
odd functionx: f x=−f−x<br />
Th<br />
How To…<br />
<br />
1. f x=f −x<br />
2. fif x=−f−x<br />
3. <br />
Example 12<br />
Determining whether a Function Is Even, Odd, or Neither<br />
f x=x +x <br />
Solution fi<br />
fl<br />
f −x=−x +−x=−x −x<br />
Th<br />
−f−x=−−x −x=x +x<br />
−f−x=f x
234<br />
CHAPTER 3 FUNCTIONS<br />
Analysis Consider the graph of fin Figure 18. Notice that the graph<br />
is symmetric about the origin. For every point (x, y) on the graph, the<br />
corresponding point −x, −y is also on the graph. For example, is on<br />
the graph of f, and the corresponding point −− is also on the graph.<br />
– – – –<br />
−−<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f x f<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 18<br />
Try It #8<br />
f s=s +s +<br />
Graphing Functions Using Stretches and Compressions<br />
<br />
ffff<br />
<br />
fiff<br />
Vertical Stretches and Compressions<br />
<br />
vertical stretch<br />
vertical compressionFigure 19<br />
<br />
y<br />
<br />
<br />
fx<br />
fx<br />
fx<br />
<br />
<br />
x<br />
Figure 19 Vertical stretch and compression<br />
vertical stretches and compressions<br />
fxgx=afxavertical stretchvertical<br />
compressionfx<br />
a ><br />
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 235<br />
How To…<br />
<br />
1. a<br />
2. a<br />
3. a >a<br />
236<br />
CHAPTER 3 FUNCTIONS<br />
How To…<br />
<br />
<br />
1. a<br />
2. a<br />
Example 14<br />
Finding a Vertical Compression of a Tabular Function<br />
fTable 10gx= __<br />
f x<br />
<br />
x <br />
f (x) <br />
Table 10<br />
Solution Thgx= __<br />
f xgf<br />
<br />
f =Th<br />
g= __<br />
f = __<br />
= __<br />
<br />
Table 11<br />
x <br />
g(x)<br />
<br />
__ <br />
__<br />
<br />
__<br />
<br />
__<br />
<br />
<br />
Table 11<br />
Analysis The result is that the function gx has been compressed vertically by __<br />
so the graph is half the original height.<br />
Try It #9<br />
fTable 12gx= __<br />
f x<br />
<br />
x <br />
f(x) <br />
Table 12<br />
. Each output value is divided in half,<br />
<br />
Example 15<br />
Recognizing a Vertical Stretch<br />
ThFigure 22f x=x gxf x<br />
n figx<br />
gx<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 22
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 237<br />
Solution ft<br />
g=f = =<br />
g <br />
__ fg= __<br />
f <br />
<br />
gx= __<br />
f x<br />
<br />
gfif<br />
gx= __<br />
f x= __<br />
x<br />
<br />
Try It #10<br />
<br />
<br />
Horizontal Stretches and Compressions<br />
<br />
<br />
horizontal stretchhorizontal<br />
compression<br />
–<br />
y = x <br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– –<br />
–<br />
<br />
<br />
y = x <br />
Figure 23<br />
<br />
y = x <br />
<br />
<br />
y =f xy =f bx<br />
y =x Figure 23Thy =x y =x <br />
. Thy =x y =x <br />
x<br />
horizontal stretches and compressions<br />
f xgx=f bxbhorizontal stretch horizontal<br />
compressionf x<br />
b >__<br />
<br />
b <br />
238<br />
CHAPTER 3 FUNCTIONS<br />
Example 16<br />
Graphing a Horizontal Compression<br />
fl<br />
R<br />
<br />
4 h<br />
Solution <br />
R=P<br />
R=P<br />
Rt=Pt<br />
Figure 24<br />
f (x)<br />
f (x)<br />
<br />
<br />
<br />
<br />
Pt<br />
<br />
<br />
<br />
<br />
Rt<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 24 (a) Original population graph (b) Compressed population graph<br />
Analysis Note that the effect on the graph is a horizontal compression where all input values are half of their original<br />
distance from the vertical axis.<br />
Example 17<br />
Finding a Horizontal Stretch for a Tabular Function<br />
f xTable 13gx=f ( __<br />
x ) <br />
x <br />
f (x) <br />
Table 13<br />
Solution Thgx=f ( __<br />
<br />
x ) g<br />
fg<br />
g=f ( __<br />
=f f g<br />
fg<br />
ċ )<br />
Table 14<br />
g=f ( __<br />
ċ ) =f =<br />
x <br />
g(x) <br />
Table 14
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 239<br />
Figure 25 <br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–––––<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
–––––<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
<br />
<br />
Figure 25<br />
Analysis Because each input value has been doubled, the result is that the function gx has been stretched horizontally<br />
by a factor of .<br />
Example 18 Recognizing a Horizontal Compression on a Graph<br />
gxf xFigure 26<br />
f (x)<br />
<br />
<br />
<br />
<br />
<br />
g<br />
f<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 26<br />
Solution Thgxf xf xgx<br />
x__<br />
<br />
( __<br />
<br />
) =g=f <br />
g=f gx=f x. Th__<br />
<br />
Analysis<br />
Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or<br />
compression. So to stretch the graph horizontally by a scale factor of , we need a coefficient of<br />
__ in our function: f ( __ x ) .<br />
This means that the input values must be four times larger to produce the same result, requiring the input to be larger,<br />
causing the horizontal stretching.<br />
Try It #11
240<br />
CHAPTER 3 FUNCTIONS<br />
Performing a Sequence of Transformations<br />
<br />
ft<br />
ft fi <br />
h fi<br />
f x+Th<br />
f xfi<br />
ft<br />
gx=f x +<br />
gff =<br />
gxgx=f x +=<br />
x +=xfift<br />
<br />
Thffi<br />
ft<br />
<br />
<br />
fbx+p=f ( b ( x + p _<br />
b) ) <br />
fx=x + <br />
fx=x+ <br />
<br />
h fi <br />
combining transformations<br />
afx+kfia<br />
k<br />
f bx−h, fi h<br />
__<br />
<br />
b <br />
f bx−h, fi_<br />
<br />
b <br />
h<br />
<br />
d fi<br />
Example 19<br />
Finding a Triple Transformation of a Tabular Function<br />
Table 15f xgx=f x+<br />
x <br />
f (x) <br />
Table 15<br />
Solution Th<br />
f x__<br />
x __<br />
Table 16<br />
<br />
x <br />
f (3x) <br />
Table 16
SECTION 3.5 TRANSFORMATION OF FUNCTIONS 241<br />
<br />
Table 17<br />
x <br />
2f (3x) <br />
Table 17<br />
ftTable 18<br />
x <br />
g(x) = 2f (3x) + 1 <br />
Table 18<br />
Example 20<br />
Finding a Triple Transformation of a Graph<br />
f xFigure 27kx=f ( __<br />
x + ) −<br />
f x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 27<br />
Solution <br />
f ( __<br />
x + ) −=f ( __<br />
<br />
x+ ) −<br />
fi__<br />
<br />
<br />
<br />
Figure 28<br />
f x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 28
242<br />
CHAPTER 3 FUNCTIONS<br />
x +Figure 29<br />
f x<br />
– – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 29<br />
−<br />
Figure 30<br />
f x<br />
– – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 30<br />
<br />
Function Transformations (http://openstaxcollege.org/l/functrans)
SECTION 3.5 SECTION EXERCISES 243<br />
3.5 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. <br />
<br />
<br />
<br />
5. <br />
<br />
2. <br />
<br />
<br />
4. <br />
<br />
x<br />
y<br />
ALGEBRAIC<br />
ft<br />
6. fx) =√ — x hift 7. fx=xhift<br />
<br />
8. fx= __<br />
<br />
x hift<br />
<br />
9. fx=<br />
__ <br />
xhift <br />
<br />
f<br />
10. y = f x − 11. y = f x + 12. y = f x + <br />
13. y = f x − 14. y = f x + 15. y = f x + <br />
16. y = f x − 17. y = f x − 18. y = f x − + <br />
19. y = f x + − <br />
s<br />
20. f x = x + − 21. gx=x+ − 22. ax=√ — −x+<br />
23. kx = −√ — x − <br />
y<br />
GRAPHICAL<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 31<br />
f<br />
<br />
<br />
<br />
<br />
x<br />
f x= x Figure 31<br />
f x<br />
24. gx = x + 25. hx = x − <br />
26. wx = x − <br />
<br />
<br />
27. f t = t + − 28. hx = |x − | + <br />
29. kx = x − − 30. mt = + √ — t +
244 CHAPTER 3 FUNCTIONS<br />
NUMERIC<br />
31. fghgxhx<br />
fx<br />
x – – <br />
f (x) – – – <br />
x – <br />
g(x) – – – <br />
x – – <br />
h(x) – – <br />
32. fghgxhx<br />
f x<br />
x – – <br />
f (x) – – <br />
x – – – <br />
g(x) – – <br />
x – – <br />
h(x) – – <br />
<br />
<br />
33. <br />
y<br />
34. <br />
y<br />
35.<br />
y<br />
f<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
f<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
f<br />
<br />
<br />
<br />
<br />
x<br />
36. <br />
y<br />
37.<br />
y<br />
38.<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
f<br />
<br />
<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
f<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
f<br />
<br />
<br />
<br />
<br />
x
SECTION 3.5 SECTION EXERCISES 245<br />
39.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
––<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
40.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
fi<br />
<br />
41.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
42.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
43.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
44.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
45.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
46.<br />
<br />
x<br />
f<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–
246 CHAPTER 3 FUNCTIONS<br />
<br />
47. fx=x 48. gx=√ — x 49. hx=<br />
__ <br />
x +x<br />
50. fx=x− 51. gx=x 52. hx=x−x <br />
<br />
f<br />
53. gx=−fx 54. gx=f−x 55. gx=fx 56. gx=fx<br />
57. gx=fx 58. gx=fx 59. gx=f ( __<br />
x ) <br />
61. gx=f−x 62. gx=−fx<br />
60. gx=f ( __<br />
x ) <br />
g<br />
<br />
63. Thfx=∣x ∣ y<br />
__<br />
<br />
65. Thfx=<br />
__ <br />
x <br />
__<br />
hift <br />
<br />
3 uni<br />
67. Thfx=x <br />
__<br />
hift<br />
<br />
64. Thfx=√ — x x<br />
<br />
66. Thfx=<br />
__ <br />
x <br />
hift<br />
<br />
68. Thfx=x <br />
hift <br />
<br />
Th<br />
<br />
69. gx=x+ − 70. gx=x+ − 71. hx=−∣x−∣+<br />
72. kx=−√ — x − 73. mx= __<br />
<br />
x<br />
74. nx= __<br />
x−<br />
<br />
75. px= ( __<br />
x ) <br />
−<br />
76. qx= ( __<br />
x ) <br />
+<br />
77. ax=√ — −x+<br />
Figure 32<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
f<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 32<br />
78. gx=fx− 79. gx=−fx 80. gx=fx+ 81. gx=fx−
SECTION 3.6 ABSOLUTE VALUE FUNCTIONS 247<br />
LEARNING OBJECTIVES<br />
In this section you will:<br />
Graph an absolute value function.<br />
Solve an absolute value equation.<br />
3.6 ABSOLUTE VALUE FUNCTIONS<br />
Figure 1 Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: “s58y”/Flickr)<br />
<br />
Th<br />
<br />
<br />
<br />
Understanding Absolute Value<br />
f x=| x |Th<br />
<br />
<br />
<br />
absolute value function<br />
fi<br />
x x ≥<br />
f x=∣ x ∣ = −x x <<br />
Example 1<br />
Using Absolute Value to Determine Resistance<br />
fi<br />
<br />
Th<br />
fift±±±<br />
±<br />
<br />
Solution fiThff<br />
R<br />
| R − |≤<br />
Try It #1<br />
20
248<br />
CHAPTER 3 FUNCTIONS<br />
Graphing an Absolute Value Function<br />
ThfiTh<br />
Figure 2<br />
y<br />
y=x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 2<br />
Figure 3y =| x −|+Thy =| x| ft<br />
ftTh<br />
y = x −<br />
<br />
y = x<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
y = x −+ <br />
<br />
<br />
<br />
y = x −<br />
<br />
x<br />
Figure 3<br />
Example 2 Writing an Equation for an Absolute Value Function Given a Graph<br />
Figure 4<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 4<br />
Solution Thft<br />
Figure 5
SECTION 3.6 ABSOLUTE VALUE FUNCTIONS 249<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
, –<br />
Figure 5<br />
fi<br />
<br />
Figure 6<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
x<br />
<br />
–<br />
–<br />
–<br />
–<br />
, −<br />
Figure 6<br />
<br />
f x=| x −|−<br />
f x=| x−|−<br />
Analysis Note that these equations are algebraically equivalent—the stretch for an absolute value function can be<br />
written interchangeably as a vertical or horizontal stretch or compression.<br />
Q & A…<br />
If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?<br />
<br />
x f x<br />
f x=a| x −|−<br />
<br />
<br />
=a| −|−<br />
<br />
=a<br />
a =<br />
Try It #2<br />
hift 2 <br />
hift
250<br />
CHAPTER 3 FUNCTIONS<br />
Q & A…<br />
Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?<br />
s. Th<br />
<br />
Th<br />
ftfl<br />
Figure 7<br />
y<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 7 (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c)<br />
The absolute value function intersects the horizontal axis at two points.<br />
Solving an Absolute Value Equation<br />
Other Types of Equations<br />
<br />
=| x −| <br />
−Th<br />
ff<br />
x −= x −=−<br />
x = x =−<br />
x = x =−<br />
<br />
fi<br />
<br />
| x|=<br />
| x −|=<br />
| x +|−=<br />
solutions to absolute value equations<br />
AB| A|=BB≥A=BA=−B<br />
B<br />
3. x
SECTION 3.6 ABSOLUTE VALUE FUNCTIONS 251<br />
Example 3<br />
Finding the Zeros of an Absolute Value Function<br />
f x=| x +|−7 , fixf x=<br />
Solution<br />
=| x +|−<br />
=| x +|<br />
=x +−=x +<br />
=x −=x<br />
x=<br />
__ = x = ___ − <br />
=−<br />
Thx =x =−Figure 8<br />
y<br />
f x<br />
<br />
<br />
x<br />
fx = <br />
<br />
<br />
<br />
<br />
− <br />
––<br />
–––<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
<br />
x<br />
fx = <br />
f<br />
Try It #3<br />
Figure 8<br />
f x=| x−|− xf x=<br />
Q & A…<br />
Should we always expect two answers when solving A= B?<br />
y fi+| x −|=<br />
<br />
Graphing Absolute Value Functions (http://openstaxcollege.org/l/graphabsvalue)<br />
Graphing Absolute Value Functions 2 (http://openstaxcollege.org/l/graphabsvalue2)
252 CHAPTER 3 FUNCTIONS<br />
3.6 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
x<br />
3. <br />
<br />
<br />
<br />
4. <br />
x<br />
<br />
ALGEBRAIC<br />
5. x<br />
<br />
<br />
7. <br />
x<br />
<br />
6. x__<br />
<br />
−<br />
<br />
8. f x<br />
f x<br />
<br />
, fixy<br />
9. fx=∣ x−∣+ 10. fx=−∣ x−∣− 11. fx=−∣ x+∣+<br />
12. fx=−∣ x+∣+ 13. fx=∣ x−∣− 14. fx=∣ −x+∣−<br />
15. fx=−∣ x−∣+<br />
GRAPHICAL<br />
t fi<br />
16. y=| x− | 17. y=| x+| 18. y=| x |+<br />
<br />
19. y=| x |− 20. y=−| x| 21. y=−| x |−<br />
22. y=−| x− |− 23. f x=−| x− |− 24. f x=−| x+ |+<br />
25. f x=| x+ |+ 26. f x=| x− |+ 27. f x=| x− |−<br />
28. f x=| x+ |+ 29. f x=−| x− |− 30. f x=−| x+ | −<br />
31. f x=<br />
__ | x+ |−
SECTION 3.6 SECTION EXERCISES 253<br />
TECHNOLOGY<br />
32. f x=| x−|<br />
<br />
<br />
33. f x=−| x|+<br />
−<br />
<br />
<br />
34. f x=−| −x|+<br />
35. f x=× ∣ x−× ∣ +× <br />
EXTENSIONS<br />
<br />
36. a<br />
xf x=| x+|+a<br />
37. a<br />
yf x=| x+|+a<br />
REAL-WORLD APPLICATIONS<br />
38. <br />
<br />
<br />
x<br />
<br />
40. <br />
<br />
x<br />
<br />
39. Thp<br />
<br />
<br />
<br />
41. <br />
<br />
x<br />
<br />
42. Th<br />
<br />
x
254<br />
CHAPTER 3 FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Verify inverse functions.<br />
Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.<br />
Find or evaluate the inverse of a function.<br />
Use the graph of a one-to-one function to graph its inverse function on the same axes.<br />
3.7 INVERSE FUNCTIONS<br />
<br />
<br />
ffi<br />
<br />
<br />
Figure 1<br />
<br />
x<br />
y<br />
f<br />
<br />
y<br />
x<br />
Figure 1 Can a function “machine” operate in reverse?<br />
Verifying That Two Functions Are Inverse Functions<br />
<br />
<br />
e fi<br />
F<br />
C=<br />
__ F−<br />
<br />
__ −≈<br />
<br />
Figure 2<br />
<br />
<br />
Figure 2
SECTION 3.7 INVERSE FUNCTIONS 255<br />
fift<br />
FftC<br />
<br />
<br />
= __<br />
F−<br />
<br />
<br />
ċ<br />
__ =F −<br />
<br />
<br />
F =ċ<br />
__ +≈<br />
ft<br />
<br />
ff<br />
Thinverse function<br />
<br />
<br />
f xf − xfxTh−<br />
−f − xnot ___<br />
<br />
f x ___ <br />
f x <br />
f <br />
Tha − <br />
a =af − ∘f<br />
<br />
f − ∘f x=f − f x=f − y=x<br />
Thxf<br />
<br />
f xg xf x<br />
g f x=xf g x=x<br />
<br />
y =xy = __<br />
x<br />
<br />
f − ∘f x=f − x= __<br />
x=x<br />
<br />
<br />
f∘f − x=f ( __ <br />
x ) = ( __<br />
<br />
x ) =x<br />
y =x−−<br />
y =<br />
__ x −−<br />
<br />
<br />
inverse function<br />
f x=yf − xinverse functionff − y=xTh<br />
f − fx=xxfff − x=xxf − <br />
f − f<br />
Thf − ff − <br />
ftf − xfx<br />
<br />
f − <br />
x≠___<br />
<br />
f x
256<br />
CHAPTER 3 FUNCTIONS<br />
Example 1<br />
Identifying an Inverse Function for a Given Input-Output Pair<br />
f =f =<br />
<br />
Solution Th<br />
f=f − =<br />
f=f − =<br />
gg =g =<br />
Analysis Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See<br />
Table 1.<br />
Try It #1<br />
(x, f (x))<br />
<br />
<br />
Table 1<br />
(x, g (x))<br />
<br />
<br />
h − =h<br />
How To…<br />
f xg x<br />
1. f gx=xgfx=x<br />
2. g =f − f =g − <br />
g ≠f − f ≠g − <br />
Example 2<br />
Testing Inverse Relationships <strong>Algebra</strong>ically<br />
<br />
f x=____<br />
<br />
x + g x= __<br />
<br />
x −g =f − <br />
<br />
Solution<br />
g fx=_<br />
−<br />
( _ <br />
x + ) <br />
=x +−<br />
=x<br />
<br />
g =f − f =g −<br />
Th<br />
<br />
f g x=_<br />
<br />
<br />
__<br />
<br />
x −+<br />
=_<br />
<br />
<br />
__<br />
<br />
x<br />
=x<br />
Analysis Notice the inverse operations are in reverse order of the operations from the original function.<br />
Try It #2<br />
fx=x −g x= √ <br />
— x− g=f −
SECTION 3.7 INVERSE FUNCTIONS 257<br />
Example 3<br />
Determining Inverse Relationships for Power Functions<br />
f x=x g x= __<br />
<br />
Solution f g x=<br />
__ x <br />
≠x<br />
<br />
xg =f − <br />
Analysis The correct inverse to the cube is, of course, the cube root √ — x = x that is, the one-third is an exponent,<br />
not a multiplier.<br />
Try It #3<br />
f x=x− g x= √ <br />
— x +g=f − <br />
Finding Domain and Range of Inverse Functions<br />
Thff − ff − <br />
ff − ff − Figure 3<br />
f<br />
f x<br />
f<br />
a<br />
b<br />
f – x<br />
f – f –<br />
Figure 3 Domain and range of a function and its inverse<br />
<br />
f x=√ — x f − x=x <br />
∞<br />
f x=√ — x <br />
f x=x <br />
<br />
<br />
−<br />
<br />
ff<br />
<br />
<br />
f x=x <br />
∞<br />
<br />
f x=x− ∞f − x=√ — x +<br />
Thf =f − =∞<br />
Thf − =f =∞<br />
Q & A…<br />
Is it possible for a function to have more than one inverse?<br />
ffghfi<br />
fg =h<br />
ff
258<br />
CHAPTER 3 FUNCTIONS<br />
domain and range of inverse functions<br />
Thf xf − xThf xf − x<br />
How To…<br />
n, fi<br />
1. <br />
<br />
2. <br />
<br />
Example 4<br />
Finding the Inverses of Toolkit Functions<br />
fi<br />
ThTable 2<br />
y<br />
Constant Identity Quadratic Cubic Reciprocal<br />
fx=c fx=x fx=x fx=x fx=_<br />
x <br />
Reciprocal squared Cube root Square root Absolute value<br />
fx=_<br />
<br />
x <br />
fx= √ <br />
— x fx=√ — x fx=x<br />
Table 2<br />
Solution Th<br />
<br />
Th∞<br />
Th∞<br />
Analysis We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in Figure<br />
4. They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes<br />
the horizontal line test, then in that restricted domain, it can have an inverse.<br />
f x<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Try It #4<br />
<br />
<br />
Figure 4 (a ) Absolute value (b ) Reciprocal squared<br />
Thf∞f−∞−
SECTION 3.7 INVERSE FUNCTIONS 259<br />
Finding and Evaluating Inverse Functions<br />
fi<br />
<br />
Inverting Tabular Functions<br />
fi<br />
<br />
<br />
<br />
<br />
Example 5<br />
Interpreting the Inverse of a Tabular Function<br />
f tTable 3tf − <br />
t (minutes) <br />
f (t) (miles) <br />
Table 3<br />
Solution Thfff − <br />
Th<br />
ff − =Th<br />
fif a=bf − b=afi<br />
f − =aaf a=tf t=<br />
t =<br />
Try It #5<br />
Table 4, fia.f b.f − <br />
tminutes <br />
ftmiles <br />
Table 4<br />
Evaluating the Inverse of a Function, Given a Graph of the Original Function<br />
Functions and Function Notation<br />
fivertical<br />
fi<br />
horizontal<br />
fi<br />
<br />
How To…<br />
fi<br />
1. y<br />
2. x
260<br />
CHAPTER 3 FUNCTIONS<br />
Example 6 Evaluating a Function and Its Inverse from a Graph at Specific Points<br />
gxFigure 5gg − <br />
g(x)<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 5<br />
Solution gfixfiyTh<br />
g=<br />
g − fig − xgx=<br />
fig=fig − =Figure 6<br />
<br />
<br />
<br />
g(x)<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Try It #6<br />
Figure 6<br />
Figure 6a.fig − b.g − <br />
Finding Inverses of Functions Represented by Formulas<br />
<br />
yxftfi<br />
xy<br />
How To…<br />
, fi<br />
1. f<br />
2. x<br />
3. xy<br />
Example 7<br />
Inverting the Fahrenheit-to-Celsius Function<br />
<br />
C= <br />
__<br />
F−
SECTION 3.7 INVERSE FUNCTIONS 261<br />
Solution<br />
C =<br />
__ F−<br />
C ċ<br />
__ =F −<br />
<br />
F =<br />
__ C +<br />
<br />
<br />
C =hF=<br />
__ F−<br />
<br />
F =h − C=<br />
__ C +<br />
<br />
h<br />
C − <br />
Try It #7<br />
xyy=<br />
__ x−<br />
Example 8<br />
Solving to Find an Inverse Function<br />
<br />
f x=____<br />
<br />
x − +<br />
<br />
Solution<br />
y =____<br />
+ <br />
x −<br />
<br />
y −=<br />
_____ <br />
x − <br />
<br />
x −=____<br />
<br />
y − <br />
<br />
x =____<br />
<br />
y − +<br />
<br />
x −y −<br />
<br />
f − <br />
y=<br />
_____ <br />
y − +f − <br />
x=____<br />
<br />
x − +<br />
Analysis The domain and range of f exclude the values and , respectively. f and f −1 are equal at two points but are<br />
not the same function, as we can see by creating Table 5.<br />
Example 9<br />
Solving to Find an Inverse with Radicals<br />
f x=+√ — x −<br />
Solution<br />
<br />
f − x=x− +<br />
x f − y<br />
f (x) y<br />
Table 5<br />
y =+√ — x −<br />
y− =x −<br />
x =y− +<br />
Thf∞f∞<br />
f − ∞<br />
Analysis The formula we found for f −1 (x) looks like it would be valid for all real x. However, f −1 itself must have an<br />
inverse (namely, f) so we have to restrict the domain of f −1 to ∞ in order to make f −1 a one-to-one function. This<br />
domain of f −1 is exactly the range of f.
262<br />
CHAPTER 3 FUNCTIONS<br />
Try It #8<br />
fx=−√ — x <br />
Finding Inverse Functions and Their Graphs<br />
fi<br />
f x=x ∞<br />
Figure 7<br />
f x<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
Figure 7 Quadratic function with domain restricted to [0, ∞).<br />
∞<br />
<br />
f − x=√ — x <br />
ff − xff − <br />
Thf − xf xfly =x<br />
Figure 8<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
f x<br />
<br />
<br />
y = x<br />
f – x<br />
x<br />
<br />
–<br />
–<br />
–<br />
–<br />
Figure 8 Square and square-root functions on the non-negative domain<br />
Th<br />
. Th<br />
Example 10<br />
Finding the Inverse of a Function Using Reflection about the Identity Line<br />
f xFigure 9f − x
SECTION 3.7 INVERSE FUNCTIONS 263<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 9<br />
Solution Th<br />
∞−∞∞−∞∞∞<br />
fly =xflfl<br />
Figure 10<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
f – x y = x<br />
f x<br />
<br />
x<br />
Figure 10 The function and its inverse, showing reflection about the identity line<br />
Try It #9<br />
ff − Example 8<br />
Q & A…<br />
Is there any function that is equal to its own inverse?<br />
f =f − f f x=xTh<br />
<br />
_<br />
<br />
<br />
_ =x<br />
x<br />
<br />
f x = c − x, c<br />
<br />
Inverse Functions (http://openstaxcollege.org/l/inversefunction)<br />
One-to-one Functions (http://openstaxcollege.org/l/onetoone)<br />
Inverse Function Values Using Graph (http://openstaxcollege.org/l/inversfuncgraph)<br />
Restricting the Domain and Finding the Inverse (http://openstaxcollege.org/l/restrictdomain)
264<br />
CHAPTER 3 FUNCTIONS<br />
3.7 SECTION EXERCISES<br />
VERBAL<br />
1. n eff<br />
<br />
2. <br />
fx=x <br />
3. 4. <br />
<br />
5. <br />
<br />
ALGEBRAIC<br />
6. fx=a−xa<br />
, fif − x<br />
7. fx=x+ 8. fx=x+ 9. fx=−x<br />
10. fx=−x x<br />
11. fx=<br />
x+ <br />
<br />
12. fx= x+ <br />
x+ <br />
fif<br />
. Thn fif<br />
13. fx=x+ 14. fx=x− 15. fx=x −<br />
x<br />
16. fx= <br />
<br />
+x gx= −x <br />
a. fgxg fx<br />
x <br />
b. fxgx<br />
f xgx<br />
17. fx= √ <br />
— x−g x=x +<br />
18. fx=−x+g x=<br />
x− <br />
− <br />
GRAPHICAL<br />
<br />
19. fx=√ — x 20. fx= √ <br />
x+<br />
21. fx=−x+ 22. fx=x −<br />
<br />
23.<br />
y<br />
24.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
––––– –<br />
–<br />
–<br />
–<br />
–<br />
f<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
f –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x
SECTION 3.7 SECTION EXERCISES 265<br />
fFigure 11<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
f<br />
<br />
<br />
<br />
<br />
x<br />
25. f<br />
26. fx=<br />
27. f − <br />
28. f − x=<br />
Figure 11<br />
Figure 12<br />
y<br />
29. f − <br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 12<br />
<br />
f<br />
<br />
<br />
x<br />
30. ff − <br />
31. f <br />
f<br />
32. f <br />
f<br />
NUMERIC<br />
f<br />
33. f= f − 34. f= f − <br />
35. f − −=− f− 36. f − −=− f−<br />
Table 6<br />
x <br />
f (x) <br />
Table 6<br />
37. f 38. fx=<br />
39. f − 40. f − x=<br />
41. fTable 7f − x<br />
x <br />
f (x) <br />
Table 7
266 CHAPTER 3 FUNCTIONS<br />
TECHNOLOGY<br />
, fi. Th<br />
<br />
42. fx=<br />
_____ <br />
x− <br />
43. fx=x −<br />
<br />
44. fx=_<br />
<br />
x−<br />
<br />
REAL-WORLD APPLICATIONS<br />
45. xy<br />
fx=<br />
__ x+<br />
<br />
<br />
46. ThC<br />
Cr=πr<br />
<br />
rCrπ<br />
47. <br />
Th<br />
tdt=t<br />
<br />
td<br />
t
CHAPTER 3 REVIEW 267<br />
CHAPTER 3 REVIEW<br />
Key Terms<br />
absolute maximum <br />
absolute minimum <br />
average rate of change e diff<br />
diff<br />
composite function <br />
<br />
decreasing function f ba<br />
dependent variable <br />
domain <br />
even function f x=f −xy<br />
function <br />
horizontal compression <br />
b ><br />
horizontal line test <br />
<br />
horizontal reflection y−<br />
horizontal shift <br />
<br />
horizontal stretch <br />
f aab<br />
b >a<br />
independent variable <br />
input <br />
interval notation <br />
<br />
<br />
inverse function f xf − xf − fx=xx<br />
ff f − x=xxf −<br />
local extrema <br />
local maximum <br />
local minimum <br />
odd function f x=−f −x<br />
<br />
one-to-one function <br />
output <br />
piecewise function <br />
range <br />
rate of change <br />
relation
268 CHAPTER 3 FUNCTIONS<br />
set-builder notation <br />
x x<br />
vertical compression <br />
<br />
Key Equations<br />
<br />
<br />
<br />
<br />
<br />
f x=cc<br />
f x=x<br />
f x=x<br />
f x=x <br />
f x=x <br />
f x=_<br />
x<br />
<br />
f x=_<br />
<br />
x <br />
f x=√ — x <br />
<br />
<br />
<br />
ft<br />
ft<br />
fl<br />
fl<br />
<br />
<br />
<br />
<br />
f x= √ <br />
— x <br />
Δy _<br />
Δx =f x −f x _<br />
<br />
x <br />
−x<br />
<br />
<br />
f∘gx=f gx<br />
gx=f x+k k ><br />
gx=f x−hh ><br />
gx=−fx<br />
gx=f −x<br />
gx=afxa><br />
gx=afx
CHAPTER 3 REVIEW 269<br />
Key Concepts<br />
3.1 Functions and Function Notation<br />
<br />
Example 1Example 2<br />
y =f xExample 3<br />
Example 4<br />
<br />
Example 5<br />
<br />
Example 6Example 7<br />
<br />
Example 8<br />
Example 9Example 10<br />
Example 11<br />
Example 12<br />
Example 13<br />
<br />
Example 14<br />
Th Example 15<br />
3.2 Domain and Range<br />
Th <br />
<br />
ThExample 1.<br />
Th<br />
Example 2Example 3Example 4<br />
<br />
Example 5<br />
Example 6Example 7<br />
Example 8<br />
Example 9Example 10<br />
Example 11Example 12<br />
Example 13<br />
3.3 Rates of Change and Behavior of Graphs<br />
. Th<br />
Example 1<br />
Example 2<br />
<br />
Example 3<br />
<br />
Example 4Example 5<br />
ThExample 6<br />
<br />
Example 7
270 CHAPTER 3 FUNCTIONS<br />
<br />
<br />
<br />
Example 8Example 9<br />
ThExample 10<br />
3.4 Composition of Functions<br />
Example 1<br />
<br />
<br />
ThExample 2Example 3<br />
Th<br />
Example 4<br />
<br />
<br />
Example 5<br />
Example 6<br />
Example 7<br />
Th<br />
Example 8Example 9<br />
<br />
<br />
ftExample 10<br />
3.5 Transformation of Functions<br />
hiftExample 1Example 2<br />
hiftExample 3Example 4Example 5<br />
<br />
Example 6<br />
ftExample 7Example 8<br />
x <br />
–<br />
y <br />
–<br />
. Th <br />
ff Example 9<br />
<br />
Example 10<br />
Example 11<br />
y<br />
f x=f −x<br />
f x=−f−x<br />
Example 12<br />
Example 13<br />
Example 14Example 15<br />
Example 16<br />
Example 17Example 18
CHAPTER 3 REVIEW 271<br />
Thh diffff <br />
<br />
Example 19Example 20<br />
3.6 Absolute Value Functions<br />
<br />
Example 1<br />
Th<br />
Example 2<br />
<br />
<br />
Example 3<br />
3.7 Inverse Functions<br />
gxf xgfx=f gx=xExample 1Example 2Example 3<br />
Example 4<br />
<br />
<br />
Example 5<br />
Th Example 6<br />
y =f xxy. Thxy<br />
Example 7Example 8Example 9<br />
Th y =x<br />
Example 10
272 CHAPTER 3 FUNCTIONS<br />
CHAPTER 3 REVIEW EXERCISES<br />
FUNCTIONS AND FUNCTION NOTATION<br />
<br />
1. abcded<br />
2. <br />
3. y +=xxy<br />
4. Figure 1<br />
<br />
f −f f −a−faf a+h<br />
5. fx=−x +x<br />
Figure <br />
<br />
–<br />
7. fx=−x+ 8. fx=x−<br />
<br />
9.<br />
y<br />
10.<br />
y<br />
11.<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
12. fx=x+ 13. fx=x −
CHAPTER 3 REVIEW 273<br />
Figure 2<br />
y<br />
14. f<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
15. f−<br />
16. fx=−x<br />
17. fx=x<br />
Figure 2<br />
ht=−t +to fi<br />
18. <br />
h−h 19. <br />
ha−h <br />
<br />
− a−<br />
DOMAIN AND RANGE<br />
, fi<br />
<br />
20. fx=<br />
<br />
x+ <br />
x−<br />
21. fx=<br />
<br />
x −x− <br />
22. fx= <br />
√— x− √ — x− <br />
23. fx= {<br />
x+ x
274 CHAPTER 3 FUNCTIONS<br />
32. Figure 3<br />
−3, 3]. Th−<br />
<br />
33. <br />
Figure 3<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 3<br />
COMPOSITION OF FUNCTIONS<br />
, fif∘gxg ∘fx<br />
34. fx=−xgx=−x 35. fx=x+gx=−x 36. fx=x +xgx=x+<br />
37. fx=√ — x+gx=_<br />
x<br />
<br />
38. fx=<br />
x+ <br />
<br />
−x <br />
gx=√—<br />
, fif∘gf∘gx<br />
39. fx= x+ <br />
x+ gx= __<br />
<br />
x <br />
<br />
40. fx= <br />
x+ gx= <br />
x− <br />
41. fx= <br />
x gx=√— x <br />
<br />
42. fx= x+<br />
x − gx=√—<br />
HfgHx=f∘gx<br />
43. Hx=<br />
√ x− <br />
x+ <br />
<br />
44. Hx= <br />
x − −<br />
<br />
TRANSFORMATION OF FUNCTIONS<br />
<br />
45. fx=x− 46. fx=x+ 47. fx=√ — x +<br />
48. fx=−x 49. fx= √ <br />
— −x <br />
50. fx=√ — −x −<br />
51. fx=x−− 52. fx=−x+ −<br />
gfFigure 4<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
53. gx=fx−<br />
54. gx=fx<br />
<br />
Figure 4
CHAPTER 3 REVIEW 275<br />
<br />
55.<br />
y<br />
56.<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
57. fx=x 58. gx=√ — x 59. hx=_<br />
x<br />
+x<br />
<br />
60.<br />
y<br />
61.<br />
y<br />
62.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
–––––<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
ABSOLUTE VALUE FUNCTIONS<br />
f x=| x |<br />
63.<br />
y<br />
64.<br />
y<br />
65.<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
66. fx=| x−| 67. fx=−| x−| 68. fx=| x−|
276 CHAPTER 3 FUNCTIONS<br />
INVERSE FUNCTIONS<br />
, fif − x<br />
x <br />
69. fx=+x 70. fx=<br />
x+ <br />
fif<br />
. Thn fif<br />
71. fx=x +<br />
72. fx=x −gx= √ <br />
— x+<br />
a. fgxgfx<br />
b. fxgx<br />
<br />
73. fx= x 74. fx=−x +x 75. f= f − <br />
76. f= f −
CHAPTER 3 PRACTICE TEST<br />
277<br />
CHAPTER 3 PRACTICE TEST<br />
<br />
1. y=x+ 2. −−<br />
f x=−x +x<br />
3. f− 4. fa<br />
5. fx=−x− +<br />
<br />
6. fx=√ — −x <br />
<br />
7. fx=x −x fa+−f 8. fx= {<br />
x+−
278 CHAPTER 3 FUNCTIONS<br />
Figure 2<br />
y<br />
25. f<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f<br />
<br />
<br />
<br />
<br />
<br />
x<br />
26. f−<br />
27. <br />
Figure 2<br />
Table 1<br />
x <br />
F (x) <br />
Table 1<br />
28. F 29. Fx=<br />
30. 31. <br />
32. F − 33. fx=−x+ f − x
4<br />
Linear Functions<br />
CHAPTER OUTLINE<br />
4.1 Linear Functions<br />
4.2 Modeling with Linear Functions<br />
4.3 Fitting Linear Models to Data<br />
Figure 1 A bamboo forest in China (credit: “JFXie”/Flickr)<br />
Introduction<br />
<br />
Th<br />
1.5 <br />
36 A <br />
<br />
Functions and Function Notation<br />
fi<br />
<br />
<br />
<br />
279
280<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Represent a linear function.<br />
Determine whether a linear function is increasing, decreasing, or constant.<br />
Interpret slope as a rate of change.<br />
Write and interpret an equation for a linear function.<br />
Graph linear functions.<br />
Determine whether lines are parallel or perpendicular.<br />
Write the equation of a line parallel or perpendicular to a given line.<br />
4.1 LINEAR FUNCTIONS<br />
Figure 1 Shanghai MagLev Train (credit: “kanegen”/Flickr)<br />
<br />
fiFigure 1<br />
<br />
<br />
<br />
<br />
<br />
Representing Linear Functions<br />
Thfi<br />
Th<br />
<br />
<br />
Representing a Linear Function in Word Form<br />
<br />
<br />
The train’s distance from the station is a function of the time during which the train moves at a constant speed plus<br />
its original distance from the station when it began moving at constant speed.<br />
Th<br />
Th<br />
<br />
. Th
SECTION 4.1 LINEAR FUNCTIONS 281<br />
Representing a Linear Function in Function Notation<br />
<br />
xmb<br />
<br />
y =mx+b<br />
f x=mx+b<br />
DtDt <br />
mThb<br />
<br />
Dt=t +<br />
Representing a Linear Function in Tabular Form<br />
The <br />
Figure 2<br />
y 1 s<br />
<br />
t<br />
D(t)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2 Tabular representation of the function D showing selected input and output values<br />
Q & A…<br />
Can the input in the previous example be any real number?<br />
Th<br />
. Th<br />
Representing a Linear Function in Graphical Form<br />
<br />
Dt=t +Figure 3<br />
<br />
ThTh<br />
yy<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 3 The graph of D(t) = 83t + 250. Graphs of linear functions are lines because the rate of change is constant.<br />
<br />
f x=x +<br />
Th
282<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
linear function<br />
linear functionslope-intercept form<br />
<br />
fx=mx+b<br />
bx =m<br />
. Thyb<br />
Example 1<br />
Using a Linear Function to Find the Pressure on a Diver<br />
ThPFigure 4<br />
dThPd=d +<br />
<br />
Figure 4 (credit: Ilse Reijs and Jan-Noud Hutten)<br />
Solution Th<br />
<br />
Analysis The initial value, , is the pressure in PSI on the diver at a depth of feet, which is the surface of the<br />
water. The rate of change, or slope, is PSI per foot. This tells us that the pressure on the diver increases PSI<br />
for each foot her depth increases.<br />
Determining Whether a Linear Function Is Increasing, Decreasing, or Constant<br />
Th<br />
<br />
Th<br />
Figure5(a)<br />
Th <br />
Figure 5(b)<br />
Figure 5(c)<br />
f x<br />
Increasing function<br />
Decreasing function<br />
f x<br />
f x<br />
Constant function<br />
f<br />
f<br />
f<br />
<br />
x x x<br />
<br />
<br />
Figure 5
SECTION 4.1 LINEAR FUNCTIONS 283<br />
increasing and decreasing functions<br />
Thincreasing linear functiondecreasing linear function<br />
<br />
fx=mx+bm ><br />
fx=mx+bm <<br />
fx=mx+bm =<br />
Example 2<br />
Deciding Whether a Function Is Increasing, Decreasing, or Constant<br />
<br />
fiTh<br />
<br />
a. ThTh<br />
<br />
b. Th<br />
<br />
c. Th<br />
<br />
Solution <br />
a. Thf x=xxTh<br />
<br />
b. Thf x=−xx<br />
<br />
ftx<br />
c. Thf x=ffTh<br />
<br />
Interpreting Slope as a Rate of Change<br />
ft<br />
x <br />
x <br />
<br />
y <br />
y <br />
x <br />
y <br />
x <br />
y <br />
m<br />
m= __<br />
<br />
=Δy _<br />
Δx =y _ −y x <br />
−x<br />
<br />
<br />
y <br />
y <br />
fy <br />
=f x <br />
y <br />
=f x <br />
<br />
<br />
m= f x −f x _ <br />
x <br />
−x<br />
<br />
<br />
Figure 6x <br />
y <br />
x <br />
y <br />
<br />
. Th
284<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
1.<br />
x – x <br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y – y <br />
<br />
x <br />
, y <br />
x , y <br />
<br />
Δy<br />
m = =<br />
Δx<br />
x<br />
y <br />
– y <br />
x <br />
– x <br />
Figure 6 The slope of a function is calculated by the change in y divided by the change in x. It does not matter which coordinate<br />
is used as the (x 2<br />
, y 2<br />
) and which is the (x 1<br />
, y 1<br />
), as long as each calculation is started with the elements from the same coordinate pair.<br />
Q & A…<br />
Are the units for slope always<br />
units for the output<br />
__<br />
units for the input ?<br />
Th<br />
<br />
calculate slope<br />
m<br />
m= __<br />
<br />
=Δy _<br />
Δx =y _ −y x <br />
−x<br />
<br />
<br />
x <br />
x <br />
y <br />
y <br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
Example 3<br />
Finding the Slope of a Linear Function<br />
f x−fi<br />
<br />
Solution Th−fi<br />
<br />
m= __<br />
<br />
=−− ________ <br />
− = __ <br />
<br />
m =. Thm ><br />
Analysis As noted earlier, the order in which we write the points does not matter when we compute the slope of the line<br />
as long as the first output value, or y-coordinate, used corresponds with the first input value, or x-coordinate, used. Note<br />
that if we had reversed them, we would have obtained the same slope.<br />
m= −− <br />
− =− <br />
− =
SECTION 4.1 LINEAR FUNCTIONS 285<br />
Try It #1<br />
fx <br />
<br />
Example 4 Finding the Population Change from a Linear Function<br />
Th<br />
<br />
Solution ThTh<br />
−=fi<br />
<br />
__<br />
<br />
= __<br />
<br />
<br />
Analysis Because we are told that the population increased, we would expect the slope to be positive. This positive slope<br />
we calculated is therefore reasonable.<br />
Try It #2<br />
Th<br />
<br />
Writing and Interpreting an Equation for a Linear Function<br />
Equations and Inequalities<br />
<br />
Th<br />
fFigure 7.<br />
f<br />
y<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 7<br />
fi<br />
<br />
m = y −y x <br />
<br />
−x<br />
<br />
<br />
= − <br />
− <br />
=− <br />
<br />
<br />
y −y <br />
=mx−x <br />
<br />
y −=− <br />
<br />
x−
286<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
d fi<br />
y −=− <br />
x−<br />
y −=− x +<br />
<br />
y =− x +<br />
<br />
fifi<br />
yThb =b<br />
mmb<br />
fx=mx+b<br />
↑ ↑<br />
− <br />
<br />
fx=− x +<br />
<br />
f x=− <br />
x +y =− x +<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
3. yy<br />
4. y<br />
Example 5<br />
Writing an Equation for a Linear Function<br />
fFigure 8<br />
y<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
f<br />
<br />
<br />
<br />
x<br />
Figure 8<br />
Solution −−<br />
m = y −y x <br />
<br />
−x<br />
<br />
<br />
= −− <br />
−− <br />
= − <br />
− <br />
=<br />
<br />
y −y <br />
=mx−x <br />
<br />
y −−=x−−<br />
y +=x+
SECTION 4.1 LINEAR FUNCTIONS 287<br />
<br />
y +=x+<br />
y +=x +<br />
y =x +<br />
Analysis This makes sense because we can see from Figure 9 that the line crosses the y-axis at the point , which is<br />
the y-intercept, so b = .<br />
y<br />
<br />
<br />
<br />
<br />
<br />
–– – – –<br />
–<br />
−− –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
Figure 9<br />
Example 6<br />
Writing an Equation for a Linear Cost Function<br />
fi<br />
ffiCCxx<br />
<br />
Solution ThfiTh<br />
ThTh<br />
Cx=+x<br />
Analysis If Ben produces items in a month, his monthly cost is found by substitution for x.<br />
C=+<br />
=<br />
So his monthly cost would be <br />
Example 7<br />
Writing an Equation for a Linear Function Given Two Points<br />
ff =−f =1 , fi<br />
Solution <br />
f =−→−<br />
f =→<br />
<br />
m = y −y x <br />
<br />
−x<br />
<br />
<br />
= −− − <br />
= <br />
<br />
<br />
y −y <br />
=mx−x <br />
<br />
y −−= <br />
<br />
x−
288<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
<br />
Try It #3<br />
y += <br />
x−<br />
y += <br />
x − <br />
<br />
y = <br />
x − <br />
<br />
fxf=−f=− <br />
<br />
Modeling Real-World Problems with Linear Functions<br />
<br />
fi<br />
fi<br />
ff<br />
How To…<br />
ff c<br />
1. <br />
2. f x=mx+b<br />
3. x =c<br />
Example 8 Using a Linear Function to Determine the Number of Songs in a Music Collection<br />
<br />
Nt<br />
<br />
Solution ThN=<br />
b =<br />
ThTh<br />
m =<br />
f x=mx+b<br />
↑ ↑<br />
<br />
<br />
Nt=t +<br />
Nt=t +<br />
Figure 10<br />
<br />
t =<br />
N=+<br />
=+<br />
=<br />
<br />
Analysis Notice that N is an increasing linear function. As the input (the number of months) increases, the output<br />
(number of songs) increases as well.
SECTION 4.1 LINEAR FUNCTIONS 289<br />
Example 9<br />
Using a Linear Function to Calculate Salary Based on Commission<br />
Th<br />
In<br />
ThIn<br />
<br />
Solution Thfi<br />
<br />
m = ________ − − <br />
<br />
<br />
=<br />
_______ <br />
<br />
=<br />
<br />
Th<br />
<br />
<br />
In=n +b<br />
=+b n =I=<br />
<br />
−=b<br />
<br />
=b<br />
Thbn =<br />
<br />
e fi<br />
In=n +<br />
fi<br />
<br />
Example 10<br />
Using Tabular Form to Write an Equation for a Linear Function<br />
Table 1<br />
Number of weeks, w <br />
Number of rats, P(w) <br />
Table 1<br />
Solution b =<br />
m<br />
. Thfi<br />
Pw=w +<br />
<br />
<br />
m = __________ − − <br />
= ___ <br />
<br />
=<br />
Q & A…<br />
Is the initial value always provided in a table of values like Table 1?<br />
<br />
<br />
fif x=mx+bb
290<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Try It #4<br />
ffTable 2<br />
xHxx<br />
<br />
x <br />
H(x) <br />
Table 2<br />
Graphing Linear Functions<br />
Th<br />
Thfi<br />
Thy<br />
f x=x<br />
Graphing a Function by Plotting Points<br />
fi<br />
Th<br />
fi<br />
f x=x<br />
<br />
<br />
ft<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
5. <br />
Example 11<br />
Graphing by Plotting Points<br />
f x=− __ x +<br />
<br />
Solution Th<br />
<br />
<br />
x =<br />
x =<br />
f =− __ <br />
+=⇒<br />
f =− __ <br />
+=⇒<br />
x = f =− __ <br />
+=⇒<br />
Figure11<br />
f x=− __ x +
SECTION 4.1 LINEAR FUNCTIONS 291<br />
f(x)<br />
<br />
f <br />
<br />
<br />
<br />
<br />
<br />
–<br />
– <br />
<br />
<br />
<br />
x<br />
Analysis<br />
Figure 11 The graph of the linear function f (x) = − 2 __<br />
3 x + 5.<br />
The graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant,<br />
which indicates a negative slope. This is also expected from the negative, constant rate of change in the equation for the function.<br />
Try It #5<br />
fx=− <br />
x+<br />
Graphing a Function Using y-intercept and Slope<br />
fi<br />
Thfiyfiy<br />
x =<br />
Th<br />
<br />
fx= x +<br />
<br />
Th <br />
Thy<br />
<br />
x =Thyy<br />
m = <br />
m = <br />
y<br />
<br />
Figure 12<br />
–<br />
<br />
<br />
–<br />
<br />
<br />
<br />
y<br />
y<br />
<br />
←=<br />
↑=<br />
<br />
<br />
<br />
Figure 12<br />
<br />
f<br />
<br />
<br />
x<br />
graphical interpretation of a linear function<br />
fx=mx+b<br />
byby<br />
m<br />
<br />
m= <br />
<br />
=Δy <br />
Δx =y −y x <br />
<br />
−x
292<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Q & A…<br />
Do all linear functions have y-intercepts?<br />
y-y-.Note y-<br />
y-<br />
How To…<br />
y<br />
1. o fiy<br />
2. <br />
3. y<br />
4. <br />
5. <br />
Example 12<br />
Graphing by Using the y-intercept and Slope<br />
f x=− x +y<br />
<br />
Solution x =fiyThx =<br />
y<br />
− <br />
Th<br />
−<br />
fiyFigure 13<br />
<br />
<br />
<br />
<br />
<br />
<br />
f(x)<br />
f<br />
<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 13 Graph of f (x ) = − 2 __<br />
3<br />
x + 5 and shows how to calculate the rise over run for the slope.<br />
Analysis The graph slants downward from left to right, which means it has a negative slope as expected.<br />
Try It #6<br />
Example 12x<br />
Graphing a Function Using Transformations<br />
f x=x<br />
ftfl<br />
Vertical Stretch or Compression<br />
f (x=mxmm<br />
flFigure14f x=x<br />
mfmm >fm<br />
SECTION 4.1 LINEAR FUNCTIONS 293<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Figure 14 Vertical stretches and compressions and reflections on the function f (x) = x.<br />
x<br />
Vertical Shift<br />
f x=mx+bbftff<br />
Figure 15bf x=xftfbb<br />
bb<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
fx = x + <br />
fx = x + <br />
fx = x<br />
fx = x – <br />
fx = x – <br />
x<br />
Figure 15 This graph illustrates vertical shifts of the function f (x) = x.<br />
ftff<br />
<br />
<br />
How To…<br />
f x=mx+b<br />
1. f x=x<br />
2. m<br />
3. b<br />
Example 13<br />
Graphing by Using Transformations<br />
f x= __ x −<br />
<br />
Solution Thm = __ <br />
__ <br />
Th<br />
b =−hift<br />
Figure16<br />
Th Figure 17
294<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
1.<br />
y<br />
y<br />
Try It #7<br />
<br />
<br />
<br />
y = x<br />
<br />
y = x<br />
<br />
<br />
<br />
y = x<br />
<br />
<br />
<br />
<br />
<br />
y = x − <br />
<br />
x<br />
x<br />
– – – – – – – <br />
–<br />
– – – – – – – <br />
–<br />
Figure 16 The function, y = x, compressed by a factor of<br />
__ 1 2 . Figure 17 The function y = __<br />
1 x, shifted down 3 units.<br />
2<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
fx=+x<br />
Q & A…<br />
In Example 15, could we have sketched the graph by reversing the order of the transformations?<br />
Th<br />
Th<br />
fi<br />
f = __ <br />
−<br />
=−<br />
=−<br />
Writing the Equation for a Function from the Graph of a Line<br />
<br />
Figure18<br />
yy<br />
y<br />
f<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 18<br />
Thfi<br />
−y<br />
<br />
m= = <br />
=<br />
y<br />
y=x +
SECTION 4.1 LINEAR FUNCTIONS 295<br />
How To…<br />
n, fi<br />
1. y<br />
2. <br />
3. y<br />
Example 14 Matching Linear Functions to Their Graphs<br />
Figure19<br />
a. f x=x + b. gx=x − c. hx=−x + d. jx= x +<br />
<br />
y<br />
– – – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 19<br />
Solution <br />
a. y<br />
<br />
gdiffy<br />
f<br />
b. y−−<br />
<br />
c. −y <br />
<br />
d. <br />
y<br />
(0, jfj<br />
e fl <br />
Figure20 <br />
y<br />
gx = x − <br />
<br />
jx =<br />
<br />
x + <br />
– – – – – –<br />
f x = x + <br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 20<br />
<br />
<br />
<br />
<br />
<br />
x<br />
hx = −x +
296<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Finding the x-intercept of a Line<br />
fiy<br />
yxx<br />
x<br />
fixf xx<br />
<br />
f x=x −<br />
x<br />
<br />
=x −<br />
<br />
=x<br />
<br />
=x<br />
x =<br />
Thx<br />
Q & A…<br />
Do all linear functions have x-intercepts?<br />
y =cc<br />
xy =xTh<br />
xFigure 21<br />
y<br />
<br />
y=<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x<br />
Figure 21<br />
x-intercept<br />
x-interceptxf x==mx+b<br />
Example 15 Finding an x-intercept<br />
xf x= x −<br />
<br />
Solution x<br />
<br />
<br />
<br />
Thx<br />
= x −<br />
<br />
= <br />
x<br />
=x<br />
x =<br />
Analysis A graph of the function is shown in Figure 22. We can see that the x-intercept is as we expected.
SECTION 4.1 LINEAR FUNCTIONS 297<br />
y<br />
– – – –<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
Figure 22<br />
Try It #8<br />
xfx= <br />
x−<br />
Describing Horizontal and Vertical Lines<br />
Th<br />
yFigure23Th<br />
m =<br />
f x=mx+bfif x=b<br />
Thf x=<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
<br />
<br />
x<br />
x − − <br />
y <br />
Figure 23 A horizontal line representing the function f (x) = 2.<br />
x<br />
<br />
<br />
fi<br />
m = __ <br />
<br />
← <br />
← <br />
<br />
Figure 24 Example of how a line has a vertical slope. 0 in the denominator of the slope.<br />
Figure25xyx =<br />
Thx =<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x <br />
y − − <br />
–<br />
–<br />
–<br />
–<br />
Figure 25 The vertical line, x = 2, which does not represent a function.
298<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
horizontal and vertical lines<br />
<br />
horizontal linefif x=b<br />
vertical linefix =a<br />
Example 16<br />
Writing the Equation of a Horizontal Line<br />
Figure26<br />
y<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
f<br />
<br />
x<br />
Figure 26<br />
Solution xy−y =−<br />
Example 17<br />
Writing the Equation of a Vertical Line<br />
Figure27<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
f<br />
<br />
x<br />
Figure 27<br />
Solution Thxx =<br />
Determining Whether Lines are Parallel or Perpendicular<br />
ThFigure28Th<br />
Thffyft<br />
y<br />
– – – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
Figure 28 Parallel lines<br />
<br />
yffff<br />
<br />
<br />
<br />
<br />
<br />
<br />
x
SECTION 4.1 LINEAR FUNCTIONS 299<br />
} }<br />
f x=−x + <br />
f x=x +<br />
f x=−x − f x=x + <br />
ThTh<br />
Figure29<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 29 Perpendicular lines<br />
Thff<br />
fiThTh<br />
m <br />
m <br />
−<br />
m <br />
m <br />
=−<br />
fi <br />
<br />
<br />
fi, fit fi<br />
<br />
Th<br />
<br />
Th−<br />
f x= <br />
x + <br />
−<br />
f x=−x +− <br />
<br />
−( <br />
) =−<br />
parallel and perpendicular lines<br />
parallel lines. Th<br />
f x=m <br />
x +b <br />
gx=m <br />
x +b <br />
m <br />
=m <br />
<br />
b <br />
=b <br />
m <br />
=m <br />
<br />
perpendicular lines<br />
f x=m <br />
x +b <br />
gx=m <br />
x +b <br />
m <br />
m <br />
=−m <br />
=− <br />
m <br />
Example 18 Identifying Parallel and Perpendicular Lines<br />
<br />
<br />
f x=x +<br />
hx=−x +<br />
gx= x −<br />
<br />
jx=x −
300<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Solution f x=x +jx=x −<br />
− <br />
<br />
gx= x −hx=−x +<br />
<br />
Analysis A graph of the lines is shown in Figure 30.<br />
hx = −x + <br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
f x = x + <br />
<br />
<br />
jx = x − <br />
<br />
<br />
gx =<br />
x − <br />
x<br />
Figure 30<br />
f x=x +jx=x −gx= x −<br />
<br />
hx=−x +<br />
Writing the Equation of a Line Parallel or Perpendicular to a Given Line<br />
<br />
<br />
Writing Equations of Parallel Lines<br />
<br />
f x=x +<br />
y<br />
f xf x<br />
gx=x + hx=x + px=x + __ <br />
<br />
fTh<br />
ft<br />
b<br />
<br />
y −y <br />
=mx−x <br />
<br />
y −=x−<br />
y −=x −<br />
y =x +<br />
gx=x +f x=x +<br />
How To…<br />
<br />
<br />
1. <br />
2. <br />
3. <br />
Example 19<br />
Finding a Line Parallel to a Given Line<br />
f x=x +
SECTION 4.1 LINEAR FUNCTIONS 301<br />
Solution Thm =x =<br />
f x=o fiy<br />
gx=x +b<br />
<br />
=+b<br />
b =−<br />
Thf xgx=x −<br />
y<br />
Analysis We can confirm that the two lines are parallel by graphing them.<br />
Figure 31 shows that the two lines will never intersect.<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – <br />
–<br />
y = x + <br />
<br />
–<br />
–<br />
– y = x − <br />
–<br />
x<br />
Figure 31<br />
Writing Equations of Perpendicular Lines<br />
<br />
<br />
f x=x +<br />
Th− __ <br />
− __ <br />
<br />
f xf x<br />
gx=− __ <br />
x + hx=− __ <br />
x + px=− __ <br />
x − __ <br />
<br />
<br />
f x<br />
− __ <br />
fiy<br />
b<br />
g x=mx+b<br />
<br />
=− __ <br />
+b<br />
<br />
=−+b<br />
<br />
=b<br />
b =<br />
Th− __ <br />
y<br />
gx=− __ x +<br />
<br />
gx=− __ x +f x=x +<br />
<br />
<br />
Q & A…<br />
A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular,<br />
but the product of their slopes is not −1. Doesn’t this fact contradict the definition of perpendicular lines?<br />
−<br />
fi
302<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
How To…<br />
<br />
<br />
1. <br />
2. <br />
3. xygx=mx+b<br />
4. b<br />
5. <br />
Example 20<br />
Finding the Equation of a Perpendicular Line<br />
f x=x +<br />
Solution Thm =<br />
− __ n <br />
fi<br />
<br />
gx=− __ x +b<br />
<br />
<br />
=− __ <br />
+b<br />
<br />
=b<br />
b =<br />
Thf xgx=− __ x +<br />
<br />
y<br />
f x = x + <br />
Analysis A graph of the two lines is shown in Figure 32.<br />
– – – – –<br />
Figure 32<br />
Note that that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear<br />
to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
<br />
gx = −<br />
<br />
x + <br />
x<br />
<br />
Try It #9<br />
hx=x −<br />
a. hx b. hx<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4.
SECTION 4.1 LINEAR FUNCTIONS 303<br />
Example 21<br />
Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point<br />
−<br />
<br />
Solution <br />
−<br />
m <br />
= <br />
−− <br />
_______ <br />
= ___ − <br />
<br />
<br />
=− __ <br />
<br />
m <br />
= _ − − <br />
__ <br />
<br />
<br />
=−( − __ <br />
) <br />
=<br />
y<br />
g x=x +b<br />
<br />
=+b<br />
<br />
=+b<br />
−=b<br />
b =−<br />
Th<br />
<br />
y =x −<br />
Try It #10<br />
−−−<br />
<br />
<br />
Linear Functions (http://openstaxcollege.org/l/linearfunctions)<br />
Finding Input of Function from the Output and Graph (http://openstaxcollege.org/l/findinginput)<br />
Graphing Functions Using Tables (http://openstaxcollege.org/l/graphwithtable)
304 CHAPTER 4 LINEAR FUNCTIONS<br />
4.1 SECTION EXERCISES<br />
VERBAL<br />
1. Et<br />
fttEt=−t<br />
<br />
<br />
2. <br />
ft<br />
ft<br />
<br />
3. <br />
<br />
<br />
ftt<br />
4. <br />
<br />
y-<br />
5. f x=a<br />
x=a<br />
<br />
<br />
ALGEBRAIC<br />
<br />
6. y= __<br />
x+ 7. y=x− 8. y=x −<br />
9. x+y= 10. x +y= 11. x+y =<br />
12. −x +y = 13. − ______<br />
x− <br />
=y<br />
<br />
14. fx=x+ 15. gx=x+ 16. ax=−x<br />
17. bx=−x 18. hx=−x+ 19. kx=−x+<br />
20. jx= __<br />
x−<br />
23. mx=− __<br />
x+<br />
21. px= __<br />
x−<br />
22. nx=− __<br />
x−<br />
, fi<br />
24. 25. 26. −<br />
27. − 28. −<br />
fi<br />
29. f−=−f= 30. f−=f=<br />
31. 32. <br />
33. − 34. −<br />
35. x−y− 36. x−y<br />
<br />
<br />
37. x−y=<br />
x +y =<br />
38. y+x=<br />
−y=x +<br />
39. y+x=<br />
−y =x +<br />
40. x−y=<br />
x +y =
SECTION 4.1 SECTION EXERCISES 305<br />
, fixy<br />
41. fx=−x+ 42. gx=x+ 43. hx=x−<br />
44. kx=−x+ 45. −x+y= 46. x+y=<br />
fi<br />
<br />
47. −<br />
−−<br />
48. −−<br />
−−<br />
49. −<br />
<br />
50. <br />
−−<br />
51. −<br />
−−<br />
<br />
52. f x=−x−<br />
−<br />
54. <br />
h t=−t+−−<br />
53. g x=x−<br />
<br />
55. <br />
p t=t+<br />
GRAPHICAL<br />
, fi<br />
56.<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
57.<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
<br />
58.<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
59.<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
60.<br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x
306 CHAPTER 4 LINEAR FUNCTIONS<br />
61.<br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
62.<br />
– –<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
63.<br />
– –<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
<br />
Figure33<br />
A B<br />
<br />
<br />
<br />
C<br />
<br />
– – – – – –<br />
–<br />
<br />
–<br />
–<br />
D<br />
–<br />
F<br />
E<br />
64. fx=−x−<br />
65. fx=−x−<br />
66. fx=− __<br />
x−<br />
67. fx=<br />
68. fx=+x<br />
69. fx=x+<br />
Figure 33<br />
<br />
70. x−y− 71. x−y<br />
72. y− __<br />
<br />
73. y__<br />
<br />
74. −−− 75. −−<br />
<br />
76. fx=−x− 77. gx=−x+ 78. hx= __<br />
x+<br />
79. kx= __<br />
x− 80. ft=+t 81. pt=−+t<br />
<br />
82. x= 83. x=− 84. rx=<br />
<br />
85.<br />
y<br />
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x
SECTION 4.1 SECTION EXERCISES 307<br />
NUMERIC<br />
fi<br />
<br />
89. x <br />
g(x) − − −<br />
90. x <br />
h(x) <br />
91. x <br />
f (x) − <br />
92. x <br />
k(x) <br />
93. x <br />
g(x) − − −<br />
94. x <br />
h (x) <br />
95. x <br />
f (x) − <br />
96. x <br />
k(x) <br />
TECHNOLOGY<br />
<br />
97. ff=f=− <br />
98. f−fx=x−<br />
x−x<br />
99. f −fx=x+<br />
100. Table 3wk<br />
ka.<br />
b.k<br />
<br />
w − b<br />
k − a −<br />
Table 3<br />
101. Table 4pq<br />
qa.<br />
b.k<br />
p b<br />
q a <br />
Table 4<br />
102. f−__<br />
y __<br />
<br />
−<br />
103. f−y<br />
−−<br />
104. ffx=ax+b−<br />
ab<br />
a. a=b= b. a=b= c. a=b=− d. a=b=−<br />
EXTENSIONS<br />
105. x<br />
<br />
x−m=<br />
107. <br />
abab+<br />
106. y<br />
<br />
ym=−<br />
108. <br />
abab+<br />
109. <br />
acd<br />
110. <br />
gx=−x+
308 CHAPTER 4 LINEAR FUNCTIONS<br />
111. gxx+<br />
f x=−x+gx=x +<br />
112. fg 113. f xg xg x<br />
f x<br />
REAL-WORLD APPLICATIONS<br />
114. <br />
<br />
<br />
n<br />
116. <br />
n<br />
p<br />
<br />
<br />
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pn=mn+bp<br />
n<br />
118. <br />
n<br />
y<br />
<br />
28 o<br />
y=mn+b<br />
n<br />
120. <br />
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Pt<br />
tft<br />
122. <br />
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FC<br />
a. <br />
<br />
b. F<br />
c. F−<br />
115. <br />
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117. <br />
Cn=+nn<br />
Cn<br />
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119. <br />
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121. <br />
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Ix=x+x<br />
ft<br />
<br />
a. <br />
b. <br />
<br />
c. <br />
<br />
d.
SECTION 4.2 MODELING WITH LINEAR FUNCTIONS 309<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Build linear models from verbal descriptions.<br />
Model a set of data with a linear function.<br />
4.2 MODELING WITH LINEAR FUNCTIONS<br />
Figure 1 (credit: EEK Photography/Flickr)<br />
<br />
<br />
x<br />
<br />
<br />
Building Linear Models from Verbal Descriptions<br />
<br />
fl<br />
1. <br />
<br />
2. <br />
<br />
3. <br />
4. . Oft<br />
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5. <br />
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7. <br />
8.
310<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Th<br />
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fi<br />
M<br />
t<br />
Mt<br />
<br />
M<br />
−<br />
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ThMt=mt+bTh<br />
<br />
Mt=mt +b<br />
↑ ↑<br />
− <br />
Mt = − t +<br />
o fix<br />
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=−t +<br />
t =____<br />
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=<br />
Thx<br />
ft<br />
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fi<br />
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ftx<br />
Th<br />
≤t ≤<br />
<br />
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<br />
<br />
Using a Given Intercept to Build a Model<br />
yy<br />
x <br />
Thy<br />
Th−<br />
<br />
f x=mx+b<br />
=−x +<br />
xo fix
SECTION 4.2 MODELING WITH LINEAR FUNCTIONS 311<br />
<br />
<br />
<br />
=−x +<br />
=x<br />
=x<br />
x =<br />
xThx<br />
<br />
Using a Given Input and Output to Build a Model<br />
<br />
<br />
fi<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
5. x<br />
6. xy<br />
7. <br />
Example 1<br />
Using a Linear Model to Investigate a Town’s Population<br />
<br />
<br />
a. <br />
b. <br />
Solution Th<br />
y<br />
<br />
fi<br />
t<br />
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t=fifi<br />
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m= __ <br />
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t =fi<br />
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312<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
m =__________<br />
− − <br />
=____<br />
<br />
=<br />
y<br />
Pt=t +<br />
t =<br />
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o fiPt=t<br />
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t ≈<br />
ft<br />
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. Thr a fi<br />
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Try It #2<br />
<br />
<br />
a. <br />
b. <br />
Using a Diagram to Model a Problem<br />
<br />
fi<br />
ThThft<br />
fiftTh<br />
<br />
Example 2<br />
Using a Diagram to Model Distance Walked<br />
<br />
Thft<br />
<br />
Solution
SECTION 4.2 MODELING WITH LINEAR FUNCTIONS 313<br />
<br />
fi<br />
t<br />
AtEt<br />
fiFigure 2<br />
<br />
<br />
<br />
<br />
Figure 2<br />
Tht =<br />
. Th<br />
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. ThAE<br />
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fi<br />
ThA<br />
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fiD<br />
ftFigure 3<br />
D<br />
tAtEtDt<br />
A<br />
E<br />
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Figure 3<br />
Figure 3n Th
314<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
n Th<br />
Dt =At +Et <br />
=t +t <br />
=t +t <br />
=t <br />
Dt=±√ — t Dt<br />
=±|t |<br />
tDt<br />
Dt=tTh<br />
D<br />
Dt=t<br />
Dt=<br />
<br />
t =<br />
t = __<br />
=<br />
Th<br />
Q & A…<br />
Should I draw diagrams when given information based on a geometric shape?<br />
e fi<br />
Example 3<br />
Using a Diagram to Model Distance Between Cities<br />
Th<br />
fi<br />
<br />
<br />
Solution Figure 4<br />
Th<br />
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Figure 4<br />
<br />
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n fi<br />
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m=______<br />
− − = __<br />
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Wx= __<br />
x<br />
m =−<br />
n fi<br />
Ex=−x +b<br />
=−+b <br />
b =<br />
Ex=−x +
SECTION 4.2 MODELING WITH LINEAR FUNCTIONS 315<br />
fifi<br />
<br />
<br />
<br />
__<br />
x =−x +<br />
<br />
__<br />
x =<br />
<br />
x =<br />
x =<br />
y =W<br />
Wx<br />
= __<br />
<br />
=<br />
Thfi<br />
<br />
= √ ——<br />
x <br />
−x <br />
+y <br />
−y <br />
<br />
=√ — − +− <br />
≈<br />
Analysis One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This<br />
means real-world applications discussing maps need linear functions to model the distances between reference points.<br />
Try It #3<br />
Th<br />
<br />
<br />
Modeling a Set of Data with Linear Functions<br />
<br />
<br />
Figure 5<br />
y y y<br />
<br />
<br />
<br />
f<br />
g<br />
x<br />
<br />
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x<br />
g<br />
f<br />
x<br />
<br />
<br />
<br />
Figure 5<br />
How To…<br />
<br />
1. <br />
2. y<br />
3. xfi
316<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Example 4<br />
Building a System of Linear Models to Choose a Truck Rental Company<br />
Thfi<br />
Th <br />
<br />
Solution Th<br />
fiTable 1<br />
Input<br />
Outputs<br />
Initial Value<br />
Rate of Change<br />
d<br />
Kd<br />
Md<br />
K=M=<br />
Kd=Pd=<br />
Table 1<br />
f x=mx+b<br />
Kd=d +<br />
Md=d +<br />
<br />
Kd<br />
<br />
Interpreting a Linear Function (http://openstaxcollege.org/l/interpretlinear)
SECTION 4.2 SECTION EXERCISES 317<br />
4.2 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
2. <br />
<br />
3. <br />
<br />
4. <br />
<br />
ALGEBRAIC<br />
5. <br />
yx=f x=+x<br />
f(x<br />
6. x<br />
f(x=–__<br />
x<br />
f(x<br />
7. y<br />
f(x=–__<br />
x<br />
f(x<br />
8. <br />
xg x=f(x=x<br />
f x<br />
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9. 10. <br />
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11. 12. <br />
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13. <br />
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14. <br />
P<br />
15. P <br />
xy<br />
16. P <br />
<br />
17. 18.
318 CHAPTER 4 LINEAR FUNCTIONS<br />
: Th. Th<br />
s fi<br />
19. <br />
W,<br />
t<br />
20. <br />
W<br />
21. W <br />
xy<br />
22. W <br />
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23. 24. <br />
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: Thffl<br />
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25. <br />
C,<br />
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29. 30. <br />
GRAPHICAL<br />
Figure 7fiy<br />
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SECTION 4.2 SECTION EXERCISES 319<br />
Figure 8fiy<br />
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Figure 8<br />
35. yyt<br />
36. y<br />
37. x<br />
38. <br />
NUMERIC<br />
fl<br />
Table 2<br />
Year Mississippi Hawaii<br />
<br />
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Table 2<br />
39. <br />
40. <br />
41. ft<br />
r? (Th<br />
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Table 3<br />
Year Indiana Alabama<br />
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Table 3<br />
42. <br />
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r? (Th
320 CHAPTER 4 LINEAR FUNCTIONS<br />
REAL-WORLD APPLICATIONS<br />
45. <br />
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47. <br />
ys a fl<br />
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x<br />
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b. y<br />
c. <br />
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48. <br />
ys a fl<br />
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a. <br />
x<br />
<br />
b. y<br />
c. <br />
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49. <br />
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a. <br />
P<br />
b. <br />
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50. <br />
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e 285. Th<br />
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a. P<br />
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51. Th<br />
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a. <br />
Rt<br />
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b. <br />
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52. <br />
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b. <br />
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SECTION 4.2 SECTION EXERCISES 321<br />
53. o diff<br />
s. Th <br />
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plus<br />
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54. o diff<br />
nies. Th <br />
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55. <br />
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58.
322<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Draw and interpret scatter plots.<br />
Use a graphing utility to find the line of best fit.<br />
Distinguish between linear and nonlinear relations.<br />
Fit a regression line to a set of data and use the linear model to make predictions.<br />
4.3 FITTING LINEAR MODELS TO DATA<br />
fi<br />
fi<br />
<br />
<br />
Drawing and Interpreting Scatter Plots<br />
<br />
<br />
Figure 1<br />
Final Exam Score<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Final Exam Score vs. Age<br />
<br />
<br />
Age<br />
Figure 1 A scatter plot of age and final exam score variables.<br />
notTh<br />
e fi<br />
Example 1<br />
Using a Scatter Plot to Investigate Cricket Chirps<br />
Table 1ff <br />
<br />
Chirps <br />
Temperature <br />
Table 1 Cricket Chirps vs Air Temperature<br />
Solution Figure 2<br />
Th
SECTION 4.3 FITTING LINEAR MODELS TO DATA 323<br />
<br />
Cricket Chirps vs. Temperature<br />
Temperature (°F)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Cricket Chirps in 15 Seconds<br />
Figure 2<br />
Finding the Line of Best Fit<br />
<br />
fiTh<br />
y<br />
<br />
Example 2<br />
Finding a Line of Best Fit<br />
t fiTable 1o fi<br />
Solution <br />
<br />
y. Th<br />
m=__<br />
=<br />
Tc=c +<br />
cTcTh<br />
Figure 3<br />
<br />
Cricket Chirps vs. Temperature<br />
<br />
T(c), Temperature (°F)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
c, Number of Chirps<br />
Analysis<br />
Figure 3<br />
This linear equation can then be used to approximate answers to various questions we might ask about the trend.
324<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Recognizing Interpolation or Extrapolation<br />
<br />
interpolation<br />
Thextrapolation<br />
<br />
Figure 4 Example 2<br />
<br />
<br />
<br />
Thff<br />
<br />
ftmodel breakdown<br />
<br />
x =<br />
s fift<br />
<br />
Cricket Chirps vs. Temperature<br />
<br />
T(c), Temperature (°F)<br />
<br />
<br />
<br />
<br />
Extrapolation<br />
Interpolation<br />
<br />
<br />
<br />
<br />
<br />
c, Number of Chirps<br />
Figure 4 Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.<br />
interpolation and extrapolation<br />
ff<br />
Thinterpolation<br />
<br />
Thextrapolation<br />
<br />
Example 3<br />
Understanding Interpolation and Extrapolation<br />
Table 1<br />
a. 15 <br />
<br />
b.
SECTION 4.3 FITTING LINEAR MODELS TO DATA 325<br />
Solution<br />
a. Th<br />
<br />
T=+<br />
=<br />
<br />
b. Th<br />
<br />
<br />
<br />
=+c<br />
=c<br />
c ≈<br />
Figure 5<br />
<br />
Cricket Chirps vs. Temperature<br />
T(c), Temperature (°F)<br />
<br />
<br />
<br />
<br />
<br />
<br />
Extrapolation<br />
Interpolation<br />
<br />
<br />
<br />
<br />
c, Number of Chirps<br />
Figure 5<br />
Analysis Our model predicts the crickets would chirp times in seconds. While this might be possible, we have<br />
no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether<br />
below around degrees.<br />
Try It #1<br />
Table 1<br />
Finding the Line of Best Fit Using a Graphing Utility<br />
fi<br />
ff least squares regression<br />
ftft <br />
fi<br />
<br />
ff
326<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
How To…<br />
n, fit fi<br />
1. List 1 (L1)<br />
2. List 2 (L2)<br />
3. Linear Regression (LinReg)<br />
Example 4<br />
Finding a Least Squares Regression Line<br />
Table 2<br />
Solution<br />
1. List 1 (L1)<br />
2. List 2 (L2)Table 2<br />
L1 <br />
L2 <br />
Table 2<br />
3. Linear Regression (LinReg)<br />
<br />
Tc=+c<br />
Analysis Notice that this line is quite similar to the equation we “eyeballed” but should fit the data better. Notice also<br />
that using this equation would change our prediction for the temperature when hearing chirps in seconds from 66<br />
degrees to:<br />
T=+<br />
=<br />
≈<br />
The graph of the scatter plot with the least squares regression line is shown in Figure 6.<br />
<br />
Number of Cricket Chirps<br />
vs. Temperature<br />
T(c), Temperature (°F)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
c, Number of Chirps<br />
Figure 6<br />
Q & A…<br />
Will there ever be a case where two different lines will serve as the best fit for the data?<br />
. Tht fi
SECTION 4.3 FITTING LINEAR MODELS TO DATA 327<br />
Distinguishing Between Linear and Non-Linear Models<br />
fi<br />
ft<br />
ffifi<br />
fiffir<br />
ffi<br />
ffi<br />
ffi<br />
rFigure 7<br />
ffi<br />
<br />
<br />
−<br />
−<br />
−<br />
<br />
− − −<br />
0 0 0<br />
Figure 7 Plotted data and related correlation coefficients. (credit: “DenisBoigelot,” Wikimedia Commons)<br />
correlation coefficient<br />
correlation coefficientr−<br />
r><br />
r<<br />
Th<br />
Th−<br />
Example 5<br />
Finding a Correlation Coefficient<br />
ffiTable 1<br />
Solution r<br />
Th<br />
effir =. Th<br />
ffi<br />
2nd>0>alphax − 1DIAGNOSTICSON<br />
Fitting a Regression Line to a Set of Data<br />
ffi<br />
<br />
t fi
328<br />
CHAPTER 4 LINEAR FUNCTIONS<br />
Example 6<br />
Using a Regression Line to Make Predictions<br />
<br />
Table 3 fi<br />
<br />
Year <br />
Consumption<br />
(billions of gallons)<br />
<br />
Table 3<br />
ThFigure 8<br />
<br />
Gas Consumption vs. Year<br />
Gas Consumption<br />
(billions of gallons)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Years After 1994<br />
Figure 8<br />
Solution t<br />
Th<br />
Ct=+t<br />
ffi<br />
<br />
t=<br />
C=+<br />
=<br />
Th<br />
Try It #2<br />
Example 6<br />
<br />
h fi<br />
Introduction to Regression Analysis (http://openstaxcollege.org/l/introregress)<br />
Linear Regression (http://openstaxcollege.org/l/linearregress)
SECTION 4.3 SECTION EXERCISES 329<br />
4.3 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
2. <br />
3. 4. e diff<br />
<br />
5. <br />
<br />
ALGEBRAIC<br />
6. <br />
<br />
xy). Th<br />
<br />
<br />
<br />
y=ax+b<br />
a=−<br />
b=<br />
r=−<br />
7. s <br />
<br />
xys). Th<br />
<br />
<br />
y=ax+b<br />
a=<br />
b=−<br />
r=−<br />
<br />
8.<br />
<br />
− − − − − −<br />
9.<br />
<br />
<br />
10.<br />
<br />
11.<br />
<br />
<br />
<br />
12. <br />
<br />
<br />
<br />
<br />
Year <br />
Population <br />
13. <br />
<br />
°<br />
<br />
<br />
Temperature, °F <br />
Time, seconds
330 CHAPTER 4 LINEAR FUNCTIONS<br />
GRAPHICAL<br />
fiFigure 9Figure 10<br />
<br />
Figure 9<br />
<br />
<br />
Figure 10<br />
<br />
14. r= 15. r=−<br />
16. r=− 17. r=−<br />
fi<br />
18.<br />
<br />
19.
SECTION 4.3 SECTION EXERCISES 331<br />
20. 21. <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
NUMERIC<br />
22. Thes. Th<br />
Table 4 <br />
<br />
Year <br />
Percent Graduates <br />
Table 4<br />
23. ThTable 5<br />
<br />
Year <br />
Imports <br />
Table 5<br />
24. Table 6<br />
<br />
<br />
Year <br />
Number Unemployed <br />
TECHNOLOGY<br />
Table 6<br />
<br />
ffi<br />
25. x <br />
y <br />
26. x <br />
y
332 CHAPTER 4 LINEAR FUNCTIONS<br />
27. x <br />
y <br />
x <br />
y <br />
28. x <br />
y <br />
29.<br />
x <br />
y − − −<br />
30. x <br />
y <br />
31. x <br />
y <br />
EXTENSIONS<br />
32. f x=x+<br />
x=−<br />
d fi<br />
<br />
33. f x=−x−<br />
x=−<br />
<br />
Thfi<br />
fi<br />
fi <br />
<br />
34. P<br />
fi<br />
<br />
35. <br />
x<br />
36. <br />
y
SECTION 4.3 SECTION EXERCISES 333<br />
REAL-WORLD APPLICATIONS<br />
Th<br />
Thfi<br />
<br />
<br />
37. y<br />
<br />
<br />
38. <br />
ThfiTh<br />
fi<br />
fifi<br />
<br />
39. y<br />
fi<br />
<br />
40. fi<br />
<br />
Thfi<br />
Thfi<br />
fifi<br />
<br />
41. y<br />
fi<br />
<br />
42. fi
334 CHAPTER 4 LINEAR FUNCTIONS<br />
CHAPTER 4 REVIEW<br />
Key Terms<br />
correlation coefficient r−<br />
e fi<br />
decreasing linear function f x=mx+bm <<br />
extrapolation <br />
horizontal line fif x=bb. Th<br />
increasing linear function f x=mx+bm ><br />
interpolation <br />
least squares regression fiff<br />
<br />
linear function <br />
model breakdown ft<br />
parallel lines <br />
perpendicular lines <br />
point-slope form y −y <br />
=mx−x <br />
<br />
slope <br />
slope-intercept form f x=mx+b<br />
vertical line fix =aa. Thfi<br />
Key Concepts<br />
4.1 Linear Functions<br />
Example 1<br />
<br />
<br />
Example 2<br />
Thdiff<br />
e diffxExample 3Example 4<br />
Example 5<br />
Thm bExample 6 <br />
Example 7<br />
diffExample 8<br />
Example 9Example 10<br />
Example 11<br />
Example 12<br />
<br />
Example 13<br />
ThExample 14<br />
xxExample 15<br />
f x=bExample 16
CHAPTER 4 REVIEW 335<br />
x =bExample 17<br />
<br />
Example 18<br />
<br />
xyf x=mx+b<br />
Example 19<br />
<br />
Example 20Example 21<br />
4.2 Modeling with Linear Functions<br />
<br />
<br />
yExample 1<br />
Example 2Example 3<br />
<br />
y<br />
◦<br />
xy =mx+b<br />
◦<br />
Thxy<br />
Example 4<br />
◦<br />
<br />
4.3 Fitting Linear Models to Data<br />
Example 1<br />
<br />
Tht fiftExample 2<br />
<br />
Example 3<br />
Th rExample 4<br />
t fiExample 5<br />
Th<br />
Example 6
336 CHAPTER 4 LINEAR FUNCTIONS<br />
CHAPTER 4<br />
REVIEW EXERCISES<br />
LINEAR FUNCTIONS<br />
1. <br />
x+y=<br />
3. <br />
fx=x−<br />
5. <br />
<br />
<br />
7. <br />
– – – – –<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
9. <br />
<br />
– – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
11. <br />
<br />
x <br />
g(x) – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
2. <br />
x −y=<br />
4. <br />
gx=−x+<br />
6. <br />
<br />
xy<br />
8. <br />
– – – – –<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
10. <br />
<br />
x – <br />
g(x) – – –<br />
12. fi<br />
<br />
ft<br />
n ft <br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
13. x−y=<br />
−x+y =<br />
14. y=__<br />
x−<br />
x +y =−<br />
, fixy<br />
15. x+y=− 16. fx=x−
CHAPTER 4 REVIEW 337<br />
fi<br />
<br />
17. <br />
−−<br />
18. −−<br />
<br />
19. <br />
fx=x−<br />
20. y<br />
− __<br />
<br />
21. ft=t− 22. <br />
x=y+<br />
x −y =<br />
23. ff<br />
<br />
<br />
<br />
MODELING WITH LINEAR FUNCTIONS<br />
24. yfx=−xf<br />
<br />
25. <br />
<br />
26. Th <br />
<br />
fflC<br />
tffl<br />
Figure 1fiy<br />
xx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
Figure 1<br />
27. yyx<br />
28. y<br />
<br />
<br />
29. <br />
a. <br />
b. <br />
c. Pt<br />
x
338 CHAPTER 4 LINEAR FUNCTIONS<br />
<br />
<br />
30. P<br />
31. <br />
Th<br />
flTable 1<br />
Year Pima Central East Valley<br />
<br />
<br />
Table 1<br />
32. <br />
33. <br />
FITTING LINEAR MODELS TO DATA<br />
34. Table 2. Th<br />
0 2 4 6 8 10<br />
– – <br />
Table 2<br />
35. Table 3<br />
<br />
Year <br />
Population <br />
Table 3<br />
36. Th<br />
Table 4t fi<br />
Predicted <br />
Actual <br />
Table 4<br />
37. t-fi<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x
CHAPTER 4 REVIEW 339<br />
Table 5<br />
5 y<br />
Year <br />
Percent Graduates <br />
Table 5<br />
38. <br />
<br />
<br />
<br />
39. <br />
40. Table 6<br />
<br />
<br />
<br />
x <br />
y <br />
Table 6<br />
41. Table 7<br />
<br />
<br />
<br />
x <br />
y <br />
Table 7<br />
Th<br />
The fi<br />
<br />
<br />
42. y<br />
<br />
<br />
43. <br />
44. <br />
<br />
45. <br />
in 2014?
340 CHAPTER 4 LINEAR FUNCTIONS<br />
CHAPTER 4<br />
PRACTICE TEST<br />
1. <br />
<br />
x+y=<br />
3. <br />
fx=x+<br />
2. <br />
fx=−x+<br />
4. <br />
<br />
−<br />
5. <br />
x−y−<br />
6. Figure 1<br />
y<br />
– – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 1<br />
<br />
<br />
<br />
<br />
<br />
x<br />
7. Figure 2<br />
y<br />
– – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 2<br />
<br />
<br />
<br />
<br />
<br />
x<br />
8. Table 1 <br />
<br />
x − <br />
g(x) <br />
Table 1<br />
9. Table 2 <br />
<br />
x <br />
g(x) <br />
Table 2<br />
10. <br />
nft<br />
<br />
<br />
11. y=__<br />
x−<br />
−x −y =<br />
13. xy<br />
x+y=−<br />
12. −x+y=<br />
x +__<br />
y =<br />
<br />
14. <br />
<br />
−−
CHAPTER 4 PRACTICE TEST<br />
341<br />
15. <br />
fx=x+<br />
16. y− __<br />
<br />
17. fx=−x+ 18. <br />
x=y+<br />
<br />
x −y =−<br />
19. ff<br />
<br />
<br />
<br />
<br />
<br />
20. y<br />
fx=−x<br />
f<br />
21. <br />
<br />
<br />
<br />
22. Th <br />
<br />
<br />
<br />
<br />
Ct<br />
<br />
Figure 3fiy<br />
xx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
Figure 3<br />
x<br />
23. yyx<br />
<br />
24. y<br />
25. <br />
<br />
a. <br />
b. <br />
c. Pt
342 CHAPTER 4 LINEAR FUNCTIONS<br />
26. Table 3<br />
Th<br />
<br />
0 2 4 6 8 10<br />
− − <br />
Table 3<br />
27. t-fi<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Table 4<br />
<br />
Year <br />
Percent Graduates <br />
Table 4<br />
28. <br />
<br />
29. <br />
30. Table 5<br />
<br />
x <br />
y <br />
Table 5<br />
ThTh<br />
<br />
fi<br />
<br />
31. y<br />
<br />
32. <br />
33.
Polynomial and Rational Functions<br />
5<br />
Figure 1 35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography.<br />
(credit “film”: modification of work by Horia Varlan; credit “memory cards”: modification of work by Paul Hudson)<br />
CHAPTER OUTLINE<br />
5.1 Quadratic Functions<br />
5.2 Power Functions and Polynomial Functions<br />
5.3 Graphs of Polynomial Functions<br />
5.4 Dividing Polynomials<br />
5.5 Zeros of Polynomial Functions<br />
5.6 Rational Functions<br />
5.7 Inverses and Radical Functions<br />
5.8 Modeling Using Variation<br />
Introduction<br />
<br />
fi<br />
<br />
fift<br />
ft<br />
eff<br />
fi<br />
<br />
343
344<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Recognize characteristics of parabolas.<br />
Understand how the graph of a parabola is related to its quadratic function.<br />
Determine a quadratic function’s minimum or maximum value.<br />
Solve problems involving a quadratic function’s minimum or maximum value.<br />
5.1 QUADRATIC FUNCTIONS<br />
Figure 1 An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<br />
Figure 1<br />
Th<br />
<br />
<br />
<br />
<br />
Recognizing Characteristics of Parabolas<br />
Th<br />
vertex<br />
<br />
Th<br />
axis of symmetry. ThFigure 2<br />
y<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
x<br />
x<br />
<br />
–<br />
y<br />
<br />
–<br />
<br />
Figure 2
SECTION 5.1 QUADRATIC FUNCTIONS 345<br />
yyThx<br />
xxzerosroots<br />
xy =<br />
Example 1<br />
Identifying the Characteristics of a Parabola<br />
yFigure 3<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – <br />
x<br />
Figure 3<br />
Solution Th<br />
x =<br />
Thxyy<br />
Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions<br />
general form of a quadratic function<br />
fx=ax +bx+c<br />
abca ≠a >a <<br />
o fi<br />
Thfix =− b <br />
a x b −ac<br />
=−b±√— <br />
a<br />
ax +bx+c =xfixx =− b <br />
a <br />
<br />
Figure 4y = x +x +<br />
<br />
a =b =c =a >Thx =− <br />
<br />
=−Th<br />
x =−Th<br />
<br />
−−Thxx<br />
−−<br />
x<br />
x<br />
– – <br />
–<br />
–<br />
<br />
<br />
<br />
y y = x + x + <br />
Figure 4
346<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
standard form of a quadratic function<br />
fx=ax−h +k<br />
hk<br />
vertex form of a quadratic function<br />
a >a <<br />
Figure 5<br />
y =−x+ +x −h =x +h =−a =−h =−k =<br />
a ftk <br />
fth >fth
SECTION 5.1 QUADRATIC FUNCTIONS 347<br />
Thfi<br />
ah +k =c<br />
k =c −ah <br />
=c −a ( __<br />
a) b <br />
=c − b <br />
a<br />
kh<br />
fh=k<br />
forms of quadratic functions<br />
f de. Th<br />
general form of a quadratic functionfx=ax +bx+cabca≠<br />
standard form of a quadratic functionfx=ax−h +ka≠<br />
Thhk<br />
h=−<br />
b <br />
a k=fh=f ( <br />
−b <br />
a ) <br />
How To…<br />
<br />
1. h k<br />
2. hkfx=ax−h +k<br />
3. xf x<br />
4. a<br />
5. a >a
348<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
−<br />
−=a+ −<br />
<br />
=a<br />
a = __<br />
<br />
g x=__<br />
<br />
x+ −<br />
<br />
gx=__<br />
<br />
x+ −<br />
=__<br />
x +x +−<br />
<br />
=__<br />
x +x +−<br />
=__<br />
x +x +−<br />
=__<br />
x +x −<br />
ft<br />
ff<br />
Analysis We can check our work using the table feature on a graphing utility. First enter Y1 = 1 __<br />
2 (x + 2) 2 − 3. Next,<br />
select TBLSET, then use TblStart = − 6 and ΔTbl = 2, and select TABLE. See Table 1.<br />
x − − − <br />
y − − − <br />
Table 1<br />
The ordered pairs in the table correspond to points on the graph.<br />
Try It #1<br />
Figure 8<br />
<br />
Figure 8 (credit: modification of work by Dan Meyer)<br />
How To…<br />
, fi<br />
1. abc<br />
2. hxabh =− b __<br />
a <br />
3. kyk =f h=f ( − b __<br />
a)
SECTION 5.1 QUADRATIC FUNCTIONS 349<br />
Example 3<br />
Finding the Vertex of a Quadratic Function<br />
f x=x −x +<br />
Solution Th<br />
h =− b <br />
a<br />
=− − <br />
<br />
Th<br />
= <br />
<br />
= <br />
<br />
k =fh<br />
=f( <br />
) <br />
=( <br />
) −( <br />
) +<br />
= <br />
<br />
afi<br />
Thfi<br />
a<br />
f x=ax +bx+c<br />
f x=x −x +<br />
Th<br />
f x=( x − <br />
) + <br />
<br />
Analysis One reason we may want to identify the vertex of the parabola is that this point will inform us where the<br />
maximum or minimum value of the output occurs, k, and where it occurs, x.<br />
Try It #2<br />
g x=+x −x<br />
Finding the Domain and Range of a Quadratic Function<br />
Th<br />
<br />
y<br />
yy<br />
<br />
domain and range of a quadratic function<br />
Th<br />
. Th fx = ax + bx + c <br />
f x ≥ f ( −<br />
__<br />
<br />
a) b [ f ( −<br />
__<br />
<br />
a) b ∞ ) <br />
Thaf x≤f ( −<br />
<br />
<br />
a) b ( −∞f ( −<br />
<br />
<br />
a) b <br />
] <br />
Thf x=ax−h +ka<br />
f x≥kaf x≤k
350<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
How To…<br />
n, fi<br />
1. <br />
2. aaa<br />
<br />
3. k<br />
4. fx≥kk∞<br />
f x≤k−∞k<br />
Example 4<br />
Finding the Domain and Range of a Quadratic Function<br />
f x=−x +x −<br />
Solution <br />
a<br />
y fix<br />
Thf h<br />
Thf x≤ <br />
( −∞ <br />
] <br />
f( <br />
h =− b <br />
a<br />
<br />
=− <br />
− <br />
= <br />
<br />
<br />
) +( ) −<br />
= <br />
<br />
) =− ( <br />
Try It #3<br />
f x=( x− <br />
) + <br />
Determining the Maximum and Minimum Values of Quadratic Functions<br />
Th<br />
Figure 9<br />
<br />
<br />
y<br />
f x = x − + <br />
–<br />
<br />
<br />
y<br />
gx = −x + + <br />
–<br />
–<br />
–<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
<br />
x = <br />
<br />
Figure 9<br />
–<br />
–<br />
–<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
<br />
x = −<br />
Thfi
SECTION 5.1 QUADRATIC FUNCTIONS 351<br />
Example 5<br />
Finding the Maximum Value of a Quadratic Function<br />
<br />
<br />
a. <br />
L<br />
b. d a<br />
Solution Figure 10<br />
W<br />
<br />
W<br />
L<br />
<br />
Figure 10<br />
a. L+W+L=L+W=<br />
WL<br />
W =−L<br />
<br />
<br />
A =LW=L−L<br />
AL=L −L <br />
LTh<br />
<br />
AL=−L +L<br />
b. <br />
<br />
<br />
aa=−b=c=<br />
o fi<br />
h =− b k =A<br />
a<br />
<br />
h =− =−<br />
−<br />
= =<br />
ThL = <br />
<br />
<br />
Analysis This problem also could be solved by graphing the quadratic function. We can see where the maximum area<br />
occurs on a graph of the quadratic function in Figure 11.<br />
Area (A)<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
A<br />
<br />
Length (L)<br />
Figure 11<br />
<br />
x
352<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
How To…<br />
o fi<br />
1. <br />
2. <br />
3. y<br />
Example 6<br />
Finding Maximum Revenue<br />
ThffTh<br />
<br />
<br />
<br />
<br />
Solution <br />
p<br />
Q=pQ<br />
fi<br />
p =Q =<br />
p =Q =fi<br />
. Th<br />
m = − _____________<br />
<br />
− <br />
=_______<br />
− <br />
=−<br />
Th<br />
y<br />
Q =−p +b Q =p =<br />
=−+b b<br />
b =<br />
ThQ =−p +<br />
<br />
<br />
=pQ<br />
<br />
=p−p +<br />
<br />
=−p +p<br />
fi<br />
n fi<br />
_________<br />
h =− − <br />
=<br />
Thfi<br />
<br />
<br />
=− +<br />
=<br />
Analysis This could also be solved by graphing the quadratic as in Figure 12. We can see the maximum revenue on a<br />
graph of the quadratic function.
SECTION 5.1 QUADRATIC FUNCTIONS 353<br />
Revenue ($1,000)<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Price (p)<br />
Figure 12<br />
x<br />
Finding the x- and y-Intercepts of a Quadratic Function<br />
fi<br />
fiyfi<br />
xFigure 13x<br />
<br />
y<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
Figure 13 Number of x-intercepts of a parabola<br />
x<br />
How To…<br />
f x, fiyx<br />
1. f o fiy<br />
2. f x=o fix<br />
Example 7<br />
Finding the y- and x-Intercepts of a Parabola<br />
yxfx=x +x −<br />
Solution e fiyf <br />
f= +−<br />
=−<br />
y−<br />
xe fifx=<br />
<br />
=x +x −<br />
<br />
<br />
=x −x +<br />
=x − =x +<br />
x =__<br />
x=−<br />
<br />
x( __<br />
) −
354<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
y<br />
f x = x<br />
Analysis By graphing the function, we can confirm that the graph crosses<br />
+ x − <br />
– <br />
the y-axis at −. We can also confirm that the graph crosses the xaxis<br />
<br />
)<br />
at ( __<br />
) and −. See Figure 14. x<br />
– – – <br />
–<br />
)<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Figure 14 <br />
Rewriting Quadratics in Standard Form<br />
Example 7<br />
y fi<br />
How To…<br />
n, fix<br />
1. abh =− b __<br />
a <br />
2. x =ho fik<br />
3. hk<br />
4. o fix<br />
Example 8<br />
Finding the x-Intercepts of a Parabola<br />
xf x=x +x −<br />
Solution <br />
=x +x −<br />
fi<br />
<br />
f x=ax−h +k<br />
a =. Thhk<br />
h =− b __<br />
a <br />
k =f −<br />
<br />
=− ___<br />
<br />
=− +−−<br />
=−<br />
=−<br />
<br />
fx=x + −<br />
<br />
<br />
=x + −<br />
<br />
=x + <br />
<br />
=x + <br />
x +=±√ — <br />
x =−±√ — <br />
Thx−−√ — −+√ —
SECTION 5.1 QUADRATIC FUNCTIONS 355<br />
<br />
x Figure 15<br />
<br />
<br />
y<br />
− <br />
– – – <br />
–<br />
–<br />
–<br />
<br />
x<br />
Figure 15<br />
Analysis We could have achieved the same results using the quadratic formula. Identify a = , b = , and c = −.<br />
<br />
<br />
<br />
<br />
<br />
b<br />
x = −b±√— −ac <br />
<br />
a<br />
<br />
x = −±√— −− <br />
<br />
<br />
x = −±√— <br />
<br />
x =<br />
−±√— <br />
x =−±√ — <br />
So the x-intercepts occur at−−√ — and−+√ — <br />
Try It #4<br />
Try Itgx=+x −x y<br />
x<br />
Example 9<br />
Applying the Vertex and x-Intercepts of a Parabola<br />
Th<br />
Ht=−t +t +<br />
a. <br />
b. <br />
c. <br />
Solution<br />
a. Th<br />
<br />
h =− <br />
− <br />
= <br />
<br />
= <br />
<br />
=<br />
Thft
356<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
b. y<br />
k =H ( − b <br />
a)<br />
=H<br />
=− ++<br />
=<br />
Th<br />
Ht=<br />
<br />
t = −±√— −− <br />
<br />
− <br />
<br />
= −±√— − <br />
<br />
<br />
<br />
t = −−√— <br />
<br />
− ≈t = <br />
<br />
−+√— − ≈−<br />
Th<br />
ftFigure 16.<br />
H<br />
<br />
<br />
<br />
Ht = −t + t + <br />
<br />
<br />
<br />
<br />
<br />
Figure 16.<br />
t<br />
Try It #5<br />
<br />
<br />
<br />
Th<br />
Ht=−t +t +<br />
a. <br />
b. <br />
c. <br />
<br />
Graphing Quadratic Functions in General Form (http://openstaxcollege.org/l/graphquadgen)<br />
Graphing Quadratic Functions in Standard Form (http://openstaxcollege.org/l/graphquadstan)<br />
Quadratic Function Review (http://openstaxcollege.org/l/quadfuncrev)<br />
Characteristics of a Quadratic Function (http://openstaxcollege.org/l/characterquad)
SECTION 5.1 SECTION EXERCISES 357<br />
5.1 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. a≠<br />
<br />
2. <br />
<br />
4. <br />
<br />
5. <br />
<br />
ALGEBRAIC<br />
<br />
6. f x=x −x+ 7. gx=x +x− 8. f x=x −x<br />
9. f x=x +x− 10. hx=x +x− 11. kx=x −x−<br />
12. f x=x −x 13. f x=x −x−<br />
<br />
<br />
14. yx=x +x+ 15. fx=x −x+ 16. fx=−x +x+<br />
17. fx=x +x− 18. ht=−t +t− 19. fx=__<br />
x +x+<br />
20. fx=− __<br />
x −x+<br />
<br />
21. fx=x − + 22. fx=−x + − 23. fx=x +x+<br />
24. fx=x −x+ 25. kx=x −x−<br />
hkxyo fi<br />
<br />
26. hk=xy= 27. hk=−−xy=− 28. hk=xy=<br />
29. hk=xy= 30. hk=−xy= 31. hk=xy=<br />
32. hk=xy= 33. hk=xy=<br />
GRAPHICAL<br />
<br />
<br />
34. fx=x −x 35. fx=x −x− 36. fx=x −x−<br />
37. fx=x −x+ 38. fx=−x +x− 39. fx=x −x−
358 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
<br />
40.<br />
y 41.<br />
y 42.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
<br />
–<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – –<br />
–<br />
<br />
–<br />
–<br />
x<br />
43.<br />
y 44. y<br />
45.<br />
y<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– –<br />
–<br />
<br />
–<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
x<br />
NUMERIC<br />
<br />
, fi<br />
46. x − − 47. x − − 48.<br />
y <br />
y <br />
x − − <br />
y − −<br />
49. x − − 50.<br />
y − − <br />
x − − <br />
y <br />
TECHNOLOGY<br />
o fi<br />
51. <br />
f x=x fx=x fx= <br />
x <br />
ffffi<br />
53. fx=x <br />
fx=x − fx − fx=x + <br />
e eff<br />
<br />
52. fx=x fx=x +<br />
fx=x fx=x +fx=x −<br />
e eff<br />
54. Th<br />
<br />
hx= − <br />
x +xx<br />
hx<br />
TRACE]<br />
<br />
<br />
55. hx=x −≤x≤<br />
∣ x ∣ hxTRACE]
SECTION 5.1 SECTION EXERCISES 359<br />
EXTENSIONS<br />
<br />
o fi<br />
56. − 57. −<br />
58. − 59. −<br />
<br />
<br />
60. fx=x <br />
y<br />
62. fx=x <br />
y<br />
64. fx=x <br />
y<br />
REAL-WORLD APPLICATIONS<br />
61. −fx=x <br />
y<br />
63. −fx=−x <br />
y<br />
65. −fx=x <br />
x−<br />
66. <br />
<br />
<br />
68. <br />
<br />
<br />
70. e diff<br />
<br />
<br />
72. <br />
<br />
ht=−t +t+<br />
<br />
74. <br />
<br />
<br />
<br />
<br />
<br />
67. <br />
<br />
<br />
69. <br />
<br />
<br />
71. <br />
<br />
p=−xx<br />
<br />
R=xċp<br />
<br />
73. <br />
<br />
<br />
ht=−t +t+<br />
<br />
75.
360<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Identify power functions.<br />
Identify end behavior of power functions.<br />
Identify polynomial functions.<br />
Identify the degree and leading coefficient of polynomial functions.<br />
5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS<br />
Figure 1 (credit: Jason Bay, Flickr)<br />
Table1<br />
Year <br />
Bird Population <br />
Table 1<br />
ThPt=−t +t +Pt<br />
tft<br />
<br />
<br />
Identifying Power Functions<br />
ffpower<br />
functioncoefficient<br />
fiffi<br />
. Thr<br />
Ar=πr <br />
r<br />
Vr= __<br />
πr <br />
ffiπ _ <br />
π<br />
r<br />
power function<br />
power function <br />
fx=kx p<br />
kpkcoefficient
SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 361<br />
Q & A…<br />
Is f (x) = 2 x a power function?<br />
fiTh<br />
. Th<br />
Example 1<br />
Identifying Power Functions<br />
<br />
f x=<br />
<br />
f x=x<br />
<br />
f x=x <br />
<br />
f x=x <br />
<br />
f x= __<br />
<br />
x<br />
<br />
f x=__<br />
<br />
x <br />
<br />
f x=√ — x <br />
f x= √ <br />
x <br />
Solution <br />
Thf x=x f x=x <br />
<br />
Thf x=x f x=x <br />
<br />
f x=x − f x=x − <br />
Th<br />
f x=x f x=x <br />
Try It #1<br />
<br />
fx=x ċx gx=−x +x hx= _ x − <br />
x + <br />
Identifying End Behavior of Power Functions<br />
Figure 2f x=x gx=x hx=x <br />
<br />
fl<br />
f x = x <br />
<br />
y<br />
gx = x <br />
<br />
hx = x <br />
<br />
<br />
– – <br />
x<br />
Figure 2 Even-power functions
362<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
fi∞<br />
fi−∞fixfi<br />
x →∞x<br />
<br />
x<br />
fif x<br />
x →±∞ f x→∞<br />
Figure 3f x=x gx=x hx=x <br />
<br />
s fl<br />
<br />
y<br />
gx = x <br />
<br />
f x = x x<br />
hx = x <br />
– – <br />
–<br />
–<br />
Figure 3 Odd-power functions<br />
Thf x=x n Figure<br />
2f x=x n nyFigure 3<br />
f x=x n n<br />
xfif xx<br />
fif x<br />
x →−∞ f x→−∞ x →∞ f x→∞<br />
Thx→−∞x→∞<br />
end behavior<br />
Figure 4 f x=kx n n<br />
<br />
Even power<br />
y<br />
Odd power<br />
y<br />
<br />
k ><br />
x<br />
x<br />
x→−∞ f x→∞<br />
x →∞ f x→∞<br />
y<br />
x→−∞ f x→∞<br />
x →∞ f x→−∞<br />
y<br />
<br />
k <<br />
x<br />
x<br />
x→−∞ f x→∞<br />
x →∞ f x→−∞<br />
Figure 4<br />
x→−∞ f x→−∞<br />
x →∞ f x→−∞
SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 363<br />
How To…<br />
f x=kx n n<br />
1. <br />
2. <br />
3. Figure 4<br />
Example 2<br />
Identifying the End Behavior of a Power Function<br />
f x=x <br />
Solution Thffix<br />
fif xx →∞f x→∞x<br />
fix →−∞f x→∞<br />
Figure 5<br />
Example 3<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 5<br />
Identifying the End Behavior of a Power Function<br />
f x=−x <br />
Solution Thffi−<br />
flxf x=x Figure 6xfi<br />
xfi<br />
<br />
x →−∞ f x→∞<br />
x →∞ f x→−∞<br />
y<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 6
364<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Analysis We can check our work by using the table feature on a graphing utility.<br />
x<br />
−<br />
−<br />
<br />
<br />
<br />
f x<br />
<br />
<br />
<br />
−<br />
−<br />
Table 2<br />
We can see from Table 2 that, when we substitute very small values for x, the output is very large, and when we substitute<br />
very large values for x, the output is very small (meaning that it is a very large negative value).<br />
Try It #2<br />
fx=−x <br />
Identifying Polynomial Functions<br />
Th<br />
8 <br />
Thrw<br />
. Th<br />
rw=+w<br />
A<br />
Ar=πr <br />
<br />
Aw=Arw<br />
=A+w<br />
=π+w <br />
<br />
Aw=π +πw+πw <br />
Thpolynomial function<br />
fiffi<br />
<br />
polynomial functions<br />
npolynomial function<br />
f x=a n<br />
x n ++ a <br />
x +a <br />
x +a <br />
Tha i<br />
ffia n<br />
<br />
=a i<br />
x i term of a polynomial function<br />
Example 4<br />
Identifying Polynomial Functions<br />
<br />
fx=x ċx + gx=−xx − hx=√ — x +<br />
Solution Thfi<br />
f x=a n<br />
x n ++a <br />
x +a <br />
x +a <br />
ffi
SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 365<br />
fxf x=x +<br />
gxgx=−x +x<br />
hx<br />
Identifying the Degree and Leading Coefficient of a Polynomial Function<br />
fi<br />
<br />
Thdegree<br />
e fi<br />
Thleading term<br />
. Thleading coefficientffi<br />
terminology of polynomial functions<br />
ft<br />
ffi<br />
<br />
fx=a n<br />
x n ++a <br />
x +a <br />
x +a <br />
<br />
<br />
How To…<br />
ffi<br />
1. x<br />
2. xo fi<br />
3. ffi<br />
Example 5<br />
Identifying the Degree and Leading Coefficient of a Polynomial Function<br />
ffi<br />
fx=+x −x <br />
gt=t −t +t<br />
hp=p −p −<br />
Solution f xxTh<br />
−x . Thffiffi−<br />
gttTh<br />
t . Thffiffi<br />
hppTh<br />
−p ffiffi−<br />
Try It #3<br />
<br />
fx=x −x +x−
366<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Identifying End Behavior of Polynomial Functions<br />
<br />
<br />
fix<br />
<br />
Table3<br />
Polynomial Function Leading Term Graph of Polynomial Function<br />
fx=x +x −x − x <br />
<br />
y<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
x<br />
fx=−x −x +x +x −x <br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
y<br />
fx=x −x +x + x <br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
y<br />
fx=−x +x +x + −x <br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Table 3
SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 367<br />
Example 6 Identifying End Behavior and Degree of a Polynomial Function<br />
Figure 7<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 7<br />
Solution xf xx<br />
f x<br />
x →−∞ f x→−∞<br />
x →∞ f x→∞<br />
xfifix<br />
fifi<br />
fl<br />
ffi<br />
Try It #4<br />
<br />
Figure 8<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 8<br />
Example 7<br />
Identifying End Behavior and Degree of a Polynomial Function<br />
f x=−x x−x+<br />
<br />
Solution f x<br />
f x=−x x−x+<br />
=−x x +x −<br />
=−x −x +x <br />
Thf x=−x −x +x Th−x <br />
Thffi−<br />
x →−∞ f x→−∞<br />
x →∞ f x→−∞
368<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Try It #5<br />
fx=x−x+x−<br />
<br />
Identifying Local Behavior of Polynomial Functions<br />
<br />
turning point<br />
<br />
y<br />
Th<br />
y<br />
a <br />
Thx<br />
xFigure 9<br />
y<br />
<br />
x<br />
x<br />
<br />
<br />
←y<br />
Figure 9<br />
intercepts and turning points of polynomial functions<br />
turning point<br />
ThyTh<br />
x<br />
How To…<br />
<br />
1. yx =d fi<br />
2. x<br />
Example 8<br />
Determining the Intercepts of a Polynomial Function<br />
f x=x−x+x−<br />
yx<br />
Solution yx<br />
f =−+−<br />
=−−<br />
=<br />
y
SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 369<br />
x<br />
<br />
=x−x+x−<br />
x −= x += x −=<br />
x = x =− x =<br />
x<br />
Figure 10<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x<br />
<br />
x<br />
−<br />
Example 9<br />
Figure 10<br />
Determining the Intercepts of a Polynomial Function with Factoring<br />
f x=x −x −yx<br />
Solution y<br />
f = − −<br />
=−<br />
y−<br />
x<br />
<br />
f x=x −x −<br />
=x −x +<br />
=x−x+x +<br />
<br />
=x−x+x +<br />
x −= x += x +=<br />
x = x =− <br />
x−<br />
Figure 11<br />
f x=f −x<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 11<br />
<br />
<br />
x<br />
−<br />
x<br />
<br />
<br />
y<br />
−<br />
Try It #6<br />
fx=x −x −xyx
370<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Comparing Smooth and Continuous Graphs<br />
Thx<br />
n th nn<br />
xThnn −Th<br />
<br />
continuous functionft<br />
smooth curveTh<br />
. Th<br />
intercepts and turning points of polynomials<br />
nnxn −<br />
Example 10<br />
Determining the Number of Intercepts and Turning Points of a Polynomial<br />
fi<br />
xf x=−x +x −x +x <br />
Solution<br />
Try It #7<br />
Thx<br />
x<br />
fx=−x −x +x +x <br />
Example 11<br />
Drawing Conclusions about a Polynomial Function from the Graph<br />
Figure 12<br />
<br />
y<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 12<br />
Solution ThFigure 13<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
x<br />
<br />
<br />
Figure 13<br />
Thx
SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 371<br />
Try It #8<br />
Figure 14 <br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 14<br />
Example 12 Drawing Conclusions about a Polynomial Function from the Factors<br />
f x=−xx+x−<br />
Solution yf <br />
f =−+−<br />
=<br />
y<br />
x<br />
<br />
=−xx+x−<br />
x = x += x −=<br />
x = x =− x =<br />
x−<br />
Th<br />
Try It #9<br />
fx=x−x+x−<br />
<br />
Find Key Information About a Given Polynomial Function (http://openstaxcollege.org/l/keyinfopoly)<br />
End Behavior of a Polynomial Function (http://openstaxcollege.org/l/endbehavior)<br />
Turning Points and x-intercepts of Polynomial Functions (http://openstaxcollege.org/l/turningpoints)<br />
Least Possible Degree of a Polynomial Function (http://openstaxcollege.org/l/leastposdegree)
372 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
5.2 SECTION EXERCISES<br />
VERBAL<br />
1. e diff <br />
<br />
3. <br />
<br />
<br />
5. <br />
<br />
x→−∞fx→−∞x→∞<br />
fx→−∞<br />
2. <br />
<br />
<br />
4. <br />
<br />
<br />
ALGEBRAIC<br />
<br />
6. fx=x 7. fx=x 8. fx=x−x <br />
9. fx=<br />
_____ x <br />
x − 10. fx=xx+x− 11. fx= x+<br />
, fiffi<br />
12. −x 13. −x 14. −x −x +x−<br />
15. x−x x+ 16. x x− <br />
<br />
17. fx=x 18. fx=x 19. fx=−x <br />
20. fx=−x 21. fx=−x −x +x− 22. fx=x +x−<br />
23. fx=x x −x+ 24. fx=−x <br />
, fi<br />
25. ft=t−t+t− 26. gn=−n−n+ 27. fx=x −<br />
28. fx=x + 29. fx=xx −x− 30. fx=x+x −<br />
GRAPHICAL<br />
<br />
31.<br />
y<br />
32.<br />
y<br />
33.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x
SECTION 5.2 SECTION EXERCISES 373<br />
34.<br />
<br />
x<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
35.<br />
<br />
x<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
36.<br />
<br />
x<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
37.<br />
<br />
x<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
38.<br />
<br />
x<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
39.<br />
x<br />
y<br />
40.<br />
x<br />
y<br />
41.<br />
x<br />
y<br />
42.<br />
x<br />
y<br />
43.<br />
x<br />
y<br />
44.<br />
x<br />
y<br />
45.<br />
x<br />
y
374 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
NUMERIC<br />
fi<br />
46. fx=−x 47. fx=x −x 48. fx=x −x <br />
49. fx=x−x−−x 50. fx=__<br />
x <br />
−x<br />
TECHNOLOGY<br />
<br />
<br />
51. fx=x x− 52. fx=xx−x+ 53. fx=x−x−x<br />
54. fx=x−x−x 55. fx=x −x 56. fx=x −<br />
57. fx=x − 58. fx=−x +x +x 59. fx=x −x −x<br />
60. fx=x −x<br />
EXTENSIONS<br />
<br />
ffi−. Th<br />
61. y−4). Thx−<br />
x→−∞<br />
fx→∞x→∞fx→∞<br />
63. ys (0, 0). Thx<br />
x→−∞<br />
fx→−∞x→∞fx→∞<br />
65. ys (0, 1). Thx<br />
x→−∞fx→∞<br />
x→∞fx→∞<br />
62. ys (0, 9). Thx−<br />
x→−∞<br />
fx→−∞x→∞fx→−∞<br />
64. ys (0, 1). Thx<br />
x→−∞fx→∞<br />
x→∞fx→−∞<br />
REAL-WORLD APPLICATIONS<br />
<br />
<br />
66. . Th<br />
<br />
d<br />
<br />
68. <br />
x<br />
<br />
x<br />
70. <br />
. Th<br />
<br />
x<br />
67. et. Th<br />
<br />
m<br />
<br />
69. <br />
x<br />
<br />
<br />
x
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 375<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Recognize characteristics of graphs of polynomial functions.<br />
Use factoring to find zeros of polynomial functions.<br />
Identify zeros and their multiplicities.<br />
Determine end behavior.<br />
Understand the relationship between degree and turning points.<br />
Graph polynomial functions.<br />
Use the Intermediate Value Theorem.<br />
5.3 GRAPHS OF POLYNOMIAL FUNCTIONS<br />
Thr a fiTable 1<br />
Year <br />
Revenues <br />
Th<br />
Table 1<br />
Rt=−t +t −t +t −<br />
Rtt =<br />
<br />
Th<br />
<br />
<br />
Recognizing Characteristics of Graphs of Polynomial Functions<br />
<br />
<br />
Figure 1<br />
<br />
y<br />
y<br />
f<br />
x<br />
x<br />
f<br />
Figure 1
376<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 1<br />
Recognizing Polynomial Functions<br />
Figure 2<br />
y<br />
y<br />
g<br />
x<br />
x<br />
f<br />
y<br />
y<br />
x<br />
x<br />
h<br />
k<br />
Figure 2<br />
Solution Thfh. Th<br />
ThgkThg<br />
Thk<br />
Q & A…<br />
Do all polynomial functions have as their domain all real numbers?<br />
<br />
Using Factoring to Find Zeros of Polynomial Functions<br />
fxf x=f<br />
<br />
fixxfi<br />
<br />
<br />
<br />
<br />
1. Th<br />
2. Th<br />
3. <br />
How To…<br />
f, fix<br />
1. f x=<br />
2. <br />
a. <br />
b. <br />
3. o fix
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 377<br />
Example 2<br />
Finding the x-Intercepts of a Polynomial Function by Factoring<br />
xf x=x −x +x <br />
Solution o fif x=<br />
x −x +x = <br />
x x −x += <br />
x x −x −= <br />
x −= x −=<br />
x = x = x =<br />
x = x =±x =±√ — <br />
Thfix−( √ — ) ( −√ — ) Figure 3<br />
y<br />
<br />
y<br />
f x = x – x + x <br />
<br />
– √<br />
√<br />
x<br />
– <br />
<br />
–<br />
– <br />
–<br />
Example 3<br />
Figure 3<br />
Finding the x-Intercepts of a Polynomial Function by Factoring<br />
xf x=x −x −x +<br />
Solution f x=<br />
x −x −x += <br />
x x−−x−= <br />
x −x−= e diff<br />
x+x−x−= <br />
x+= x −= x −=<br />
x=− x = x =<br />
Thx−Figure 4<br />
y<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
f x = x − x − x + <br />
x<br />
–<br />
–<br />
–<br />
Figure 4
378<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 4<br />
Finding the y- and x-Intercepts of a Polynomial in Factored Form<br />
yxgx=x− x +<br />
Solution yg<br />
g=− +<br />
=<br />
y<br />
xgx=<br />
x− x +=<br />
x− = x +=<br />
x( − <br />
) <br />
x −= x =− __<br />
<br />
x =<br />
Analysis We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial<br />
as shown in Figure 5.<br />
y<br />
gx = (x − ) (x + )<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
<br />
– – – – <br />
–<br />
x<br />
Figure 5<br />
Example 5<br />
Finding the x-Intercepts of a Polynomial Function Using a Graph<br />
xhx=x +x +x −<br />
Solution Th<br />
fi<br />
ffi<br />
Figure 6xx =−−<br />
<br />
y hx = x + x + x − <br />
– – <br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 6
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 379<br />
xg t<br />
h−=h−=h=<br />
hx=x +x +x −<br />
h−=− +− +−−=−+−−=<br />
h−=− +− +−−=−+−−=<br />
h= + +−=++−=<br />
x<br />
<br />
hx=x +x +x −<br />
=x+x+x−<br />
Try It #1<br />
yxfx=x −x +x<br />
Identifying Zeros and Their Multiplicities<br />
ffx<br />
ff<br />
<br />
fx=x+x− x+ <br />
Figure 7x <br />
<br />
<br />
<br />
<br />
– – <br />
–<br />
–<br />
–<br />
–<br />
y<br />
f x = (x + )(x − ) (x + ) <br />
x<br />
Figure 7 Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.<br />
xx =−x+=Thx<br />
x =−Th<br />
<br />
xx =x− =Th<br />
The <br />
<br />
x− =x−x−<br />
Thx−Th<br />
multiplicityThx =<br />
x−
380<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
xx =−x+ =Th<br />
flfiThe <br />
f x=x <br />
<br />
touchx<br />
crossxFigure 8<br />
<br />
y<br />
y<br />
y<br />
p = p = p = <br />
x<br />
x<br />
x<br />
<br />
<br />
Figure 8<br />
<br />
<br />
r flx<br />
<br />
r flx<br />
graphical behavior of polynomials at x-intercepts<br />
x−h p x h<br />
px =h multiplicityp<br />
Thx. Th<br />
x<br />
Th<br />
How To…<br />
n<br />
1. x<br />
2. x <br />
3. x<br />
4. Thn<br />
Example 6<br />
Identifying Zeros and Their Multiplicities<br />
Figure 9
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 381<br />
y<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
Figure 9<br />
Solution Th. Th<br />
ftfix =−Thx<br />
. Th−<br />
Thx =−. Th. Th<br />
Thx =Thx<br />
<br />
Try It #2<br />
Figure 10<br />
y<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
Figure 10<br />
Determining End Behavior<br />
<br />
fx=a n<br />
x n +a n−<br />
x n − ++ a <br />
x +a <br />
xx<br />
ThTh<br />
−−<br />
end behavior<br />
a n<br />
x n x<br />
f xx<br />
f xxf x<br />
Figure 11
382<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Even Degree<br />
ffia n<br />
><br />
y<br />
Odd Degree<br />
ffia n<br />
><br />
y<br />
x<br />
x<br />
<br />
x→ ∞f x→ ∞<br />
x→ −∞f x→ ∞<br />
ffia n<br />
<<br />
y<br />
<br />
x→ ∞f x→ ∞<br />
x→ −∞f x→ ∞<br />
ffia n<br />
<<br />
y<br />
x<br />
x<br />
<br />
x→ ∞f x→ −∞<br />
x→ −∞f x→ −∞<br />
<br />
x→ ∞f x→ −∞<br />
x→ −∞f x→ ∞<br />
Figure 11<br />
Understanding the Relationship Between Degree and Turning Points<br />
<br />
<br />
f x=x −x −x +xFigure 12Th<br />
<br />
y<br />
<br />
<br />
<br />
<br />
x<br />
<br />
Figure 12<br />
Thf Th
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 383<br />
interpreting turning points<br />
<br />
<br />
nn −<br />
Example 7<br />
Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<br />
<br />
a. f x=−x +x −x + b. f x=−x− +x <br />
Solution<br />
a. f x=−x +x −x +<br />
f x=x −x −x +<br />
<br />
Th−=<br />
b. f x=−x− +x <br />
<br />
f x=−x− +x <br />
a n<br />
=−x x −x <br />
Th. Th<br />
Th−=<br />
Graphing Polynomial Functions<br />
<br />
<br />
How To…<br />
<br />
1. <br />
2. y<br />
f −x=f xf −x=−fx<br />
3. x<br />
4. <br />
5. <br />
6. <br />
7. <br />
Example 8<br />
Sketching the Graph of a Polynomial Function<br />
f x=−x+ x−<br />
Solution Thxx =−Th<br />
xx =<br />
<br />
yf <br />
y<br />
f =−+ −<br />
=−ċċ−<br />
=
384<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
−x <br />
flfi<br />
fiFigure 13<br />
y<br />
x<br />
Figure 13<br />
<br />
x →−∞f x→∞<br />
x<br />
f −x=−−x+ −x−f x<br />
− x<br />
yyFigure 14<br />
y<br />
<br />
−<br />
x<br />
Figure 14<br />
ft<br />
Figure 15<br />
y<br />
<br />
−<br />
<br />
x<br />
Figure 15<br />
x →∞f x→−∞
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 385<br />
Figure 16<br />
Figure 15<br />
<br />
y<br />
f x = −(x + ) (x − )<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
x<br />
–<br />
–<br />
–<br />
Figure 16 The complete graph of the polynomial function f (x ) = −2(x + 3) 2 (x − 5)<br />
Try It #3<br />
fx= <br />
xx− x+ <br />
Using the Intermediate Value Theorem<br />
<br />
xfif<br />
ThIntermediate Value Theoremabfa
386<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 9<br />
Using the Intermediate Value Theorem<br />
f x=x −x +x +x =x =<br />
Solution f xx =Table 2<br />
x <br />
f x − <br />
Table 2<br />
x =f f Th<br />
<br />
x =x =<br />
Analysis We can also see on the graph of the function in Figure 18 that there are two real zeros between x = and<br />
x = .<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f = <br />
<br />
<br />
<br />
<br />
<br />
f =−<br />
f = <br />
Figure 18<br />
Try It #4<br />
fx=x −x −x x=x=<br />
Writing Formulas for Polynomial Functions<br />
fi<br />
x<br />
x<br />
factored form of polynomials<br />
e px =x <br />
x <br />
x n<br />
n <br />
f x=ax−x p x−x p x−x n p n pi <br />
a<br />
x<br />
How To…<br />
<br />
1. xo fi<br />
2. x<br />
3. s s<br />
4. y
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 387<br />
Example 10 Writing a Formula for a Polynomial Function from the Graph<br />
Figure 19<br />
y<br />
<br />
<br />
<br />
– – – <br />
–<br />
–<br />
–<br />
<br />
x<br />
Figure 19<br />
Solution Thxx =−Thy, −x =−x =<br />
<br />
x =e <br />
<br />
f x=ax+x− x−<br />
y−a<br />
f =a+− −<br />
−=a+− −<br />
−=−a<br />
a =__<br />
<br />
<br />
Thf x= <br />
x+x− x−<br />
Try It #5<br />
Figure 20<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
Figure 20<br />
Using Local and Global Extrema<br />
fifi<br />
fi<br />
fi<br />
<br />
<br />
global maximumglobal minimum<br />
Th
388<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
local and global extrema<br />
x =a <br />
x =a<br />
af a≥f xxx =aa<br />
f a≤f xxx =a<br />
global maximumglobal minimum<br />
s a gaf a≥f xxaf a≤f x<br />
x<br />
ffFigure 21<br />
– ––<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 21<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Q & A…<br />
Do all polynomial functions have a global minimum or maximum?<br />
f x=x<br />
<br />
Example 11<br />
Using Local Extrema to Solve Applications<br />
4 <br />
<br />
Solution Figure 22<br />
w<br />
w<br />
w<br />
w<br />
w<br />
w<br />
w<br />
w<br />
w<br />
Figure 22<br />
ft−w−w<br />
w. Th<br />
Vw=−w−ww<br />
=w −w +w
SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 389<br />
w−w−w<br />
Th <br />
wTh<br />
390 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
5.3 SECTION EXERCISES<br />
VERBAL<br />
1. e diffx<br />
f<br />
3. ue Th<br />
<br />
5. x<br />
<br />
<br />
2. nn<br />
<br />
<br />
4. <br />
<br />
ALGEBRAIC<br />
, fixt<br />
6. Ct=t−t+t− 7. Ct=t+t−t+ 8. Ct=tt− t+<br />
9. Ct=tt−t+ 10. Ct=t −t +t 11. Ct=t +t −t <br />
12. fx=x −x 13. fx=x +x −x 14. fx=x +x −x<br />
15. fx=x +x −x− 16. fx=x +x −x− 17. fx=x −x −x+<br />
18. fx=x −x − 19. fx=x +x − 20. fx=x −x −x+<br />
21. fx=x −x −x 22. fx=x −x −x 23. fx=x −x +x<br />
Thfi<br />
<br />
24. fx=x −xx=−x=− 25. fx=x −xx=x=<br />
26. fx=x −xx=x= 27. fx=−x +x=x=<br />
28. fx=−x −xx=−x= 29. fx=x −x+x=x=<br />
, fi<br />
30. fx=x+ x− 31. fx=x x+ x− <br />
32. fx=x x− x+ 33. fx=x x +x+<br />
34. fx=x+ x −x+ 35. fx=x+ x −x+<br />
36. fx=xx −x+x +x+ 37. fx=x −x −x <br />
38. fx=x +x +x 39. fx=x −x +x <br />
40. fx=x x −x +x 41. fx=x x −x +x <br />
GRAPHICAL<br />
xy<br />
42. fx=x+ x− 43. gx=x+x− 44. hx=x− x+ <br />
45. kx=x− x− 46. mx=−xx−x+ 47. nx=−xx+x−
SECTION 5.3 SECTION EXERCISES 391<br />
<br />
48. <br />
fx<br />
49. <br />
fx<br />
50. <br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
51. <br />
fx<br />
52. <br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
53. <br />
y<br />
54. <br />
y<br />
55. <br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
56. <br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
<br />
57. x=−x=x=<br />
y−<br />
59. x=x=<br />
x=−y<br />
<br />
58. x=−x=−x=<br />
y<br />
60. x=<br />
x=x=−y<br />
−
392 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
61. x=<br />
x =<br />
63. x=−x=−x=<br />
y<br />
65. x=_ <br />
<br />
x=x=−y<br />
<br />
62. x=x=x=<br />
y−<br />
64. x=−x=<br />
x=−y<br />
66. x=−x=<br />
<br />
TECHNOLOGY<br />
<br />
<br />
67. fx=x −x− 68. fx=x −x− 69. fx=x +x<br />
70. fx=−x +x− 71. fx=x −x +<br />
EXTENSIONS<br />
<br />
72.<br />
74. <br />
–<br />
<br />
–<br />
–<br />
–<br />
f x<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
fx<br />
<br />
<br />
– – – – <br />
– –<br />
x<br />
x<br />
73.<br />
<br />
<br />
<br />
<br />
<br />
<br />
fx<br />
– – – <br />
– <br />
– <br />
– <br />
– <br />
– <br />
– <br />
– <br />
<br />
<br />
x<br />
– <br />
– <br />
– <br />
REAL-WORLD APPLICATIONS<br />
<br />
75. <br />
xx<br />
<br />
<br />
x<br />
77. x+<br />
x+<br />
<br />
x<br />
79. x+<br />
<br />
n. Th<br />
V=_<br />
<br />
πr hrh<br />
76. <br />
xx<br />
<br />
x<br />
78. x+
SECTION 5.4 DIVIDING POLYNOMIALS 393<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Use long division to divide polynomials.<br />
Use synthetic division to divide polynomials.<br />
5.4 DIVIDING POLYNOMIALS<br />
Figure 1 Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)<br />
Th<br />
[] y fi<br />
V =l ċw ċh<br />
=ċċ<br />
=<br />
fi<br />
<br />
h = V ____<br />
l ċw<br />
_______<br />
= ċ <br />
=<br />
fifi<br />
<br />
x −x −x +xTh<br />
xx −fi<br />
<br />
Using Long Division to Divide Polynomials<br />
<br />
<br />
<br />
Long Division<br />
×=−=<br />
<br />
− ×=−=<br />
<br />
R <br />
−
394<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Th<br />
<br />
<br />
=ċ+<br />
<br />
=ċ+<br />
=+<br />
=<br />
Division Algorithmft<br />
<br />
Th<br />
Th<br />
x −x +x +x +<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x+x −x +x +<br />
x <br />
x+x −x +x +<br />
x <br />
x+x −x +x +<br />
−x +x <br />
−x +x<br />
x −x<br />
x+x −x +x +<br />
−x +x <br />
−x +x<br />
−−x +x<br />
x +<br />
x −x +<br />
x+x −x +x +<br />
−x +x <br />
−x +x<br />
−−x +x<br />
x +<br />
−x +<br />
−<br />
<br />
x xx <br />
x +x <br />
<br />
<br />
−x x−x<br />
x +−x<br />
<br />
xx<br />
x +<br />
<br />
x −x +x + <br />
<br />
x + =x −x +− <br />
x + <br />
<br />
<br />
<br />
x −x +x + <br />
<br />
x + =x+x −x +−<br />
<br />
x – x + x + = x+ x – x + + –
SECTION 5.4 DIVIDING POLYNOMIALS 395<br />
the Division Algorithm<br />
Division Algorithmf xdx<br />
dxf xqxrx<br />
fx=dxqx+rx<br />
qxrx. Th<br />
dx<br />
rx=dxf x. Thdxqxf x<br />
How To…<br />
<br />
1. <br />
2. fi<br />
<br />
3. <br />
4. <br />
5. <br />
6. <br />
7. <br />
Example 1<br />
x +x −x +<br />
Solution<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Using Long Division to Divide a Second-Degree Polynomial<br />
x+x +x −<br />
x<br />
x+x +x −<br />
x<br />
x+x +x −<br />
−x +x<br />
−x −<br />
x −<br />
x+x +x −<br />
−x +x<br />
−x −<br />
−−x −<br />
<br />
Thx −. Th<br />
<br />
x xx<br />
x +x<br />
<br />
−xx−<br />
x +−<br />
<br />
<br />
<br />
<br />
__________<br />
x +x − <br />
x + =x −<br />
x +x −=x+x −<br />
Analysis This division problem had a remainder of . This tells us that the dividend is divided evenly by the divisor,<br />
and that the divisor is a factor of the dividend.
396<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 2<br />
Using Long Division to Divide a Third-Degree Polynomial<br />
x +x −x +x −<br />
Solution<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x +x −<br />
x −)x +x −x +<br />
−x −x <br />
x −x<br />
−x −x<br />
−x +<br />
−−x +<br />
<br />
Th<br />
x xx <br />
x − x <br />
x xx<br />
x −x<br />
−xx−<br />
x −−<br />
. Th<br />
<br />
__________________<br />
x +x −x + <br />
x − =x +x<br />
_____ <br />
−+ <br />
x − <br />
Analysis We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<br />
<br />
x −x +x −+=x +x −x +<br />
Notice, as we write our result,<br />
the dividend isx +x −x +<br />
the divisor isx −<br />
the quotient isx +x −<br />
the remainder is <br />
Try It #1<br />
x −x +x−x+<br />
Using Synthetic Division to Divide Polynomials<br />
Synthetic division<br />
ffi<br />
<br />
<br />
x −x +x +x +<br />
The fi<br />
x +x +<br />
x+x −x +x +<br />
−x +x <br />
−x +x<br />
−−x −x<br />
x +<br />
−x +<br />
−<br />
Thffi
SECTION 5.4 DIVIDING POLYNOMIALS 397<br />
− <br />
− −<br />
− <br />
−<br />
−<br />
fi<br />
fi<br />
−Th<br />
ffi<br />
− − <br />
− −<br />
− −<br />
ftTh<br />
ffi<br />
x −x +−. ThExample 3<br />
synthetic division<br />
x −kk <br />
synthetic divisione cffi<br />
How To…<br />
<br />
1. k<br />
2. ffi<br />
3. ffi<br />
4. ffik<br />
5. <br />
6. k<br />
7. <br />
8. Th<br />
<br />
Example 3<br />
Using Synthetic Division to Divide a Second-Degree Polynomial<br />
x −x −x −<br />
Solution kffi<br />
− −<br />
ffiffik<br />
− −<br />
<br />
<br />
k<br />
. Th<br />
− −<br />
<br />
<br />
Thx +. Thx −
398<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Analysis Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the<br />
remainder.<br />
x−x ++=x −x −<br />
Example 4<br />
Using Synthetic Division to Divide a Third-Degree Polynomial<br />
x +x −x −x +<br />
Solution Thx +k =−−<br />
<br />
− − −<br />
− − <br />
− <br />
Thx +x −. Th. Thx +x +x −x −<br />
Analysis<br />
The graph of the polynomial function f x = x + x − x − in Figure 2 shows a zero at x = k = −.<br />
This confirms that x + is a factor of 4x + x − x − .<br />
y<br />
− −<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 2<br />
Example 5<br />
Using Synthetic Division to Divide a Fourth-Degree Polynomial<br />
−x +x +x −x −<br />
Solution xffi<br />
Th−x +x +x<br />
_ <br />
++ <br />
x − <br />
Try It #2<br />
− −<br />
− <br />
− <br />
x +x −x+x+<br />
Using Polynomial Division to Solve Application Problems
SECTION 5.4 DIVIDING POLYNOMIALS 399<br />
Example 6<br />
Using Polynomial Division in an Application Problem<br />
Thx −x −x +xTh<br />
xx −t<br />
Solution Th<br />
Figure 3<br />
<br />
x<br />
x− <br />
Figure 3<br />
<br />
V =l ċw ċh<br />
<br />
x −x −x +x =x ċx−ċh<br />
h, fix<br />
<br />
<br />
x ċx−ċh<br />
x<br />
= x −x −x +x<br />
x<br />
<br />
x−h=x −x −x +<br />
h<br />
h = x −x −x + <br />
x − <br />
<br />
− − <br />
−<br />
− <br />
Thx +x −. Thx +x −<br />
Try It #3<br />
Thx +x −x+Thx+<br />
<br />
<br />
Dividing a Trinomial by a Binomial Using Long Division (http://openstaxcollege.org/l/dividetribild)<br />
Dividing a Polynomial by a Binomial Using Long Division (http://openstaxcollege.org/l/dividepolybild)<br />
Ex 2: Dividing a Polynomial by a Binomial Using Synthetic Division (http://openstaxcollege.org/l/dividepolybisd2)<br />
Ex 4: Dividing a Polynomial by a Binomial Using Synthetic Division (http://openstaxcollege.org/l/dividepolybisd4)
400 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
5.4 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
2. n<br />
<br />
ALGEBRAIC<br />
<br />
3. x +x−÷x− 4. x −x−÷x− 5. x +x+÷x+<br />
6. x −x+÷x+ 7. x −x−÷x+ 8. −x −÷x+<br />
9. x −x+÷x+ 10. x −÷x− 11. x −x+÷x+<br />
12. x −x +x−÷x− 13. x +x −x+÷x+<br />
o fi<br />
14. x −x +x−÷x+ 15. x −x −x+÷x−<br />
16. x −x −x−÷x+ 17. x −x −x−÷x+<br />
18. x −x +x+÷x− 19. x −x +x−÷x+<br />
20. −x +x −÷x− 21. x +x −x−÷x−<br />
22. x −x +x+÷x+ 23. x −x +÷x+<br />
24. x −x+÷x+ 25. x −x +x−÷x−<br />
26. x −x +x−÷x− 27. x −x+÷x−<br />
28. x −x +x+÷x+ 29. x +x −x −x+÷x+<br />
30. x −x +÷x− 31. x +x −x +x+÷x+<br />
32. x −x +x −x+÷x− 33. x −x +x −x+÷x−<br />
34. x +x −x −x+÷x+ 35. x −x +x −x+÷x−<br />
36. x −x −x+÷x− 37. x +x −x +x+÷x+<br />
e fi<br />
<br />
38. x−x −x −x+ 39. x−x −x −x+ 40. x+−x +x +<br />
41. x−x −x − 42. x− <br />
x −x +x−<br />
43. x+ <br />
x +x −x+<br />
GRAPHICAL<br />
<br />
. Thffi<br />
44. x −x+<br />
y<br />
45. x +x+<br />
y<br />
46. x +x+<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–
SECTION 5.4 SECTION EXERCISES 401<br />
47. x +x+<br />
y<br />
<br />
<br />
<br />
48. x +x+<br />
y<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
– <br />
–<br />
–<br />
–<br />
–<br />
o fi<br />
49. _______<br />
x − x− <br />
50. + x _______<br />
x+ <br />
52. ____________<br />
−x −x <br />
− x+ 53. ______ x − <br />
x+ <br />
51. +x− x __________<br />
x− <br />
TECHNOLOGY<br />
<br />
54. xk − <br />
x− k=<br />
k=<br />
56. x −k <br />
x−k k=<br />
k=<br />
x k<br />
58. <br />
x− k=<br />
k=<br />
EXTENSIONS<br />
55. xk + <br />
x+ k=<br />
k=<br />
x k<br />
57. <br />
x+ k=<br />
k=<br />
<br />
59. _____ x+ <br />
x−i<br />
62. _____ x + <br />
x+i<br />
REAL-WORLD APPLICATIONS<br />
60. _____ x + <br />
x−i<br />
63. _____ x + <br />
x−i<br />
61. _____ x+ <br />
x+i<br />
<br />
64. x+x +x− 65. x+x +x +x+<br />
66. x−x −x +x −x−<br />
<br />
<br />
67. x +x −x−<br />
x+x−<br />
69. x +x +x−x−<br />
x+<br />
68. x −x −x+<br />
x−x−<br />
70. x +x −x−<br />
x+<br />
<br />
<br />
71. πx −x −x−x+ 72. πx +x −x−x+<br />
73. πx +x +x −x−<br />
x+
402<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Evaluate a polynomial using the Remainder Theorem.<br />
Use the Factor Theorem to solve a polynomial equation.<br />
Use the Rational Zero Theorem to find rational zeros.<br />
Find zeros of a polynomial function.<br />
Use the Linear Factorization Theorem to find polynomials with given zeros.<br />
Use Decartes' Rule of Signs.<br />
Solve real-world applications of polynomial equations.<br />
5.5 ZEROS OF POLYNOMIAL FUNCTIONS<br />
ffTh<br />
ThTh<br />
<br />
<br />
Th<br />
<br />
Evaluating a Polynomial Using the Remainder Theorem<br />
<br />
Remainder Theoremx −k<br />
kf k<br />
f xdx<br />
dxf xqxrx<br />
f x=dxqx+rx<br />
dxx −k<br />
fx=x−kqx+r<br />
x −krx =k<br />
f k=k−kqk+r<br />
=ċqk+r<br />
=r<br />
f kf xx −k<br />
the Remainder Theorem<br />
f xx −kf k<br />
How To…<br />
ff xx =kTh<br />
1. x −k<br />
2. Thf k
SECTION 5.5 ZEROS OF POLYNOMIAL FUNCTIONS 403<br />
Example 1<br />
Using the Remainder Theorem to Evaluate a Polynomial<br />
r Thf x=x −x −x +x −x =<br />
Solution fiTh<br />
x −<br />
Th. Thf =<br />
Analysis We can check our answer by evaluating f ().<br />
− − −<br />
<br />
<br />
<br />
f x = x − x −x + x − <br />
f = − − + −<br />
= <br />
Try It #1<br />
der Thfx=x −x −x +x +x=−<br />
Using the Factor Theorem to Solve a Polynomial Equation<br />
Factor Theorem<br />
<br />
fx=x−kqx+r<br />
krf k=f x=x−kqx+f x=x−kqx<br />
x −kf xkf xx −kf x<br />
x −kf xf x) =x−kqx+rTh<br />
k<br />
ThThn<br />
nThn<br />
<br />
the Factor Theorem<br />
Factor Theoremkf xx−kf x<br />
How To…<br />
r Th<br />
1. x−k<br />
2. fi<br />
3. x−k<br />
4. <br />
5. <br />
Example 2 Using the Factor Theorem to Solve a Polynomial Equation<br />
x+x −x −x +
404<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Solution x+<br />
− − − <br />
− −<br />
− <br />
Thx+<br />
<br />
x+x −x +<br />
<br />
x+x−x−<br />
r Thx −x −x +−<br />
Try It #2<br />
r Th fx=x +x −x−x−<br />
Using the Rational Zero Theorem to Find Rational Zeros<br />
Th<br />
fiThRational ZeroTheorem<br />
ffi<br />
<br />
x = <br />
x = <br />
Th<br />
<br />
<br />
x −__<br />
=x − __<br />
= <br />
x −=x −= <br />
f x=x −x −<br />
f x=x −x +<br />
<br />
<br />
f x=ċx −x +ċ<br />
<br />
ffi<br />
<br />
<br />
ffiThTh<br />
<br />
the Rational Zero Theorem<br />
Rational Zero Theoremf x=a n<br />
x n +a n−<br />
x n− ++a <br />
x +a <br />
<br />
ffif x p _<br />
q<br />
pa <br />
q<br />
ffia n<br />
<br />
ffi
SECTION 5.5 ZEROS OF POLYNOMIAL FUNCTIONS 405<br />
How To…<br />
f xo Tho fi<br />
1. ffi<br />
2. p _<br />
q pqffi<br />
<br />
3. f ( p _<br />
q) <br />
Example 3<br />
Listing All Possible Rational Zeros<br />
f x=x −x +x −<br />
Solution Thf x, −<br />
ffi<br />
Th−−p =±±±<br />
Thffiq =±±<br />
−<br />
<br />
p<br />
_<br />
q =± __<br />
± __<br />
<br />
p _<br />
q =± __<br />
± __<br />
<br />
p _<br />
q =± __<br />
± __<br />
<br />
<br />
= <br />
=<br />
<br />
p _<br />
q = __<br />
<br />
=±±±± __<br />
<br />
Example 4<br />
Using the Rational Zero Theorem to Find Rational Zeros<br />
o Tho fif x=x +x −x +<br />
Solution<br />
Tho Th p _<br />
q<br />
f xpq<br />
p _<br />
q = ___<br />
<br />
ffi <br />
_<br />
= <br />
Th±±±Th p _<br />
q ±± _ <br />
Th<br />
<br />
xf x<br />
f −=− +− −−+=<br />
f = + −+=<br />
f ( − __<br />
<br />
) = ( − __<br />
<br />
f ( __<br />
<br />
) = ( __<br />
<br />
) +( − __<br />
) −( − __<br />
) +=<br />
) +( __<br />
) −( __<br />
) +=− _<br />
<br />
−− <br />
f <br />
xf x<br />
<br />
Try It #3<br />
o Th fx=x −x +x+
406<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Finding the Zeros of Polynomial Functions<br />
ThTh<br />
zeros<br />
How To…<br />
fo fi<br />
1. o Th<br />
2. <br />
<br />
3. <br />
<br />
4. <br />
<br />
Example 5<br />
Finding the Zeros of a Polynomial Function with Repeated Real Zeros<br />
f x=x −x −<br />
Solution Tho Th p _<br />
q<br />
f xp−q<br />
<br />
p _<br />
q = ___<br />
<br />
ffi <br />
__<br />
= − <br />
<br />
Th−±±±±Th p _<br />
q ±± <br />
± <br />
<br />
<br />
fi<br />
− −<br />
<br />
<br />
x−. Th<br />
x−x +x +<br />
Thf x<br />
x−x + <br />
Thfi<br />
<br />
<br />
x +=<br />
x =− __<br />
<br />
Th− <br />
<br />
Analysis Look at the graph of the function f in Figure 1. Notice, at x = −, the graph bounces off the x-axis, indicating<br />
the even multiplicity for the zero − At x = , the graph crosses the x-axis, indicating the odd multiplicity<br />
for the zero x =.<br />
y<br />
<br />
<br />
<br />
<br />
–2.––––<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 1<br />
<br />
<br />
x
SECTION 5.5 ZEROS OF POLYNOMIAL FUNCTIONS 407<br />
Using the Fundamental Theorem of <strong>Algebra</strong><br />
fi<br />
ThFundamental Theorem of <strong>Algebra</strong><br />
. Th<br />
ff x=ThTh<br />
c <br />
Thf xx −c <br />
<br />
x −c <br />
<br />
Th<br />
c <br />
x −c <br />
<br />
ThTh<br />
f x<br />
the Fundamental Theorem of <strong>Algebra</strong><br />
ThThf xn >f x<br />
<br />
f xn >a<br />
f xn<br />
fx=ax−c <br />
x−c <br />
x−c n<br />
<br />
c <br />
c <br />
c n<br />
Thf xn<br />
Q & A…<br />
Does every polynomial have at least one imaginary zero?<br />
<br />
<br />
Example 6<br />
Finding the Zeros of a Polynomial Function with Complex Zeros<br />
f x=x +x +x +<br />
Solution Tho Th p _<br />
q<br />
f xpq<br />
<br />
p _<br />
q = ___<br />
<br />
ffi <br />
_<br />
= <br />
Th±±Th p _<br />
q ,<br />
±±± <br />
fi<br />
−<br />
− <br />
− −<br />
<br />
x+−. Th<br />
x+x +<br />
o fi<br />
<br />
x +=<br />
x =− __<br />
<br />
Thf x−± _<br />
i √— <br />
x =±<br />
√ − =±i _ √—
408<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Analysis Look at the graph of the function f in Figure 2. Notice that, at x = −, the graph crosses the x-axis, indicating an<br />
odd multiplicity () for the zero x = −. Also note the presence of the two turning points. This means that, since there is a<br />
3 rd degree polynomial, we are looking at the maximum number of turning points. So, the end behavior of increasing without<br />
bound to the right and decreasing without bound to the left will continue. Thus, all the x-intercepts for the function are shown.<br />
So either the multiplicity of x = − is and there are two complex solutions, which is what we found, or the multiplicity at<br />
x = − is three. Either way, our result is correct.<br />
<br />
<br />
y<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
Figure 2<br />
Try It #4<br />
fx=x +x −x+<br />
Using the Linear Factorization Theorem to Find Polynomials with Given Zeros<br />
Th<br />
nnTh<br />
nThLinear Factorization Theorem<br />
x−cc<br />
fffia +bib ≠f xTh<br />
Thx −a+bif xfffix −a−bif xTh<br />
x −a−bix −a+bi<br />
ffi<br />
fffia +bi<br />
a −bif x. ThTh<br />
complex conjugate theorem<br />
Linear Factorization Theorem,<br />
x−cc<br />
fffia +bi<br />
a −bi<br />
How To…<br />
fcf cfTh<br />
o fi<br />
1. <br />
2. <br />
3. cf cffi<br />
4.
Example 7<br />
SECTION 5.5 ZEROS OF POLYNOMIAL FUNCTIONS 409<br />
Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros<br />
ffi−if −=<br />
Solution x =iThx =−iTh<br />
x+x−x−ix+i<br />
<br />
f x=ax+x−x−ix+i<br />
f x=ax +x −x +<br />
f x=ax +x −x +x −<br />
o fiaf −=x =−f =f x<br />
<br />
=a− +− −− +−−<br />
<br />
=a−<br />
−=a<br />
<br />
f x=−x +x −x +x −<br />
<br />
f x=−x −x +x −x +<br />
Analysis We found that both i and −i were zeros, but only one of these zeros needed to be given. If i is a zero of a<br />
polynomial with real coefficients, then −i must also be a zero of the polynomial because −i is the complex conjugate of i.<br />
Q & A…<br />
If 2 + 3i were given as a zero of a polynomial with real coefficients, would 2 − 3i also need to be a zero?<br />
ffi<br />
<br />
Try It #5<br />
<br />
−if=<br />
Using Descartes’ Rule of Signs<br />
Th<br />
Descartes’ Rule of Signs<br />
f x<br />
<br />
f x=x +x +x +x−<br />
Th<br />
Thf −x<br />
f −x=−x +−x +−x +−x−<br />
f −x=+x −x + x −x −<br />
f −x. Thf x
410<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Descartes’ Rule of Signs<br />
Descartes’ Rule of Signs,f x=a n<br />
x n +a n−<br />
x n− ++a <br />
x +a <br />
<br />
ffi<br />
Thes f x<br />
<br />
Thf −x<br />
<br />
Example 8 Using Descartes’ Rule of Signs<br />
<br />
f x=−x −x +x −x −<br />
Solution <br />
f x=−x −x +x − x −<br />
Figure 3<br />
Thf −x<br />
<br />
f −x=−−x −−x +−x −−x−<br />
f −x=−x +x +x +x −<br />
f −x=−x +x +x +x−<br />
Figure 4<br />
<br />
ThTable 1<br />
Positive Real Zeros Negative Real Zeros Complex Zeros Total Zeros<br />
<br />
<br />
<br />
<br />
Table 1<br />
Analysis We can confirm the numbers of positive and<br />
negative real roots by examining a graph of the function. See<br />
Figure 5. We can see from the graph that the function has <br />
positive real roots and negative real roots.<br />
Try It #6<br />
x = −<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– – – – –<br />
–<br />
–<br />
–<br />
fx = − x − x + x − x − <br />
x = −<br />
<br />
<br />
<br />
Figure 5<br />
<br />
fx=x −x +x −x+<br />
<br />
Solving Real-World Applications<br />
<br />
<br />
<br />
<br />
x
SECTION 5.5 ZEROS OF POLYNOMIAL FUNCTIONS 411<br />
Example 9<br />
Solving Polynomial Equations<br />
ffTh<br />
ThTh<br />
<br />
<br />
Solution ThV =lwh<br />
l =w +<br />
<br />
h = <br />
w<br />
V =w+w( __<br />
w )<br />
V =__<br />
w +__<br />
w <br />
<br />
<br />
=__<br />
w +__<br />
w V<br />
=w +w <br />
=w +w − <br />
ThTh<br />
±±±±±±±±±±<br />
<br />
x =<br />
x =<br />
−<br />
<br />
−<br />
−<br />
<br />
−<br />
x =<br />
−<br />
<br />
<br />
<br />
<br />
l=w +=+=h = __<br />
w = __<br />
=<br />
Th<br />
Try It #7<br />
s. Th<br />
<br />
<br />
<br />
Real Zeros, Factors, and Graphs of Polynomial Functions (http://openstaxcollege.org/l/realzeros)<br />
Complex Factorization Theorem (http://openstaxcollege.org/l/factortheorem)<br />
Find the Zeros of a Polynomial Function (http://openstaxcollege.org/l/findthezeros)<br />
Find the Zeros of a Polynomial Function 2 (http://openstaxcollege.org/l/findthezeros2)<br />
Find the Zeros of a Polynomial Function 3 (http://openstaxcollege.org/l/findthezeros3)
412 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
5.5 SECTION EXERCISES<br />
VERBAL<br />
1. der Th 2. o Th<br />
<br />
3. e diff 4. <br />
<br />
<br />
<br />
5. <br />
<br />
ALGEBRAIC<br />
r Tho fi<br />
6. x −x +÷x− 7. x −x +x−÷x+<br />
8. x +x −x−÷x+ 9. −x +x+÷x−<br />
10. x −x +x −x +x−÷x+ 11. x −÷x−<br />
12. x +x −x+÷x− 13. x +x −x+÷x+<br />
Thfi<br />
<br />
14. fx=x −x +x−x− 15. fx=x +x −x+x+<br />
16. fx=x +x −x+x+ 17. fx=x +x +x+x+<br />
18. fx=−x +x −x− 19. x +x +x+x+<br />
20. x −x+x− 21. x +x −x−x+<br />
o Tho fi<br />
22. x −x −x+= 23. x +x −x−= 24. x +x −x−=<br />
25. x +x −x−= 26. x −x −x+= 27. x −x −x−=<br />
28. x +x −x−= 29. x −x −x+= 30. x −x −x−=<br />
31. x −x +x−= 32. x −x +x+= 33. x −x −x +x+=<br />
34. x +x −x −x+= 35. x +x −x −x+= 36. x −x −x +x−=<br />
37. x +x −x −x−= 38. x −x+= 39. x +x +x +x+<br />
, fi<br />
40. x +x +x+= 41. x −x +x−= 42. x +x +x+=<br />
43. x −x +x+= 44. x +x +x +x−= 45. x −x +x+=<br />
GRAPHICAL<br />
Thfi<br />
<br />
46. fx=x − 47. fx=x −x − 48. fx=x −x −x+<br />
49. fx=x −x +x− 50. fx=x +x −x +x− 51. fx=x +x +x+
SECTION 5.5 SECTION EXERCISES 413<br />
52. fx=x −x −x+ 53. fx=x −x −x +x+ 54. fx=x −x −x +x+<br />
55. fx=x −x +<br />
NUMERIC<br />
<br />
56. fx=x +x −x+ 57. fx=x +x −x+ 58. fx=x +x −x+<br />
59. fx=x −x +x+ 60. fx=x −x +x +x −<br />
TECHNOLOGY<br />
fi<br />
<br />
61. fx=x −x + 62. fx=x −x −x−<br />
63. fx=x −x −x+ 64. fx=x +x +x −x+<br />
65. fx=x −x +x −x+<br />
EXTENSIONS<br />
<br />
66. −f= 67. −<br />
f=<br />
68. − <br />
<br />
−f−=−<br />
70. −−−f−=−<br />
69. − <br />
<br />
−f−=−<br />
REAL-WORLD APPLICATIONS<br />
, fi<br />
71. Thh. Th<br />
h. Th<br />
<br />
73. Th<br />
t. Th<br />
<br />
75. Thh. Th<br />
t. Th<br />
<br />
72. Th<br />
s. Th<br />
74. Th<br />
h. Th<br />
<br />
, fi<br />
76. Tht. Th<br />
π<br />
78. Tht diff. Th<br />
π<br />
80. Th t. <br />
Th<br />
<br />
<br />
π π<br />
77. Th<br />
Thπ<br />
79. Tht diff<br />
Thπ
414<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Use arrow notation.<br />
Solve applied problems involving rational functions.<br />
Find the domains of rational functions.<br />
Identify vertical asymptotes.<br />
Identify horizontal asymptotes.<br />
Graph rational functions.<br />
5.6 RATIONAL FUNCTIONS<br />
xTh<br />
Cx=x −x +x<br />
x<br />
Thx<br />
f x=__________________<br />
x −x + <br />
x<br />
<br />
<br />
<br />
<br />
Using Arrow Notation<br />
<br />
Figure 1<br />
Graphs of Toolkit Functions<br />
y<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f x =<br />
<br />
x<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
f x =<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 1<br />
f x=_ x<br />
<br />
1. xy=x →−∞<br />
2. x= h zer<br />
<br />
3. xy=x→∞<br />
arrow notationxf xTable 1
SECTION 5.6 RATIONAL FUNCTIONS 415<br />
Symbol<br />
x→a −<br />
x→a + <br />
x→∞<br />
x→−∞<br />
f x→∞<br />
f x→−∞<br />
fx→a<br />
Meaning<br />
xa xaa<br />
x x<br />
x x<br />
Th <br />
Th <br />
Tha<br />
Table 1 Arrow Notation<br />
Local Behavior of f (x) =<br />
__ 1 x<br />
f x= _ x<br />
<br />
fix = <br />
fi<br />
Table 2<br />
x − − − −<br />
f (x) = 1 __<br />
x<br />
− − − −<br />
<br />
Table 2<br />
x → − f x→−∞<br />
<br />
fiTable 3<br />
x <br />
f (x) = 1 __<br />
x<br />
<br />
<br />
Figure 2<br />
Table 3<br />
x → + f x→∞<br />
x → +<br />
f x → ∞<br />
y<br />
x → −∞<br />
f x → <br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x → ∞<br />
f x → <br />
x → –<br />
f x → −∞<br />
Figure 2
416<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Thvertical asymptote<br />
x =Figure 3<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x = <br />
<br />
<br />
<br />
x<br />
Figure 3<br />
vertical asymptote<br />
vertical asymptotex =afi<br />
a<br />
x →af x→∞x →af x→−∞<br />
End Behavior of f(x ) = 1 _<br />
x<br />
xfixfi<br />
Figure 4<br />
x →∞f x→x →−∞f x→<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
x → −∞<br />
–<br />
f x → –<br />
–<br />
–<br />
y<br />
x → +<br />
f x → ∞<br />
x → ∞<br />
f x → <br />
x<br />
<br />
x → –<br />
f x → −∞<br />
Figure 4<br />
<br />
ffThhorizontal asymptote<br />
<br />
y =Figure 5<br />
y<br />
y = <br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x = <br />
<br />
<br />
<br />
x<br />
Figure 5
SECTION 5.6 RATIONAL FUNCTIONS 417<br />
horizontal asymptote<br />
horizontal asymptotey =b<br />
<br />
x →∞x →−∞f x→b<br />
Example 1<br />
Using Arrow Notation<br />
Figure 6<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 6<br />
Solution x =fi<br />
x =<br />
x → − f x→−∞x → + f x→∞<br />
<br />
y = <br />
x →∞f x→x →−∞f x→<br />
Try It #1<br />
<br />
Example 2<br />
<br />
<br />
<br />
Using Transformations to Graph a Rational Function<br />
ft <br />
<br />
Solution ft <br />
<br />
fx=_____<br />
<br />
x + +<br />
<br />
f x=______<br />
x + <br />
x + <br />
ThftFigure 7<br />
x = −<br />
– – – – – –<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
Figure 7<br />
<br />
<br />
<br />
x<br />
y = <br />
x
418<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
fix =−x =−<br />
x →− − f x→−∞x →− + f x→∞<br />
<br />
y =<br />
x →±∞f x→<br />
Analysis Notice that horizontal and vertical asymptotes are shifted left and up along with the function.<br />
Try It #2<br />
<br />
hift<br />
Solving Applied Problems Involving Rational Functions<br />
Example2ftf x) = x + <br />
x + Th<br />
rational function<br />
fi<br />
ft<br />
rational function<br />
rational functionPxQx<br />
fx= Px <br />
Qx =a x p +a<br />
p p−<br />
x p − ++a <br />
x +a<br />
___ <br />
Qx≠<br />
b q<br />
x q +b q−<br />
x q − ++b <br />
x +b <br />
Example 3 Solving an Applied Problem Involving a Rational Function<br />
<br />
<br />
r ft<br />
<br />
Solution t<br />
Th<br />
<br />
<br />
Wt=+t<br />
<br />
St=+t<br />
ThC<br />
+t<br />
Ct=________<br />
+t<br />
ThftCtt =<br />
+<br />
C=__________<br />
<br />
+ <br />
=___<br />
<br />
<br />
Th<br />
<br />
+<br />
C=_________<br />
<br />
+ <br />
=__<br />
<br />
<br />
___<br />
<br />
≈> __<br />
<br />
=ft
SECTION 5.6 RATIONAL FUNCTIONS 419<br />
Try It #3<br />
Thft.m., 20 f<br />
y fio s<br />
Finding the Domains of Rational Functions<br />
fi<br />
<br />
o fi<br />
domain of a rational function<br />
Th<br />
How To…<br />
n, fi<br />
1. <br />
2. o fix<br />
3. Th<br />
Example 4<br />
Finding the Domain of a Rational Function<br />
f x=_<br />
x + x − <br />
Solution <br />
x −=<br />
x =<br />
x =±<br />
Thx =±. Thx =±<br />
Analysis A graph of this function, as shown in Figure 8, confirms that the function is not defined when x = ±.<br />
y<br />
y = <br />
– – – – –<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
x = <br />
x<br />
Figure 8<br />
There is a vertical asymptote atx = and a hole in the graph atx =−We will discuss these types of holes in greater<br />
detail later in this section<br />
Try It #4<br />
x<br />
fx= <br />
x−x−
420<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Identifying Vertical Asymptotes of Rational Functions<br />
<br />
<br />
<br />
Vertical Asymptotes<br />
Th<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. fiTh<br />
<br />
5. Th<br />
Example 5<br />
Identifying Vertical Asymptotes<br />
+x<br />
kx=<br />
<br />
−x −x <br />
Solution <br />
<br />
+x<br />
kx=________<br />
<br />
−x −x <br />
+x<br />
=___________<br />
<br />
+x−x <br />
fifi<br />
<br />
<br />
+x−x=<br />
x =−<br />
x =−x =Th<br />
Figure 9fi<br />
y<br />
x = −<br />
<br />
x = <br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
<br />
y = −<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
Figure 9
SECTION 5.6 RATIONAL FUNCTIONS 421<br />
Removable Discontinuities<br />
fi<br />
removable discontinuity<br />
x<br />
f x=<br />
− <br />
x −x − <br />
fx= x+x− <br />
<br />
x+x− <br />
x +Thx =−<br />
x −<br />
Thx =Figure 10<br />
<br />
y<br />
<br />
<br />
x = −<br />
<br />
<br />
<br />
– – <br />
–<br />
x<br />
–<br />
–<br />
x = <br />
Figure 10<br />
removable discontinuities of rational functions<br />
A removable discontinuityx =aa<br />
e <br />
fiTh<br />
Th<br />
<br />
t v<br />
Example 6<br />
Identifying Vertical Asymptotes and Removable Discontinuities for a Graph<br />
kx= x − x − <br />
Solution <br />
x −<br />
kx=___________<br />
<br />
x−x+ <br />
x −Thx =<br />
Th<br />
x +Thx =−<br />
Thx =−Figure 11
422<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
y<br />
x = −<br />
<br />
<br />
<br />
– – – – <br />
–<br />
x<br />
–<br />
–<br />
Figure 11<br />
Thx =−x =<br />
Try It #5<br />
x<br />
fx=<br />
− <br />
x −x +x <br />
Identifying Horizontal Asymptotes of Rational Functions<br />
<br />
output<br />
input<br />
<br />
<br />
Th<br />
Case 1:>y =<br />
x +<br />
f x=_________<br />
<br />
x +x − <br />
f x≈_<br />
x<br />
x = _<br />
x<br />
Th<br />
gx=_ x<br />
<br />
y =Figure 12<br />
<br />
<br />
<br />
y<br />
y = <br />
– – – – <br />
–<br />
–<br />
x<br />
x = −<br />
–<br />
x = <br />
Figure 12 Horizontal asymptote y = 0 when f(x ) = p(x ____ ) , q(x ) ≠ 0 where degree of p < degree of q.<br />
q(x )<br />
Case 2:<<br />
f x=__________<br />
x −x + x − <br />
f x≈_ x<br />
x<br />
=xTh<br />
gx=xff<br />
gx=xf<br />
gy =x. Th
SECTION 5.6 RATIONAL FUNCTIONS 423<br />
fi x −x +<br />
Thx + <br />
x − <br />
gx=x +e Figure 13<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – <br />
–<br />
–<br />
–<br />
y<br />
y = x + <br />
x = <br />
Figure 13 Slant asymptote when f( x ) = p( ____ x ) , q( x ) ≠ 0 where degree of p > degree of q by 1<br />
q( x )<br />
Case 3:=y = a _ n<br />
a<br />
b n<br />
b n<br />
<br />
ffipxqxf x= px <br />
n<br />
qx qx≠<br />
x<br />
f x=_________<br />
+ <br />
x +x − <br />
f x≈_<br />
x <br />
x =Th<br />
gx=x →±∞f x→y =<br />
Figure 14<br />
– – – – – <br />
–<br />
–<br />
x = − x = <br />
Figure 14 Horizontal asymptote when f ( x ) = p( _<br />
x ) , q( x ) ≠ 0 where degree of p = degree of q.<br />
q( x )<br />
<br />
<br />
<br />
<br />
<br />
<br />
fx=_______<br />
x −x <br />
x + <br />
<br />
f x≈___<br />
x <br />
x =x <br />
ffi<br />
x→±∞f x→∞<br />
<br />
<br />
<br />
<br />
–<br />
y<br />
x<br />
x<br />
y = <br />
horizontal asymptotes of rational functions<br />
The <br />
<br />
is less thany =<br />
is greater than degree of denominator by one<br />
is equal to
424<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 7<br />
Identifying Horizontal and Slant Asymptotes<br />
<br />
a. gx= x −x ________<br />
x +x <br />
b. −x + hx=x _________<br />
x + c. +x<br />
kx=x ______<br />
x − <br />
Solution f x= px <br />
qx qx≠<br />
a. gx=_<br />
x −x<br />
x +x: Thp =q = <br />
s. Thy = _ y =<br />
b. hx=_ x −x +<br />
Thp =q =p >q<br />
x +<br />
_ x −x +<br />
<br />
<br />
− <br />
x +<br />
− <br />
− <br />
Thx −s 13. Thy =x −<br />
c. kx=_<br />
x +x<br />
: Thp =
SECTION 5.6 RATIONAL FUNCTIONS 425<br />
Try It #6<br />
<br />
fx= _____________<br />
x−x+ <br />
x−x+ <br />
intercepts of rational functions<br />
yfi<br />
yfi<br />
x<br />
n ix<br />
o z<br />
Example 10<br />
Finding the Intercepts of a Rational Function<br />
x−x+<br />
f x= <br />
<br />
x−x+x− <br />
Solution n fiy<br />
−+<br />
f = _________________<br />
<br />
−+− <br />
=___<br />
− <br />
<br />
=− <br />
<br />
=−<br />
x<br />
x−x+<br />
<br />
=_________________<br />
<br />
x−x+x− <br />
<br />
=x−x+<br />
x =−<br />
y−x−Figure 16<br />
(−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x = −<br />
<br />
(<br />
<br />
(–<br />
x = <br />
<br />
Figure 16<br />
<br />
x = <br />
x y = <br />
Try It #7<br />
hiftt 3 uni<br />
Th xy
426<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Graphing Rational Functions<br />
Example 9x<br />
<br />
ff<br />
<br />
Th<br />
<br />
fifi<br />
Figure 17<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
y = x<br />
<br />
<br />
x = <br />
<br />
<br />
<br />
x<br />
Figure 17<br />
<br />
fifi<br />
Figure 18<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
y = x<br />
<br />
<br />
x = <br />
<br />
<br />
<br />
x<br />
Figure 18<br />
f x= x+ x− Figure <br />
19<br />
x+ x−<br />
<br />
y<br />
f x = x + x −<br />
x + x −<br />
<br />
<br />
– – – – <br />
(− – (<br />
–<br />
y = <br />
x<br />
x = −<br />
–<br />
x = <br />
Figure 19
SECTION 5.6 RATIONAL FUNCTIONS 427<br />
xx =−x+ <br />
<br />
xx =x−<br />
<br />
x =−x+ e <br />
f x=_<br />
<br />
x <br />
x =x−<br />
e <br />
f x= <br />
x <br />
How To…<br />
<br />
1. o fiy<br />
2. <br />
3. <br />
o fix<br />
4. x<br />
5. <br />
, fi<br />
6. fi<br />
<br />
7. <br />
8. <br />
Example 11<br />
Graphing a Rational Function<br />
f x= x+x− <br />
<br />
x+ x− <br />
Solution <br />
l fiy<br />
f=__<br />
+− <br />
+ − <br />
=<br />
fixfi<br />
xx =−x =<br />
<br />
yx−<br />
fiThx +=<br />
x −=x =−x =<br />
ThTh<br />
<br />
y =<br />
x<br />
y<br />
s fiFigure 20
428<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
Figure 20<br />
Thx =−<br />
Thfi<br />
fi <br />
x =<br />
Figure 21ftx <br />
<br />
y = <br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
x = –<br />
Figure 21<br />
<br />
<br />
<br />
x = <br />
<br />
x<br />
Try It #8<br />
fx= x+ x− <br />
x− x− <br />
<br />
Writing Rational Functions<br />
<br />
<br />
x<br />
x<br />
<br />
<br />
<br />
writing rational functions from intercepts and asymptotes<br />
xx =x <br />
x <br />
x n<br />
x =v <br />
v <br />
v m<br />
x i<br />
=v j<br />
<br />
<br />
fx=a x−x p x−x <br />
p x−x n<br />
p n<br />
___<br />
x−v <br />
q x−v <br />
q x−v m<br />
q n<br />
p i<br />
q i<br />
<br />
a<br />
x
SECTION 5.6 RATIONAL FUNCTIONS 429<br />
How To…<br />
<br />
1. x<br />
Thfi<br />
ffi<br />
2. <br />
<br />
3. o fi<br />
Example 12 Writing a Rational Function from Intercepts and Asymptotes<br />
Figure 22<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
Figure 22<br />
Solution Thxx =−x =<br />
ThThx =−<br />
<br />
x fifi<br />
Thx =_<br />
<br />
x<br />
Figure 23<br />
fi<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – <br />
–<br />
x<br />
–<br />
x<br />
–<br />
Figure 23<br />
<br />
x+x−<br />
fx=a<br />
__ <br />
x+x−
430<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
o fiy−<br />
−=a__<br />
+− <br />
+− <br />
−<br />
−=a<br />
_ <br />
<br />
a<br />
_<br />
= − − = _<br />
<br />
Ths a fif x= x+x− <br />
<br />
x+x− <br />
<br />
Graphing Rational Functions (http://openstaxcollege.org/l/graphrational)<br />
Find the Equation of a Rational Function (http://openstaxcollege.org/l/equatrational)<br />
Determining Vertical and Horizontal Asymptotes (http://openstaxcollege.org/l/asymptote)<br />
Find the Intercepts, Asymptotes, and Hole of a Rational Function (http://openstaxcollege.org/l/interasymptote)
SECTION 5.6 SECTION EXERCISES 431<br />
5.6 SECTION EXERCISES<br />
VERBAL<br />
1. l diff<br />
<br />
<br />
3. <br />
<br />
<br />
5. <br />
x<br />
2. l diff<br />
<br />
4. <br />
<br />
ALGEBRAIC<br />
, fi<br />
6. fx=_____<br />
x− <br />
x+ <br />
9. fx=<br />
_________<br />
x +x− <br />
x −x + <br />
7. fx=x+ _____<br />
x − <br />
8. fx=_________<br />
x + <br />
x −x− <br />
, fi<br />
10. fx=____<br />
<br />
x− <br />
13. fx=__________<br />
x<br />
x +x− <br />
11. fx=_____<br />
<br />
x+ <br />
14. fx=______<br />
+x<br />
x − <br />
12. fx=_____<br />
x<br />
x − <br />
15. fx= _______ x− <br />
x −x<br />
16. fx=___________<br />
x − <br />
x +x +x<br />
19. fx= −x ______<br />
x− <br />
17. fx=______<br />
x+ x − <br />
18. fx=x− _____<br />
<br />
x− <br />
, fixy<br />
20. fx=_____<br />
x+ x + <br />
21. fx=_____<br />
x<br />
x −x<br />
22. fx=<br />
___________<br />
x +x+ x +x+ <br />
23. fx=<br />
___________<br />
x +x+ x −x+ <br />
24. fx=−x _______<br />
<br />
x − <br />
<br />
25. fx=_____<br />
x<br />
x+ <br />
26. fx=_____<br />
x<br />
x− <br />
27. fx=−x<br />
_____<br />
x− <br />
28. fx=<br />
_________<br />
x −x+ <br />
x −x− <br />
29. fx=___________<br />
x − <br />
x +x− <br />
, fi<br />
30. fx= x +x ________<br />
x+ <br />
31. − fx=x _______ <br />
x− <br />
32. − fx=x ________<br />
x− <br />
33. fx= x −x _______<br />
x + <br />
34. +x+ fx=x _________<br />
x−
432 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
GRAPHICAL<br />
<br />
<br />
35. Thhift 36. Thhift<br />
<br />
37. Thhift<br />
<br />
38. Thhift<br />
<br />
, fi<br />
<br />
39. px=______<br />
x− <br />
x+ <br />
40. qx=x− _____<br />
x− <br />
41. sx=______<br />
<br />
x− <br />
42. rx=______<br />
<br />
x+ <br />
43. fx=x __<br />
−x− <br />
x +x− <br />
44. gx=x __<br />
+x− <br />
x −+ <br />
45. ax=_<br />
x +x− x − 46. bx=x _<br />
−x− x − 47. hx=x _ +x−<br />
<br />
x−<br />
48. kx=__<br />
x −x− 49. wx=__<br />
x−x+x− 50. zx=__<br />
x+ x− <br />
x− x+ x− x−x+x+ <br />
<br />
51. x=<br />
x=−x<br />
−y<br />
52. x=−<br />
x=−x<br />
y<br />
53. x=−<br />
x=−x<br />
−<br />
y=<br />
54. x=−<br />
x=x−<br />
<br />
y=−<br />
55. x=−<br />
x=y<br />
<br />
56. x=<br />
x=y<br />
<br />
<br />
57. y<br />
58. y<br />
59. y<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
– <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x
SECTION 5.6 SECTION EXERCISES 433<br />
60. y<br />
61. y<br />
62. y<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
63. y<br />
64. y<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
–– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
NUMERIC<br />
fl<br />
<br />
65. fx=_____<br />
<br />
x− <br />
66. fx=_____<br />
x<br />
x− <br />
67. fx=_____<br />
x<br />
x+ <br />
68. fx=______<br />
x<br />
x− <br />
69. fx=_________<br />
x <br />
x +x+ <br />
TECHNOLOGY<br />
f xf x><br />
70. fx=_____<br />
<br />
x+ <br />
71. fx=_____<br />
<br />
x− <br />
72. fx=___________<br />
<br />
x−x+ <br />
73. fx=___________<br />
x+ <br />
x−x− <br />
74. fx=____________<br />
x+ <br />
x− x+ <br />
EXTENSIONS<br />
<br />
75. fx=_____<br />
x − <br />
x− <br />
76. + fx=x _____ <br />
x+ <br />
77. +x− fx=x ________<br />
x− <br />
78. fx=<br />
__________<br />
x +x− x+ 79. +x <br />
fx=x ______<br />
<br />
x+
434 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
REAL-WORLD APPLICATIONS<br />
<br />
80. <br />
<br />
<br />
<br />
<br />
<br />
ftt<br />
<br />
81. <br />
<br />
<br />
<br />
<br />
<br />
ftt<br />
82. ThC<br />
tft<br />
t<br />
Ct=_<br />
+t <br />
t<br />
83. ThC<br />
tft<br />
t<br />
Ct=_<br />
t + <br />
s hig<br />
Th<br />
<br />
84. <br />
<br />
x=<br />
<br />
86. <br />
<br />
x=<br />
85. <br />
et. Th<br />
t. Th<br />
t. Th<br />
<br />
x=<br />
<br />
87. <br />
<br />
x=<br />
88. <br />
<br />
o co<br />
<br />
x=
SECTION 5.7 INVERSES AND RADICAL FUNCTIONS 435<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the inverse of an invertible polynomial function.<br />
Restrict the domain to find the inverse of a polynomial function.<br />
5.7 INVERSES AND RADICAL FUNCTIONS<br />
<br />
Figure 1<br />
Th<br />
V =__<br />
πr h <br />
=__<br />
πr r <br />
=__<br />
πr <br />
<br />
Vr<br />
fi<br />
<br />
√ <br />
r = <br />
V<br />
π<br />
ThVr<br />
<br />
<br />
Finding the Inverse of a Polynomial Function<br />
fgfab<br />
gba<br />
<br />
<br />
Figure 2<br />
fifi<br />
<br />
<br />
<br />
<br />
Figure 2
436<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
<br />
xy<br />
Figure 3<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – – –<br />
<br />
–<br />
–<br />
<br />
x<br />
Figure 3<br />
fi<br />
yx=ax <br />
a<br />
<br />
=a <br />
a =__<br />
<br />
=__<br />
<br />
<br />
<br />
yx=__<br />
x <br />
<br />
<br />
yxx<br />
fi<br />
<br />
fiis<br />
xfi<br />
<br />
y =__<br />
<br />
x<br />
<br />
y =x <br />
x =±√ — y<br />
Thx<br />
<br />
y =__<br />
x x ><br />
<br />
xx<br />
Th<br />
<br />
=l ċw<br />
=ċx<br />
=x<br />
=√ — y<br />
Th<br />
1. <br />
2.Thdiff
SECTION 5.7 INVERSES AND RADICAL FUNCTIONS 437<br />
ftfi<br />
invertible functions<br />
f − x<br />
f − xf xTh−<br />
f x f x − =_<br />
<br />
f x <br />
f − f<br />
ff − fxf − <br />
<br />
f − f x=xxf<br />
<br />
f f − x=xxf −<br />
<br />
verifying two functions are inverses of one another<br />
fgxfg<br />
g f x=f gx=x<br />
How To…<br />
fi<br />
<br />
1. f xy<br />
2. xy<br />
3. yf − x<br />
Example 1 Verifying Inverse Functions<br />
<br />
f x=_<br />
<br />
x + f − x= −x ≠−<br />
x<br />
Solution f − f x=xf f − x=x<br />
f − fx=f ( − _____ <br />
x + ) <br />
<br />
=_<br />
−<br />
<br />
<br />
_____ <br />
x + <br />
=x+−<br />
=x<br />
f f x=f( − __<br />
<br />
x − ) <br />
<br />
=__<br />
<br />
( __<br />
<br />
x − ) +<br />
=_<br />
<br />
<br />
__ <br />
x<br />
=x<br />
<br />
Thf x=<br />
<br />
x + f − x= <br />
x −<br />
Try It #1<br />
fx= x+ <br />
f − x=x−
438<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 2<br />
Finding the Inverse of a Cubic Function<br />
f x=x +<br />
Solution Th<br />
g fx<br />
y =x +<br />
x =y +<br />
x −=y <br />
_____ x − <br />
=y<br />
√<br />
—<br />
f − x= _____ x −<br />
<br />
<br />
Analysis Look at the graph of f and f − . Notice that one graph is the reflection of the other about the liney = x. This is<br />
always the case when graphing a function and its inverse function.<br />
Also, since the method involved interchanging x and y, notice corresponding points. If a, b is on the graph of f , then (b, a) is on<br />
the graph of f − . Since , is on the graph of f, then , is on the graph of f − . Similarly, since , is on the graph of f, then<br />
, is on the graph of f − . See Figure 4<br />
<br />
y<br />
f x= x + 1<br />
<br />
y = x<br />
<br />
<br />
x<br />
– – – <br />
–<br />
<br />
–<br />
–<br />
Figure 4<br />
Try It #2<br />
fx= √ — x+<br />
Restricting the Domain to Find the Inverse of a Polynomial Function<br />
fi<br />
<br />
Th<br />
<br />
o fi<br />
restricting the domain
SECTION 5.7 INVERSES AND RADICAL FUNCTIONS 439<br />
How To…<br />
n fi<br />
1. <br />
2. f xy<br />
3. xy<br />
4. yf − x<br />
5. f − x<br />
<br />
Example 3<br />
Restricting the Domain to Find the Inverse of a Polynomial Function<br />
f<br />
a. f x=x− x ≥ b. f x=x− x ≤<br />
Solution Thf x=x− x ≥<br />
x ≤Figure 5<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – –<br />
–<br />
f x = x − x ≥ <br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – –<br />
–<br />
f x = x − x ≤ <br />
<br />
x<br />
Figure 5<br />
o fif xy<br />
y =x− xy<br />
x =y− <br />
±√ — x =y − <br />
<br />
±√ — x =y<br />
Th<br />
xyf x<br />
xxyy<br />
x ≥y ≥<br />
a. Thx ≥<br />
f x≥e +<br />
f − x=+√ — x<br />
b. Thx ≤<br />
f x≤−<br />
f − x=−√ — x<br />
Analysis On the graphs in Figure 6, we see the original function graphed on the same set of axes as its inverse function.<br />
Notice that together the graphs show symmetry about the line y = x. The coordinate pair is on the graph of f and the<br />
coordinate pair is on the graph of f − . For any coordinate pair, if a, b is on the graph of f, then b, a is on the graph of<br />
f − . Finally, observe that the graph of f intersects the graph of f − on the line y = x. Points of intersection for the graphs<br />
of f and f − will always lie on the line y = x.
440<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Example 4<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – –<br />
–<br />
y<br />
<br />
f x<br />
<br />
<br />
<br />
y = x<br />
f – x<br />
x<br />
Figure 6<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – –<br />
–<br />
y<br />
f x<br />
<br />
<br />
<br />
<br />
y = x<br />
f – x<br />
x<br />
Finding the Inverse of a Quadratic Function When the Restriction Is Not Specified<br />
n fi<br />
f x=x− −<br />
Solution −<br />
<br />
x ≥<br />
fif xy<br />
x<br />
y =x− − xy<br />
x =y− − <br />
x +=y− <br />
± √ — x +=y − <br />
± √ — x +=y <br />
<br />
f − x=± √ — x +<br />
x ≥<br />
+<br />
f − x=+√ — x +<br />
fiTh<br />
<br />
Analysis Notice that we arbitrarily decided to restrict the domain on x ≥ . We could just have easily opted to restrict the<br />
domain on x ≤ , in which casef − (x) = − √ — x + . Observe the original function graphed on the same set of axes as its<br />
inverse function in Figure 7. Notice that both graphs show symmetry about the line y = x. The coordinate pair (, −) is on<br />
the graph of f and the coordinate pair (−, ) is on the graph of f − . Observe from the graph of both functions on the same set<br />
of axes that<br />
<br />
<br />
<br />
f =f − =∞<br />
f − =f =−∞<br />
ff − y =x<br />
y<br />
y = x<br />
−<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f x<br />
f – x<br />
<br />
−<br />
x<br />
Figure 7
SECTION 5.7 INVERSES AND RADICAL FUNCTIONS 441<br />
Try It #3<br />
fx=x +x ≥<br />
Solving Applications of Radical Functions<br />
<br />
fi<br />
<br />
How To…<br />
n, fi<br />
1. <br />
2. f xyx<br />
3. <br />
Example 5<br />
Finding the Inverse of a Radical Function<br />
f x=√ — x −n fi<br />
Solution f x≥f xyx<br />
y =√ — x −<br />
x =√ — y −<br />
x =√ — y −<br />
x =y −<br />
x +=y<br />
f − x=x +<br />
f xy<br />
xy<br />
<br />
<br />
f − x<br />
<br />
f − x=x +x ≥<br />
Analysis Notice in Figure 8 that the inverse is a reflection of the original function over the line y = x. Because the<br />
original function has only positive outputs, the inverse function has only positive inputs.<br />
<br />
<br />
<br />
<br />
y<br />
f – x<br />
y = x<br />
<br />
<br />
<br />
f x<br />
x<br />
Figure 8<br />
Try It #4<br />
fx=√ — x+
442<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Solving Applications of Radical Functions<br />
<br />
<br />
Example 6 Solving an Application with a Cubic Function<br />
Th<br />
<br />
V =__<br />
πr <br />
<br />
V = <br />
πr Vr<br />
Thπ =<br />
Solution Vr ><br />
<br />
V ≥rV<br />
fi<br />
<br />
V =__<br />
πr <br />
<br />
r =___<br />
V<br />
π r <br />
r =√ <br />
<br />
___ V r<br />
π<br />
ThV =π =<br />
Th3 ft<br />
r =<br />
<br />
√ V <br />
= <br />
π<br />
√ <br />
______ ċ <br />
ċ <br />
≈ √ <br />
— <br />
≈<br />
Determining the Domain of a Radical Function Composed with Other Functions<br />
<br />
Example 7<br />
Finding the Domain of a Radical Function Composed with a Rational Function<br />
f x= √ ___________ x+x− <br />
<br />
x− <br />
Solution<br />
<br />
fi<br />
<br />
x+x− <br />
<br />
x− ≥Th<br />
xx =−<br />
<br />
Figure 9<br />
<br />
<br />
<br />
<br />
− <br />
– – – – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
x= <br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 9
SECTION 5.7 INVERSES AND RADICAL FUNCTIONS 443<br />
ThxxTh<br />
<br />
Th<br />
y√ — <br />
yxx =− <br />
<br />
<br />
f xfif x−≤x
444 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
5.7 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
2. <br />
<br />
4. Th<br />
<br />
ALGEBRAIC<br />
, fi<br />
5. fx=x− ∞ 6. fx=x+ −∞ 7. fx=x+ −−∞<br />
8. fx=x +−∞ 9. fx=−x ∞ 10. fx=−x ∞<br />
11. fx=x +∞<br />
, fi<br />
12. fx=x + 13. fx=x + 14. fx=−x <br />
15. fx=−x <br />
, fi<br />
16. fx=√ — x+ 17. fx=√ — −x 18. fx=+√ — x−<br />
19. fx=√ — x−+ 20. fx=+ √ <br />
— x<br />
21. fx=− √ <br />
— x<br />
22. fx=_____<br />
<br />
x+ <br />
25. fx=_____<br />
x− <br />
x+ <br />
23. fx=_____<br />
<br />
x− <br />
26. fx=x+ ______<br />
<br />
−x<br />
24. fx=x+ _____<br />
<br />
x+ <br />
27. fx=______<br />
x+ <br />
−x<br />
28. fx=x +x−∞ 29. fx=x +x+−∞ 30. fx=x −x+∞<br />
GRAPHICAL<br />
, fi<br />
31. fx=x +x≥ 32. fx=−x x≥ 33. fx=x+ x≥−<br />
34. fx=x− x≥ 35. fx=x + 36. fx=−x <br />
37. fx=x +xx≥− 38. fx=x −x+x≥ 39. fx= __<br />
x<br />
40. fx=__<br />
<br />
x x≥<br />
<br />
41. fx= √ __<br />
x+x− x 42. fx= √ __<br />
x+x− <br />
43. fx= x− √ _ xx+ x−<br />
44. fx= √ _ x −x−<br />
45. fx= x− √ _ −x <br />
x+
SECTION 5.7 SECTION EXERCISES 445<br />
TECHNOLOGY<br />
Th<br />
y<br />
46. fx=x −x−y= 47. fx=x +x−y= 48. fx=x +x−y=<br />
49. fx=x +x−y=− 50. fx=x +x+y=−<br />
EXTENSIONS<br />
, fiabc<br />
51. fx=ax +b 52. fx=x +bx 53. fx=√ — ax +b<br />
54. fx= √ — ax+b<br />
55. fx= ax+b ______<br />
x+c<br />
REAL-WORLD APPLICATIONS<br />
<br />
56. <br />
htftt<br />
ht=−t t<br />
h <br />
50 m<br />
58. ThVr<br />
Vr= <br />
πr r<br />
V <br />
<br />
60. <br />
n<br />
Cn= +n <br />
+n <br />
C<br />
nnC<br />
<br />
<br />
62. ThVr<br />
hV=πr h<br />
<br />
V <br />
<br />
64. ThV<br />
rhV=_ πr h <br />
rh<br />
12 f <br />
<br />
57. <br />
htftt<br />
ht=−t t<br />
h <br />
<br />
59. ThA<br />
rAr=πr r<br />
V <br />
<br />
61. ThT<br />
l<br />
Tl=π<br />
√ <br />
l <br />
lT<br />
<br />
<br />
63. ThA<br />
rhA=πr +πrh<br />
<br />
V <br />
<br />
65. <br />
rV
446<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Solve direct variation problems.<br />
Solve inverse variation problems.<br />
Solve problems involving joint variation.<br />
5.8 MODELING USING VARIATION<br />
ffThff<br />
<br />
ff<br />
<br />
Solving Direct Variation Problems<br />
Th<br />
e =se<br />
<br />
Table 1<br />
s, sales price e = 0.16s Interpretation<br />
e== a <br />
e== a <br />
e== a <br />
Table 1<br />
<br />
<br />
<br />
direct variationvaries directly<br />
Figure 1<br />
Thy =kx n Thk<br />
constant of variationk =n =<br />
<br />
<br />
<br />
e, Earnings, $<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
s, Sales Prices in Dollars<br />
Figure 1
SECTION 5.8 MODELING USING VARIATION 447<br />
direct variation<br />
xy<br />
y=kx n<br />
direct variationyvaries directlyn<br />
xk = y _<br />
x<br />
n k<br />
constant of variationfi<br />
How To…<br />
<br />
1. xy<br />
2. yfix<br />
<br />
3. <br />
4. o fi<br />
Example 1<br />
Solving a Direct Variation Problem<br />
Thyxy =x =, fiyx<br />
Solution Thy =kx Thy<br />
x<br />
k = y _<br />
x <br />
=__<br />
<br />
<br />
=__<br />
<br />
<br />
<br />
x =y<br />
y =__<br />
<br />
x<br />
y =__<br />
<br />
<br />
=<br />
Analysis The graph of this equation is a simple cubic, as shown in Figure 2.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2<br />
x
448<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Q & A…<br />
Do the graphs of all direct variation equations look like Example 1?<br />
<br />
<br />
Try It #1<br />
Thyxy=x= yx<br />
Solving Inverse Variation Problems<br />
<br />
T = <br />
<br />
d<br />
°<br />
Table 2<br />
d, depth T = 14,000 _<br />
d<br />
Interpretation<br />
0 ft<br />
_ =<br />
<br />
0 ft°<br />
ft<br />
_<br />
<br />
=<br />
0 ft°<br />
ft<br />
_<br />
<br />
=<br />
0 ft°<br />
Table 2<br />
Th<br />
inversely proportionalvaries inversely<br />
inverse variations<br />
Figure 3<br />
e Thy = k _<br />
x<br />
<br />
k =<br />
<br />
Temperature, T<br />
(°Fahrenheit)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Depth, d (ft)<br />
Figure 3<br />
inverse variation<br />
xy<br />
y=__<br />
k x n<br />
kyvaries inverselynxinversely<br />
proportionalinverse variationsk =x n y
SECTION 5.8 MODELING USING VARIATION 449<br />
Example 2<br />
Writing a Formula for an Inversely Proportional Relationship<br />
<br />
<br />
Solution tv<br />
vt=fi<br />
vt=t = v<br />
tv<br />
tv=___<br />
<br />
v<br />
=v −<br />
<br />
<br />
How To…<br />
<br />
1. xy<br />
2. yfix<br />
<br />
3. <br />
4. o fi<br />
Example 3<br />
Solving an Inverse Variation Problem<br />
yxy =x =, fiyx<br />
Solution Thy = k <br />
x<br />
yx<br />
Th<br />
k =x y<br />
= ċ<br />
=<br />
<br />
y=__<br />
k xk =<br />
y =___<br />
<br />
x <br />
x =y<br />
y =___<br />
<br />
<br />
=__<br />
<br />
Analysis The graph of this equation is a rational function, as shown in Figure 4.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 4<br />
<br />
Try It #2<br />
yxy=x= yx<br />
x
450<br />
CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
Solving Problems Involving Joint Variation<br />
ft<br />
<br />
joint variation<br />
Thc<br />
nd<br />
joint variation<br />
<br />
xyzx =kyzxy<br />
zx = ky <br />
z <br />
Example 4<br />
Solving Problems Involving Joint Variation<br />
xyzx =y =z =<br />
fixy =z =<br />
Solution <br />
x =_<br />
ky <br />
√ <br />
— z<br />
x =y =z =o fik<br />
<br />
=_<br />
k <br />
√ <br />
— <br />
<br />
=__<br />
k<br />
<br />
<br />
=k<br />
<br />
x =_<br />
y <br />
√ <br />
— z<br />
o fixy =z =yz<br />
Try It #3<br />
x =_<br />
√ <br />
— <br />
=<br />
xyzx=y=z= x<br />
y=z=<br />
<br />
Direct Variation (http://openstaxcollege.org/l/directvariation)<br />
Inverse Variation (http://openstaxcollege.org/l/inversevariatio)<br />
Direct and Inverse Variation (http://openstaxcollege.org/l/directinverse)
SECTION 5.8 SECTION EXERCISES 451<br />
5.8 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
2. <br />
<br />
ALGEBRAIC<br />
<br />
4. yxx=y= 5. yxx=y=<br />
6. yx 7. yxx=y=<br />
x=y=<br />
8. yx 9. yxx=<br />
x=y=<br />
y=<br />
10. yxx=y= 11. yxx = 3, y=<br />
12. yx 13. yx<br />
x=y=<br />
x=y=<br />
14. yx 15. yx<br />
x=y=<br />
x=y=<br />
16. yxzx= 17. yxzwx=z=<br />
z=y=<br />
w=y=<br />
18. yx<br />
zx=z=y=<br />
20. yxz<br />
wx=z=<br />
w=y=<br />
22. yx<br />
zw<br />
x=z=w=y=<br />
19. yxz<br />
x=z=y=<br />
21. yxzwx=<br />
z=w=y=<br />
23. yxz<br />
wtx=z=w=<br />
t=y=<br />
NUMERIC<br />
o fi<br />
24. yxx=y= 25. yxx=<br />
yx=<br />
y=yx=<br />
26. yxx= 27. yx<br />
y=yx=<br />
x=y=yx=<br />
28. yx<br />
29. yxx=y=<br />
x=y=yx=<br />
yx=<br />
30. yxx=<br />
y=yx=<br />
32. yx<br />
x=y=yx=<br />
34. yxzx=z=<br />
y=yx=z=<br />
36. yxz<br />
x=z=y=yx=<br />
z=<br />
38. yxzw<br />
x=z=w=y=y<br />
x=z=w=<br />
31. yxx=<br />
y=yx=<br />
33. yx<br />
x=y=yx=<br />
35. yxzwx=z=<br />
w=y=yx=z=w=<br />
37. yx<br />
zx=z=y=y<br />
x=z=<br />
39. yxz<br />
wx=<br />
z=w=y=yx=<br />
z=w=
452 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
40. yxzwtx=z=<br />
w=t=y=yx=z=w=t=<br />
TECHNOLOGY<br />
<br />
41. yxx= 42. yxx=y=<br />
y=<br />
43. yx 44. yxx=y=<br />
x=y=<br />
45. yx<br />
x=y=<br />
EXTENSIONS<br />
T<br />
a<br />
46. <br />
Ta<br />
48. <br />
Ta<br />
50. <br />
<br />
<br />
REAL-WORLD APPLICATIONS<br />
<br />
51. Ths<br />
t<br />
<br />
53. Th<br />
<br />
<br />
<br />
<br />
55. Th<br />
<br />
<br />
<br />
<br />
57. Th<br />
<br />
<br />
<br />
59. Th <br />
<br />
<br />
<br />
<br />
<br />
47. <br />
<br />
<br />
49. <br />
<br />
<br />
52. Thv<br />
tft<br />
ft<br />
54. Th<br />
<br />
<br />
<br />
<br />
<br />
56. Th<br />
<br />
<br />
f 3 m<br />
<br />
58. Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
60. ThK<br />
mv
CHAPTER 5 REVIEW 453<br />
CHAPTER 5<br />
REVIEW<br />
Key Terms<br />
arrow notation <br />
<br />
axis of symmetry <br />
x =− b <br />
a <br />
coefficient <br />
constant of variation kfi<br />
<br />
continuous function lift<br />
<br />
degree <br />
Descartes’ Rule of Signs <br />
f xf −x<br />
direct variation <br />
<br />
Division Algorithm f xdx<br />
dxf xqxrxfx=dx<br />
qx+rxqxrxTh<br />
dx<br />
end behavior <br />
Factor Theorem kf xx−kf x<br />
Fundamental Theorem of <strong>Algebra</strong> <br />
general form of a quadratic function f x=ax +bx+c<br />
abca ≠<br />
global maximum f af a≥f xx<br />
global minimum f af a≤f xx<br />
horizontal asymptote y =b<br />
<br />
Intermediate Value Theorem abfa
454 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
polynomial function <br />
<br />
power function f x=kx p k<br />
p<br />
rational function <br />
Rational Zero Theorem p _<br />
q<br />
p<br />
q <br />
Remainder Theorem f xx −kf k<br />
removable discontinuity efifi<br />
<br />
roots xy =<br />
smooth curve <br />
standard form of a quadratic function f x=ax−h +k<br />
hk<br />
synthetic division x −k<br />
term of a polynomial function a i<br />
x i f x=a n<br />
x n ++a <br />
x +a <br />
x +a <br />
turning point <br />
varies directly <br />
varies inversely <br />
vertex <br />
<br />
vertex form of a quadratic function <br />
vertical asymptote x =a a<br />
zeros xy =<br />
Key Equations<br />
<br />
<br />
<br />
<br />
f x=ax +bx+c<br />
f x=ax−h +k<br />
f x=a n<br />
x n ++a <br />
x +a <br />
x +a <br />
f x=dxqx+rxqx≠<br />
<br />
<br />
<br />
f x= Px <br />
Qx = a x p p−<br />
+a<br />
p p−<br />
x ++a <br />
x +a<br />
___ <br />
Qx≠<br />
q q−<br />
b q<br />
x <br />
+b q −<br />
x <br />
++b <br />
x +b <br />
y =kx n<br />
k<br />
y = k _<br />
x n k
CHAPTER 5 REVIEW 455<br />
Key Concepts<br />
5.1 Quadratic Functions<br />
<br />
Th<br />
ThThx<br />
xThyyExample 1<br />
Example 7Example 8<br />
ft<br />
Example 2<br />
ThExample 3<br />
Th. ThExample 4<br />
y<br />
Th<br />
Example 5<br />
Example 6<br />
Th Example 9.<br />
5.2 Power Functions and Polynomial Functions<br />
Example 1<br />
Th<br />
ThExample 2Example 3<br />
<br />
Example 4<br />
ThTh<br />
Thffi<br />
ffiExample 5<br />
Th<br />
Example 6Example 7<br />
nn xn −Example 8<br />
Example 9Example 10Example 11Example 12<br />
5.3 Graphs of Polynomial Functions<br />
Example 1<br />
fi<br />
Example 2Example 3Example 4<br />
fix<br />
xExample 5<br />
ThxExample 6<br />
Th<br />
Th<br />
Th<br />
Th<br />
nn −Example 7<br />
, fi<br />
e fin −Example 8Example 10<br />
Example 11
456 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
ThThf af b<br />
cabf c=Example 9<br />
5.4 Dividing Polynomials<br />
<br />
Example 1Example 2<br />
Th<br />
<br />
x −kExample 3<br />
Example 4Example 5<br />
Example 6<br />
5.5 Zeros of Polynomial Functions<br />
fif kf xx −k<br />
er ThExample 1<br />
r Thkf xx−kf xExample 2<br />
Thffi<br />
ffi Example 3Example 4<br />
ffi<br />
o fiExample 5<br />
l ThExample 6<br />
<br />
<br />
x−ccExample 7<br />
Th<br />
<br />
Thf −x<br />
Example 8<br />
<br />
Example 9<br />
5.6 Rational Functions<br />
f x= _<br />
x f x= _<br />
<br />
x <br />
Example 1<br />
<br />
Example 2<br />
ftExample 3<br />
Th<br />
Example 4<br />
Th<br />
Example 5<br />
<br />
Example 6<br />
<br />
Example 7Example 8Example 9Example 10<br />
fi<br />
Example 11<br />
xx =x <br />
x <br />
x n<br />
x =v <br />
v <br />
v m<br />
<br />
x i<br />
=v j
CHAPTER 5 REVIEW 457<br />
Example 12<br />
5.7 Inverses and Radical Functions<br />
fx=a x−x p x−x p x−x n p ___ n<br />
<br />
x−v <br />
q x−v q x−v m q n<br />
Th<br />
f − fff − Example 1<br />
fi<br />
Example 2<br />
fi<br />
Example 3Example 4<br />
fiExample 5<br />
Example 7<br />
Example 6Example 8<br />
5.8 Modeling Using Variation<br />
Example 1<br />
<br />
Example 2<br />
Example 3<br />
<br />
Example 4
458 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
CHAPTER 5 REVIEW EXERCISES<br />
QUADRATIC FUNCTIONS<br />
Th<br />
<br />
1. fx=x −x− 2. fx=−x −x<br />
, fi<br />
3. Th− 4. Th−<br />
<br />
<br />
5. <br />
<br />
<br />
<br />
<br />
6. <br />
<br />
h<br />
xhx= − +x<br />
x<br />
<br />
POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS<br />
<br />
effi<br />
7. fx=x −x +x− 8. fx= x+ −x 9. fx=x −x+x <br />
<br />
10. fx=x +x −x + 11. fx=x −x + 12. fx=x +x−x <br />
GRAPHS OF POLYNOMIAL FUNCTIONS<br />
, fi<br />
13. fx=x+ x−x+ 14. fx=x +x +x 15. fx=x −x +x−<br />
<br />
16.<br />
y<br />
17.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
––<br />
– – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x
CHAPTER 5 REVIEW 459<br />
18. ue Th<br />
fx=x −x+<br />
DIVIDING POLYNOMIALS<br />
o fi<br />
19. ______________<br />
x −x +x+ <br />
x− 20. _______________<br />
−x +x+ x <br />
x+ <br />
fi<br />
<br />
21. ______________<br />
x −x +x− <br />
x+ 22. __________<br />
x +x+ x− <br />
23. _________________<br />
x +x −x− x+ 24. _______________<br />
+x +x+ x x+ <br />
ZEROS OF POLYNOMIAL FUNCTIONS<br />
o Th<br />
25. x −x −x−= 26. x +x +x−=<br />
27. x −x +x −x+= 28. x +x +x +x+=<br />
fi<br />
29. x −x −x+= 30. x −x +x −x+=<br />
RATIONAL FUNCTIONS<br />
fi<br />
<br />
31. fx=_____<br />
x+ <br />
x− <br />
33. fx=________<br />
x − x +x− <br />
32. + fx=x _____ <br />
x − <br />
34. fx=x+ _____<br />
x − <br />
, fi<br />
35. fx=_____<br />
x − <br />
x+ <br />
36. −x + fx=x __________<br />
x + <br />
INVERSES AND RADICAL FUNCTIONS<br />
<br />
37. fx=x− x≥ 38. fx=x+ −x≥− 39. fx=x +x−x≥−<br />
40. fx=x − 41. fx=√ — x+− 42. fx=_____<br />
x− x+
460 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
MODELING USING VARIATION<br />
, fi<br />
43. yxx=<br />
y= yx=<br />
45. yxz<br />
x=z=y= yx=z=<br />
44. yx<br />
x =y= yx=<br />
46. yxz<br />
wx=z=w=<br />
y = yx=z=w=<br />
<br />
47. Th<br />
<br />
e e<br />
n t<br />
<br />
<br />
48. ThV<br />
TP
CHAPTER 5 PRACTICE TEST<br />
461<br />
CHAPTER 5 PRACTICE TEST<br />
ffi<br />
1. fx=x −x −x <br />
<br />
2. fx=x −x +x− 3. fx=−x −x−x <br />
<br />
4. fx=x +x−<br />
, fi<br />
5. <br />
<br />
6. <br />
<br />
<br />
<br />
7. fx=x− x−x − 8. fx=x −x +x <br />
<br />
9.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
o fi<br />
10. __________<br />
x +x− x+ <br />
o fi<br />
11. __________<br />
x +x − x− 12. ________________<br />
+x −x− x x+ <br />
o Thu fi<br />
13. fx=x +x −x− 14. fx=x +x +x +x+
462 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS<br />
15. fx=x +x +x −x− 16. fx=x +x +x +x +x+<br />
n, fi<br />
17. x=x=<br />
x=−y<br />
18. x= <br />
<br />
x=−<br />
<br />
19. x −x +=<br />
, fi<br />
20. fx=_________<br />
x+ <br />
x −x− <br />
21. +x− fx=x _________<br />
x − <br />
<br />
22. fx=_________<br />
x +x− x− <br />
<br />
23. fx=√ — x−+ 24. fx=x −<br />
25. fx=______<br />
x+ <br />
x− <br />
<br />
26. yxx=<br />
y=yx=<br />
27. yxz<br />
x=z=y= yx=z=<br />
<br />
28. Th
Exponential and Logarithmic<br />
Functions<br />
6<br />
Figure 1 Electron micrograph of E. Coli bacteria (credit: “Mattosaurus,” Wikimedia Commons)<br />
CHAPTER OUTLINE<br />
6.1 Exponential Functions 6.5 Logarithmic Properties<br />
6.2 Graphs of Exponential Functions 6.6 Exponential and Logarithmic Equations<br />
6.3 Logarithmic Functions 6.7 Exponential and Logarithmic Models<br />
6.4 Graphs of Logarithmic Functions 6.8 Fitting Exponential Models to Data<br />
Introduction<br />
<br />
Th<br />
<br />
<br />
fi<br />
<br />
ft <br />
Table 1<br />
<br />
y<br />
<br />
Hour <br />
Bacteria <br />
Table 1<br />
<br />
<br />
<br />
<br />
fb<br />
463
464<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Evaluate exponential functions.<br />
Find the equation of an exponential function.<br />
Use compound interest formulas.<br />
Evaluate exponential functions with base e.<br />
6.1 EXPONENTIAL FUNCTIONS<br />
Th<br />
<br />
ft<br />
exponential growth<br />
exponential functions<br />
<br />
Identifying Exponential Functions<br />
<br />
f x=x +<br />
Thff<br />
percent<br />
Defining an Exponential Function<br />
<br />
<br />
fi<br />
<br />
grow exponentiallydoublepercent increase<br />
Th<br />
<br />
Percent change changepercent<br />
Exponential growthincrease<br />
percent<br />
Exponential decaydecrease<br />
percent<br />
<br />
Thfi<br />
Th<br />
Table 1<br />
x f (x)=2 x gx=2x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 1
SECTION 6.1 EXPONENTIAL FUNCTIONS 465<br />
Table 2<br />
Exponential growthsame percentage<br />
<br />
Linear growthsame amount<br />
<br />
fffi<br />
<br />
<br />
g 2 t<br />
Thf x=ab x ab<br />
<br />
b ><br />
466<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
exponential function<br />
x<br />
f(x=ab x<br />
<br />
a<br />
bb ≠<br />
Thf<br />
Thfa ><br />
Thfa <<br />
yay =<br />
Example 1<br />
Identifying Exponential Functions<br />
not<br />
fx= x− g x=x h x=( <br />
) x j x=− x<br />
Solution fi<br />
Thgx=x gx=x <br />
<br />
bb ≠Thjx=− x <br />
−<br />
Try It #1<br />
<br />
f(x=x −x +<br />
gx= x<br />
hx=x +<br />
jx= −x<br />
Evaluating Exponential Functions<br />
<br />
b<br />
b =−x = <br />
Thf x=f ( <br />
<br />
) =− =√ — −<br />
<br />
<br />
b =1. Thf x= x =x<br />
f x=b x x <br />
<br />
f x= x f <br />
f x= x<br />
f = x =<br />
=
SECTION 6.1 EXPONENTIAL FUNCTIONS 467<br />
<br />
f x= x f <br />
f x= x<br />
f = x =<br />
= <br />
= <br />
<br />
f = ≠ =<br />
Example 2 Evaluating Exponential Functions<br />
f x= x + f <br />
Solution <br />
f x= x + <br />
f = + x =<br />
= <br />
= <br />
= <br />
Try It #2<br />
f x= x− f <br />
Defining Exponential Growth<br />
ft<br />
fi<br />
<br />
<br />
exponential growth<br />
exponential growth<br />
xabb ≠<br />
f x=ab<br />
<br />
x<br />
a<br />
bx<br />
exponential function<br />
ff<br />
Ax=+x<br />
<br />
Bx=+ x <br />
Table 3<br />
Year, x Stores, Company A Stores, Company B<br />
+= + =<br />
+= + =<br />
+= + =<br />
+= + =<br />
x Ax=+x Bx=+ x<br />
Table 3
468<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
ThfiFigure 2<br />
<br />
Number of Stores<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
Bx= + x<br />
<br />
<br />
Years<br />
Ax= + x<br />
Figure 2 The graph shows the numbers of stores Companies A and B opened over a five-year period.<br />
∞∞ft<br />
<br />
Bx=+ x <br />
<br />
+=Bx= x <br />
basexexponent<br />
Example 3<br />
Evaluating a Real-World Exponential Model<br />
<br />
ThPt= t t<br />
<br />
Solution t =ft<br />
<br />
P= ≈<br />
Th<br />
Try It #3<br />
Th <br />
Pt= t t<br />
<br />
Example 3<br />
Finding Equations of Exponential Functions<br />
<br />
<br />
o fiab<br />
How To…<br />
<br />
1. aaa<br />
f x=ab x b<br />
2. af x=ab x <br />
o fiab<br />
3. abf x=ab x <br />
<br />
<br />
x
SECTION 6.1 EXPONENTIAL FUNCTIONS 469<br />
Example 4<br />
Writing an Exponential Model When the Initial Value Is Known<br />
Th<br />
NtNt<br />
Solution tftTh<br />
<br />
fta =<br />
Nt=b t o fib<br />
Nt=b t<br />
=b <br />
<br />
__<br />
=b <br />
b =( __<br />
) b<br />
b ≈ <br />
NOTE:Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places<br />
for the remainder of this section.<br />
ThNt= t <br />
<br />
<br />
<br />
Figure 3<br />
∞∞<br />
Try It #4<br />
Deer Population<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Nt<br />
<br />
<br />
<br />
<br />
Years<br />
Figure 3 Graph showing the population of deer over time, N(t) = 80(1.1447) t , t years after 2006.<br />
<br />
236 <br />
Nt<br />
Example 5<br />
Writing an Exponential Model When the Initial Value is Not Known<br />
−<br />
Solution f x=ab x <br />
ab<br />
−=ab −<br />
=ab <br />
t
470<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
e fiab<br />
<br />
=ab −<br />
<br />
b −=a<br />
<br />
a =b <br />
ab<br />
<br />
=ab <br />
=b b =b a<br />
b =( <br />
) b<br />
b ≈ <br />
be fia<br />
a =b ≈ ≈<br />
Thf x= x <br />
Figure 4<br />
−. Th<br />
fx<br />
−<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
Try It #5<br />
Figure 4 The graph of f (x) = 2.4492(0.6389) x models exponential decay.<br />
<br />
<br />
Q & A…<br />
Do two points always determine a unique exponential function?<br />
x xff x<br />
<br />
<br />
x<br />
How To…<br />
<br />
1. y<br />
<br />
2. yaaa<br />
f x=ab x b<br />
3. af x=ab x <br />
o fiab<br />
4. f x=ab x
SECTION 6.1 EXPONENTIAL FUNCTIONS 471<br />
Example 6<br />
Writing an Exponential Function Given Its Graph<br />
Figure 5<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – ––<br />
– ––<br />
<br />
–<br />
x<br />
Figure 5<br />
Solution yfiTha =<br />
<br />
y =ab x <br />
y =b x a<br />
=b y x<br />
=b <br />
b =± <br />
bb =ab<br />
f (x= x <br />
Try It #6<br />
Figure 6<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
√<br />
<br />
<br />
x<br />
Figure 6<br />
How To…<br />
o fi<br />
1. [STAT]<br />
2. L1L2<br />
3. L1x<br />
4. L2y<br />
5. [STAT]CALCExpReg (Exponential Regression)[ENTER]<br />
6. Thab y =a ⋅b x<br />
Example 7<br />
Using a Graphing Calculator to Find an Exponential Function<br />
o fi<br />
Solution [STAT][EDIT][1: Edit…]L1L2<br />
L1xL2y<br />
[STAT][CALC][0: ExpReg][ENTER]Tha =b =Th<br />
y =⋅ x <br />
Try It #7
472<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Applying the Compound-Interest Formula<br />
<br />
compound interestThcompounding<br />
<br />
annual percentage rate (APR)nominal rate<br />
Thnominal<br />
ffgreater<br />
! Th<br />
<br />
tPrn<br />
At=P ( +r <br />
n) nt<br />
Table 4<br />
<br />
Frequency<br />
<br />
<br />
<br />
<br />
<br />
Table 4<br />
the compound interest formula<br />
Compound interest<br />
At=P ( +r _<br />
n) nt<br />
Value after 1 year<br />
<br />
<br />
<br />
<br />
<br />
<br />
At<br />
t<br />
Pft<br />
r<br />
n<br />
Example 8<br />
Calculating Compound Interest<br />
<br />
<br />
Solution P =r =<br />
n =<br />
At =<br />
At=P ( +r <br />
n) nt <br />
A=( + ____<br />
) ⋅ <br />
≈ <br />
Th<br />
Try It #8
SECTION 6.1 EXPONENTIAL FUNCTIONS 473<br />
Example 9<br />
Using the Compound Interest Formula to Solve for the Principal<br />
<br />
<br />
<br />
<br />
Solution Thr =n =<br />
fiP<br />
P<br />
<br />
<br />
At=P ( +r _<br />
n) nt <br />
=P ( + ____<br />
) <br />
=P <br />
Arnt<br />
<br />
______<br />
<br />
=P P<br />
P ≈ <br />
<br />
Try It #9<br />
Example 9.<br />
Evaluating Functions with Base e<br />
Table 5<br />
<br />
g. Th<br />
Table 5<br />
Frequency<br />
A(t) = ( 1 +<br />
_<br />
n) 1 n Value<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
( + <br />
) <br />
( + <br />
) <br />
( + <br />
) <br />
( + <br />
) <br />
( + <br />
) <br />
( +<br />
<br />
) <br />
( +<br />
<br />
) <br />
( +<br />
<br />
) <br />
Table 5
474<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Thnn<br />
( + n) <br />
n e<br />
Th<br />
<br />
the number e<br />
The<br />
( + __<br />
<br />
n) n n<br />
Thee<br />
e ≈<br />
<br />
Example 10<br />
Using a Calculator to Find Powers of e<br />
e o fi<br />
Solution [e x ]Th[e^(]<br />
[)][ENTER]e ≈fiExp<br />
fio fie<br />
Try It #10<br />
e − o fi<br />
Investigating Continuous Growth<br />
e<br />
econtinuous growth or<br />
decay modelsfi<br />
fl<br />
the continuous growth/decay formula<br />
tar<br />
At=ae rt<br />
<br />
a<br />
r<br />
t<br />
r>r <<br />
<br />
<br />
At=Pe rt<br />
<br />
P<br />
r<br />
t
SECTION 6.1 EXPONENTIAL FUNCTIONS 475<br />
How To…<br />
t<br />
1. a<br />
2. r<br />
a. r ><br />
b. r <<br />
3. t<br />
4. At<br />
Example 11<br />
Calculating Continuous Growth<br />
<br />
<br />
Solution r =<br />
ThP =fift<br />
t =<br />
At=Pe rt <br />
=e <br />
≈<br />
Thft<br />
Prt<br />
<br />
Try It #11<br />
<br />
<br />
Example 12 Calculating Continuous Decay<br />
<br />
Solution r =−Th<br />
a =o fiftt =<br />
At=ae rt <br />
=e − <br />
≈<br />
<br />
art<br />
<br />
Try It #12<br />
Example 12ft<br />
<br />
Exponential Growth Function (http://openstaxcollege.org/l/expgrowth)<br />
Compound Interest (http://openstaxcollege.org/l/compoundint)
476 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
6.1 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. Th nominal<br />
<br />
<br />
<br />
nominal rate<br />
<br />
<br />
2. <br />
<br />
<br />
<br />
ALGEBRAIC<br />
<br />
4. Th<br />
f w<br />
6. Th<br />
<br />
8. Tht<br />
ht=−t +t+<br />
5. <br />
e<br />
<br />
7. <br />
<br />
t<br />
At= t <br />
Bt= t <br />
9. 10. <br />
<br />
11. <br />
<br />
ft<br />
<br />
13. <br />
<br />
<br />
<br />
<br />
t infl<br />
<br />
12. <br />
<br />
ft<br />
<br />
<br />
<br />
14. y=−t 15. y= x<br />
16. y= x<br />
<br />
17. y= t<br />
, fi<br />
18. 19. 20. ( − <br />
) <br />
21. − 22. <br />
fi
SECTION 6.1 SECTION EXERCISES 477<br />
<br />
, fi<br />
23.<br />
x <br />
24.<br />
x <br />
<br />
f(x) −<br />
h(x) <br />
25.<br />
x <br />
26.<br />
m (x) <br />
27. x <br />
g(x) − <br />
x <br />
f(x) <br />
At=P ( + r <br />
n) nt <br />
28. ft<br />
<br />
10, 250 ( + <br />
) <br />
<br />
30. <br />
<br />
32. <br />
<br />
<br />
34. <br />
<br />
ft<br />
<br />
36. <br />
r<br />
38. <br />
<br />
<br />
ft<br />
29. <br />
<br />
31. <br />
<br />
<br />
33. <br />
P<br />
35. <br />
<br />
5 m<br />
37. <br />
<br />
<br />
f $9,000 aft<br />
<br />
<br />
39. y=e t 40. y=e <br />
t 41. y=e −t<br />
42. <br />
<br />
<br />
ft<br />
43. <br />
ft<br />
<br />
NUMERIC<br />
<br />
44. fx= x f − 45. fx=− x+ f − 46. fx=e x f <br />
47. fx=−e x− f − 48. fx= −x+ +f − 49. fx=e x −f <br />
50. fx=− <br />
−x + f
478 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
TECHNOLOGY<br />
fi<br />
<br />
51. 52. 53. <br />
54. 55. <br />
EXTENSIONS<br />
56. annual percentage yield<br />
<br />
<br />
<br />
<br />
=( + r <br />
) −<br />
58. <br />
f(x=ab x ab<br />
b≠<br />
bb=e n n<br />
<br />
es a b<br />
60. ThA<br />
t wr<br />
tAt=ae rt a<br />
<br />
<br />
t<br />
It=e rt −<br />
REAL-WORLD APPLICATIONS<br />
61. Th<br />
<br />
<br />
<br />
63. <br />
<br />
<br />
<br />
<br />
<br />
65. <br />
<br />
, in o<br />
<br />
67. <br />
<br />
<br />
<br />
<br />
57. <br />
<br />
<br />
In<br />
t con<br />
59. <br />
d 1. Th<br />
b><br />
fx=a( <br />
b) x <br />
b=e n <br />
fx=ae −nx <br />
n<br />
62. <br />
ft<br />
<br />
ft<br />
64. <br />
<br />
<br />
<br />
66. <br />
<br />
<br />
<br />
<br />
<br />
Hint : <br />
<br />
68. <br />
f 7% wa<br />
ft
SECTION 6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 479<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Graph exponential functions.<br />
Graph exponential functions using transformations.<br />
6.2 GRAPHS OF EXPONENTIAL FUNCTIONS<br />
<br />
fi<br />
<br />
<br />
<br />
Graphing Exponential Functions<br />
<br />
f x=b x f x= x <br />
Table 1<br />
x − − − <br />
f (x) = 2x<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 1<br />
constantratio<br />
f x=ab x bTh<br />
a<br />
<br />
x<br />
x<br />
x<br />
Figure 1f x= x <br />
fx<br />
−<br />
−<br />
− <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – <br />
<br />
<br />
<br />
<br />
<br />
f x= x<br />
x<br />
Figure 1 Notice that the graph gets close to the x-axis, but never touches it.<br />
Thf x= x ∞y =<br />
f x=b x <br />
gx=( <br />
) x Table 2<br />
<br />
<br />
<br />
<br />
<br />
x
480<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
x − − − <br />
g(x)=( 1 2 ) x <br />
<br />
Table 2<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
Figure 2gx=( <br />
) x <br />
gx= <br />
−<br />
–<br />
x<br />
–<br />
<br />
– – – –<br />
–<br />
gx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 2<br />
Thgx=( <br />
) x ∞y =<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
characteristics of the graph of the parent function f (x) = b x<br />
f x=b x b >b ≠<br />
<br />
fx<br />
y =<br />
f x= b x<br />
–∞∞<br />
b > <br />
∞<br />
x<br />
y<br />
b<br />
b ><br />
<br />
b <<br />
x<br />
Figure 3<br />
<br />
Figure 3<br />
<br />
fx<br />
f x= b x<br />
< b < <br />
b<br />
x<br />
How To…<br />
f x=b x <br />
1. <br />
2. y<br />
3. <br />
4. −∞∞∞y =
SECTION 6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 481<br />
Example 1<br />
Sketching the Graph of an Exponential Function of the Form f (x) = b <br />
f x= x <br />
Solution <br />
b =Th <br />
y =<br />
Table 3<br />
x − − − <br />
f (x) = 0.25 x <br />
Table 3<br />
y−<br />
Figure 4<br />
fx= x<br />
−<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Figure 4<br />
<br />
<br />
<br />
Th−∞∞∞y =<br />
Try It #1<br />
f x= x <br />
Graphing Transformations of Exponential Functions<br />
<br />
ftfl<br />
f x=b x <br />
ftfl<br />
<br />
Graphing a Vertical Shift<br />
Thfidf x=b x d<br />
f x= x <br />
ftd =ftgx= x +fthx= x −<br />
ftFigure 5<br />
g(x) = 2 x + 3<br />
f (x) = 2 x<br />
h(x) = 2 x – – –<br />
− 3<br />
– – – – <br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 5<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
y = 3<br />
y = 0<br />
y = −3
482<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
ftf x= x <br />
Th−∞∞<br />
hiftgx= x +<br />
◦ y <br />
◦ Th y =<br />
◦ Th∞<br />
hifthx= x −<br />
◦ y −<br />
◦ Th y =−<br />
◦ Th−∞<br />
Graphing a Horizontal Shift<br />
Thcf x=b x <br />
ftcopposite<br />
f x= x ftc = ftgx= x + ft<br />
h x= x − ftFigure 6<br />
y<br />
<br />
<br />
<br />
<br />
<br />
g(x) = 2 x + 3 f (x) = 2 x h(x) = 2 x − 3<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
y = 0<br />
x<br />
ftf x= x <br />
Th−∞∞<br />
Thy =<br />
y <br />
Figure 6<br />
◦ hift gx= x + y x + = x <br />
<br />
◦ hifthx= y( x − <br />
) =( x − <br />
) x <br />
<br />
<br />
shifts of the parent function f (x) = b x<br />
cdf x=b x +c +dftf x=b x<br />
dsamed<br />
coppositec<br />
y(b c +d)<br />
Thy =d<br />
Thd∞<br />
Th−∞∞
SECTION 6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 483<br />
How To…<br />
f x=b x +c +d<br />
1. y =d<br />
2. −cd f x=b x ftcccc<br />
3. f x=b x dddd<br />
4. −∞∞d∞y =d<br />
Example 2<br />
Graphing a Shift of an Exponential Function<br />
f x= x + −<br />
Solution f x=b x + c +db =c =d =−<br />
y =dy =−<br />
−cd −−<br />
f x=b x <br />
<br />
<br />
<br />
− <br />
<br />
– – – – – – –<br />
–<br />
−− –<br />
–<br />
–<br />
f x<br />
f (x) = 2 x + − 3<br />
<br />
<br />
x<br />
<br />
y = −3<br />
Figure 7<br />
Th−∞∞−∞y =−<br />
Try It #2<br />
f x= x − +<br />
How To…<br />
f x=b x +c +dx<br />
1. [Y=]Y 1<br />
=<br />
2. f xY 2<br />
=<br />
3. [WINDOW]yY 2<br />
=<br />
4. [GRAPH]fif x<br />
5. fix[2ND][CALC]<br />
[ENTER]. Thx<br />
Example 3<br />
Approximating the Solution of an Exponential Equation<br />
= x +<br />
Solution [Y=] x +Y 1<br />
=ThY 2<br />
=<br />
−x−y[GRAPH]. Thx =<br />
[2ND][CALC][5: intersect][ENTER]Th<br />
xffff<br />
ffGuess?x ≈
484<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Try It #3<br />
= x −<br />
Graphing a Stretch or Compression<br />
ft<br />
f x=b x ∣ a ∣ ><br />
f x= x a =gx= x <br />
Figure 8a = <br />
hx= <br />
x Figure 8<br />
– –<br />
<br />
y g(x) = 2) x<br />
f (x) = 2 x<br />
<br />
<br />
<br />
<br />
<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
y f (x) = 2 x<br />
hx= 2) x<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
<br />
– – – – – <br />
y = 0 –<br />
y = 0<br />
–<br />
<br />
<br />
<br />
Figure 8 (a) g(x) = 3(2) x stretches the graph of f (x) = 2 x vertically by a factor of 3.<br />
(b) h(x) =<br />
__ 1 3 (2)x compresses the graph of f (x) = 2 x vertically by a factor of<br />
__ 1 3 .<br />
stretches and compressions of the parent function f ( x ) = b x<br />
a >f x=ab x<br />
a∣ a ∣ ><br />
a∣ a ∣ <<br />
ya<br />
y =∞−∞∞<br />
<br />
Example 4<br />
Graphing the Stretch of an Exponential Function<br />
f x=( <br />
) x <br />
Solution <br />
b = <br />
x<br />
<br />
xx<br />
a =f x= ( <br />
) x <br />
Table 4<br />
x − − − <br />
f (x)=4( 1 <br />
2 ) x <br />
Table 4<br />
y−
SECTION 6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 485<br />
Figure 9<br />
fx<br />
y = 0<br />
<br />
– <br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
<br />
<br />
<br />
<br />
<br />
f x= x<br />
<br />
x<br />
<br />
–<br />
Figure 9<br />
Th−∞∞∞y =<br />
Try It #4<br />
f x= <br />
x <br />
Graphing Reflections<br />
ftflxy<br />
f x=b x −flx<br />
−flyf x= x <br />
flThflxgx=− x Figure<br />
10flyhx= −x Figure 10<br />
x<br />
y<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
f (x) = 2 x<br />
f (x) = 2 x<br />
h(x) = 2 −x <br />
<br />
<br />
y = 0 <br />
y = 0<br />
x<br />
x<br />
<br />
– – – – – <br />
–<br />
–<br />
–<br />
–<br />
g(x) = −2 x<br />
–<br />
Figure 10 (a) g(x) = −2 x reflects the graph of f (x) = 2 x about the x-axis. (b) g(x) = 2 −x reflects the graph of f (x) = 2 x about the y-axis.<br />
reflections of the parent function f (x) = b x<br />
Thf x=−b x<br />
f x=b x x<br />
y−<br />
−∞<br />
y =−∞∞<br />
Thf x=b −x<br />
f x=b x y<br />
yy =∞−∞∞
486<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 5<br />
Writing and Graphing the Reflection of an Exponential Function<br />
g xflf x=( <br />
) x x<br />
<br />
Solution flf x=( <br />
g x=− ( <br />
) x Table 5<br />
) x xf x−<br />
x − − − <br />
g(x)= −<br />
__<br />
( 4) 1 x − − − − − − −<br />
Table 5<br />
y−−−−<br />
<br />
gx<br />
<br />
<br />
<br />
<br />
y = 0 <br />
– – – – – –<br />
−− –<br />
–<br />
–<br />
–<br />
gx= − x<br />
<br />
−<br />
<br />
−<br />
x<br />
Figure 11<br />
Th−∞∞−∞y =<br />
Try It #5<br />
gx f x= x y<br />
<br />
Summarizing Translations of the Exponential Function<br />
Table<br />
6<br />
Translations of the Parent Function f (x) = b x<br />
Translation<br />
Form<br />
ft<br />
ce left<br />
fx=b x +c +d<br />
d<br />
<br />
| a |><br />
SECTION 6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 487<br />
translations of exponential functions<br />
<br />
f x=ab x +c +d<br />
y =b x b ><br />
hiftc<br />
∣ a ∣ ∣ a ∣ ><br />
∣ a ∣
488 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
6 .2 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
ALGEBRAIC<br />
3. Thf x= x y<br />
f 4. W<br />
gx<br />
y<br />
5. Thf x= x x<br />
hift<br />
e ngxy<br />
<br />
7. Thf x=( <br />
) − x hift<br />
hift <br />
x<br />
gx<br />
y<br />
2. <br />
<br />
<br />
4. Thf x= ( <br />
) −x <br />
y <br />
<br />
gx<br />
y<br />
6. Thf x= x hift<br />
<br />
xhift<br />
gx<br />
y<br />
<br />
GRAPHICAL<br />
fly<br />
y<br />
8. fx=( <br />
) x 9. gx=− x 10. hx= −x<br />
<br />
11. fx=( <br />
) x gx= x hx= x<br />
12. fx= <br />
x gx= x hx= x<br />
Figure 12<br />
B C D E<br />
A<br />
F<br />
Figure 12<br />
13. fx= x 14. fx= x 15. fx= x<br />
16. fx= x 17. fx= x 18. fx= x
SECTION 6.2 SECTION EXERCISES 489<br />
Figure 13fx=ab x <br />
y<br />
B C D<br />
A<br />
E<br />
F<br />
x<br />
Figure 13<br />
19. b 20. b<br />
21. a 22. a<br />
flx<br />
23. fx= <br />
x 24. fx=x − 25. fx=− x +<br />
f x= x <br />
<br />
26. fx= −x 27. hx= x + 28. fx= x−<br />
<br />
29. fx=− x − 30. fx=( <br />
) x −<br />
31. fx= −x +<br />
f x= x Th<br />
<br />
32. fx 33. fx 34. fxts left<br />
35. fx 36. fxx 37. fxy<br />
y = x <br />
38.<br />
y<br />
39.<br />
y<br />
40.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–
490 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
, fi<br />
41.<br />
y<br />
42.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
NUMERIC<br />
x<br />
43. gx= <br />
x− g 44. f x= x− −f 45. hx=− <br />
( <br />
) x +h−<br />
TECHNOLOGY<br />
<br />
f x=ab x +d<br />
46. −=− ( <br />
) −x<br />
49. =( <br />
) x− −<br />
47. = <br />
( <br />
) x<br />
50. −=− x+ +<br />
48. = x +<br />
EXTENSIONS<br />
51. fx=b x <br />
gx= ( <br />
b) x . Th<br />
<br />
b x ( <br />
b) x b><br />
53. fx= x <br />
gx= hx=( x− ) x . Th<br />
<br />
b ( x <br />
b ) b x n<br />
n<br />
b><br />
52. <br />
54.
SECTION 6.3 LOGARITHMIC FUNCTIONS 491<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Convert from logarithmic to exponential form.<br />
Convert from exponential to logarithmic form.<br />
Evaluate logarithms.<br />
Use common logarithms.<br />
Use natural logarithms.<br />
6.3 LOGARITHMIC FUNCTIONS<br />
Figure 1 Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)<br />
<br />
<br />
Figure 1<br />
Th<br />
Th <br />
f <br />
− = =<br />
<br />
Converting from Logarithmic to Exponential Form<br />
ff<br />
<br />
<br />
ffTh x =xff<br />
x<br />
<br />
ffi x = = =x<br />
2 y = x Figure 2<br />
y<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
<br />
<br />
<br />
y= x<br />
y= <br />
x<br />
Figure 2
492<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
fi<br />
Figure 2Thy =b x <br />
x =b y xy<br />
yfiyxy = b<br />
x<br />
Thblogarithmb<br />
Thbxyfibx<br />
s ybyx<br />
s = <br />
=<br />
<br />
<br />
b<br />
x=y ⇔b y =xb >b ≠<br />
b<br />
<br />
=<br />
b<br />
x=y<br />
<br />
Th<br />
by =x<br />
b<br />
x<br />
f x<br />
b<br />
x<br />
x<br />
<br />
=<br />
b<br />
c=a b a =c<br />
<br />
<br />
<br />
Th y = b<br />
x y =b x <br />
definition of the logarithmic function<br />
logarithm b x fifi<br />
x >b>b≠<br />
y= b<br />
xb y = x<br />
<br />
b<br />
x b x b x<br />
y b x<br />
x y <br />
n. Th<br />
b ∞<br />
b −∞∞<br />
Q & A…<br />
Can we take the logarithm of a negative number?<br />
<br />
<br />
<br />
How To…<br />
b<br />
x=y<br />
1. y = b<br />
xbyx<br />
2. b<br />
x= y b y =x
SECTION 6.3 LOGARITHMIC FUNCTIONS 493<br />
Example 1<br />
Converting from Logarithmic Form to Exponential Form<br />
<br />
a. <br />
√ — = <br />
b. <br />
=<br />
Solution byx. Thb y =x<br />
a. <br />
√ — = <br />
<br />
b. <br />
=<br />
b = y = <br />
x =√— Th <br />
√ — = <br />
<br />
=√ — <br />
b = y = x =9. Th <br />
= =<br />
Try It #1<br />
<br />
a. <br />
= b. <br />
=<br />
Converting from Exponential to Logarithmic Form<br />
bx<br />
y. Th x = b<br />
y<br />
Example 2<br />
Converting from Exponential Form to Logarithmic Form<br />
<br />
<br />
a. = b. = c. − =<br />
<br />
<br />
Solution byx. Th x = b<br />
y<br />
a. = b = x = y =8. Th = <br />
=<br />
b. =<br />
b = x = y =Th = <br />
=<br />
<br />
c. − = <br />
<br />
Try It #20<br />
<br />
<br />
b = x =− y =<br />
<br />
Th− =<br />
<br />
<br />
<br />
( <br />
<br />
) =−<br />
<br />
a. = b. = c. − = <br />
<br />
Evaluating Logarithms<br />
<br />
<br />
=<br />
<br />
=<br />
<br />
<br />
<br />
=49. Th <br />
=<br />
=27. Th <br />
=<br />
<br />
( <br />
) <br />
<br />
<br />
= =( <br />
) = <br />
<br />
Th ( <br />
) =
494<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
How To…<br />
y = b<br />
x<br />
1. x b b y =x<br />
2. b y b x<br />
Example 3<br />
Solving Logarithms Mentally<br />
y = <br />
<br />
Solution y =<br />
<br />
= <br />
=<br />
Try It #3<br />
y = <br />
<br />
Example 4<br />
Evaluating the Logarithm of a Reciprocal<br />
y = <br />
( <br />
) <br />
Solution y = <br />
<br />
<br />
<br />
= <br />
b−a = <br />
b a<br />
<br />
<br />
− =<br />
__ <br />
<br />
Th <br />
( <br />
) =−<br />
Try It #4<br />
y = <br />
( ) <br />
=__<br />
<br />
<br />
Using Common Logarithms<br />
<br />
x <br />
xcommon logarithm<br />
<br />
<br />
definition of the common logarithm<br />
common logarithm <br />
xxTh<br />
x fifi<br />
x ><br />
y=x y = x<br />
x xx<br />
Th y x
SECTION 6.3 LOGARITHMIC FUNCTIONS 495<br />
How To…<br />
y =x<br />
1. x y =x<br />
2. y <br />
x<br />
Example 5<br />
Finding the Value of a Common Logarithm Mentally<br />
y =<br />
Solution y =<br />
==<br />
Try It #5<br />
y =<br />
How To…<br />
y =x<br />
1. [LOG]<br />
2. x[ ) ]<br />
3. [ENTER]<br />
Example 6<br />
Finding the Value of a Common Logarithm Using a Calculator<br />
y =<br />
Solution<br />
[LOG]<br />
[ ) ]<br />
[ENTER]<br />
≈<br />
Analysis Note that = and that = . Since is between and , we know that log must be<br />
between log and log. This gives us the following:<br />
< < <br />
<br />
< < <br />
Try It #6<br />
y =<br />
Example 7<br />
Rewriting and Solving a Real-World Exponential Model<br />
Th<br />
Th x = x ff<br />
ff<br />
Solution <br />
<br />
x =<br />
= x
496<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
<br />
[LOG]<br />
[ ) ]<br />
[ENTER]<br />
≈<br />
Thff<br />
Try It #7<br />
Th<br />
Th x = x diff<br />
e diff<br />
Using Natural Logarithms<br />
Theefi<br />
natural logarithms. The e<br />
xx<br />
xTh<br />
=fi<br />
o fie<br />
definition of the natural logarithm<br />
natural logarithme e<br />
xx. Th<br />
x fifi<br />
x ><br />
y=xe y = x<br />
xe xx.<br />
Th y ex<br />
y =e y =xe x = x x e = x x ><br />
How To…<br />
y =x<br />
1. [LN]<br />
2. x[ ) ]<br />
3. [ENTER]<br />
Example 8<br />
Evaluating a Natural Logarithm Using a Calculator<br />
y =<br />
Solution<br />
[LN]<br />
[ ) ]<br />
[ENTER]<br />
≈<br />
Try It #8<br />
−<br />
<br />
Introduction to Logarithms (http://openstaxcollege.org/l/intrologarithms)
SECTION 6.3 SECTION EXERCISES 497<br />
6.3 SECTION EXERCISES<br />
VERBAL<br />
1. b <br />
<br />
b y = x b<br />
x = y b > b ≠<br />
3. b<br />
x = y <br />
x <br />
5. <br />
b<br />
n diff<br />
2. f x= b<br />
x<br />
gx=b x <br />
<br />
4. <br />
b<br />
n diff<br />
ALGEBRAIC<br />
<br />
6. <br />
q=m 7. <br />
b=c 8. <br />
y=x 9. x<br />
=y<br />
10. y<br />
x=− 11. <br />
a=b 12. y<br />
=x 13. <br />
=a<br />
14. v=t 15. w=n<br />
<br />
16. x =y 17. c d =k 18. m − =n 19. x =y<br />
20. x − <br />
=y 21. n = 22. ( <br />
24. a =b 25. e k =h<br />
) m =n<br />
23. y x = <br />
x <br />
26. <br />
x= 27. <br />
x=− 28. <br />
x= 29. <br />
x=<br />
30. <br />
x= 31. <br />
x= <br />
32. <br />
x= 33. <br />
x=−<br />
34. x= 35. x=<br />
fi<br />
36. 37. 38. 39. e <br />
40. e − 41. e +<br />
NUMERIC<br />
b <br />
42. <br />
( ) 43. √ — 44. <br />
( <br />
) +<br />
45. <br />
<br />
<br />
46. 47. 48. + 49. −
498 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
<br />
50. ( e ) 51. 52. e − − 53. ( e ) <br />
TECHNOLOGY<br />
<br />
54. 55. 56. ( <br />
) 57. √— 58. √ — <br />
EXTENSIONS<br />
59. x =f(x=x<br />
x =<br />
<br />
61. x x=<br />
<br />
63. e <br />
=<br />
<br />
60. f(x=fx=x<br />
x<br />
62. <br />
<br />
( =−<br />
<br />
) <br />
REAL-WORLD APPLICATIONS<br />
64. ThEI<br />
<br />
EI= <br />
( f ) <br />
t<br />
ft<br />
<br />
<br />
<br />
66. ThI<br />
<br />
I =M<br />
I <br />
−M <br />
<br />
<br />
M<br />
<br />
<br />
<br />
<br />
<br />
<br />
65. <br />
EI−
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 499<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Identify the domain of a logarithmic function.<br />
Graph logarithmic functions.<br />
6.4 GRAPHS OF LOGARITHMIC FUNCTIONS<br />
Graphs of Exponential Functions<br />
<br />
<br />
<br />
causeeffect<br />
ff<br />
tA =e t <br />
<br />
<br />
<br />
Figure 1<br />
Years<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Logarithmic Model Showing Years as a Function of the Balance in the Account<br />
<br />
<br />
<br />
<br />
<br />
Account balance<br />
Figure 1<br />
fi<br />
<br />
Finding the Domain of a Logarithmic Function<br />
<br />
fi<br />
fiy =b x xb >b ≠<br />
Thy−∞∞<br />
Thy∞<br />
y = b<br />
xy =b x <br />
<br />
Thy = b<br />
xy =b x ∞<br />
Thy = b<br />
xy =b x −∞∞
500<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
y = b<br />
x<br />
ftfl<br />
<br />
Graphs of Exponential Functionsrangey =b x <br />
y = b<br />
xdomainfi<br />
only of positive real numbers<br />
Th<br />
f x= <br />
x −Thfix<br />
x −o fix<br />
x −> <br />
x > <br />
x > <br />
f x= <br />
x −∞<br />
How To…<br />
<br />
1. <br />
2. x<br />
3. <br />
Example 1 Identifying the Domain of a Logarithmic Shift<br />
f x= <br />
x+<br />
Solution Thfifix +><br />
<br />
x +> Th<br />
x >− <br />
Thf x= <br />
x+−∞<br />
Try It #1<br />
f x= <br />
x−+<br />
Example 2<br />
Identifying the Domain of a Logarithmic Shift and Reflection<br />
f x=−x<br />
Solution Thfifi<br />
−x ><br />
<br />
−x > Th<br />
−x >− <br />
x
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 501<br />
Try It #2<br />
f x=x−+<br />
Graphing Logarithmic Functions<br />
fi<br />
Thy = b<br />
x<br />
ftfl<br />
y = b<br />
x<br />
y =b x fly =x<br />
y = x <br />
x = <br />
yTable 1<br />
x − − − <br />
2 x = y <br />
<br />
<br />
<br />
<br />
<br />
log 2<br />
(y) = x − − − <br />
Table 1<br />
Table 1<br />
f x= x gx= <br />
xTable 2<br />
f (x) = 2 x<br />
( − <br />
) <br />
( − <br />
) <br />
( − <br />
) <br />
g(x) = log 2<br />
(x)<br />
( <br />
− ) <br />
( <br />
− ) <br />
( <br />
− ) <br />
Table 2<br />
xyFigure 2fg<br />
f x= x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
y = x<br />
gx= x<br />
<br />
x<br />
Figure 2 Notice that the graphs of f (x) = 2 x and g(x) = log 2<br />
(x) are reflections about the line y = x.<br />
<br />
fx= x ygx= <br />
xx<br />
Thf x= x −∞∞gx= <br />
x<br />
Thf x= x ∞gx= <br />
x
502<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
characteristics of the graph of the parent function, f (x) = log b<br />
(x)<br />
xb >b ≠e t<br />
f x= b<br />
x<br />
<br />
f(x)<br />
f(x)<br />
x =<br />
∞<br />
−∞∞<br />
x<br />
b<br />
ye<br />
b ><br />
<br />
Figure 4bf x= b<br />
xff<br />
<br />
Notex<br />
e ≈<br />
x<br />
Figure 3<br />
x = <br />
–––– –<br />
–<br />
b<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
x = <br />
f x= b x<br />
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 503<br />
f x<br />
–– – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x= <br />
<br />
<br />
<br />
f x= x<br />
x<br />
Figure 5<br />
Th∞−∞∞x =<br />
Try It #3<br />
f x= x<br />
<br />
Graphing Transformations of Logarithmic Functions<br />
<br />
ftfly = b<br />
x<br />
Graphing a Horizontal Shift of f (x ) = log b<br />
(x )<br />
cf x= b<br />
x c<br />
oppositecft<br />
f x= b<br />
xc >ftftgx = b<br />
x+cfthx= b<br />
x−c<br />
Figure 6<br />
Shift left<br />
g (x) = log b<br />
(x + c)<br />
y<br />
x= –c<br />
x= gx= b x + c<br />
x= <br />
y<br />
Shift right<br />
h(x) = log b<br />
(x − c)<br />
x= c<br />
f x= b x<br />
b − c<br />
− c<br />
<br />
b<br />
f x= b x<br />
x<br />
b<br />
<br />
b + c<br />
+ c<br />
hx= b x − c<br />
x<br />
Thx =−c.<br />
−c, ∞<br />
−∞∞<br />
Figure 6<br />
Thx =c.<br />
c, ∞<br />
−∞∞<br />
horizontal shifts of the parent function y = log b<br />
x<br />
cf x= b<br />
x+c<br />
y = b<br />
x cc ><br />
y = b<br />
xcc <<br />
x =−c<br />
−c∞<br />
−∞∞
504<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
How To…<br />
f x= b<br />
x+c<br />
1. ft<br />
a. c > f x= b<br />
x c<br />
b. c < f x= b<br />
xc<br />
2. x =−c<br />
3. ft<br />
cx<br />
4. <br />
5. Th−c∞−∞∞x =−c<br />
Example 4 Graphing a Horizontal Shift of the Parent Function y = log b<br />
( x)<br />
f x= <br />
x−<br />
<br />
Solution f x= <br />
x−x +−=x −<br />
c =−c
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 505<br />
Shift up<br />
g (x) = log b<br />
(x) + d<br />
Shift down<br />
h(x) = log b<br />
(x) − d<br />
y<br />
y<br />
x= <br />
gx= b x+ d<br />
x= <br />
f x= b x<br />
b − d <br />
b −d <br />
b<br />
<br />
f x= b x<br />
x<br />
b<br />
<br />
b +d <br />
b d <br />
hx= b x− d<br />
x<br />
Thx =<br />
∞<br />
−∞∞<br />
Thx =<br />
∞<br />
−∞∞<br />
Figure 8<br />
vertical shifts of the parent function y = log b<br />
(x)<br />
df x= b<br />
x+d<br />
y = b<br />
xdd ><br />
y = b<br />
xdd <<br />
x =<br />
∞<br />
−∞∞<br />
How To…<br />
f x= b<br />
x+d<br />
1. ft<br />
a. d > f x= b<br />
xd<br />
b. d < f x= b<br />
xd<br />
2. x =<br />
3. ftd<br />
y<br />
4. <br />
5. Th∞−∞∞x =<br />
Example 5<br />
Graphing a Vertical Shift of the Parent Function y = log b<br />
(x)<br />
f x= <br />
x−<br />
<br />
Solution f x= <br />
x−d =−. Thd <<br />
Th f x= <br />
x<br />
Thx =
506<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
( <br />
− ) <br />
Thy<br />
( <br />
− ) −−<br />
Th∞−∞∞x =<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– –<br />
–<br />
–<br />
–<br />
–<br />
<br />
y= <br />
x<br />
<br />
x<br />
<br />
− f x= <br />
x − <br />
−<br />
x= <br />
Figure 9<br />
Th∞−∞∞x =<br />
Try It #5<br />
f x= <br />
x+<br />
<br />
Graphing Stretches and Compressions of y = log b<br />
(x )<br />
f x= b<br />
xa ><br />
a ><br />
f x= b<br />
xg x=a b<br />
xhx= a x b<br />
Figure 10<br />
Vertical Stretch<br />
g (x) = alog b<br />
(x), a > 1<br />
y<br />
gx= a b x<br />
x= <br />
y<br />
Vertical Compression<br />
h(x) = <br />
_<br />
a 1 log (x), a > 1<br />
b<br />
x= <br />
a<br />
b <br />
b<br />
<br />
fx= b x<br />
x<br />
b<br />
<br />
fx= b x<br />
<br />
hx= a b x<br />
b a <br />
x<br />
Thx =.<br />
x<br />
∞<br />
−∞∞.<br />
Thx =.<br />
x<br />
∞<br />
−∞∞<br />
Figure 10
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 507<br />
vertical stretches and compressions of the parent function y = log b<br />
(x)<br />
a ><br />
f x = alog b<br />
x<br />
y = b<br />
xaa ><br />
y = b<br />
xa<br />
1. <br />
a. ∣ a ∣ >f x= b<br />
xa<br />
b. ∣ a ∣
508<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Try It #6<br />
f x= <br />
x<br />
<br />
<br />
Example 7 Combining a Shift and a Stretch<br />
f x=x+<br />
Solution fi <br />
Figure 12Thftx =−Thx<br />
−Th−∞−, <br />
x =xx =x +=<br />
y<br />
–– – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x= −<br />
Figure 12<br />
<br />
y= x + <br />
y= x + <br />
x<br />
y= x<br />
Th−∞−∞∞x =−<br />
Try It #7<br />
f x=x−+<br />
Graphing Reflections of f (x ) = logb(x )<br />
f x= b<br />
x−flxinput<br />
−flt yflb ><br />
e f x= b<br />
xflxgx=− b<br />
x<br />
flyhx= b<br />
−x<br />
Reflection about the x-axis<br />
g (x) = log b<br />
(x), b > 1<br />
Reflection about the y-axis<br />
h(x) = log b<br />
(−x), b > 1<br />
y<br />
y<br />
x= <br />
x= <br />
b − <br />
<br />
b<br />
f x= b x<br />
gx= − b x<br />
x<br />
hx= b −x<br />
−b<br />
−<br />
f x= b x<br />
<br />
b<br />
x<br />
x<br />
<br />
Thx =.<br />
x<br />
Thb − , <br />
∞<br />
−∞∞<br />
Figure 13<br />
x<br />
<br />
Thx =.<br />
x−<br />
Th−b<br />
−∞<br />
−∞∞
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 509<br />
reflections of the parent function y = log b<br />
x<br />
Thf x=− b<br />
x<br />
y = b<br />
xx<br />
∞−∞∞x =<br />
<br />
Thf x= b<br />
−x<br />
y = b<br />
xy<br />
−∞<br />
−∞∞x =<br />
How To…<br />
f x= b<br />
x<br />
Iff (x)=−log b<br />
(x)<br />
x =<br />
x<br />
flf x= b<br />
x<br />
x<br />
<br />
∞−∞∞<br />
x =<br />
Table 3<br />
If f (x)=log b<br />
(−x)<br />
x =<br />
x<br />
flf x= b<br />
x<br />
y<br />
<br />
−∞−∞∞<br />
x =<br />
Example 8<br />
Graphing a Reflection of a Logarithmic Function<br />
f x=−x<br />
<br />
Solution f x=−x<br />
b =input<br />
−f yThf x=−xx<br />
x =<br />
x−<br />
<br />
y<br />
x= <br />
–<br />
f x= −x<br />
y= x<br />
<br />
–<br />
<br />
x<br />
Figure 14<br />
Th−∞−∞∞x =<br />
Try It #8<br />
f x=−−x
510<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
How To…<br />
<br />
1. [Y=]Y <br />
=Y <br />
=<br />
2. [GRAPH][WINDOW]fi<br />
s<br />
3. fix[2ND][CALC]<br />
[ENTER]. Thx<br />
Example 9<br />
Approximating the Solution of a Logarithmic Equation<br />
x+=−x−<br />
Solution [Y=]+Y 1<br />
=Th−x−Y 2<br />
=<br />
x–yGRAPHThx =<br />
[2ND][CALC]5: intersect[ENTER]Th<br />
xffff<br />
ffGuess?x ≈<br />
Try It #9<br />
x+=−x<br />
Summarizing Translations of the Logarithmic Function<br />
Table<br />
4<br />
Translations of the Parent Functiony =log b<br />
(x)<br />
Translation<br />
Form<br />
ft<br />
ce left y= b<br />
x+c+d<br />
d<br />
<br />
∣ a ∣ ><br />
∣ a ∣ <<br />
flx<br />
fly<br />
<br />
Table 4<br />
y=a b<br />
x<br />
y=− b<br />
x<br />
y= b<br />
−x<br />
y=a b<br />
x+c+d<br />
translations of logarithmic functions<br />
y = b<br />
x<br />
fx=a b<br />
x+c+d<br />
y = b<br />
xb ><br />
hiftd<br />
hift c<br />
∣ a ∣ ∣ a ∣ ><br />
∣ a ∣
SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 511<br />
Example 10<br />
Finding the Vertical Asymptote of a Logarithm Graph<br />
f x=− <br />
x++<br />
Solution Thx =−<br />
Analysis The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve units<br />
to the left shifts the vertical asymptote to x = −.<br />
Try It #10<br />
f x=+x−<br />
Example 11<br />
Finding the Equation from a Graph<br />
Figure 15<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 15<br />
Solution Thx =−fl<br />
<br />
fx=−ax++k<br />
−−−<br />
<br />
=−a−++k −<br />
=−a+k <br />
=k =<br />
<br />
−=−a++ −<br />
−=−a <br />
<br />
a =_____<br />
<br />
<br />
a<br />
<br />
Thf x=−<br />
<br />
x++<br />
Analysis You can verify this answer by comparing the function values in Table 5 with the points on the graph in<br />
Figure 15.<br />
x − <br />
f (x) − − −<br />
x <br />
f (x) − − − − −<br />
Table 5
512<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Try It #11<br />
Figure 16<br />
fx<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
Figure 16<br />
Q & A…<br />
Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<br />
Figure 16Th<br />
x =−x =−<br />
x|x >−Th<br />
s <br />
. Thx → − + f x→−∞x →∞f x→∞<br />
<br />
<br />
Graph an Exponential Function and Logarithmic Function (http://openstaxcollege.org/l/graphexplog)<br />
Match Graphs with Exponential and Logarithmic Functions (http://openstaxcollege.org/l/matchexplog)<br />
Find the Domain of Logarithmic Functions (http://openstaxcollege.org/l/domainlog)
SECTION 6.4 SECTION EXERCISES 513<br />
6.4 SECTION EXERCISES<br />
VERBAL<br />
1. Th<br />
<br />
<br />
<br />
3. ff<br />
<br />
2. ff<br />
<br />
4. <br />
f (x) = b<br />
x. x<br />
5. <br />
<br />
ALGEBRAIC<br />
<br />
6. f x= <br />
x+ 7. hx=( <br />
−x )<br />
9. hx=x+− 10. fx= <br />
−x−<br />
8. gx= <br />
x+−<br />
<br />
11. fx= b<br />
x− 12. gx=−x 13. fx=x+<br />
14. fx=−x+ 15. gx=−x+−<br />
<br />
16. fx=−x 17. fx=( x− <br />
) <br />
19. gx=x+− 20. fx= <br />
−x+<br />
18. hx= −x−+ <br />
xy<br />
<br />
21. hx= <br />
x−+ 22. fx=x++ 23. gx=−x−<br />
24. fx= <br />
x+− 25. hx=x−<br />
GRAPHICAL<br />
Figure 17<br />
<br />
y<br />
A<br />
dx=x<br />
<br />
<br />
<br />
<br />
<br />
B<br />
C<br />
D<br />
E<br />
x<br />
fx=x<br />
gx= <br />
x<br />
–<br />
hx= <br />
x<br />
–<br />
jx= <br />
x<br />
Figure 17
514 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Figure 18<br />
–5 –4 –3 –2<br />
–1<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
–1<br />
–2<br />
–3<br />
–4<br />
–5<br />
–6<br />
y<br />
A<br />
1 2 3 4 5<br />
B<br />
C<br />
x<br />
fx= x<br />
<br />
gx= <br />
x<br />
hx= x<br />
<br />
<br />
<br />
<br />
<br />
Figure 18<br />
<br />
3. fx=xgx= x<br />
3. fx= <br />
xgx=x<br />
3. fx=xgx=<br />
<br />
<br />
<br />
<br />
3. fx=e x gx=x<br />
Figure 19<br />
y<br />
<br />
<br />
B<br />
–5 –4 –3 –2<br />
–1<br />
5<br />
4<br />
3<br />
2<br />
1<br />
–1<br />
–2<br />
–3<br />
–4<br />
–5<br />
A<br />
1 2 3 4 5<br />
C<br />
x<br />
Figure 19<br />
. f x= <br />
−x+ . gx=− <br />
x+ 4. hx= <br />
x+<br />
<br />
4. fx= <br />
x+ 4. fx=x 4. f x=−x<br />
4. gx=x++4. gx=−x+ 4. hx=− <br />
x+−<br />
<br />
4. y= <br />
x<br />
y<br />
. fx= <br />
x<br />
y<br />
–5 –4 –3 –2<br />
–1<br />
5<br />
4<br />
3<br />
2<br />
1<br />
–1<br />
–2<br />
–3<br />
–4<br />
–5<br />
1 2 3 4 5<br />
x<br />
–5 –4 –3 –2<br />
–1<br />
5<br />
4<br />
3<br />
2<br />
1<br />
–1<br />
–2<br />
–3<br />
–4<br />
–5<br />
1 2 3 4 5<br />
x
SECTION 6.4 SECTION EXERCISES 515<br />
. fx= <br />
x<br />
y<br />
5. fx= <br />
x<br />
y<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
– – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
TECHNOLOGY<br />
o fi<br />
5. x−+=x−+<br />
5. x−+=−x−+ 5. x−=−x+<br />
5. x+= <br />
−x+ 5.<br />
<br />
−x=x++ <br />
EXTENSIONS<br />
5. bb≠<br />
b<br />
<br />
. <br />
5. fx=<br />
x<br />
<br />
gx) = − <br />
x<br />
<br />
. <br />
fx=( x+ <br />
x− ) <br />
6. x<br />
fx=x +x+
516<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Use the product rule for logarithms.<br />
Use the quotient rule for logarithms.<br />
Use the power rule for logarithms.<br />
Expand logarithmic expressions.<br />
Condense logarithmic expressions.<br />
Use the change-of-base formula for logarithms.<br />
6.5 LOGARITHMIC PROPERTIES<br />
Figure 1 The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)<br />
Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
fi<br />
Thfia<br />
<br />
<br />
=−H + <br />
<br />
=( _____ <br />
H + ) <br />
Th−H + <br />
( <br />
<br />
H + ) <br />
Using the Product Rule for Logarithms<br />
Th<br />
<br />
<br />
<br />
b<br />
=<br />
<br />
b<br />
b=<br />
<br />
= = <br />
= =
SECTION 6.5 LOGARITHMIC PROPERTIES 517<br />
<br />
<br />
b<br />
b x =x<br />
b x b =xx ><br />
<br />
b<br />
<br />
b x =x <br />
=<br />
e e e b b x =xe e =<br />
<br />
<br />
b<br />
M= b<br />
NM =N<br />
<br />
x= <br />
x +x<br />
x<br />
x =x + <br />
x = x<br />
<br />
x+ <br />
x +=Th<br />
<br />
product rule of exponentsx a x b =x a +b <br />
product rule for logarithms<br />
<br />
<br />
xMNbb ≠<br />
<br />
b<br />
MN= b<br />
M+ b<br />
N<br />
m = b<br />
Mn = b<br />
Nb m =Mb n =N<br />
b<br />
MN= b<br />
b m b n MN<br />
= b<br />
b m +n <br />
=m +n <br />
= b<br />
M+ b<br />
N mn<br />
<br />
b<br />
wxyz<br />
<br />
b<br />
wxyz= b<br />
w+ b<br />
x+ b<br />
y+ b<br />
z<br />
the product rule for logarithms<br />
product rule for logarithms<br />
<br />
b<br />
MN= b<br />
M+ b<br />
Nb ><br />
How To…<br />
<br />
1. <br />
2.
518<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 1<br />
<br />
xx +<br />
Using the Product Rule for Logarithms<br />
Solution <br />
<br />
<br />
xx += <br />
xx +<br />
<br />
<br />
<br />
xx += <br />
+ <br />
+ <br />
+ <br />
x+ <br />
x +<br />
Try It #1<br />
b<br />
k<br />
Using the Quotient Rule for Logarithms<br />
quotient rule of exponents<br />
<br />
x a <br />
b =x a−b Thquotient rule for logarithms<br />
ff<br />
<br />
xMNbb ≠<br />
<br />
b<br />
( M <br />
N ) = b M− b N<br />
m = b<br />
Mn = b<br />
Nb m =Mb n =N<br />
<br />
b<br />
( M <br />
N ) = b ( bm<br />
b )<br />
n<br />
= b<br />
b m −n <br />
=m −n<br />
MN<br />
<br />
<br />
= b<br />
M− b<br />
N mn<br />
( _<br />
x +x<br />
x + ) fi<br />
<br />
<br />
( _______<br />
x +x<br />
x + )=( xx+ ________<br />
x+ ) <br />
=( x __<br />
) <br />
<br />
<br />
<br />
Th<br />
<br />
( x __<br />
) =x−<br />
=+x−<br />
the quotient rule for logarithms<br />
quotient rule for logarithms ff<br />
<br />
b<br />
( M <br />
N ) = b M− b N<br />
How To…<br />
ff<br />
1. <br />
2. <br />
<br />
3.
SECTION 6.5 LOGARITHMIC PROPERTIES 519<br />
Example 2<br />
Using the Quotient Rule for Logarithms<br />
<br />
( xx−<br />
<br />
x +−x )<br />
<br />
Solution <br />
xx−<br />
<br />
(<br />
<br />
<br />
x +−x )<br />
= <br />
xx−− <br />
x +−x<br />
<br />
<br />
<br />
xx−− <br />
x +−x= <br />
+ <br />
+ <br />
x+ <br />
x−− <br />
x ++ <br />
−x<br />
= <br />
+ <br />
+ <br />
x+ <br />
x−− <br />
x +− <br />
−x<br />
Analysis There are exceptions to consider in this and later examples. First, because denominators must never be zero,<br />
this expression is not defined for x = −<br />
_ and x = . Also, since the argument of a logarithm must be positive, we note<br />
<br />
as we observe the expanded logarithm, that x > , x > , x > −<br />
_ , and x < . Combining these conditions is beyond the<br />
<br />
scope of this section, and we will not consider them here or in subsequent exercises.<br />
Try It #2<br />
<br />
(<br />
x<br />
<br />
+x<br />
<br />
xx−x− ) <br />
<br />
Using the Power Rule for Logarithms<br />
x <br />
<br />
<br />
b<br />
x = b<br />
x⋅x<br />
= b<br />
x+ b<br />
x<br />
=<br />
b x<br />
fi<br />
power rule for logarithms<br />
<br />
<br />
= √ — = _<br />
e<br />
=e−<br />
the power rule for logarithms<br />
power rule for logarithms<br />
<br />
b<br />
M n =n b<br />
M<br />
How To…<br />
<br />
<br />
1. <br />
2.
520<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 3<br />
<br />
x <br />
Expanding a Logarithm with Powers<br />
Solution Thx<br />
<br />
<br />
<br />
x = <br />
x<br />
Try It #3<br />
x <br />
Example 4<br />
Rewriting an Expression as a Power before Using the Power Rule<br />
<br />
<br />
Solution <br />
= <br />
<br />
<br />
<br />
<br />
<br />
= <br />
<br />
Try It #4<br />
( <br />
x ) <br />
Example 5<br />
Using the Power Rule in Reverse<br />
xffi<br />
Solution <br />
x<br />
x<br />
x=x <br />
Try It #5<br />
<br />
<br />
<br />
Expanding Logarithmic Expressions<br />
ft<br />
<br />
<br />
b ( x _<br />
y ) = b x− b y<br />
= b<br />
+ b<br />
x− b<br />
y<br />
<br />
<br />
<br />
b<br />
( A <br />
C ) = b AC − <br />
= b<br />
A+ b<br />
C − <br />
= b<br />
A+− b<br />
C<br />
= b<br />
A− b<br />
C<br />
ff<br />
fi
SECTION 6.5 LOGARITHMIC PROPERTIES 521<br />
Example 6 Expanding Logarithms Using Product, Quotient, and Power Rules<br />
( x y<br />
_<br />
)ff<br />
Solution t r<br />
<br />
( x y<br />
<br />
)=x y−<br />
The fi<br />
<br />
x y−=x +y−<br />
e fi<br />
<br />
x +y−=x+y−<br />
Try It #6<br />
( x y <br />
z )<br />
Example 7<br />
√ — x <br />
Solution<br />
<br />
Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression<br />
√ — x =x <br />
Try It #7<br />
√ <br />
— x <br />
=<br />
__ x<br />
Q & A…<br />
Can we expand ln(x 2 + y 2 )?<br />
. Thff<br />
Example 8<br />
Expanding Complex Logarithmic Expressions<br />
<br />
( x x + x − ) <br />
Solution <br />
Try It #8<br />
<br />
( x x + x − ) = <br />
+ <br />
x + <br />
x +− <br />
x − <br />
= <br />
+ <br />
x + <br />
x +− <br />
x − <br />
(<br />
x−x +<br />
√— <br />
x − ) <br />
<br />
Condensing Logarithmic Expressions<br />
= <br />
+ <br />
x+ <br />
x +− <br />
x −<br />
<br />
ff
522<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
How To…<br />
ff<br />
<br />
1. fi<br />
<br />
2. <br />
3. ff<br />
Example 9<br />
Using the Product and Quotient Rules to Combine Logarithms<br />
<br />
+ <br />
− <br />
<br />
Solution <br />
<br />
+ <br />
= <br />
= <br />
<br />
Th<br />
<br />
− <br />
<br />
Th<br />
Try It #9<br />
−+−<br />
<br />
− <br />
= <br />
( __<br />
<br />
) = <br />
<br />
Example 10<br />
Condensing Complex Logarithmic Expressions<br />
<br />
x + <br />
x−− x+ <br />
Solution e fi<br />
<br />
<br />
x +<br />
__ x−− x+ = <br />
x + <br />
√ — x −− <br />
x+ <br />
<br />
<br />
x + <br />
√ — x −− <br />
x+ = <br />
x √ — x −− <br />
x+ <br />
ff<br />
<br />
<br />
x √ — x −− <br />
x+ = <br />
( x √ — ________ x − <br />
x+ )<br />
Try It #10<br />
+x−x −+x−<br />
Example 11<br />
Rewriting as a Single Logarithm<br />
x−x++ x x +<br />
Solution e fi<br />
<br />
x−x++ x x +=x −x+ +( x + x −1 )<br />
<br />
x −x+ +( x + x −1 ) =x −( x+ x + x −1 )<br />
ff<br />
<br />
x −( x+ x + x −1 ) =<br />
( <br />
x <br />
x+ x + x −1 )
SECTION 6.5 LOGARITHMIC PROPERTIES 523<br />
Try It #11<br />
x+x+−x +<br />
Example 12<br />
Applying of the Laws of Logs<br />
=−H + <br />
ff<br />
Solution CPTh<br />
P =−CC. Th<br />
<br />
=−C<br />
<br />
=−C=−+C=−−C<br />
P =−C<br />
<br />
=P −≈P −<br />
<br />
Try It #12<br />
<br />
Using the Change-of-Base Formula for Logarithms<br />
<br />
echange-of-base formula<br />
<br />
power rule for logarithms<br />
Mbnn ≠b ≠<br />
<br />
b<br />
M= M n n<br />
b <br />
y = b<br />
Mnb y =M<br />
<br />
n<br />
b y = n<br />
M <br />
<br />
y n<br />
b= n<br />
M<br />
y = n M <br />
n<br />
b <br />
b<br />
M= n M <br />
n<br />
b <br />
<br />
y<br />
y<br />
5<br />
fi<br />
<br />
<br />
5<br />
= <br />
≈
524<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
the change-of-base formula<br />
change-of-base formula<br />
Mbnn ≠b ≠<br />
b<br />
M= M n<br />
<br />
n<br />
b <br />
<br />
<br />
b<br />
M= M _<br />
b<br />
<br />
b<br />
M= n M <br />
n<br />
b<br />
How To…<br />
b<br />
M<br />
nn ≠<br />
1. nxx<br />
e<br />
2. <br />
a. ThnM<br />
b. Thnb<br />
Example 13<br />
Changing Logarithmic Expressions to Expressions Involving Only Natural Logs<br />
<br />
<br />
Solution <br />
n =e<br />
Th<br />
. Th<br />
b<br />
M= M _<br />
b<br />
Try It #13<br />
<br />
<br />
Q & A…<br />
Can we change common logarithms to natural logarithms?<br />
<br />
=<br />
_<br />
<br />
Example 14<br />
Using the Change-of-Base Formula with a Calculator<br />
<br />
<br />
<br />
= <br />
<br />
Solution <br />
e<br />
Try It #14<br />
<br />
= e<br />
≈ <br />
<br />
<br />
<br />
The Properties of Logarithms (http://openstaxcollege.org/l/proplog)<br />
Expand Logarithmic Expressions (http://openstaxcollege.org/l/expandlog)<br />
Evaluate a Natural Logarithmic Expression (http://openstaxcollege.org/l/evaluatelog)
SECTION 6.5 SECTION EXERCISES 525<br />
6.5 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
n √ — x <br />
2. <br />
<br />
ALGEBRAIC<br />
ff<br />
<br />
3. b<br />
xy 4. abc 5. b ( <br />
) <br />
6. (<br />
x <br />
<br />
w z )<br />
7. ( <br />
) <br />
8. <br />
y x <br />
<br />
9. +x+y 10. <br />
+ <br />
a+ <br />
+ <br />
b 11. b<br />
− b<br />
<br />
12. a−d−c 13. − b ( <br />
) <br />
14. <br />
<br />
<br />
ff<br />
15. ( x y z )<br />
16. ( a− <br />
y<br />
<br />
) −y<br />
b − c ) 17. √— x y − 18. ( y √ <br />
19. x y √ <br />
— x y <br />
<br />
20. x +x 21. x −x 22. x+x+<br />
23. x− <br />
y+z 24. <br />
c+ a <br />
+ b <br />
<br />
<br />
25. <br />
e 26. <br />
<br />
<br />
=a <br />
=b<br />
ab<br />
27. <br />
28. <br />
29. <br />
( ) <br />
NUMERIC<br />
<br />
30. <br />
( <br />
) − <br />
<br />
31. <br />
+ <br />
<br />
<br />
32. <br />
− <br />
+ <br />
( <br />
) <br />
<br />
o fi<br />
33. <br />
34. <br />
35. <br />
36. <br />
( <br />
) <br />
EXTENSIONS<br />
38. x<br />
<br />
x++ <br />
x+=<br />
<br />
40. <br />
<br />
b x =m<br />
<br />
42. <br />
= <br />
<br />
<br />
37. <br />
<br />
39. <br />
<br />
x+− <br />
x−=<br />
<br />
<br />
41. b<br />
n=<br />
<br />
n<br />
b <br />
b>n>
526<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Use like bases to solve exponential equations.<br />
Use logarithms to solve exponential equations.<br />
Use the definition of a logarithm to solve logarithmic equations.<br />
Use the one-to-one property of logarithms to solve logarithmic equations.<br />
Solve applied problems involving exponential and logarithmic equations.<br />
6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS<br />
Figure 1 Wild rabbits in Australia. The rabbit population grew so quickly in Australia that the event became known as the “rabbit plague.” (credit: Richard Taylor, Flickr)<br />
Th<br />
<br />
<br />
<br />
<br />
<br />
Using Like Bases to Solve Exponential Equations<br />
Thfi<br />
bSTb >b ≠b S =b T S =T<br />
Th<br />
Th<br />
Th<br />
<br />
x − = x<br />
<br />
, Th<br />
x<br />
<br />
x − =___<br />
x<br />
<br />
<br />
x − =___<br />
x<br />
<br />
x − = x − <br />
x −=x − <br />
x = x<br />
x =
SECTION 6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS 527<br />
using the one-to-one property of exponential functions to solve exponential equations<br />
STb ≠<br />
b S =b T S =<br />
How To…<br />
b S =b T ST<br />
<br />
1. b S =b T <br />
2. <br />
3. S =T<br />
Example 1<br />
x − = x − <br />
Solving an Exponential Equation with a Common Base<br />
Solution x − = x − Th<br />
x −=x − <br />
x =<br />
x<br />
Try It #1<br />
x = x + <br />
Rewriting Equations So All Powers Have the Same Base<br />
<br />
<br />
= x − Th<br />
x<br />
<br />
= x −<br />
= x− <br />
= x − <br />
=x − <br />
=x <br />
x =<br />
<br />
How To…<br />
<br />
1. <br />
2. b S =b T <br />
3. <br />
4. S =T<br />
Example 2<br />
x + = x + <br />
Solving Equations by Rewriting Them to Have a Common Base<br />
Solution<br />
x + = x +<br />
x+ = x+ <br />
x + = x + <br />
x +=x + <br />
x =<br />
x
528<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Try It #2<br />
x = x + <br />
Example 3<br />
x =√ — <br />
Solution<br />
Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base<br />
x = <br />
<br />
<br />
<br />
Try It #3<br />
x =√ — <br />
x =__<br />
<br />
x =__<br />
<br />
<br />
<br />
x<br />
Q & A…<br />
Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problemsolving<br />
process?<br />
<br />
fi<br />
Example 4<br />
x + =−<br />
Solving an Equation with Positive and Negative Powers<br />
Solution ThTh x<br />
<br />
Analysis Figure 2 shows that the two graphs do not cross so the left side is never equal to the right side. Thus the<br />
equation has no solution.<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
y<br />
y = x + <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
y = −<br />
Figure 2<br />
Try It #4<br />
x =−<br />
Solving Exponential Equations Using Logarithms<br />
<br />
a=ba =b
SECTION 6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS 529<br />
How To…<br />
<br />
1. <br />
a.<br />
b.<br />
2. <br />
Example 5<br />
x + = x <br />
Solving an Equation Containing Powers of Different Bases<br />
Solution x + = x Th<br />
x + = x <br />
x+=x <br />
x+=x <br />
x−x=− xx<br />
x−=− x<br />
x( __<br />
<br />
__<br />
) <br />
<br />
Try It #5<br />
x = x + <br />
Q & A…<br />
Is there any way to solve 2 x = 3 x ?<br />
. Th<br />
Equations Containing e<br />
) = ( <br />
( __ <br />
) <br />
x = <br />
_<br />
<br />
x<br />
( __<br />
) <br />
eTh<br />
fi e <br />
<br />
How To…<br />
y =Ae kt t<br />
1. A<br />
2. <br />
3. k<br />
Example 6<br />
=e t <br />
Solution<br />
Solve an Equation of the Form y = Ae k t<br />
=e t<br />
=e t <br />
=t xe x <br />
<br />
t =___<br />
t
530<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Analysis Using laws of logs, we can also write this answer in the form t = ln√ — . If we want a decimal approximation<br />
of the answer, we use a calculator.<br />
Try It #6<br />
e t =<br />
Q & A…<br />
Does every equation of the form y = Ae kt have a solution?<br />
Thk ≠yA<br />
=−e t <br />
Example 7<br />
e x +=<br />
Solving an Equation That Can Be Simplified to the Form y = Ae k t<br />
Solution<br />
e x +=<br />
e x = <br />
e x =__<br />
<br />
<br />
<br />
<br />
x =( __<br />
) <br />
x =__<br />
( __<br />
) <br />
x<br />
Try It #7<br />
+ t =e t <br />
Extraneous Solutions<br />
extraneous solution<br />
<br />
<br />
<br />
Example 8<br />
x −e x =<br />
Solution<br />
Solving Exponential Functions in Quadratic Form<br />
e x − x =<br />
e x −e x −= <br />
e x +e x −= <br />
e x +=e x −= <br />
e x =−e x = <br />
e x = <br />
x = <br />
Analysis When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because<br />
zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation<br />
e x = − because a positive number never equals a negative number. The solution ln(−) is not a real number, and in the<br />
real number system this solution is rejected as an extraneous solution.
SECTION 6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS 531<br />
Try It #8<br />
e x =e x +<br />
Q & A…<br />
Does every logarithmic equation have a solution?<br />
<br />
Using the Definition of a Logarithm to Solve Logarithmic Equations<br />
b<br />
x=yb y =x<br />
<br />
<br />
<br />
+ <br />
x −=<br />
fix<br />
<br />
<br />
+ <br />
x −=<br />
<br />
x −= <br />
<br />
x −= <br />
=x − <br />
=x − <br />
=x <br />
x = <br />
using the definition of a logarithm to solve logarithmic equations<br />
Sb cb >b ≠<br />
b<br />
S=cb c =S<br />
Example 9 Using <strong>Algebra</strong> to Solve a Logarithmic Equation<br />
x+=<br />
Solution<br />
x+=<br />
x= <br />
x= <br />
x =e <br />
Try It #9<br />
+x=<br />
Example 10<br />
x=<br />
Solution<br />
<br />
<br />
Using <strong>Algebra</strong> Before and After Using the Definition of the Natural Logarithm<br />
x=<br />
x=__<br />
<br />
x =e <br />
x =<br />
__ e
532<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Try It #10<br />
x+=<br />
Example 11<br />
x=<br />
Using a Graph to Understand the Solution to a Logarithmic Equation<br />
Solution<br />
x=<br />
x =e <br />
Figure 3x<br />
e ≈e ≈<br />
y<br />
<br />
<br />
y = x<br />
y = <br />
e ≈ <br />
<br />
–<br />
–<br />
<br />
x<br />
Figure 3 The graphs of y = ln(x ) and y = 3 cross at the point (e 3 , 3), which is approximately (20.0855, 3).<br />
Try It #11<br />
x =<br />
<br />
Using the One-to-One Property of Logarithms to Solve Logarithmic Equations<br />
Th<br />
x >S >T ><br />
bb ≠<br />
b<br />
S= b<br />
TS =T<br />
<br />
<br />
x−= <br />
x −=<br />
x −=xx =x =<br />
<br />
−= <br />
=<br />
ThTh<br />
Th<br />
<br />
<br />
x −−=x+<br />
x<br />
<br />
x −−=x+<br />
<br />
( x ______ − ) =x+ <br />
<br />
______<br />
x − =x +<br />
<br />
<br />
x −=x + <br />
x = x
SECTION 6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS 533<br />
x =x −−=x+<br />
<br />
<br />
<br />
−−=+<br />
−=<br />
( __<br />
<br />
) =<br />
Th<br />
using the one-to-one property of logarithms to solve logarithmic equations<br />
STbb ≠<br />
b<br />
S= b<br />
TS =T<br />
<br />
<br />
How To…<br />
<br />
1. <br />
b<br />
S= b<br />
T<br />
2. <br />
3. S =T<br />
Example 12<br />
Solving an Equation Using the One-to-One Property of Logarithms<br />
x =x +<br />
Solution<br />
<br />
x =x +<br />
x =x + <br />
x −x −= <br />
x−x+= <br />
x −=x += <br />
x =x =− x<br />
Analysis There are two solutions: or −. The solution − is negative, but it checks when substituted into the original<br />
equation because the argument of the logarithm functions is still positive.<br />
Try It #12<br />
x =<br />
Solving Applied Problems Using Exponential and Logarithmic Equations<br />
<br />
<br />
<br />
<br />
<br />
Table 1
534<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Substance Use Half-life<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 1<br />
<br />
ftfi<br />
<br />
At=A <br />
e T t <br />
<br />
At=A <br />
e t <br />
<br />
At=A <br />
e t T At=A <br />
( __<br />
<br />
) t T <br />
<br />
A <br />
<br />
T<br />
t<br />
yftt<br />
Example 13<br />
Using the Formula for Radioactive Decay to Find the Quantity of a Substance<br />
<br />
<br />
T Solution<br />
y =e <br />
<br />
t <br />
<br />
<br />
=e <br />
=e <br />
<br />
t <br />
<br />
t <br />
ft<br />
<br />
<br />
<br />
<br />
=( e <br />
t ) <br />
<br />
= <br />
t<br />
<br />
<br />
eM =M<br />
<br />
t =× <br />
<br />
t ≈<br />
t<br />
Analysis Ten percent of grams is grams. If grams decay, the amount of uranium- remaining is <br />
grams.<br />
Try It #13<br />
<br />
<br />
Solving Logarithmic Equations (http://openstaxcollege.org/l/solvelogeq)<br />
Solving Exponential Equations with Logarithms (http://openstaxcollege.org/l/solveexplog)
SECTION 6.6 SECTION EXERCISES 535<br />
6.6 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
<br />
3. <br />
<br />
ALGEBRAIC<br />
<br />
4. −v− = −v 5. ⋅ x = 6. x+ ⋅ x =<br />
7. −n ⋅ <br />
=n+ 8. ⋅ x+ = 9. b<br />
<br />
= −b<br />
b<br />
10. ( ) n ⋅= <br />
<br />
11. x− = 12. e x = 13. e r+ −=−<br />
14. ⋅ a = 15. −⋅ p+ −=− 16. e n− +=−<br />
17. e −k += 18. −e x− −=− 19. −e x+ +=−<br />
20. x+ = x− 21. e x −e x −= 22. e x+ −=−<br />
23. e x+ += 24. e x+ −= 25. e −x− −=−<br />
26. x+ = x − 27. e x −e x −= 28. e −x +=−<br />
fi<br />
29. ( <br />
) =−<br />
30. <br />
= <br />
<br />
fi<br />
31. <br />
n= 32. − <br />
x= 33. + <br />
k=<br />
34. n++= 35. −−x=<br />
<br />
36. −x=−x 37. <br />
n−= <br />
−n 38. x+−x=<br />
39. −x=x −x 40. <br />
−m= <br />
m 41. x−−x=<br />
42. <br />
n −n= <br />
−+n 43. x −+=<br />
x<br />
44. x+=x+ 45. x+x−=x 46. <br />
x+=<br />
47. +−x = 48. <br />
x+− <br />
x= <br />
49. −−x=<br />
50. <br />
x− <br />
= <br />
<br />
GRAPHICAL<br />
xTh<br />
<br />
51. <br />
x−=− 52. <br />
x+= 53. x=<br />
54. x−= 55. +−x= 56. −+ <br />
−x=−<br />
57. x−−=− 58. −x=−x 59. <br />
−x −x= <br />
x−<br />
60. x+=−x 61. <br />
−x= <br />
x− 62. x +=x+<br />
<br />
63. <br />
<br />
<br />
−x−= 64. x−x+=
536 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
<br />
65. <br />
<br />
ft<br />
67. Th<br />
=e t t<br />
<br />
<br />
66. Th<br />
D D=<br />
( I <br />
I <br />
<br />
) <br />
I<br />
I <br />
= − <br />
<br />
<br />
⋅ <br />
TECHNOLOGY<br />
<br />
Th<br />
68. t = 69. e x =<br />
70. t = 71. x− =<br />
72. e −t =<br />
<br />
<br />
<br />
73. e x − += 74. +x+= 75. −x−=+<br />
76. P<br />
P=e −x x<br />
<br />
<br />
<br />
Hint<br />
77. ThM<br />
M= <br />
( E E <br />
) E<br />
<br />
E <br />
= <br />
<br />
<br />
· <br />
EXTENSIONS<br />
78. <br />
b b x =x<br />
80. A=a ( + r <br />
k) kt <br />
<br />
t<br />
79. <br />
y=Ae kt <br />
<br />
tt<br />
<br />
81. <br />
T<br />
T=T s<br />
+ T <br />
−T s<br />
e −kt T s<br />
<br />
T <br />
<br />
k<br />
<br />
<br />
tt
SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 537<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Model exponential growth and decay.<br />
Use Newton’s Law of Cooling.<br />
Use logistic-growth models.<br />
Choose an appropriate model for data.<br />
Express an exponential model in base e.<br />
6. 7 EXPONENTIAL AND LOGARITHMIC MODELS<br />
Figure 1 A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology campus. (credit: Georgia Tech Research Institute)<br />
<br />
<br />
Modeling Exponential Growth and Decay<br />
<br />
<br />
<br />
y=A <br />
e kt<br />
A <br />
ek<br />
doubling time<br />
fi<br />
<br />
s fi<br />
<br />
y =A <br />
e kt A <br />
e<br />
k<br />
half-life
538<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
ft<br />
<br />
Figure 2Figure 3<br />
xx<br />
y<br />
y = e x<br />
<br />
<br />
<br />
e<br />
<br />
<br />
− <br />
e <br />
y = <br />
x<br />
– – – – – <br />
–<br />
–<br />
y = <br />
y = e −x<br />
− <br />
<br />
e<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
e<br />
<br />
x<br />
Figure 2 A graph showing exponential<br />
growth. The equation is y = 2e 3x .<br />
Figure 3 A graph showing exponential<br />
decay. The equation is y = 3e −2x .<br />
ftft<br />
Thorder of magnitudefi<br />
<br />
fi× <br />
<br />
characteristics of the exponential function, y = A 0<br />
e kt<br />
y =A <br />
e kt <br />
<br />
y<br />
y<br />
y =<br />
<br />
–∞∞<br />
k A <br />
e<br />
–<br />
k A e<br />
∞<br />
x<br />
y = A e kt<br />
k > <br />
y = A e kt<br />
k < <br />
yA <br />
<br />
A <br />
A <br />
k >Figure 4<br />
A<br />
– <br />
k e<br />
k 0 and exponential decay when k < 0.<br />
Example 1<br />
Graphing Exponential Growth<br />
<br />
<br />
Solution fifiA <br />
<br />
A <br />
A <br />
=fikftt=<br />
. Th<br />
<br />
=e k ⋅<br />
=e k <br />
=k <br />
k =. Thy =e t =e t = t . ThFigure 5
SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 539<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
y= e t<br />
<br />
t<br />
Figure 5 The graph of y = 10e (ln2)t .<br />
Analysis The population of bacteria after ten hours is We could describe this amount is being of the order of<br />
magnitude . The population of bacteria after twenty hours is which is of the order of magnitude , so we<br />
could say that the population has increased by three orders of magnitude in ten hours.<br />
Half-Life<br />
<br />
half-life<br />
<br />
<br />
o fi<br />
<br />
__ A =A e kt<br />
0<br />
e fikA <br />
<br />
Th<br />
<br />
__<br />
A =A e kt<br />
0<br />
<br />
<br />
__<br />
kt<br />
=e A<br />
<br />
<br />
( <br />
)=kt<br />
− =kt<br />
<br />
<br />
<br />
<br />
−<br />
____ =t<br />
k<br />
k<br />
tk. Th<br />
How To…<br />
, fi<br />
1. A =A <br />
e kt <br />
2. A <br />
A t<br />
<br />
3. o fikk<br />
t =− ____<br />
<br />
k<br />
Note: It is also possible to find the decay rate usingk =− <br />
t
540<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 2<br />
Finding the Function that Describes Radioactive Decay<br />
Tht<br />
Solution Th<br />
A =A <br />
e kt Th<br />
A <br />
=A <br />
e k ⋅ A <br />
ft<br />
=e k A <br />
<br />
=k <br />
k =______<br />
<br />
<br />
k<br />
A =A <br />
e ( <br />
) t k<br />
Thft=A <br />
e ( <br />
) t . t<br />
<br />
≈−×− <br />
Try It #1<br />
Th <br />
<br />
Radiocarbon Dating<br />
Th<br />
<br />
ff<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
ftt<br />
A ≈A <br />
e ( <br />
) t<br />
<br />
A<br />
A <br />
<br />
Th<br />
A =A <br />
e kt Th<br />
A <br />
=A <br />
e k ⋅ tA <br />
f t<br />
=e k A <br />
<br />
=k <br />
k =______<br />
k<br />
<br />
A =A <br />
e ( <br />
) t r<br />
o fit<br />
( A <br />
t = A<br />
<br />
)<br />
−
SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 541<br />
<br />
Th<br />
r<br />
A ≈A <br />
e −t <br />
r = A ≈e A −t <br />
t<br />
<br />
r<br />
t =<br />
<br />
−<br />
How To…<br />
<br />
1. k<br />
r<br />
2. kt =<br />
<br />
− t<br />
Example 3<br />
Finding the Age of a Bone<br />
<br />
Solution =kt<br />
r<br />
t =<br />
<br />
−<br />
<br />
<br />
=<br />
− r<br />
≈<br />
<br />
Th<br />
Analysis The instruments that measure the percentage of carbon- are e xtremely sensitive and, as we mention above,<br />
a scientist will need to do much more work than we did in order to be satisfied. Even so, carbon dating is only accurate<br />
to about , so this age should be given as years ± or years ± years.<br />
Try It #<br />
<br />
<br />
Calculating Doubling Time<br />
<br />
fi<br />
doubling time<br />
A =A <br />
e kt <br />
A <br />
=A <br />
e kt <br />
Th<br />
A <br />
=A <br />
e kt<br />
Th<br />
=e kt <br />
=kt<br />
t =<br />
<br />
k<br />
t = <br />
k <br />
A <br />
<br />
<br />
<br />
t
542<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 4 Finding a Function That Describes Exponential Growth<br />
<br />
<br />
Solution Th<br />
t =___<br />
<br />
k<br />
Th<br />
=___<br />
<br />
k<br />
<br />
k =___<br />
<br />
k<br />
<br />
A =A <br />
e <br />
t k<br />
ThA <br />
e <br />
t .<br />
Try It #<br />
<br />
<br />
<br />
Using Newton’s Law of Cooling<br />
<br />
<br />
<br />
<br />
ThNewton’s Law of Coolingfi<br />
<br />
Tt=Ae kt +T s<br />
Th<br />
Tt=Ab ct +T s<br />
Tt=Ae bct <br />
+T s<br />
Tt=Ae ctb +T s<br />
<br />
Tt=Ae kt +T s<br />
<br />
<br />
<br />
cb)k<br />
Newton’s law of cooling<br />
ThTT s<br />
<br />
Tt=Ae kt +T s<br />
<br />
<br />
e diff<br />
<br />
How To…<br />
<br />
1. T s<br />
y<br />
2. Tt=Ae kt +T s<br />
o fiAk<br />
3. o fio fi
SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 543<br />
Example 5<br />
Using Newton’s Law of Cooling<br />
°°<br />
ft°<br />
°<br />
Solution <br />
<br />
Tt=Ae kt +<br />
T=<br />
=Ae k +<br />
<br />
A =<br />
A<br />
T=k<br />
=e k + <br />
=e k <br />
<br />
___<br />
k<br />
=e <br />
<br />
( ___<br />
) =k<br />
<br />
( ___<br />
)<br />
k =<br />
_<br />
≈− k<br />
<br />
ThTt=e −t +<br />
<br />
=e −t + Tt<br />
=e −t <br />
<br />
<br />
___ <br />
=e−t<br />
<br />
( ___ <br />
) =−t<br />
<br />
( ___ <br />
)<br />
t =<br />
_<br />
≈ t<br />
−<br />
°<br />
Try It #<br />
70 <br />
o 45 dego 60 deg<br />
Using Logistic Growth Models<br />
<br />
<br />
<br />
<br />
<br />
ft
544<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Thlogistic growth modelfi<br />
carrying capacityabc<br />
x<br />
c<br />
f(x=_______<br />
+ae −b x<br />
ThFigure 6Th <br />
<br />
<br />
f x<br />
<br />
y = c<br />
<br />
c<br />
+a<br />
a) c<br />
b <br />
f x=<br />
<br />
<br />
c<br />
+ ae –bx<br />
x<br />
logistic growth<br />
Th<br />
<br />
<br />
c <br />
+a <br />
ccarrying capacity, or limiting value<br />
b<br />
Figure 6<br />
c_<br />
f(x=<br />
+ae −b x<br />
Example 6<br />
Using the Logistic-Growth Model<br />
flTh<br />
fl<br />
Th<br />
<br />
t =fl<br />
flfifl<br />
b =flft<br />
s flft<br />
Solution <br />
c<br />
f x=_______<br />
+ae −b x<br />
fl<br />
c =fiat =<br />
c =a =<br />
+a<br />
<br />
Thftflf (x=<br />
≈<br />
−x<br />
+e<br />
fl<br />
e flu ic =<br />
Analysis Remember that, because we are dealing with a virus, we cannot predict with certainty the number of people<br />
infected. The model only approximates the number of people infected and will not give us exact or actual values. The graph in<br />
Figure 7 gives a good picture of how this model fits the data.
SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 545<br />
<br />
<br />
<br />
<br />
<br />
Cases<br />
<br />
<br />
<br />
<br />
<br />
<br />
y = <br />
<br />
<br />
<br />
<br />
<br />
<br />
Days<br />
<br />
Figure 7 The graph of f (x) =<br />
1000 __<br />
1 + 999e −0.6030x<br />
Try It #<br />
Example 6f fl<br />
Choosing an Appropriate Model for Data<br />
<br />
fl<br />
fi<br />
<br />
ft<br />
<br />
<br />
<br />
Thft<br />
<br />
ft<br />
<br />
Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
fl<br />
ftfi
546<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 7<br />
Choosing a Mathematical Model<br />
fiTable1<br />
<br />
Solution<br />
<br />
x <br />
y <br />
Table 1<br />
Figure 8fi<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
Figure 8<br />
<br />
<br />
y =ab xe fi=ab<br />
a =b=Thb =y =ax<br />
a<br />
y =ax<br />
<br />
=a<br />
a =_____<br />
<br />
<br />
a = ≈y <br />
=x<br />
<br />
Figure 9<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
x=<br />
y = (x)<br />
<br />
x<br />
Figure 9 The graph of y = 2lnx.<br />
d fi<br />
Figure 9y =x Figure 10
SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 547<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
x=<br />
y = (x )<br />
<br />
Figure 10 The graph of y = ln(x 2 )<br />
Thx >fiy =x =xx ><br />
x
548<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Example 8 Changing to base e<br />
y = x y =A <br />
e kx <br />
Solution<br />
Th<br />
y = x<br />
=e x <br />
<br />
=e x <br />
=e x <br />
Try It #<br />
y = x e<br />
<br />
<br />
<br />
<br />
<br />
<br />
Logarithm Application – pH (http://openstaxcollege.org/l/logph)<br />
Exponential Model – Age Using Half-Life (http://openstaxcollege.org/l/expmodelhalf)<br />
Newton’s Law of Cooling (http://openstaxcollege.org/l/newtoncooling)<br />
Exponential Growth Given Doubling Time (http://openstaxcollege.org/l/expgrowthdbl)<br />
Exponential Growth – Find Initial Amount Given Doubling Time (http://openstaxcollege.org/l/initialdouble)
SECTION 6.7 SECTION EXERCISES 549<br />
6.7 SECTION EXERCISES<br />
VERBAL<br />
1. halflife<br />
<br />
<br />
3. <br />
doubling time<br />
<br />
5. <br />
<br />
2. <br />
<br />
4. . Th<br />
<br />
<br />
NUMERIC<br />
6. Thftt<br />
Tt=e −t +ft<br />
<br />
f x= <br />
+e<br />
<br />
−x<br />
7. f 8. f <br />
9. 10. <br />
11. <br />
<br />
hmic. Th<br />
<br />
12. f(x= x <br />
eo fi <br />
x <br />
f(x) <br />
TECHNOLOGY<br />
<br />
<br />
13. x <br />
f(x) <br />
14.<br />
15.<br />
16.<br />
x <br />
f(x) <br />
x <br />
f(x) <br />
x <br />
f(x) <br />
fit<br />
<br />
Pt=<br />
<br />
+e −t<br />
17. 18. f fi<br />
19. <br />
e fi<br />
21. <br />
<br />
20. e fi<br />
ft<br />
22. e fi<br />
P
550 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
EXTENSIONS<br />
23. <br />
<br />
<br />
<br />
<br />
25. <br />
M= <br />
( S S <br />
) <br />
<br />
S<br />
27. b x =e xb b≠<br />
24. Th<br />
Pt=P <br />
e rt P <br />
<br />
r>t <br />
M<br />
26. y<br />
c<br />
y=<br />
<br />
+ae −rx<br />
<br />
REAL-WORLD APPLICATIONS<br />
<br />
<br />
28. 29. <br />
<br />
ftts. Th <br />
<br />
ft<br />
<br />
30. <br />
f <br />
<br />
<br />
<br />
31. <br />
<br />
33. <br />
ft<br />
o fi <br />
<br />
<br />
<br />
35. Th<br />
<br />
<br />
<br />
37. <br />
<br />
Th<br />
o fi <br />
<br />
<br />
ft<br />
32. <br />
ft<br />
tys. Th <br />
ft<br />
<br />
34. Th<br />
<br />
<br />
<br />
36. <br />
<br />
<br />
t? (Th
SECTION 6.7 SECTION EXERCISES 551<br />
ft<br />
ft<br />
38. <br />
<br />
39. <br />
<br />
<br />
<br />
<br />
ftr fift<br />
40. <br />
<br />
42. <br />
ft<br />
41. <br />
<br />
<br />
ft<br />
43. 44. <br />
ft<br />
45. <br />
<br />
fi<br />
<br />
46.<br />
x<br />
47.<br />
x<br />
– – – – – <br />
<br />
– – – – – <br />
<br />
48. <br />
− <br />
___ W<br />
m <br />
− <br />
___ W <br />
m W <br />
m <br />
49. <br />
M= <br />
( __ S <br />
S <br />
) <br />
<br />
<br />
<br />
<br />
<br />
ThNt=<br />
<br />
−t<br />
+e<br />
ftt<br />
50. 51. <br />
ft<br />
52. tNt<br />
<br />
<br />
53. ft<br />
<br />
<br />
<br />
a. f t= t b. f t=e t c. f t=e −t d. f t=__________<br />
<br />
+e −t
552<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Build an exponential model from data.<br />
Build a logarithmic model from data.<br />
Build a logistic model from data.<br />
6.8 FITTING EXPONENTIAL MODELS TO DATA<br />
<br />
Thfifi<br />
regression analysisfi<br />
Th<br />
o fit fi. Th<br />
modelftfunctionequationmodel<br />
fiThmodel<br />
<br />
<br />
fi<br />
<br />
fi<br />
fl<br />
<br />
Building an Exponential Model from Data<br />
<br />
<br />
<br />
way<br />
<br />
<br />
y =ab x y =A <br />
e kx <br />
y =ab x fl<br />
y =ab x a ><br />
b<br />
Th y =a<br />
b >x<br />
<br />
SECTION 6.8 FITTING EXPONENTIAL MODELS TO DATA 553<br />
exponential regression<br />
Exponential regression<br />
<br />
ExpRegfiTh<br />
y =ab x<br />
<br />
b<br />
b ><br />
554<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Solution<br />
a. STATEDITBAC<br />
ThSTATPLOTFigure 1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
x<br />
Figure 1<br />
ExpRegSTATCALC<br />
y= x<br />
<br />
y= x<br />
r ≈fi<br />
d fiFigure 2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
x<br />
Figure 2<br />
b. xy<br />
y = x <br />
<br />
= x<br />
≈<br />
<br />
ft
SECTION 6.8 FITTING EXPONENTIAL MODELS TO DATA 555<br />
Try It #1<br />
Table 2ft<br />
Month <br />
Debt ($) <br />
a. o fi<br />
b. ft<br />
Table 2<br />
Q & A…<br />
Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?<br />
<br />
<br />
<br />
<br />
Building a Logarithmic Model from Data<br />
<br />
<br />
<br />
way<br />
fi<br />
fl<br />
fl<br />
y =a +bx<br />
x<br />
Tha<br />
b > <br />
b < <br />
logarithmic regression<br />
Logarithmic regressionfi<br />
fi<br />
Th<br />
y=a +bx<br />
<br />
x<br />
b ><br />
b <<br />
How To…<br />
<br />
1. STATEDIT<br />
a. <br />
b. <br />
c.
556<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
2. STATPLOT<br />
a. ZOOM9o fi<br />
b. <br />
3. <br />
a. LnRegSTATCALC<br />
b. ab y =a +bx<br />
4. d fi<br />
Example 2<br />
Using Logarithmic Regression to Fit a Model to Data<br />
<br />
<br />
Table 3 <br />
Year <br />
Life Expectancy (Years) <br />
Year <br />
Life Expectancy (Years) <br />
Table 3<br />
a. xx =x = y<br />
o fi<br />
b. <br />
Solution<br />
a. STAT EDIT<br />
Th STATPLOT<br />
Figure 3<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
x<br />
Figure 3<br />
LnRegSTATCALC<br />
y=+x<br />
d fiFigure 4<br />
Center for Disease Control and Prevention
SECTION 6.8 FITTING EXPONENTIAL MODELS TO DATA 557<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 4<br />
b. x= y<br />
y =+x <br />
=+ x<br />
≈ <br />
<br />
<br />
Try It #2<br />
fiTable 4<br />
<br />
Year<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Number Sold (Thousands)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 4<br />
a. x x= y <br />
<br />
b.
558<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Building a Logistic Model from Data<br />
ff<br />
<br />
limiting value<br />
<br />
Th<br />
<br />
<br />
<br />
c<br />
y=_______<br />
+ae −b x<br />
<br />
c<br />
<br />
<br />
+a <br />
b > ( a <br />
b<br />
c ) <br />
y =c<br />
ccarrying capacity<br />
logistic regression<br />
Logistic regressiont fi<br />
o fi<br />
. Th<br />
c<br />
y=_______<br />
+ae<br />
<br />
−b x<br />
Th<br />
c <br />
+a <br />
y =c<br />
How To…<br />
<br />
1. STATEDIT<br />
a. <br />
b. <br />
c. <br />
2. STATPLOT<br />
a. ZOOM9o fi<br />
b. <br />
3. <br />
a. LogisticSTATCALC<br />
b. abc y =<br />
+ae −b x<br />
c <br />
4. d fi<br />
Example 3<br />
Using Logistic Regression to Fit a Model to Data<br />
<br />
Table 5 <br />
The World Bankn
SECTION 6.8 FITTING EXPONENTIAL MODELS TO DATA 559<br />
Year Americans with Cellular Service (%) Year Americans with Cellular Service (%)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 5<br />
a. xx = y <br />
o fi<br />
b. <br />
<br />
c. c<br />
<br />
Solution<br />
a. STATEDIT<br />
ThSTATPLOT<br />
Figure 5<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 5<br />
LogisticSTATCALC <br />
<br />
y=<br />
___________________ <br />
+e −x<br />
Figure 6d fi<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
Figure 6<br />
x<br />
x
560<br />
CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
b. x =<br />
y<br />
<br />
y =____________________<br />
<br />
+e <br />
−x<br />
<br />
=_____________________<br />
<br />
+e<br />
−<br />
x<br />
≈ <br />
<br />
c. Th <br />
<br />
c =<br />
ft<br />
Try It #3<br />
Table 6<br />
Year<br />
Seal Population<br />
(Thousands)<br />
Year<br />
Seal Population<br />
(Thousands)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 6<br />
a. xx=y<br />
o fi<br />
b. <br />
c. <br />
<br />
Exponential Regression on a Calculator (http://openstaxcollege.org/l/pregresscalc)
SECTION 6.8 SECTION EXERCISES 561<br />
6.8 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
d fi<br />
3. <br />
<br />
<br />
2. <br />
<br />
<br />
4. <br />
<br />
<br />
5. y<br />
<br />
<br />
GRAPHICAL<br />
fiFigure 7 <br />
Figure 11<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
Figure 7<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
Figure 8<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
x<br />
<br />
Figure 9<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 10<br />
Figure 11<br />
6. y=e −x 7. y=−x 8. y= x<br />
9. y=+x<br />
<br />
10. y=__________<br />
<br />
+e −x
562 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
NUMERIC<br />
11. <br />
<br />
<br />
Pt=<br />
<br />
+ −t<br />
13. <br />
hp=−p<br />
php=<br />
12. At= x <br />
e<br />
<br />
14. <br />
<br />
Pt=<br />
<br />
−t<br />
+e<br />
tPt=<br />
15. y<br />
<br />
TECHNOLOGY<br />
io: ThPx<br />
<br />
Px=<br />
<br />
+e −x<br />
16. <br />
<br />
18. ft<br />
<br />
20. <br />
<br />
<br />
17. <br />
19. <br />
<br />
io: ThP<br />
<br />
Px=<br />
x<br />
−x<br />
+e<br />
21. <br />
<br />
22. <br />
<br />
23. ft 24. <br />
<br />
25. <br />
<br />
<br />
Table 7<br />
x <br />
f (x) <br />
Table 7<br />
26. <br />
<br />
28. <br />
e<br />
27. <br />
t fi<br />
29.
SECTION 6.8 SECTION EXERCISES 563<br />
30. x<br />
f x=<br />
Table 8<br />
x <br />
f (x) <br />
Table 8<br />
31. <br />
e d<br />
33. <br />
<br />
35. x<br />
fx=<br />
32. <br />
t fi<br />
34. <br />
<br />
Table 9<br />
x <br />
f (x) <br />
Table 9<br />
36. <br />
<br />
38. <br />
x=<br />
40. x<br />
fx=<br />
Table 10<br />
37. LOGREG<br />
<br />
y=a+bxt fi<br />
39. <br />
<br />
x <br />
f (x) <br />
Table 10<br />
41. <br />
<br />
43. <br />
x=<br />
45. x<br />
fx=<br />
42. LOGREG<br />
<br />
y= a+bxt fi<br />
44. <br />
<br />
Table 11<br />
x <br />
f (x) <br />
Table 11
564 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
46. <br />
<br />
47. LOGISTIC <br />
y =<br />
c <br />
−b x<br />
t fi<br />
+ae<br />
48. 49. <br />
<br />
50. x<br />
<br />
Table 12<br />
x <br />
f (x) <br />
Table 12<br />
51. <br />
<br />
52. LOGISTIC <br />
c<br />
h m y =<br />
<br />
−b x<br />
+ae<br />
t fi<br />
53. 54. <br />
<br />
55. x<br />
<br />
EXTENSIONS<br />
56. <br />
c<br />
Pt=<br />
<br />
<br />
+ae −bt<br />
t=<br />
P=P <br />
<br />
c−Pt <br />
Pt =c−P e −bt <br />
P <br />
57. <br />
fx<br />
x<br />
<br />
y =x<br />
<br />
<br />
58. <br />
<br />
<br />
59. f − x<br />
c<br />
fx=<br />
<br />
−b x<br />
+ae<br />
60. <br />
<br />
Pt=<br />
<br />
−t<br />
+e
CHAPTER 6 REVIEW 565<br />
CHAPTER 6 REVIEW<br />
Key Terms<br />
annual percentage rate (APR) nominal rate<br />
carrying capacity <br />
change-of-base formula er b<br />
common logarithm x <br />
xx<br />
compound interest <br />
doubling time <br />
exponential growth <br />
extraneous solution <br />
half-life <br />
logarithm bxy = b<br />
x<br />
c<br />
logistic growth model f x=<br />
<br />
c_<br />
−b x<br />
+ae +a c<br />
b<br />
natural logarithm ex e<br />
xx<br />
Newton’s Law of Cooling <br />
<br />
nominal rate annual percentage rate<br />
order of magnitude <br />
<br />
power rule for logarithms <br />
<br />
product rule for logarithms <br />
quotient rule for logarithms o a diff<br />
Key Equations<br />
definition of the exponential function<br />
definition of exponential growth<br />
compound interest formula<br />
continuous growth formula<br />
General Form for the Translation of the<br />
Parent Function f (x)=b x<br />
Definition of the logarithmic function<br />
Definition of the common logarithm<br />
f x=b x b >b ≠<br />
f x=ab x a >b >b ≠<br />
A =a ( + r <br />
k) kt <br />
Att<br />
t<br />
Pft<br />
r<br />
n<br />
At=ae rt <br />
t<br />
aa<br />
P<br />
ee ≈<br />
f x=ab x +c +d<br />
x >b >b ≠ y = b<br />
xb y =x<br />
x >y =x y =x
566 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
Definition of the natural logarithm<br />
General Form for the Translation of the<br />
Parent Logarithmic Function f (x)=log b<br />
(x)<br />
The Product Rule for Logarithms<br />
x >y =xe y =x<br />
f x=a b<br />
x+c+d<br />
b<br />
MN= b<br />
M+ b<br />
N<br />
The Quotient Rule for Logarithms<br />
The Power Rule for Logarithms<br />
b ( M __<br />
N ) = b M− b N<br />
b<br />
M n =n b<br />
M<br />
The Change-of-Base Formula<br />
b<br />
M = M _ n<br />
n >n ≠b ≠<br />
n<br />
b<br />
One-to-one property for exponential functions<br />
<br />
ST<br />
bb S =b T S =T<br />
Definition of a logarithm<br />
Sbc<br />
b ≠<br />
<br />
b<br />
S=cb c =S<br />
One-to-one property for logarithmic functions<br />
<br />
ST<br />
bb ≠<br />
<br />
b<br />
S= b<br />
TS =T<br />
Half-life formula <br />
A =A <br />
e kt k t = k <br />
Tt=Ae kt +T s<br />
T s<br />
A =T−T s<br />
<br />
k<br />
<br />
Example 1<br />
Example 2Example 3<br />
Example 4<br />
Example 5<br />
Example 6<br />
Example 7<br />
Tht<br />
Example 8<br />
<br />
Example 9<br />
Theft<br />
e ≈<br />
[e x ][exp(x)]eExample 10<br />
e<br />
Example 11Example 12
CHAPTER 6 REVIEW 567<br />
6.2 Graphs of Exponential Functions<br />
Th f x=b x y−∞∞∞<br />
y =Example 1<br />
b >Th y =<br />
<br />
<br />
<br />
cc <<br />
Example 4<br />
Th f x= b<br />
x+d y = b<br />
x<br />
dd ><br />
dd <<br />
Example 5
568 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
a > f x=a b<br />
x<br />
y = b<br />
xa∣ a ∣ ><br />
y = b<br />
xa∣ a ∣ <<br />
Example 6Example 7<br />
y = b<br />
x− x<br />
− y<br />
Thf x=− b<br />
x x<br />
Th f x= b<br />
−x y<br />
Example 8<br />
Example 9<br />
f x=a b<br />
x+c+d<br />
Table 4<br />
f x=a b<br />
x+c+d<br />
x =−cExample 10<br />
f x=a b<br />
x+c+d<br />
Example 11<br />
6.5 Logarithmic Properties<br />
<br />
Example 1<br />
ff<br />
Example 2<br />
<br />
Example 3Example 4Example 5<br />
<br />
Example 6Example 7Example 8<br />
Thdiff<br />
Example 9Example 10Example 11Example 12<br />
<br />
Example 13<br />
Thfte<br />
n logs. ThExample 14<br />
6.6 Exponential and Logarithmic Equations<br />
<br />
Th<br />
<br />
<br />
Example 1<br />
not<br />
<br />
Example 2Example 3Example 4<br />
<br />
Example 5<br />
e<br />
Example 6Example 7<br />
ft <br />
Example 8
CHAPTER 6 REVIEW 569<br />
b<br />
S=cS <br />
b c =S<br />
Example 9Example 10<br />
b<br />
S=cy = b<br />
S<br />
y =cx<br />
Example 11<br />
b<br />
S= b<br />
TST<br />
S =TExample 12<br />
<br />
Example 13<br />
6.7 Exponential and Logarithmic Models<br />
f x=ab x b >k
570 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
CHAPTER 6 REVIEW EXERCISES<br />
EXPONENTIAL FUNCTIONS<br />
1. y= t <br />
<br />
<br />
2. Th<br />
At= t t<br />
<br />
ft<br />
3. <br />
4. Table 1<br />
<br />
5. <br />
<br />
<br />
<br />
7. y=e −t <br />
<br />
<br />
GRAPHS OF EXPONENTIAL FUNCTIONS<br />
9. fx= x <br />
y<br />
x <br />
f(x) <br />
Table 1<br />
6. <br />
<br />
<br />
<br />
<br />
8. <br />
<br />
<br />
ft<br />
10. fx=( <br />
) x <br />
y<br />
y<br />
11. Th fx= x y<br />
gxy<br />
12. Th fx= x <br />
<br />
LOGARITHMIC FUNCTIONS<br />
13. <br />
=x<br />
<br />
15. a − =b<br />
<br />
17. x <br />
x=( <br />
) <br />
– – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – <br />
–<br />
–<br />
–<br />
<br />
Figure 1<br />
14. s=t<br />
<br />
16. e − =h<br />
<br />
18. <br />
( <br />
) <br />
19. 20. <br />
<br />
<br />
<br />
<br />
<br />
<br />
x
CHAPTER 6 REVIEW 571<br />
21. e − 22. ( √ <br />
— )<br />
<br />
GRAPHS OF LOGARITHMIC FUNCTIONS<br />
23. gx=x+− 24. hx=−x+<br />
25. <br />
gx=x+− <br />
LOGARITHMIC PROPERTIES<br />
26. rst 27. <br />
x+ <br />
+ <br />
y+ <br />
<br />
<br />
28. m ( <br />
) <br />
30. ( <br />
x ) <br />
32. ( r s t )<br />
34. b+c+ −a <br />
<br />
29. z–x–y<br />
31. − y ( ) <br />
33. <br />
( b √ b+ <br />
b− )<br />
35. <br />
v+ <br />
w− u _ <br />
<br />
<br />
36. <br />
e 37. x− =hm. Th<br />
x<br />
<br />
EXPONENTIAL AND LOGARITHMIC EQUATIONS<br />
38. x x = x + 39. <br />
<br />
( <br />
<br />
<br />
) −x−= <br />
40. <br />
−x −=no<br />
solution<br />
42. e x −=<br />
no solution<br />
41. <br />
e n − +=−no<br />
solution<br />
43. e x− −=−<br />
no solution<br />
44. 2x − = x+ <br />
no solution<br />
46. <br />
− <br />
n=<br />
48. <br />
<br />
+ <br />
−x= <br />
<br />
f tno solution<br />
50. Th<br />
D D=<br />
( I <br />
I <br />
<br />
) I<br />
I <br />
= − <br />
<br />
<br />
− <br />
52. f − <br />
fx=e x + −<br />
45. e x −e x −=<br />
no solution<br />
47. <br />
+a+=<br />
49. <br />
+x −=<br />
no solution<br />
51. Th<br />
Pt=e t t<br />
<br />
<br />
<br />
53. f − <br />
fx= <br />
x +
572 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
EXPONENTIAL AND LOGARITHMIC MODELS<br />
<br />
<br />
54. <br />
55. ftt<br />
Th ft<br />
<br />
ff<br />
ftr fift<br />
56. <br />
<br />
57. <br />
<br />
<br />
ThNt=<br />
<br />
−t<br />
+e<br />
ftt<br />
58. <br />
59. <br />
60. <br />
<br />
<br />
<br />
61. x <br />
f(x) <br />
62. x <br />
f(x) <br />
63. −d (0, 4). Th<br />
e<br />
FITTING EXPONENTIAL MODELS TO DATA<br />
<br />
64. Pt=<br />
<br />
−t<br />
+e<br />
<br />
<br />
65. ThPt=<br />
t<br />
−t<br />
+e<br />
<br />
<br />
<br />
Thfi<br />
o fi<br />
66. x <br />
f (x) <br />
67. x <br />
f (x) <br />
68. x <br />
f (x)
CHAPTER 6 PRACTICE TEST<br />
573<br />
CHAPTER 6 PRACTICE TEST<br />
1. Th<br />
At= t t<br />
<br />
ft<br />
3. <br />
<br />
<br />
, co<br />
<br />
5. fx= −x <br />
y<br />
y<br />
2. <br />
<br />
4. <br />
<br />
<br />
ft<br />
6. Th<br />
fx=( <br />
) x <br />
<br />
y<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x<br />
7. <br />
=a<br />
<br />
9. x<br />
x=<br />
<br />
11. <br />
<br />
13. <br />
fx= <br />
−x+<br />
<br />
8. e =m<br />
<br />
10. <br />
12. gx=−x+<br />
14. ab<br />
<br />
15. t<br />
− t<br />
16. <br />
( a <br />
b ) <br />
17. <br />
y z √ <br />
— x−<br />
19. x− =hm. Th<br />
x<br />
<br />
21. <br />
−e a− −=−<br />
no solution<br />
23. −e −x− −=<br />
no solution<br />
25. e x −e x −=<br />
no solution<br />
18. <br />
c+d+ a <br />
+b+ <br />
<br />
<br />
20. ( ) x <br />
= ( <br />
)<br />
−x− <br />
<br />
22. e x+ +=<br />
no solution<br />
24. x− = x− <br />
no solution<br />
26. <br />
n−=−
574 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS<br />
27. <br />
x −+=<br />
no solution<br />
28. Th<br />
D <br />
D=<br />
( I __<br />
I <br />
<br />
) <br />
I<br />
I <br />
= − <br />
<br />
<br />
− <br />
29. <br />
ft<br />
o fi<br />
<br />
<br />
<br />
31. <br />
<br />
<br />
ft<br />
<br />
<br />
<br />
ft<br />
30. <br />
e<br />
o fi <br />
32. Th<br />
<br />
Pt=<br />
t<br />
−t<br />
+e<br />
<br />
<br />
<br />
33. Table 2<br />
<br />
x <br />
f(x) <br />
Table 2<br />
<br />
34. Thf fiPt=<br />
t<br />
−t<br />
+e<br />
<br />
<br />
<br />
. Tho fi<br />
o fi<br />
35. x <br />
f(x) <br />
36. x <br />
f(x) <br />
37. x <br />
f(x)
7<br />
Systems of Equations and Inequalities<br />
CHAPTER OUTLINE<br />
Introduction<br />
Figure 1 Enigma machines like this one, once owned by Italian dictator Benito Mussolini, were used by government<br />
and military officials for enciphering and deciphering top-secret communications during World War II. (credit: Dave Addey, Flickr)<br />
7.1 Systems of Linear Equations: Two Variables<br />
7.2 Systems of Linear Equations: Three Variables<br />
7.3 Systems of Nonlinear Equations and Inequalities: Two Variables<br />
7.4 Partial Fractions<br />
7.5 Matrices and Matrix Operations<br />
7.6 Solving Systems with Gaussian Elimination<br />
7.7 Solving Systems with Inverses<br />
7.8 Solving Systems with Cramer's Rule<br />
<br />
<br />
<br />
<br />
<br />
<br />
Th<br />
<br />
Th<br />
<br />
fi<br />
<br />
<br />
<br />
<br />
<br />
575
576<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Solve systems of equations by graphing.<br />
Solve systems of equations by substitution.<br />
Solve systems of equations by addition.<br />
Identify inconsistent systems of equations containing two variables.<br />
Express the solution of a system of dependent equations containing two variables.<br />
7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
Figure 1 (credit: Thomas Sørenes)<br />
Th<br />
<br />
fi<br />
fi<br />
<br />
Introduction to Systems of Equations<br />
<br />
system of linear equations<br />
<br />
fifi<br />
<br />
fi<br />
<br />
<br />
ff<br />
<br />
x +y =<br />
<br />
x −y =<br />
solution<br />
<br />
fi<br />
f fi<br />
+= <br />
−= <br />
<br />
consistent system<br />
independent systemTh
SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
577<br />
ffdependent system<br />
y<br />
fiTh<br />
fi<br />
inconsistent system<br />
ThffyTh<br />
<br />
types of linear systems<br />
Th<br />
independent systemxyTh<br />
<br />
inconsistent system<br />
dependent systemfi. Th. Th<br />
<br />
Figure 2<br />
y<br />
y<br />
y<br />
<br />
−<br />
<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – – – – – – –<br />
–<br />
–<br />
–<br />
–– –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
Figure 2<br />
How To…<br />
<br />
1. <br />
2. <br />
<br />
Example 1<br />
Determining Whether an Ordered Pair Is a Solution to a System of Equations<br />
<br />
<br />
x +y =<br />
<br />
x −=y<br />
Solution <br />
<br />
+=<br />
= <br />
<br />
−=<br />
= <br />
Thfi<br />
Analysis We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that<br />
satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines. See Figure 3.
578<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
x +y=<br />
– – <br />
–<br />
–<br />
x<br />
–<br />
− =y<br />
–<br />
–<br />
x<br />
Figure 3<br />
Try It #1<br />
<br />
<br />
x −y =<br />
<br />
x+=y<br />
Solving Systems of Equations by Graphing<br />
Th<br />
<br />
<br />
Example 2<br />
Solving a System of Equations in Two Variables by Graphing<br />
<br />
<br />
x +y =−<br />
<br />
x −y =−<br />
Solution e fiy<br />
<br />
<br />
y<br />
<br />
<br />
x +y =−<br />
y =−x −<br />
x −y =−<br />
y =x +<br />
Figure 4<br />
y<br />
y = −x−<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
y = x+<br />
x<br />
<br />
–<br />
–<br />
−− –<br />
–<br />
Figure 4
SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
579<br />
Th−−<br />
<br />
<br />
−+−=−<br />
<br />
−=−<br />
<br />
−−−=−<br />
<br />
−=−<br />
Th−−<br />
Try It #2<br />
<br />
<br />
x −y =−<br />
<br />
−x+y=<br />
Q & A…<br />
Can graphing be used if the system is inconsistent or dependent?<br />
<br />
fi<br />
<br />
Solving Systems of Equations by Substitution<br />
<br />
<br />
<br />
substitution method<br />
<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
3. fifi<br />
<br />
4. <br />
Example 3 Solving a System of Equations in Two Variables by Substitution<br />
<br />
<br />
−x+y =−<br />
<br />
x −y =<br />
Solution e fiy<br />
<br />
−x+y =−<br />
<br />
y =x −
580<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
x −y<br />
<br />
x −y =<br />
<br />
x −x−=<br />
<br />
x −x +=<br />
<br />
−x =−<br />
<br />
x =<br />
x =e fiy<br />
<br />
−+y =−<br />
<br />
y =<br />
<br />
<br />
<br />
−x+y =−<br />
−+=− <br />
<br />
x −y =<br />
−= <br />
Try It #3<br />
<br />
x =y +<br />
<br />
=x −y<br />
Q & A…<br />
Can the substitution method be used to solve any linear system in two variables?<br />
ffi−<br />
<br />
Solving Systems of Equations in Two Variables by the Addition Method<br />
addition method<br />
ffi<br />
ffift<br />
<br />
How To…<br />
<br />
1. xy <br />
2. <br />
ffi<br />
<br />
ffi<br />
3. <br />
4. <br />
5.
SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
581<br />
Example 4 Solving a System by the Addition Method<br />
<br />
<br />
x +y =−<br />
<br />
−x+y =<br />
Solution ffix<br />
−ffixfix<br />
<br />
<br />
x +y =−<br />
<br />
−x+y =<br />
<br />
y =<br />
xy<br />
<br />
y =<br />
<br />
y = <br />
<br />
Thyx<br />
<br />
−x+y =<br />
<br />
<br />
<br />
<br />
Th( − <br />
<br />
) <br />
e fi<br />
<br />
−x+ <br />
=<br />
−x=− <br />
<br />
−x= <br />
<br />
x =− <br />
<br />
x +y =−<br />
<br />
<br />
( − <br />
) + ( <br />
) =−<br />
− <br />
+ <br />
=−<br />
<br />
− <br />
=−<br />
−=− <br />
Analysis We gain an important perspective on systems of equations by looking at the graphical representation. See<br />
Figure 5 to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution<br />
because observing the graph confirms that the system has exactly one solution.<br />
y<br />
x + y = −<br />
−x + y =<br />
<br />
–<br />
<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 5
582<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 5<br />
Using the Addition Method When Multiplication of One Equation Is Required<br />
<br />
<br />
x +y =−<br />
<br />
x −y =<br />
Solution fi<br />
xx−x<br />
<br />
x −y =<br />
−x−y=− −<br />
−x +y =− <br />
<br />
<br />
x +y =−<br />
<br />
−x +y =−<br />
<br />
y =−<br />
<br />
y =−<br />
y =−x<br />
<br />
x +y =−<br />
<br />
x +−=−<br />
<br />
x −=−<br />
<br />
x =<br />
<br />
x =<br />
−Figure 6<br />
<br />
x −y =<br />
<br />
−−=+<br />
= <br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x − y= <br />
x + y = −<br />
Figure 6<br />
Try It #4<br />
<br />
<br />
<br />
x −y =<br />
x+y=−<br />
Example 6 Using the Addition Method When Multiplication of Both Equations Is Required<br />
<br />
<br />
x +y =−<br />
<br />
x −y =
SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
583<br />
Solution xxThx<br />
xfi<br />
−<br />
<br />
−x +y=−−<br />
<br />
−x −y =<br />
<br />
x −y=<br />
<br />
x −y =<br />
Th<br />
<br />
−x −y =<br />
<br />
x −y =<br />
<br />
−y =<br />
<br />
y =−<br />
y =−l fi<br />
<br />
x +−=−<br />
<br />
x −=−<br />
<br />
x =−<br />
<br />
x =−<br />
Th−−<br />
<br />
x −y =<br />
<br />
−−−=<br />
<br />
−+=<br />
<br />
=<br />
Figure 7<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
x + y= −<br />
–<br />
–<br />
−− –<br />
–<br />
x − y = <br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Example 7<br />
Figure 7<br />
Using the Addition Method in Systems of Equations Containing Fractions<br />
<br />
<br />
<br />
x <br />
+y <br />
=<br />
x <br />
−y <br />
=<br />
Solution <br />
<br />
<br />
( x <br />
+y <br />
) =<br />
<br />
<br />
<br />
x +y =<br />
( x <br />
−y <br />
) =<br />
x −y =
584<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
−x<br />
<br />
−x −y=−<br />
<br />
−x +y =−<br />
x<br />
<br />
x +y =<br />
<br />
−x +y =−<br />
<br />
y =<br />
<br />
y =<br />
y =e fi<br />
<br />
x +=<br />
<br />
x =<br />
<br />
x = <br />
<br />
<br />
=<br />
Th( <br />
) <br />
<br />
x <br />
−y <br />
=<br />
<br />
<br />
<br />
Try It #5<br />
<br />
<br />
<br />
<br />
<br />
<br />
− <br />
=<br />
<br />
− <br />
=<br />
<br />
=<br />
x +y =<br />
x+y=<br />
Identifying Inconsistent Systems of Equations Containing Two Variables<br />
<br />
ffy<br />
Th<br />
=<br />
Example 8<br />
Solving an Inconsistent System of Equations<br />
<br />
<br />
<br />
x =−y<br />
x +y =<br />
Solution x<br />
<br />
<br />
x +y =<br />
<br />
−y+y =<br />
<br />
+y =<br />
<br />
=<br />
≠. Th
SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
585<br />
Thfi<br />
e fi<br />
<br />
x =−y<br />
<br />
y =−x +<br />
<br />
y =− <br />
x + <br />
<br />
<br />
<br />
x +y =<br />
<br />
y =−x +<br />
<br />
y =− <br />
x + <br />
<br />
ffyTh<br />
<br />
<br />
y =− <br />
x + <br />
<br />
<br />
y =− <br />
x + <br />
<br />
Analysis Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will<br />
intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in<br />
common. The graphs of the equations in this example are shown in Figure 8.<br />
Try It #6<br />
<br />
<br />
y = − x +<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–––<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Figure 8<br />
<br />
<br />
y −x =<br />
<br />
y−x=<br />
<br />
y = −<br />
x +<br />
<br />
x<br />
<br />
Expressing the Solution of a System of Dependent Equations Containing Two Variables<br />
<br />
fi<br />
ft=<br />
Example 9<br />
Finding a Solution to a Dependent System of Linear Equations<br />
<br />
<br />
x +y =<br />
<br />
x +y =<br />
Solution <br />
xfi−<br />
x
586<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
x +y =<br />
<br />
−x+y=−<br />
<br />
−x −y =−<br />
<br />
<br />
−x −y =−<br />
<br />
+x +y =<br />
<br />
=<br />
fi<br />
Analysis If we rewrote both equations in the slope-intercept form, we might know what the solution would look like<br />
before adding. Let’s look at what happens when we convert the system to slope-intercept form.<br />
<br />
x +y =<br />
<br />
y =−x +<br />
<br />
<br />
<br />
y =− <br />
x + <br />
<br />
x +y =<br />
y =−x +<br />
<br />
y =− <br />
x + <br />
<br />
<br />
y =− <br />
x + <br />
<br />
See Figure 9. Notice the results are the same. The general solution to the system is( x− <br />
x + <br />
) <br />
y<br />
<br />
<br />
<br />
<br />
x + y= <br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
<br />
x + y = <br />
Figure 9<br />
Try It #7<br />
<br />
y −x =<br />
−y+x=−<br />
Using Systems of Equations to Investigate Profits<br />
<br />
Threvenue function<br />
R =xpx =<br />
p =. ThFigure 10<br />
Thcost functionfi<br />
ThFigure 10Thx<br />
. Thy
SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
587<br />
Money (in hundreds of dollars)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Quantity (in hundreds of units)<br />
Figure 10<br />
Thbreak-even point<br />
<br />
. Th<br />
Thfi<br />
Th ffThprofit function<br />
Px=Rx−Cx<br />
<br />
Example 10<br />
Finding the Break-Even Point and the Profit Function Using Substitution<br />
Cx=x +Rx=xfi<br />
rofi<br />
Solution y<br />
<br />
y =x +<br />
<br />
y =x<br />
x +e fix<br />
<br />
x +=x<br />
<br />
=x<br />
<br />
=x<br />
Thx =<br />
<br />
=<br />
Th<br />
ThfiPx=Rx−Cx<br />
<br />
Px=x −x +<br />
<br />
=x −<br />
ThfiPx=x −<br />
Analysis The cost to produce units is , and the revenue from the sales of units is also . To<br />
make a profit, the business must produce and sell more than units. See Figure 11.<br />
We see from the graph in Figure 12 that the profit function has a negative value until x = , when the graph crosses<br />
the x-axis. Then, the graph emerges into positive y-values and continues on this path as the profit function is a straight<br />
line. This illustrates that the break-even point for businesses occurs when the profit function is . The area to the left of<br />
the break-even point represents operating at a loss.
588<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Dollars<br />
<br />
<br />
<br />
<br />
<br />
<br />
Example 11<br />
<br />
Quantity<br />
<br />
<br />
<br />
<br />
Cx=x+<br />
<br />
Rx=x<br />
Figure 11<br />
Dollars profit<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Writing and Solving a System of Equations in Two Variables<br />
<br />
<br />
<br />
<br />
<br />
Profit<br />
<br />
Px=x−<br />
Figure 12<br />
Quantity<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Th<br />
<br />
Solution c=a=<br />
Th<br />
<br />
c +a =<br />
ThcTh<br />
a. Th<br />
<br />
<br />
c +a =<br />
<br />
<br />
c +a =<br />
<br />
c +a =<br />
fiffifica<br />
a<br />
<br />
c +a =<br />
<br />
a =−c<br />
−cac<br />
<br />
c +−=<br />
<br />
c +−c =<br />
<br />
−c =−<br />
<br />
c =<br />
c =e fia<br />
<br />
+a =<br />
<br />
a =<br />
e fi<br />
Try It #8<br />
<br />
<br />
<br />
Solving Systems of Equations Using Substitution (http://openstaxcollege.org/l/syssubst)<br />
Solving Systems of Equations Using Elimination (http://openstaxcollege.org/l/syselim)<br />
Applications of Systems of Equations (http://openstaxcollege.org/l/sysapp)
SECTION 7.1 SECTION EXERCISES 589<br />
7.1 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
<br />
5. <br />
diff<br />
2. <br />
<br />
<br />
fi<br />
4. <br />
<br />
<br />
<br />
ALGEBRAIC<br />
<br />
6. x−y=<br />
x+y=<br />
9. −x+y=<br />
x+y=−<br />
7. −x−y=<br />
−x+y=−<br />
10. x+y=<br />
x −y=−<br />
<br />
8. x+y=<br />
x+y=<br />
11. x+y=<br />
x+y=<br />
12. x −y=<br />
x+y=−<br />
13. x+y=−<br />
x+y=<br />
14. x+y=−<br />
x −y=<br />
15. −x+y=<br />
−x−y=<br />
16. x−y=<br />
−x+y=<br />
17. x+y=<br />
x+y=−<br />
18. −x+y=<br />
x −y=−<br />
19. <br />
x+ <br />
y=<br />
20. − <br />
x+ <br />
y=<br />
<br />
x+ <br />
y=<br />
− <br />
x+ <br />
y=<br />
<br />
21. −x+y=−<br />
x+y=<br />
22. x−y=−<br />
x+y=<br />
23. x−y=−<br />
−x−y=<br />
24. x−y=<br />
x+y=<br />
25. −x+y=−<br />
x−y=<br />
29. −x+y=<br />
x−y=−<br />
26. x+y=<br />
−x−y=−<br />
30. −x+y=<br />
x−y=<br />
27. <br />
x+ y =<br />
<br />
<br />
x− <br />
y=− <br />
28. <br />
x+ <br />
y= <br />
<br />
− <br />
x+ <br />
y=− <br />
<br />
<br />
31. x+y=<br />
x+y=<br />
32. x −y=−<br />
x+y=<br />
33. x−y=<br />
x−y=<br />
34. x− <br />
y=− <br />
<br />
−x+ <br />
y=
590 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
35. x−y= <br />
<br />
x+y= <br />
<br />
36. x+y=<br />
x+y=<br />
37. <br />
x− <br />
y=<br />
− <br />
x+ <br />
y=−<br />
38. <br />
x+ <br />
y= <br />
<br />
<br />
x+ <br />
y=− <br />
<br />
39. x+y=−<br />
x+y=<br />
40. x+y=<br />
x −y=<br />
GRAPHICAL<br />
<br />
fi<br />
41. x−y=<br />
x−y=<br />
44. x−y=<br />
x−y=<br />
TECHNOLOGY<br />
42. −x+y=<br />
x−y=<br />
45. x−y=<br />
−x+y=−<br />
43. x+y=<br />
x+y=<br />
<br />
<br />
46. x+y=<br />
−x+y=<br />
47. −x+y=<br />
x+y=<br />
48. x+y=<br />
x −y=<br />
49. x+y=<br />
−x+y=<br />
50. −x+y=<br />
x+y=<br />
EXTENSIONS<br />
ABCDEFAF<br />
A≠BAE≠BD<br />
51. x+y=A<br />
x−y=B<br />
52. x+Ay=<br />
x+By=<br />
53. Ax+y=<br />
Bx+y=<br />
54. Ax+By=C<br />
x+y=<br />
55. Ax+By=C<br />
Dx+Ey=F<br />
REAL-WORLD APPLICATIONS<br />
<br />
56. uff<br />
C=x+<br />
R=x<br />
58. <br />
Cx=x+<br />
Rx=x<br />
60. <br />
Cx=x+<br />
ft<br />
<br />
<br />
57. <br />
Cx=x+Rx=x<br />
fi<br />
59. Cx=x+x<br />
t. Th<br />
ft
SECTION 7.1 SECTION EXERCISES 591<br />
<br />
61. d diff<br />
<br />
63. Th<br />
<br />
ft<br />
<br />
65. <br />
. Th<br />
<br />
<br />
<br />
67. Th<br />
<br />
<br />
69. <br />
<br />
<br />
<br />
71. <br />
<br />
<br />
<br />
<br />
<br />
fi<br />
d de<br />
73. <br />
<br />
<br />
<br />
<br />
75. s. Th<br />
<br />
<br />
<br />
<br />
77. <br />
<br />
g a diff<br />
<br />
t t<br />
62. <br />
<br />
<br />
64. es a fl<br />
<br />
<br />
<br />
<br />
l cos<br />
66. <br />
<br />
<br />
<br />
<br />
68. <br />
<br />
Th<br />
n t<br />
ft<br />
<br />
<br />
<br />
70. fi<br />
<br />
fi<br />
h y<br />
72. <br />
<br />
<br />
<br />
<br />
<br />
74. <br />
<br />
<br />
76. <br />
ftt. Th<br />
<br />
Th
592<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Solve systems of three equations in three variables.<br />
Identify inconsistent systems of equations containing three variables.<br />
Express the solution of a system of dependent equations containing three variables.<br />
7.2 SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES<br />
Figure 1 (credit: “Elembis,” Wikimedia Commons)<br />
<br />
<br />
fi<br />
<br />
fi<br />
<br />
<br />
fi<br />
Solving Systems of Three Equations in Three Variables<br />
<br />
ftfi<br />
fifi<br />
Th<br />
<br />
fixyz<br />
<br />
Ax+By+Cz=D<br />
Ey+Fz=G<br />
Hz=K<br />
Thzfiyx<br />
<br />
1. <br />
2. <br />
3.
SECTION 7.2 SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES<br />
593<br />
solution setxyzfi<br />
<br />
fifl<br />
e fl<br />
number of possible solutions<br />
Figure 2Figure 3<br />
ftsolution set<br />
xyz <br />
ft<br />
= <br />
<br />
ft<br />
=<br />
<br />
Figure 2 ( a)Three planes intersect at a single point, representing a three-by-three system with a single solution.<br />
( b) Three planes intersect in a line, representing a three-by-three system with infinite solutions.<br />
<br />
<br />
Figure 3 All three figures represent three-by-three systems with no solution. ( a) The three planes intersect with each other, but not at a common point.<br />
(b) Two of the planes are parallel and intersect with the third plane, but not with each other.<br />
( c) All three planes are parallel, so there is no point of intersection.<br />
Example 1 Determining Whether an Ordered Triple Is a Solution to a System<br />
−<br />
<br />
x +y +z =<br />
<br />
x −y +z =<br />
<br />
x +y +z =<br />
Solution xyz<br />
<br />
x +y +z =<br />
<br />
+−+=<br />
<br />
<br />
<br />
x −y +z =<br />
<br />
−−+=<br />
<br />
++=
594<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
x +y +z =<br />
<br />
+−+=<br />
<br />
−+=<br />
<br />
<br />
Th−<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 2<br />
Solving a System of Three Equations in Three Variables by Elimination<br />
<br />
x −y +z = <br />
−x+y −z =− <br />
x −y +z = <br />
Solution Thfi<br />
x<br />
x −y +z = <br />
−x+y −z =− <br />
y +z = <br />
Th−Th<br />
x<br />
−x +y −z =− −<br />
x −y +z = <br />
−y −z =− <br />
z<br />
y +z = <br />
−y−z =− <br />
z = <br />
<br />
x −y +z = <br />
y +z = <br />
z = <br />
z =y<br />
<br />
y +=<br />
<br />
y +=<br />
<br />
y =−<br />
z =y =−. Thx<br />
<br />
x −−+=<br />
<br />
x ++=<br />
<br />
x =
SECTION 7.2 SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES<br />
595<br />
Th−Figure 4<br />
−<br />
x = <br />
z = 2<br />
Example 3<br />
y = −2<br />
Figure 4<br />
Solving a Real-World Problem Using a System of Three Equations in Three Variables<br />
ff<br />
<br />
<br />
Th<br />
Solution <br />
<br />
<br />
x =<br />
<br />
y =<br />
<br />
z =<br />
The fi<br />
<br />
x +y +z =<br />
<br />
<br />
<br />
z =y +<br />
Th<br />
<br />
x +y +z =<br />
Th<br />
<br />
x +y +z =<br />
<br />
−y +z =<br />
<br />
x +y +z =<br />
. Th<br />
x +y +z = <br />
−y +z = <br />
x +y +z = <br />
<br />
<br />
x +y +z =<br />
<br />
x +y +z =<br />
<br />
−y +z =<br />
−<br />
<br />
x +y +z =<br />
<br />
y +z =<br />
<br />
−y +z =
596<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
<br />
x +y +z =<br />
<br />
y +z =<br />
<br />
z =<br />
zyTh<br />
zyx<br />
<br />
z =<br />
<br />
z =<br />
<br />
y +=<br />
<br />
y =<br />
<br />
x ++=<br />
<br />
x =<br />
<br />
Try It #1<br />
<br />
<br />
x +y −z =−<br />
<br />
x −y −z =<br />
x−y+z=<br />
Identifying Inconsistent Systems of Equations Containing Three Variables<br />
<br />
fiTh<br />
<br />
Th=<br />
<br />
Example 4<br />
Solving an Inconsistent System of Three Equations in Three Variables<br />
<br />
x −y +z = <br />
−x+y −z = <br />
x −y +z = <br />
Solution ffixx<br />
x −y +z = <br />
−x+y −z = <br />
−y −z = <br />
−<br />
−x +y −z =− −<br />
x −y +z = <br />
y +z =− <br />
Th<br />
−y −z = <br />
y +z =− <br />
<br />
=
SECTION 7.2 SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES<br />
597<br />
Thfi=<br />
<br />
Analysis In this system, each plane intersects the other two, but not at the same location. Therefore, the system is<br />
inconsistent.<br />
Try It #2<br />
<br />
<br />
x +y +z =<br />
y −z =<br />
x+y+z=<br />
Expressing the Solution of a System of Dependent Equations Containing Three Variables<br />
fi<br />
Thfi<br />
Th<br />
ff<br />
fi<br />
Example 5<br />
Finding the Solution to a Dependent System of Equations<br />
<br />
x +y −z = <br />
x +y −z = <br />
x −y +z = <br />
Solution −<br />
−x −y +z = −<br />
x +y −z = <br />
<br />
=<br />
Th=<br />
fiTh−<br />
d fi=<br />
fi<br />
<br />
<br />
<br />
x +y −z =<br />
x −y +z =<br />
x −z =<br />
z<br />
<br />
x −z =<br />
<br />
z = <br />
x<br />
zy
598<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
<br />
<br />
<br />
x +y −( <br />
x ) =<br />
x +y − x =<br />
<br />
y = x −x<br />
<br />
y = <br />
x<br />
( x <br />
x <br />
x ) xThyz<br />
x<br />
Analysis As shown in Figure 5, two of the planes are the same and they intersect the third plane on a line. The solution<br />
set is infinite, as all points along the intersection line will satisfy all three equations.<br />
x − y + z = 0<br />
−4x − 2y + 6z = 0<br />
4x + 2y − 6z = 0<br />
Figure 5<br />
Q & A…<br />
Does the generic solution to a dependent system always have to be written in terms of x?<br />
x<br />
xy<br />
Try It #3<br />
<br />
<br />
x +y +z =<br />
x −y −z =<br />
x+y+z=<br />
<br />
Ex 1: System of Three Equations with Three Unknowns Using Elimination (http://openstaxcollege.org/l/systhree)<br />
Ex. 2: System of Three Equations with Three Unknowns Using Elimination (http://openstaxcollege.org/l/systhelim)
SECTION 7.2 SECTION EXERCISES 599<br />
7.2 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
<br />
2. <br />
<br />
<br />
4. <br />
<br />
5. <br />
<br />
f y<br />
f n<br />
ALGEBRAIC<br />
<br />
6. x −y+z=−<br />
x+y+z=− −<br />
−x+y+z=−<br />
7. x−y+z=<br />
x+y+z=<br />
x+y=<br />
−−<br />
8. x −y+z=<br />
−x−y+z=<br />
x+y−z=<br />
−<br />
9. x−y=<br />
x−z= −<br />
x−y+z=−<br />
10. −x−y+z=<br />
x+y −z= −<br />
−x+y −z=−<br />
<br />
11. x−y+z=−<br />
x+y+z=<br />
x+y+z=<br />
14. x−y+z=<br />
−x+y+z=<br />
x+y−z=−<br />
12. x−y+z=<br />
x−y−z=−<br />
x+y−z=<br />
15. x−y+z=<br />
−x+y−z=−<br />
x+y−z=<br />
13. x+y+z=<br />
−x+y+z=<br />
x−y+z=−<br />
16. x+y+z=<br />
−x+y−z=<br />
x−y+z=−<br />
<br />
17. x−y+z=<br />
−x+y−z=−<br />
y+z=−<br />
20. x+y−z=<br />
x+y−z=<br />
−x−y+z=−<br />
23. x+y+z=<br />
y+z=−<br />
−y−z=−<br />
18. x−y+z=<br />
−x+y=<br />
x−z=<br />
21. x+y−z=<br />
−x+y+z=<br />
−x+y+z=<br />
24. x−y+z=−<br />
−x+y−z=<br />
−x+y+z=−<br />
19. x+y−z=<br />
−x−y+z=−<br />
x+y+z=<br />
22. x+y−z=<br />
−x−y−z=−<br />
−x−y+z=−<br />
25. x+y+z=<br />
x−y+z=<br />
x−z=
600 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
26. x+y−z=<br />
x−y+z=−<br />
x+y−z=<br />
29. x−y+z=<br />
<br />
x− <br />
y+ <br />
z=<br />
−x− <br />
y−z=−<br />
27. x+y+z=<br />
x−y+z=<br />
x−z=<br />
30. <br />
x− <br />
y+ <br />
z=− <br />
<br />
<br />
x− <br />
y− <br />
z=− <br />
− <br />
x− <br />
y− <br />
z=− <br />
<br />
28. x− <br />
y−z=− <br />
<br />
x+z=<br />
−x+ <br />
y= <br />
<br />
31. − <br />
x− <br />
y− <br />
z= <br />
<br />
− <br />
x− <br />
y− <br />
z=<br />
− <br />
x− <br />
y− <br />
z=− <br />
<br />
32. <br />
x− <br />
y+ <br />
z=<br />
<br />
x− <br />
y+ <br />
z=−<br />
<br />
x+ <br />
y− <br />
z=<br />
33. <br />
x− <br />
y+ <br />
z=<br />
− <br />
x− <br />
y+ <br />
z=−<br />
− <br />
x− <br />
y+ <br />
z=−<br />
34. − <br />
x− <br />
y+ <br />
z=− <br />
<br />
− <br />
x− <br />
y+ <br />
z=− <br />
<br />
− <br />
x− <br />
y+ <br />
z=<br />
35. − <br />
x− <br />
y+ <br />
z=−<br />
− <br />
x− <br />
y+ <br />
z= <br />
<br />
− <br />
x− <br />
y+ <br />
z= <br />
<br />
38. x+y−z=<br />
x+y−z=<br />
x+y−z=<br />
41. x+y+z=<br />
x−y−z=−<br />
x−y−z=−<br />
44. x+y+z=<br />
x+y+z=<br />
x+y+z=<br />
36. <br />
x+ <br />
y+ <br />
z= <br />
− <br />
x− <br />
y− <br />
z=− <br />
<br />
<br />
x+ <br />
y+ <br />
z= <br />
39. x+y−z=−<br />
x+y−z=−<br />
x+y−z=−<br />
42. x−y−z=<br />
x−y−z=<br />
x−y−z=−<br />
45. x+y+z=<br />
x−y+z=<br />
x+y+z=<br />
37. x−y+z=<br />
x−y+z=<br />
x−y+z=<br />
40. x−y+z=<br />
x−y+z=<br />
x−y+z=<br />
43. x+y−z=<br />
x−y+z=<br />
x−y−z=<br />
EXTENSIONS<br />
xyz<br />
46. x+y+z=<br />
x− <br />
+y− <br />
+z+ <br />
=<br />
x− <br />
+y+ <br />
+z− <br />
= <br />
<br />
47. x−y− z+ <br />
= <br />
<br />
x+ y− <br />
+z=−<br />
x+ <br />
−y+z=<br />
48. x+ <br />
−y− <br />
+z+ <br />
=<br />
x− <br />
+y+ <br />
−z+ <br />
=<br />
x+ <br />
−y+ <br />
+z+ <br />
=<br />
49. x− <br />
+y+ <br />
−z− <br />
=<br />
x+ <br />
+y− <br />
+z+ <br />
=<br />
x+ <br />
−y− <br />
+z+=<br />
50. x− <br />
+y+ <br />
+z+ <br />
=<br />
x+y−z=<br />
x+y−z=
SECTION 7.2 SECTION EXERCISES 601<br />
REAL-WORLD APPLICATIONS<br />
51. o 108. Th<br />
<br />
<br />
<br />
53. <br />
<br />
. Th. Th<br />
<br />
<br />
<br />
55. ff<br />
. Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
57. . Th<br />
<br />
. Th<br />
<br />
Th <br />
. Th<br />
<br />
<br />
<br />
59. t. Th<br />
f $28,112.50. Th<br />
<br />
<br />
<br />
<br />
61. <br />
<br />
<br />
o infl<br />
$151,830. Th<br />
<br />
<br />
<br />
<br />
52. o 147. Th<br />
<br />
<br />
54. <br />
<br />
<br />
<br />
<br />
56. ff<br />
<br />
<br />
diffax. Th<br />
<br />
<br />
ax. Th<br />
<br />
<br />
<br />
<br />
58. <br />
. Th<br />
<br />
s $15.25. Th<br />
<br />
<br />
<br />
60. h $19.50. Th<br />
<br />
<br />
<br />
<br />
62. <br />
<br />
ft
602 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
63. <br />
. Th <br />
<br />
<br />
ft<br />
<br />
<br />
<br />
65. Th<br />
<br />
<br />
<br />
. Th<br />
<br />
Th<br />
<br />
<br />
<br />
67. Th<br />
<br />
a. Th<br />
ts. Th<br />
<br />
<br />
<br />
<br />
69. <br />
t. Th<br />
d fi<br />
<br />
n fid fi<br />
<br />
<br />
d fi<br />
64. <br />
. Th <br />
<br />
<br />
ft<br />
t fi<br />
<br />
<br />
66. Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
68. Th<br />
<br />
ka. Th<br />
n. Th<br />
<br />
<br />
<br />
<br />
70. <br />
<br />
fif fi<br />
<br />
<br />
<br />
ffi<br />
ffi<br />
ffi<br />
: Th<br />
Th
SECTION 7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES<br />
603<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Solve a system of nonlinear equations using substitution.<br />
Solve a system of nonlinear equations using elimination.<br />
Graph a nonlinear inequality.<br />
Graph a system of nonlinear inequalities.<br />
7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES<br />
Figure 1<br />
ThTh<br />
ff<br />
Figure 1 Halley’s Comet (credit: "NASA Blueshift"/Flickr)<br />
<br />
Th<br />
Solving a System of Nonlinear Equations Using Substitution<br />
system of nonlinear equations<br />
Ax+By+C =<br />
. Th<br />
<br />
. Th<br />
Intersection of a Parabola and a Line<br />
Th<br />
possible types of solutions for points of intersection of a parabola and a line<br />
Figure 2<br />
n. Th<br />
n. Th<br />
s. Th<br />
<br />
y<br />
y<br />
y<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
– – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
Figure 2<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
x
604<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
How To…<br />
, fi<br />
1. <br />
2. <br />
3. <br />
4. <br />
Example 1<br />
<br />
<br />
<br />
Solving a System of Nonlinear Equations Representing a Parabola and a Line<br />
x −y =−<br />
y =x +<br />
Solution e fix<br />
<br />
x −y =−<br />
x =y − x<br />
<br />
y =x +<br />
y =y− + x<br />
<br />
<br />
y =y− <br />
<br />
=y −y ++<br />
<br />
=y −y +<br />
<br />
=y −y +<br />
<br />
=y−y−<br />
yy =y =yfix<br />
<br />
<br />
x −y =−<br />
<br />
x −=−<br />
<br />
x =<br />
<br />
x −=−<br />
<br />
x =<br />
Thfixy<br />
Figure 3<br />
– –<br />
y =x<br />
<br />
+ <br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
Figure 3<br />
x − y = −<br />
<br />
<br />
<br />
x
SECTION 7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES<br />
605<br />
Q & A…<br />
Could we have substituted values for y into the second equation to solve for x in Example 1?<br />
xx<br />
y =<br />
<br />
y =x +<br />
<br />
=x +<br />
<br />
x =<br />
<br />
x =±√ — =<br />
Th<br />
y =<br />
<br />
y =x +<br />
<br />
=x +<br />
<br />
x =<br />
<br />
x =±√ — =±<br />
−<br />
Try It #1<br />
<br />
<br />
x −y =−<br />
<br />
x −y=<br />
Intersection of a Circle and a Line<br />
<br />
<br />
possible types of solutions for the points of intersection of a circle and a line<br />
Figure 4<br />
n. Th<br />
n. Th<br />
s. Th<br />
<br />
Figure 4<br />
How To…<br />
, fi<br />
1. <br />
2. <br />
3. <br />
4.
606<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 2<br />
Finding the Intersection of a Circle and a Line by Substitution<br />
<br />
<br />
x +y =<br />
<br />
y =x −<br />
Solution yy =x −<br />
<br />
<br />
x +x − =<br />
<br />
x +x −x +=<br />
<br />
x −x +=<br />
x<br />
<br />
x −x +=<br />
<br />
x−x−=<br />
<br />
x =<br />
<br />
x =<br />
xy<br />
<br />
y =−<br />
<br />
=<br />
y =−<br />
<br />
=−<br />
Th−fixy<br />
Figure 5<br />
<br />
x y <br />
<br />
<br />
<br />
y x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 5<br />
Try It #2<br />
<br />
<br />
x +y =<br />
x−y=−<br />
Solving a System of Nonlinear Equations Using Elimination<br />
ft<br />
<br />
ft
SECTION 7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES<br />
607<br />
possible types of solutions for the points of intersection of a circle and an ellipse<br />
Figure 6<br />
Th<br />
<br />
n. Th<br />
s. Th<br />
s. Th<br />
s. Th<br />
Thr <br />
Figure 6<br />
Example 3<br />
Solving a System of Nonlinear Equations Representing a Circle and an Ellipse<br />
<br />
x +y = <br />
x +y = <br />
Solution −<br />
<br />
−x +y =−<br />
<br />
−x −y =−<br />
<br />
x +y =<br />
<br />
y =<br />
fty<br />
<br />
y =<br />
<br />
y =±√ — =±<br />
y =±x<br />
x + = x +− =<br />
<br />
x +=<br />
x +=<br />
<br />
x =<br />
<br />
x ==±<br />
<br />
x =±√ — =±<br />
Th−−−−Figure 7<br />
−<br />
– –<br />
−−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
Figure 7<br />
<br />
<br />
<br />
x<br />
<br />
−
608<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Try It #3<br />
<br />
x +y =<br />
x +y =<br />
Graphing a Nonlinear Inequality<br />
<br />
<br />
<br />
nonlinear inequality<br />
<br />
y >ay a; (b) an example of y ≥ a; (c) an example of y < a; (d) an example of y ≤ a<br />
How To…<br />
<br />
1. . Th<br />
2. ≤≥<br />
3. <br />
4. fi<br />
<br />
<br />
5. <br />
Example 4 Graphing an Inequality for a Parabola<br />
y >x +<br />
Solution y =x +y >x +<br />
Th<br />
<br />
y >x +<br />
> +<br />
> <br />
> +<br />
>
SECTION 7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES<br />
609<br />
ThFigure 9<br />
<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
– –<br />
<br />
<br />
<br />
x<br />
Graphing a System of Nonlinear Inequalities<br />
Figure 9<br />
<br />
system of nonlinear inequalities<br />
<br />
Thfffi<br />
Th<br />
feasible region<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
4. <br />
<br />
Example 5<br />
Graphing a System of Inequalities<br />
<br />
<br />
<br />
x −y ≤<br />
x +y ≤<br />
Solution Thfi<br />
y. Thx<br />
<br />
x −y =<br />
<br />
x +y =<br />
<br />
x =<br />
<br />
x =<br />
<br />
x =±<br />
xy<br />
<br />
x −y =<br />
<br />
−y =<br />
<br />
−y =<br />
<br />
y =<br />
<br />
− −y =<br />
<br />
−y =<br />
<br />
y =
610<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Th−<br />
<br />
x −y ≤<br />
<br />
x ≤y<br />
<br />
y ≥x <br />
<br />
x +y ≤<br />
<br />
y ≤−x +<br />
Figure 10Th<br />
x +y ≤x −y ≤<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– <br />
<br />
– –<br />
– – –<br />
– –<br />
<br />
Figure 10<br />
<br />
<br />
<br />
<br />
x<br />
Try It #4<br />
<br />
y ≥x −<br />
x−y≥−<br />
<br />
Solve a System of Nonlinear Equations Using Substitution (http://openstaxcollege.org/l/nonlinsub)<br />
Solve a System of Nonlinear Equations Using Elimination (http://openstaxcollege.org/l/nonlinelim)
SECTION 7.3 SECTION EXERCISES 611<br />
7.3 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
<br />
<br />
3. <br />
<br />
<br />
<br />
<br />
5. <br />
<br />
xfie n<br />
2. <br />
<br />
<br />
4. <br />
fi<br />
ALGEBRAIC<br />
<br />
6. x+y=<br />
x +y =<br />
9. y=−x<br />
x +y =<br />
7. y=x −<br />
x +y =<br />
10. x=<br />
x −y =<br />
8. y=x<br />
x +y =<br />
<br />
11. x −y =<br />
x +y =<br />
14. y −x =<br />
x +y =<br />
12. x +y =<br />
x −y =<br />
15. x +y + <br />
=<br />
y=x <br />
13. x +y =<br />
x −y =x −<br />
<br />
16. −x +y=−<br />
x−y=<br />
19. x +y =<br />
y=−x <br />
22. x −x =y<br />
x +y=<br />
17. −x +y=<br />
−x+y=<br />
20. x −x =y<br />
y= <br />
−x<br />
23. x −x =y<br />
x +y=<br />
18. x +y =<br />
y=x −<br />
21. x +y =<br />
x − +y =<br />
<br />
24. x +y =<br />
y=−x <br />
27. x −y =<br />
x−y=<br />
30. x +y =<br />
x −y =<br />
25. x −y =<br />
x=<br />
28. −x +y=<br />
−x+y=−<br />
31. x +y =<br />
y =x <br />
26. x −y =<br />
y=<br />
29. −x +y=<br />
y=−x<br />
32. x −y +=<br />
y +x =
612 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
33. x −y =<br />
x − +y =<br />
36. x −y −x−y−=<br />
−x +y =<br />
34. x −y =<br />
x − +y =<br />
37. x +y −y=<br />
x +y=<br />
35. x −y =<br />
x +y =<br />
38. x +y =<br />
xy=<br />
GRAPHICAL<br />
<br />
39. x +y< 40. x +y <<br />
<br />
41. x +y<<br />
y>x<br />
44. x −y >−<br />
x +y <<br />
42. x +yx+<br />
45. x +y ><br />
x −y <<br />
43. x +y <<br />
x −y ><br />
EXTENSIONS<br />
<br />
46. y≥e x <br />
y≤x+<br />
47. y≤−x<br />
y≤e x <br />
, fi<br />
48. x + <br />
y =<br />
x − <br />
y +=<br />
49. x − <br />
y =<br />
x − <br />
y = <br />
<br />
50. x −xy+y −=<br />
x+y=<br />
51. x −xy −y −=<br />
x +y =<br />
52. x +xy −y −=<br />
x=y+<br />
TECHNOLOGY<br />
fi<br />
53. xy<<br />
y>√ — x <br />
54. x +y<<br />
y>x<br />
REAL-WORLD APPLICATIONS<br />
<br />
<br />
55. <br />
<br />
<br />
56. Tho 360. Th<br />
<br />
<br />
57. <br />
Cx=x −x+<br />
Rx=−x +x+<br />
fi<br />
<br />
fi<br />
58. <br />
Cx=x − x+<br />
Rx=−x +x<br />
fi<br />
fi
SECTION 7.4 PARTIAL FRACTIONS<br />
613<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Decompose P (x )Q (x ), where Q (x ) has only nonrepeated linear factors.<br />
Decompose P (x )Q (x ), where Q (x ) has repeated linear factors.<br />
Decompose P (x )Q (x ), where Q (x ) has a nonrepeated irreducible quadratic factor.<br />
Decompose P (x )Q (x ), where Q (x ) has a repeated irreducible quadratic factor.<br />
7.4 PARTIAL FRACTIONS<br />
<br />
<br />
<br />
Th<br />
<br />
Decomposing P(x ) ____<br />
Q(x )<br />
Where Q(x ) Has Only Nonrepeated Linear Factors<br />
Thfi<br />
fffi<br />
<br />
fi<br />
partial fractions<br />
<br />
<br />
x− + − x+ <br />
d fio fix+x−<br />
d fi<br />
<br />
<br />
x− ( x+ <br />
x+ ) + − x+ ( x− <br />
x− ) =x+−x+ <br />
<br />
x+x− =<br />
x+ <br />
x −x− <br />
<br />
<br />
x+ <br />
<br />
x −x − = <br />
<br />
fi <br />
x − + − x+ <br />
<br />
<br />
<br />
e fi<br />
fi<br />
<br />
x −x −x−x+<br />
<br />
fiTh
614<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
partial fraction decomposition of<br />
____ P(x) : Q(x) has nonrepeated linear factors<br />
Q(x)<br />
Th ____ Px Qx Qx<br />
PxQx<br />
Px <br />
Qx =<br />
A <br />
<br />
( a <br />
x+b <br />
) +<br />
A <br />
<br />
( a <br />
x+b <br />
) +<br />
A <br />
<br />
( a <br />
x+b <br />
) ++<br />
A n<br />
<br />
<br />
( a n<br />
x+b n<br />
) <br />
<br />
How To…<br />
<br />
1. ABC<br />
fiA n<br />
<br />
Px <br />
Qx =<br />
A <br />
<br />
( a <br />
x+b <br />
) +<br />
A <br />
<br />
( a <br />
x+b <br />
) ++<br />
A n<br />
<br />
( a n<br />
x+b n<br />
) <br />
2. <br />
3. <br />
4. ffi <br />
<br />
Example 1 Decomposing a Rational Function with Distinct Linear Factors<br />
<br />
<br />
x<br />
<br />
x+x− <br />
Solution ABC<br />
<br />
x<br />
<br />
x+x− =<br />
A<br />
<br />
x+ +<br />
B<br />
<br />
x − <br />
<br />
<br />
x+x−<br />
[ <br />
x<br />
<br />
x+x − ] = x+x −<br />
[ <br />
A<br />
x+ ] +x+<br />
−<br />
[ <br />
B<br />
x − ] <br />
Th<br />
<br />
x=Ax−+Bx+<br />
<br />
<br />
x=Ax−A+Bx+B<br />
<br />
x=A+Bx−A+B<br />
ffi<br />
<br />
=A+B<br />
<br />
=−A+B<br />
B<br />
<br />
=+B<br />
<br />
=−A+B<br />
<br />
=+B<br />
<br />
=B<br />
B=<br />
<br />
=A+<br />
<br />
=A<br />
Th<br />
<br />
x<br />
<br />
x+x− =<br />
<br />
x+ +<br />
<br />
x−
SECTION 7.4 PARTIAL FRACTIONS<br />
615<br />
AB<br />
xABx=A<br />
B<br />
<br />
x=Ax−+Bx+<br />
<br />
=A−+B+<br />
<br />
=+B<br />
<br />
=B<br />
B=ABx=−<br />
<br />
<br />
x=Ax−+Bx+<br />
<br />
−=A−−+B−+<br />
<br />
−=−A+<br />
<br />
− <br />
− =A<br />
<br />
=A<br />
AB<br />
<br />
x<br />
<br />
x+x− =<br />
<br />
x+ +<br />
<br />
x− <br />
ft<br />
ft<br />
<br />
Try It #1<br />
<br />
x<br />
<br />
x−x− <br />
Decomposing ____ P(x )<br />
Q(x )<br />
Where Q(x ) Has Repeated Linear Factors<br />
<br />
<br />
partial fraction decomposition of<br />
____ P(x) : Q(x) has repeated linear factors<br />
Q(x)<br />
Th Px <br />
Qx Qxn<br />
PxQx<br />
Px <br />
Qx =<br />
A <br />
<br />
ax+b +<br />
A <br />
<br />
ax+b +<br />
A <br />
<br />
ax+b ++<br />
A n<br />
<br />
ax+b n <br />
<br />
How To…<br />
<br />
1. ABC<br />
Px <br />
Qx =<br />
A <br />
<br />
ax+b +<br />
A <br />
<br />
ax+b ++<br />
A n<br />
<br />
ax+b n <br />
2. <br />
3. <br />
4. ffi
616<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 2<br />
Decomposing with Repeated Linear Factors<br />
<br />
<br />
−x +x+ <br />
x −x +x <br />
Solution Thxx− x−<br />
xx−x− <br />
<br />
−x +x+ <br />
<br />
x −x +x =A x<br />
+<br />
B<br />
<br />
x− +<br />
C<br />
<br />
x− <br />
<br />
<br />
<br />
xx− [<br />
−x +x+ <br />
<br />
xx− <br />
]<br />
=<br />
[ A <br />
x +<br />
B<br />
<br />
x− +<br />
C<br />
x− ] xx− <br />
−x +x+=Ax − +Bxx−+Cx<br />
<br />
<br />
−x +x+=Ax −x++Bx −x+Cx<br />
<br />
=Ax −Ax+A+Bx −Bx+Cx<br />
<br />
=A+Bx +−A−B+Cx+A<br />
ffi. Th<br />
<br />
−x +x+=A+Bx +−A −B+Cx+A<br />
A+B=− <br />
−A −B+C= <br />
A= <br />
A<br />
<br />
A=<br />
<br />
A=<br />
A=<br />
<br />
A+B=−<br />
<br />
+B=−<br />
<br />
B=−<br />
ThCAB<br />
<br />
<br />
<br />
<br />
<br />
<br />
−A−B+C=<br />
−−−+C=<br />
−++C=<br />
C=<br />
−x +x+ <br />
x <br />
−x +x = x −<br />
<br />
x− +<br />
<br />
x− <br />
Try It #2<br />
<br />
<br />
x − <br />
x−
SECTION 7.4 PARTIAL FRACTIONS<br />
617<br />
Decomposing P(x ) ____<br />
Q(x )<br />
Where Q(x ) Has a Nonrepeated Irreducible Quadratic Factor<br />
<br />
ABC<br />
Th<br />
Ax+BBx+C<br />
decomposition of<br />
_____ P(x) Q(x) has a nonrepeated irreducible quadratic factor<br />
Q(x):<br />
Th Px <br />
Qx Qx<br />
degree of PxQxwritten as<br />
Px <br />
Qx =<br />
A <br />
x + B <br />
<br />
<br />
<br />
a <br />
x +b <br />
x +c <br />
+<br />
A <br />
x + B <br />
<br />
<br />
<br />
a <br />
x +b <br />
x +c <br />
++<br />
A n<br />
x + B n<br />
<br />
<br />
a n<br />
x +b n<br />
x +c n<br />
<br />
Th<br />
ABC<br />
How To…<br />
<br />
1. ABCA <br />
<br />
x+B <br />
A <br />
x+B <br />
<br />
Px <br />
Qx =<br />
A<br />
<br />
ax+b +<br />
A <br />
x + B <br />
<br />
<br />
<br />
a <br />
x +b <br />
x +c <br />
+<br />
A <br />
x + B <br />
<br />
<br />
<br />
a <br />
x +b <br />
x +c <br />
++<br />
A n<br />
x + B n<br />
<br />
<br />
a n<br />
x +b n<br />
x +c n<br />
<br />
2. <br />
3. <br />
4. ffi <br />
<br />
Example 3<br />
Decomposing ____ P(x) When Q(x) Contains a Nonrepeated Irreducible Quadratic Factor<br />
Q(x)<br />
<br />
<br />
x<br />
+x− <br />
x+x +x+ <br />
Solution <br />
. Th<br />
<br />
x<br />
+x− <br />
x+x +x+ =<br />
A<br />
<br />
x+ +<br />
Bx+C<br />
<br />
x +x+ <br />
<br />
<br />
<br />
<br />
x+x +x+ [<br />
x<br />
+x − <br />
x+x +x+ ]<br />
=<br />
[ <br />
A<br />
<br />
x+ +<br />
Bx+C<br />
x +x+ ] x+x +x+<br />
x +x−=Ax +x++Bx+Cx+
618<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
AxBx+Cx=−<br />
<br />
<br />
x +x−=Ax +x++Bx+Cx+<br />
<br />
− +−−=A− +−++B−+C−+<br />
<br />
=A<br />
A=<br />
ATh<br />
<br />
<br />
<br />
x +x −=x +x++Bx+Cx+<br />
x +x −=x +x++Bx +B+Cx+C<br />
x +x −=+Bx ++B+Cx++C<br />
ffiffi <br />
<br />
+B= <br />
+B+C= <br />
+C=− <br />
BC<br />
+B= <br />
<br />
B=<br />
+C=− <br />
<br />
C=−<br />
<br />
C=−<br />
Th<br />
<br />
x<br />
+x− <br />
x+x +x+ =<br />
<br />
x+ +<br />
x− <br />
x +x+ <br />
Q & A…<br />
Could we have just set up a system of equations to solve Example 3?<br />
AfiTh<br />
<br />
<br />
x +x −=Ax +Ax+A+Bx +B+Cx+C<br />
<br />
x +x −=A+Bx +A+B+Cx+A+C<br />
<br />
A+B=<br />
A+B+C=<br />
<br />
A+C=−<br />
Try It #3<br />
<br />
x −x+ <br />
x−x +
SECTION 7.4 PARTIAL FRACTIONS<br />
619<br />
Decomposing P(x ) ____<br />
Q(x )<br />
Where Q(x ) Has a Repeated Irreducible Quadratic Factor<br />
fi<br />
fi<br />
Th<br />
<br />
decomposition of<br />
____ P(x) : when Q(x) has a repeated irreducible quadratic factor<br />
Q(x)<br />
Th Px <br />
Qx Qx<br />
PxQx<br />
Px<br />
<br />
ax <br />
+bx +c n =<br />
A <br />
x + B <br />
<br />
<br />
<br />
ax +bx +c +<br />
A <br />
x + B <br />
<br />
<br />
<br />
ax +bx +c +<br />
A <br />
x + B <br />
<br />
<br />
<br />
ax +bx +c ++<br />
A n<br />
x + B n<br />
<br />
<br />
<br />
ax +bx +c n <br />
<br />
How To…<br />
<br />
1. ABCA <br />
x+B <br />
<br />
A <br />
x+B <br />
<br />
Px <br />
Qx =<br />
A<br />
<br />
ax+b +<br />
A <br />
x + B <br />
<br />
<br />
<br />
ax +bx +c +<br />
A <br />
x + B <br />
<br />
<br />
<br />
ax +bx +c ++<br />
A n<br />
x + B n<br />
<br />
<br />
<br />
ax +bx +c n <br />
2. <br />
3. <br />
4. ffi <br />
<br />
Example 4<br />
Decomposing a Rational Function with a Repeated Irreducible Quadratic Factor in the Denominator<br />
<br />
x +x +x −x+ <br />
xx + <br />
Solution Thxx +x + <br />
Ax+B<br />
<br />
<br />
x +x +x −x+ <br />
<br />
xx + =A x<br />
+Bx+C +<br />
Dx+E<br />
<br />
x + x + <br />
xx + <br />
<br />
x +x +x −x+=Ax + +Bx+Cxx ++Dx+Ex<br />
<br />
x +x +x −x+=Ax +x ++Bx +Bx +Cx +Cx+Dx +Ex<br />
<br />
=Ax +Ax +A+Bx +Bx +Cx +Cx+Dx +Ex<br />
<br />
x +x +x −x+=A+Bx +Cx +A+B+Dx +C+Ex+A
620<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
ffi<br />
<br />
A+B=<br />
<br />
C=<br />
<br />
A+B+D=<br />
<br />
C+E=−<br />
<br />
A=<br />
A=e fi<br />
<br />
+B=<br />
<br />
B=<br />
A=B=<br />
<br />
++D=<br />
<br />
D=−<br />
C=<br />
<br />
+E=−<br />
<br />
E=−<br />
A=B=C=D=−<br />
E=−<br />
x +x +x −x+ <br />
<br />
xx + = x +<br />
<br />
x + −<br />
x+ <br />
x + <br />
Try It #4<br />
<br />
<br />
x −x +x− <br />
x −x+ <br />
<br />
Partial Fraction Decomposition (http://openstaxcollege.org/l/partdecomp)<br />
Partial Fraction Decomposition With Repeated Linear Factors (http://openstaxcollege.org/l/partdecomprlf)<br />
Partial Fraction Decomposition With Linear and Quadratic Factors (http://openstaxcollege.org/l/partdecomlqu)
SECTION 7.4 SECTION EXERCISES 621<br />
7.4 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. <br />
<br />
2. <br />
<br />
<br />
4. <br />
<br />
<br />
5. <br />
<br />
<br />
x+<br />
<br />
x +x+ =<br />
A<br />
<br />
x+ +<br />
B<br />
<br />
x+ <br />
x+=Ax++Bx+<br />
<br />
xAB<br />
AB<br />
ALGEBRAIC<br />
, fi<br />
x+<br />
6. <br />
x +x+ <br />
7. <br />
x − <br />
x −x− <br />
8. <br />
−x− <br />
x −x− <br />
x+<br />
9. <br />
x +x+ <br />
x+<br />
12. <br />
x +x+ <br />
15. x<br />
<br />
x − <br />
x+<br />
18. <br />
x +x+ <br />
10. <br />
x<br />
<br />
x +x+ <br />
13. x<br />
<br />
x − <br />
16. <br />
x− <br />
x −x+ <br />
19. <br />
x− <br />
x −x+ <br />
11. <br />
x−<br />
<br />
x −x+ <br />
14. <br />
x<br />
<br />
x − <br />
17. x− x −x− <br />
, fi<br />
20. −x− <br />
x+ <br />
21. <br />
x<br />
<br />
x− <br />
22. x+ <br />
x+ <br />
23. −x− <br />
x+ <br />
24. −x− <br />
x− <br />
25. <br />
−x<br />
<br />
x− <br />
x+<br />
26. <br />
x +x+ <br />
27. x +x+ <br />
xx+ 28. x +x+ <br />
xx+ <br />
29. x +x +x+ <br />
x x+ 30. x −x +x+ <br />
x x +x+
622 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
fi<br />
<br />
x<br />
31. +x+ <br />
x+x +x+ <br />
x<br />
34. +x+ <br />
x+x +x− <br />
32. <br />
x +x+ <br />
x−x +x+ <br />
35. <br />
x +x− <br />
x+x +x+ <br />
33. <br />
−x +x+ <br />
x−x +x+ <br />
36. <br />
x<br />
<br />
<br />
x+x +x− <br />
37. x +x+ x − 38. −x +x− <br />
x + 39. x −x+ <br />
x + <br />
40. x +x+ x − 41. x +x+ <br />
x − 42. −x +x− <br />
x − <br />
43. −x −x +x+ <br />
<br />
x +x<br />
fi<br />
44. x +x +x+ <br />
x + 45. x +x +x+ <br />
x + 46. x −x +x− <br />
x − <br />
47. x +x+ x+ 48. x +x +x<br />
<br />
x +x+ <br />
49. <br />
x +<br />
<br />
x +x+ <br />
50. x +x+x+ <br />
x +x+ 51. <br />
x+ <br />
xx + <br />
52. x +x +x +x+ <br />
xx + <br />
53. x − <br />
x −x <br />
54. x −x+ x +x <br />
EXTENSIONS<br />
, fi<br />
55. x + <br />
x+ <br />
56. x −x +x+ <br />
x − <br />
n fi<br />
57. <br />
x+ +<br />
<br />
x − −<br />
x − <br />
x −x − <br />
58. <br />
<br />
x − −<br />
<br />
x+ −<br />
x+ <br />
x +x − <br />
59. x<br />
<br />
x − −<br />
−x<br />
<br />
x +x+ − x− x −x
SECTION 7.5 MATRICES AND MATRIX OPERATIONS<br />
623<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the sum and difference of two matrices.<br />
Find scalar multiples of a matrix.<br />
Find the product of two matrices.<br />
7.5 MATRICES AND MATRIX OPERATIONS<br />
Figure 1 (credit: “SD Dirk,” Flickr)<br />
<br />
Table 1<br />
Wildcats Mud Cats<br />
<br />
<br />
<br />
Table 1<br />
fi<br />
<br />
. Th<br />
Finding the Sum and Difference of Two Matrices<br />
<br />
<br />
<br />
ABC<br />
<br />
A= [ <br />
] B =[ <br />
− <br />
] C = <br />
− <br />
[ <br />
]
624<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Describing Matrices<br />
ftm ×nmn<br />
fifiAfia ij<br />
<br />
ijAa <br />
<br />
<br />
a <br />
a <br />
a <br />
A= a<br />
<br />
[ <br />
a <br />
a<br />
a <br />
a <br />
a <br />
] <br />
n ×nTh<br />
×<br />
×n<br />
a <br />
a <br />
a <br />
<br />
m ×<br />
<br />
a <br />
<br />
a [ <br />
a <br />
] <br />
ffi<br />
ft<br />
fi<br />
<br />
matrices<br />
matrixABC<br />
a ij<br />
ijft<br />
m ×nmn<br />
Example 1<br />
Finding the Dimensions of the Given Matrix and Locating Entries<br />
A<br />
a. A<br />
b. a <br />
a <br />
<br />
<br />
A<br />
<br />
=[ <br />
] −<br />
Solution<br />
a. Th×<br />
b. a <br />
Tha <br />
<br />
<br />
Adding and Subtracting Matrices<br />
<br />
<br />
Thaddition and subtraction of matrices is only possible when the<br />
matrices have the same dimensions××<br />
××
SECTION 7.5 MATRICES AND MATRIX OPERATIONS<br />
625<br />
adding and subtracting matrices<br />
ABABC<br />
D<br />
+B =a ij<br />
+b ij<br />
=c ij<br />
<br />
A−B =Da ij<br />
−b ij<br />
=d ij<br />
<br />
<br />
A +B =B +A<br />
<br />
<br />
A+B+C =A +B+C<br />
Example 2<br />
AB<br />
Finding the Sum of Matrices<br />
A = [ a b <br />
Solution <br />
A +B = [ a b <br />
h ]<br />
<br />
c d ] B = [<br />
e f<br />
g<br />
h ]<br />
<br />
c d ] + [<br />
e f<br />
g<br />
= [<br />
a + e b + f<br />
c + g d + h ]<br />
<br />
Example 3<br />
AB<br />
Adding Matrix A and Matrix B<br />
A = [ <br />
<br />
] B = [ ] <br />
Solution a <br />
A<br />
n 1b <br />
B<br />
A +B = [ <br />
<br />
] + [ ] <br />
= [<br />
+ + + + ]<br />
<br />
= [ ] <br />
Example 4<br />
ffAB<br />
Finding the Difference of Two Matrices<br />
A = [ − <br />
Solution <br />
A −B = [ − <br />
<br />
] B = [ ] <br />
<br />
] − [ ] <br />
= [ −− − <br />
− − ] <br />
=[ − − − ]
626<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 5 Finding the Sum and Difference of Two 3 x 3 Matrices<br />
AB<br />
a. <br />
b. e diff<br />
<br />
− −<br />
A=[ <br />
<br />
<br />
− <br />
] [ B = <br />
<br />
<br />
−<br />
− − ] <br />
<br />
− −<br />
a. <br />
−<br />
A +B =[<br />
−<br />
<br />
<br />
<br />
] <br />
[ +<br />
<br />
<br />
− −<br />
− ] <br />
<br />
− − −<br />
+ −+<br />
<br />
−−<br />
=[ <br />
+ − <br />
−<br />
<br />
− − + −<br />
−<br />
<br />
=[ <br />
<br />
−<br />
] <br />
− −<br />
b. <br />
−<br />
A −B =[<br />
−<br />
<br />
<br />
<br />
] <br />
[ −<br />
<br />
<br />
− −<br />
− ] <br />
<br />
− − −<br />
− −−<br />
<br />
−+<br />
=[ <br />
− + <br />
+<br />
<br />
+ −− +<br />
− − <br />
<br />
<br />
=[ <br />
<br />
<br />
− <br />
] <br />
] <br />
] <br />
Try It #1<br />
AB<br />
<br />
A= [<br />
<br />
<br />
<br />
<br />
− ] <br />
B= [<br />
<br />
−<br />
−<br />
<br />
] Finding Scalar Multiples of a Matrix<br />
<br />
<br />
Th<br />
scalar multiple<br />
<br />
Th<br />
. Th Table 2<br />
Lab A<br />
Lab B<br />
<br />
<br />
<br />
Table 2
SECTION 7.5 MATRICES AND MATRIX OPERATIONS<br />
627<br />
<br />
<br />
C <br />
<br />
=[ <br />
] <br />
C<br />
<br />
C <br />
<br />
=[ <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
[ ] <br />
<br />
<br />
] [ + <br />
<br />
<br />
] [ = ] <br />
<br />
[ <br />
<br />
<br />
Th<br />
<br />
C <br />
<br />
=[ <br />
] <br />
<br />
] [ = <br />
] ThAB<br />
<br />
scalar multiplication<br />
s fi<br />
A =<br />
[ a a <br />
a <br />
a <br />
] <br />
cA<br />
cA=c<br />
[ a a <br />
a <br />
a <br />
] <br />
=<br />
[ ca ca <br />
ca <br />
ca <br />
] <br />
ABCab<br />
<br />
aA+B=aA+aB<br />
<br />
a+bA=aA+bA<br />
<br />
<br />
Example 6 Multiplying the Matrix by a Scalar<br />
A<br />
A = [ ] <br />
Solution A<br />
A = [ ] <br />
= [ ċ ċ ċ ċ ] <br />
=[ ]
628<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Try It #2<br />
B, fi−B<br />
A= [ ] <br />
Example 7 Finding the Sum of Scalar Multiples<br />
A +B<br />
<br />
− <br />
A=[ − <br />
−<br />
Solution , fiAB<br />
<br />
ċ −<br />
A<br />
ċ<br />
=[ ċ<br />
− ċ<br />
] <br />
<br />
ċ ċ −<br />
<br />
− <br />
<br />
=[ − <br />
] <br />
−<br />
<br />
− ċ<br />
B<br />
ċ<br />
=[ ċ<br />
− ċ<br />
<br />
ċ ċ −<br />
− <br />
] [ B = −<br />
] ] <br />
<br />
<br />
−<br />
<br />
− <br />
<br />
<br />
=[ − <br />
<br />
−<br />
A +B<br />
<br />
− −<br />
<br />
A +B =[ − <br />
] [ + <br />
<br />
− <br />
] − −<br />
<br />
− −+ +<br />
<br />
=[ + −−<br />
+<br />
] <br />
<br />
+ + −−<br />
<br />
− <br />
<br />
=[ − <br />
] <br />
−<br />
Finding the Product of Two Matrices<br />
] <br />
<br />
fi<br />
Am ×rBr ×n<br />
ABm ×nABA<br />
Bfi<br />
A ⋅ B<br />
× ×<br />
<br />
ABfiTh<br />
<br />
iABiAjB<br />
ABA×B×AB<br />
a 2 ×<br />
A =<br />
[ a a b<br />
<br />
a <br />
b <br />
b <br />
<br />
<br />
a <br />
a <br />
a <br />
] [ B = <br />
b <br />
b <br />
b<br />
<br />
b <br />
b <br />
b <br />
]
SECTION 7.5 MATRICES AND MATRIX OPERATIONS<br />
629<br />
e fiAB<br />
1. AB A B<br />
b <br />
<br />
a <br />
a <br />
a <br />
<br />
[ <br />
b<br />
<br />
b <br />
] =a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
<br />
2. AB AB<br />
b <br />
<br />
a <br />
a <br />
a <br />
<br />
[ <br />
b<br />
<br />
b <br />
] =a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
<br />
3. AB AB<br />
b <br />
<br />
a <br />
a <br />
a <br />
<br />
[ <br />
b<br />
<br />
b <br />
] =a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
<br />
ABABA<br />
BAB<br />
AB= [ a ⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
<br />
<br />
<br />
<br />
<br />
<br />
a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
a <br />
⋅b <br />
+a <br />
⋅b <br />
+a <br />
⋅b <br />
] <br />
properties of matrix multiplication<br />
ABC<br />
<br />
ABC=ABC<br />
CA + B = CA + CB,<br />
<br />
A + BC = AC + BC.<br />
<br />
Example 8 Multiplying Two Matrices<br />
AB<br />
= [ <br />
<br />
] B = [ ] <br />
Solution A×B<br />
×ThTh×<br />
<br />
AB= [ ] [ ] <br />
= [<br />
+ + <br />
+ + ]<br />
<br />
= [ ] <br />
Example 9<br />
Multiplying Two Matrices<br />
AB<br />
a. AB b. BA<br />
A =[ − <br />
] B = [ <br />
] <br />
<br />
−<br />
−
630<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Solution<br />
a. A×B×<br />
ABTh×<br />
AB<br />
AB=[ − −<br />
][ <br />
<br />
−<br />
<br />
] <br />
= [<br />
−+−+ −−++ <br />
<br />
+−+ −++ ]<br />
<br />
=[ − ] <br />
b. ThB×A×Th<br />
×<br />
−<br />
BA=[ <br />
<br />
−<br />
<br />
] [ − <br />
] <br />
−+− +−<br />
<br />
+−<br />
=[ <br />
−−+<br />
−+ −+<br />
<br />
<br />
−+ + +<br />
− <br />
<br />
<br />
=[ <br />
− <br />
−<br />
<br />
Analysis Notice that the products AB and BA are not equal.<br />
AB=[ − − <br />
] [ ≠ <br />
− −<br />
] =BA<br />
<br />
This illustrates the fact that matrix multiplication is not commutative.<br />
] <br />
] <br />
Q & A…<br />
Is it possible for AB to be defined but not BA?<br />
A×B×AB<br />
fiBA<br />
fi<br />
Example 10<br />
Using Matrices in Real-World Problems<br />
Table 3,<br />
<br />
Wildcats Mud Cats<br />
<br />
<br />
<br />
Table 3<br />
Table 4.<br />
Goals<br />
Balls<br />
Jerseys<br />
Table 4
SECTION 7.5 MATRICES AND MATRIX OPERATIONS<br />
631<br />
. Th<br />
<br />
<br />
E =[ <br />
] <br />
Th<br />
C = <br />
<br />
] <br />
<br />
CE= [ <br />
<br />
<br />
<br />
=++++<br />
<br />
=<br />
Th<br />
How To…<br />
<br />
1. ABC<br />
2. <br />
3. fifi<br />
<br />
Example 11<br />
Using a Calculator to Perform Matrix Operations<br />
AB−C<br />
<br />
− <br />
A=[ − <br />
−<br />
−<br />
] B = [ <br />
<br />
−<br />
− <br />
− −<br />
− − −<br />
] [ C = <br />
− <br />
] <br />
<br />
− −<br />
Solution AAB<br />
BCC<br />
<br />
A×B−C<br />
Th<br />
[ <br />
] <br />
−<br />
<br />
− <br />
<br />
−<br />
<br />
− <br />
<br />
Dimensions of a Matrix (http://openstaxcollege.org/l/matrixdimen)<br />
Matrix Addition and Subtraction (http://openstaxcollege.org/l/matrixaddsub)<br />
Matrix Operations (http://openstaxcollege.org/l/matrixoper)<br />
Matrix Multiplication (http://openstaxcollege.org/l/matrixmult)
632 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
7.5 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. ABBA <br />
<br />
5. e? Th<br />
AB=BA<br />
<br />
2. <br />
<br />
4. <br />
<br />
<br />
<br />
ALGEBRAIC<br />
<br />
fi<br />
A=[ <br />
<br />
] B = [ ] [ C = <br />
<br />
<br />
<br />
<br />
] [ D = <br />
] E = <br />
[ <br />
<br />
<br />
<br />
] F = [ <br />
] <br />
<br />
<br />
<br />
<br />
<br />
6. A+B 7. C+D 8. A+C 9. B−E 10. C+ 11. D−B<br />
<br />
A=[ <br />
<br />
] B = [ <br />
<br />
<br />
<br />
<br />
<br />
] C = <br />
[ <br />
<br />
<br />
<br />
<br />
] D = [ <br />
] <br />
12. A 13. B 14. −B 15. −C 16. <br />
C<br />
17. D<br />
<br />
A=[ − <br />
<br />
<br />
] B = [ − ] [ C = <br />
<br />
<br />
− <br />
<br />
<br />
] D = [ <br />
] <br />
<br />
− <br />
<br />
−<br />
18. AB 19. BC 20. CA 21. BD 22. DC 23. CB<br />
<br />
<br />
A=[ − <br />
] B = [ − <br />
− ] C =[ <br />
<br />
<br />
<br />
] D = [ − −<br />
<br />
] [ E = <br />
− −<br />
] <br />
24. A + B − C 25. A + D 26. C + B 27. D + E 28. C − D 29. D − E
SECTION 7.5 SECTION EXERCISES 633<br />
<br />
A =A ċA<br />
A=[ − <br />
<br />
<br />
− <br />
<br />
− <br />
] B = <br />
[ − ] [ C = <br />
] <br />
30. AB 31. BA 32. CA 33. BC 34. A 35. B <br />
36. C 37. B A 38. A B 39. AB 40. BA <br />
<br />
A =A ċA<br />
A=[ <br />
<br />
] B = [ − − − ] [ C = <br />
<br />
<br />
<br />
− <br />
] D = [ <br />
] <br />
<br />
−<br />
− <br />
<br />
41. AB 42. BA 43. BD 44. DC 45. D 46. A <br />
47. D 48. ABC 49. ABC<br />
TECHNOLOGY<br />
<br />
<br />
<br />
− <br />
A=[ −<br />
<br />
] B = [ <br />
<br />
<br />
− <br />
<br />
<br />
] [ C = <br />
] <br />
50. AB 51. BA 52. CA 53. BC 54. ABC<br />
EXTENSIONS<br />
<br />
<br />
= [<br />
<br />
] 55. B 56. B 57. B 58. B <br />
59. B n B B
634<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Write the augmented matrix of a system of equations.<br />
Write the system of equations from an augmented matrix.<br />
Perform row operations on a matrix.<br />
Solve a system of linear equations using matrices.<br />
7. 6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION<br />
Figure 1 German mathematician Carl Friedrich Gauss (1777–1855).<br />
<br />
fifi<br />
ff<br />
<br />
fiSystems of Linear Equations: Two Variables<br />
<br />
Writing the Augmented Matrix of a System of Equations<br />
<br />
ffi<br />
ffi<br />
augmented matrix<br />
×<br />
<br />
x +y =<br />
<br />
x −y =<br />
<br />
[ − ∣ ] <br />
ffi. Thcoefficient matrix<br />
[ − ]
SECTION 7.6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION<br />
635<br />
<br />
<br />
<br />
<br />
ffi<br />
<br />
[ <br />
x −y −z =<br />
[ <br />
x +y =<br />
x −z =<br />
] <br />
<br />
− −<br />
<br />
−<br />
] <br />
<br />
− − <br />
<br />
<br />
∣<br />
− <br />
xfi<br />
yz<br />
ax+by+cz=d<br />
ffi<br />
How To…<br />
<br />
1. ffixe fi<br />
2. ffiy<br />
3. zffi<br />
4. <br />
Example 1 Writing the Augmented Matrix for a System of Equations<br />
<br />
<br />
x +y −z =<br />
<br />
x −y +z =<br />
<br />
x −y +z =<br />
Solution Thffi<br />
<br />
−<br />
[ − <br />
<br />
∣ <br />
] <br />
− <br />
Try It #1<br />
<br />
<br />
x −y =<br />
<br />
x+y=<br />
Writing a System of Equations from an Augmented Matrix<br />
<br />
<br />
fi
636<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 2 Writing a System of Equations from an Augmented Matrix Form<br />
<br />
[ <br />
] <br />
<br />
− − −<br />
− −<br />
<br />
∣ <br />
− <br />
Solution xyz<br />
<br />
− −<br />
[ − −<br />
<br />
−<br />
∣ <br />
] →<br />
<br />
x −y −z =−<br />
x −y −z =<br />
<br />
<br />
− −x +y +z =<br />
Try It #2<br />
<br />
[ <br />
] <br />
<br />
− <br />
− <br />
<br />
<br />
∣<br />
<br />
−<br />
Performing Row Operations on a Matrix<br />
row operations<br />
<br />
<br />
row-echelon formmain diagonal<br />
<br />
<br />
a b<br />
[ <br />
<br />
d<br />
] <br />
<br />
row-equivalent<br />
<br />
1. leading<br />
2. <br />
3. <br />
4. <br />
ffi<br />
o fi<br />
1. R i<br />
↔R j<br />
<br />
2. cR i<br />
<br />
3. R i<br />
+cR j<br />
<br />
<br />
<br />
fi<br />
fi
SECTION 7.6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION<br />
637<br />
Gaussian elimination<br />
Gaussian eliminationTh<br />
A<br />
<br />
a <br />
a <br />
a <br />
A= a<br />
<br />
[ <br />
a <br />
a<br />
] =[ ft A <br />
<br />
b <br />
b <br />
<br />
b<br />
a <br />
a <br />
a <br />
] <br />
<br />
<br />
<br />
<br />
How To…<br />
<br />
1. The fiffi<br />
2. e fie fi<br />
3. <br />
4. <br />
5. <br />
6. <br />
<br />
7. <br />
Example 3<br />
Solving a 2 × 2 System by Gaussian Elimination<br />
<br />
<br />
x +y =<br />
<br />
x −y = <br />
<br />
Solution <br />
<br />
[ −<br />
]<br />
<br />
. Th<br />
∣ <br />
R <br />
↔R <br />
→ [ − ∣ <br />
] <br />
fiTh<br />
−<br />
−R <br />
+R <br />
=R <br />
→ [ − ∣ <br />
] <br />
<br />
<br />
<br />
<br />
R <br />
=R <br />
→ [ − ∣ <br />
] <br />
. Thy =y =e fi<br />
<br />
<br />
Th( <br />
) <br />
x −= <br />
<br />
x =
638<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Try It #3<br />
<br />
<br />
x +y =<br />
x−=−<br />
Example 4<br />
Using Gaussian Elimination to Solve a System of Equations<br />
×<br />
<br />
x +y =<br />
<br />
x +y =<br />
Solution <br />
[ ∣ ] <br />
. The fi <br />
<br />
<br />
<br />
R <br />
=R <br />
→[ <br />
∣ <br />
] <br />
−<br />
−R <br />
+R <br />
=R <br />
→[ <br />
∣ <br />
] <br />
Th=. Th<br />
Example 5<br />
<br />
<br />
<br />
Solving a Dependent System<br />
x +y =<br />
x +y =<br />
Solution <br />
A =[ ∣ ] <br />
<br />
− <br />
R <br />
+R <br />
=R <br />
→ [ ∣ ] <br />
R <br />
↔R <br />
→[ ∣ ] <br />
Thy =Thfi<br />
fio fiy<br />
<br />
x +y =<br />
<br />
y =−x<br />
<br />
( x− <br />
x ) <br />
y =− <br />
x<br />
Example 6 Performing Row Operations on a 3×3 Augmented Matrix to Obtain Row-Echelon Form<br />
<br />
[ <br />
] <br />
<br />
− <br />
− <br />
∣<br />
− <br />
Solution ThfiTh−w
SECTION 7.6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION<br />
639<br />
Th<br />
− <br />
−R <br />
+R <br />
=R <br />
→ [ <br />
<br />
<br />
−<br />
<br />
∣<br />
<br />
] <br />
− <br />
<br />
<br />
−<br />
R <br />
+R <br />
=R <br />
→<br />
<br />
[ −<br />
<br />
<br />
∣<br />
<br />
] <br />
− <br />
<br />
<br />
− <br />
<br />
R <br />
+R <br />
=R <br />
→ [ −<br />
<br />
<br />
∣<br />
<br />
] <br />
<br />
Th<br />
<br />
− <br />
<br />
R <br />
=R <br />
→[ <br />
−<br />
<br />
<br />
∣<br />
− <br />
<br />
] <br />
<br />
Try It #4<br />
<br />
x −y +z =<br />
<br />
−x+y =−<br />
<br />
x−y+z=<br />
Solving a System of Linear Equations Using Matrices<br />
<br />
<br />
Th<br />
<br />
Example 7<br />
Solving a System of Linear Equations Using Matrices<br />
<br />
<br />
x −y +z =<br />
<br />
x +y −z =−<br />
<br />
x −y −z =<br />
Solution <br />
<br />
− <br />
−<br />
<br />
<br />
∣<br />
− <br />
− − <br />
<br />
− <br />
−R <br />
+R <br />
=R <br />
→ [ <br />
<br />
−<br />
<br />
<br />
∣<br />
− <br />
− − <br />
ThR <br />
R <br />
<br />
− <br />
R <br />
R <br />
<br />
→[ <br />
− −<br />
− −<br />
[ <br />
] <br />
] −R <br />
+R <br />
=R <br />
→ [ <br />
] <br />
] <br />
<br />
− <br />
<br />
−<br />
<br />
<br />
∣ − <br />
<br />
− −
640<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
<br />
− <br />
−R <br />
+R <br />
=R <br />
→[ <br />
−<br />
<br />
<br />
∣<br />
− ] − R <br />
=R <br />
→[ <br />
<br />
− <br />
<br />
−<br />
<br />
<br />
∣<br />
− <br />
<br />
<br />
Th<br />
<br />
x −y +z =<br />
<br />
y −z =−<br />
<br />
z =<br />
−<br />
Example 8<br />
Solving a Dependent System of Linear Equations Using Matrices<br />
<br />
<br />
−x−y +z =−<br />
<br />
x +y =<br />
<br />
y −z =<br />
Solution <br />
− − <br />
[ <br />
−<br />
<br />
<br />
<br />
∣ <br />
− <br />
−. Th<br />
<br />
− − −<br />
−R <br />
→<br />
<br />
[ <br />
<br />
<br />
∣ <br />
] <br />
− <br />
<br />
<br />
R <br />
↔R <br />
→<br />
−<br />
[ −<br />
<br />
∣<br />
<br />
<br />
<br />
−R <br />
+R <br />
=R <br />
→<br />
−<br />
[ −<br />
<br />
∣<br />
− <br />
−<br />
<br />
R <br />
+R <br />
=R <br />
→[ <br />
<br />
<br />
−<br />
<br />
∣ <br />
<br />
Th<br />
<br />
x +y −z =<br />
<br />
y −z =<br />
<br />
=<br />
=fifi<br />
yfiz<br />
x<br />
<br />
x +y −z =<br />
<br />
y =<br />
<br />
x +z−z =<br />
<br />
x +z =<br />
<br />
] <br />
z = −x<br />
<br />
<br />
] <br />
] <br />
] <br />
]
SECTION 7.6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION<br />
641<br />
zyx<br />
<br />
y −z =<br />
<br />
z = −x<br />
<br />
<br />
<br />
y −( −x<br />
) =<br />
<br />
y = −x<br />
<br />
Th( <br />
<br />
x−x <br />
−x ) <br />
Try It #5<br />
<br />
x +y −z =<br />
<br />
x +y +z =<br />
x+y−z=<br />
Q & A…<br />
Can any system of linear equations be solved by Gaussian elimination?<br />
<br />
How To…<br />
<br />
1. ABC<br />
2. ref(<br />
Example 9<br />
<br />
<br />
<br />
<br />
Solving Systems of Equations with Matrices Using a Calculator<br />
x +y +z =−<br />
−x +y −z =−<br />
−x−y +z =<br />
Solution <br />
<br />
−<br />
<br />
− −<br />
<br />
∣<br />
− <br />
− − −<br />
A<br />
−<br />
A=[ <br />
<br />
− − −<br />
] <br />
<br />
− − <br />
ref(A<br />
<br />
A<br />
<br />
[<br />
<br />
]<br />
x + y + z =− <br />
<br />
− → y + <br />
z =− <br />
− <br />
z =− <br />
( <br />
− <br />
− <br />
) <br />
[ <br />
]
642<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 10<br />
Applying 2 × 2 Matrices to Finance<br />
<br />
Th<br />
Solution x =<br />
y =<br />
<br />
x +y =<br />
<br />
x +y =<br />
<br />
[ <br />
∣ ] <br />
−<br />
[ ∣ ] <br />
<br />
<br />
y =<br />
<br />
y =<br />
−=<br />
Th<br />
Example 11 Applying 3 × 3 Matrices to Finance<br />
<br />
ThTh<br />
<br />
Solution xy<br />
z. Th<br />
<br />
x +y +z =<br />
<br />
x +y +z =<br />
<br />
x −z =<br />
<br />
<br />
<br />
[ <br />
<br />
∣<br />
<br />
] − <br />
<br />
<br />
<br />
−R <br />
+R <br />
=R <br />
<br />
→[ <br />
<br />
∣ <br />
] − <br />
<br />
<br />
−R <br />
+R <br />
=R <br />
→<br />
<br />
[ <br />
<br />
<br />
∣<br />
<br />
− − −<br />
<br />
<br />
<br />
<br />
R <br />
=R <br />
→[ <br />
<br />
∣ <br />
− − −<br />
[<br />
<br />
<br />
<br />
R <br />
+R <br />
=R <br />
→ <br />
<br />
∣<br />
− <br />
<br />
] −<br />
Th− z =−z =<br />
<br />
] <br />
]
SECTION 7.6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION<br />
643<br />
Thy + z =z =<br />
<br />
<br />
y + <br />
=<br />
<br />
y +=<br />
<br />
y =<br />
The fix +y +z =y =z =<br />
<br />
x ++=<br />
<br />
x =<br />
Th<br />
Try It #6<br />
<br />
t 10%. Th<br />
<br />
<br />
<br />
<br />
Solve a System of Two Equations Using an Augmented Matrix (http://openstaxcollege.org/l/system2augmat)<br />
Solve a System of Three Equations Using an Augmented Matrix (http://openstaxcollege.org/l/system3augmat)<br />
Augmented Matrices on the Calculator (http://openstaxcollege.org/l/augmatcalc)
644 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
7.6 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. <br />
o diff<br />
<br />
2. <br />
<br />
<br />
4. <br />
<br />
<br />
[ − ∣ ] <br />
5. <br />
<br />
ALGEBRAIC<br />
<br />
6. x−y=<br />
x+y=<br />
9. x+y+z=<br />
x+y=<br />
x+y+z=−<br />
7. y=<br />
x−y=<br />
10. x+y+z=<br />
x−y+z=−<br />
x+y=−<br />
8. x+y+z=<br />
−x+y+z=<br />
x+z=<br />
<br />
<br />
∣ ] <br />
11. [ − − ∣ ] 12. [ <br />
<br />
14. [ <br />
<br />
<br />
− <br />
<br />
∣<br />
<br />
] 15. [ <br />
] <br />
<br />
−<br />
<br />
<br />
<br />
∣<br />
<br />
− −<br />
<br />
16. <br />
[ ∣ <br />
<br />
<br />
<br />
] <br />
[ 13. <br />
<br />
− − <br />
<br />
<br />
∣<br />
− <br />
<br />
− ∣ − ] <br />
] 17. <br />
[ ∣ ] 18. <br />
[ ∣ ] 19. <br />
[ − <br />
<br />
20. <br />
[ − ∣ − ] <br />
21. x−y=−<br />
x+y=<br />
22. x+y=−<br />
x+y=−<br />
23. x+y=<br />
x+y=<br />
24. −x−y=−<br />
x−y=−<br />
25. −x+y=<br />
x+y=<br />
26. x+y=<br />
−x−y=−<br />
27. −x+y=<br />
x−y=−<br />
28. x+y=<br />
x+y=<br />
32. <br />
x− <br />
y=−<br />
<br />
x+ <br />
y=<br />
29. x−y=<br />
x+y=<br />
33. <br />
[ <br />
<br />
<br />
∣<br />
<br />
] 34. <br />
30. −x−y=<br />
−x−y=<br />
<br />
<br />
[ ∣<br />
<br />
<br />
<br />
−<br />
] 35. <br />
31. <br />
x− <br />
y=<br />
<br />
x+ <br />
y=<br />
<br />
<br />
[ ∣<br />
]
SECTION 7.6 SECTION EXERCISES 645<br />
36. <br />
<br />
− − 37. −x+y−z=<br />
[ − <br />
<br />
∣<br />
<br />
] x+y−z=<br />
−<br />
x−y+z=−<br />
39. x+y+z=<br />
−x−y−z=−<br />
x+y+z=<br />
40. x+y−z=<br />
−x−y+z=−<br />
x+y−z=<br />
38. x+y−z=−<br />
x−y−z=<br />
x+y+z=<br />
41. x+y−z=<br />
−x−y+z=−<br />
x+y−z=<br />
42. x+y=<br />
x+z=<br />
−y−z=−<br />
45. − <br />
x+ <br />
y+ <br />
z=− <br />
<br />
<br />
x− <br />
y+ <br />
z=<br />
<br />
x+ <br />
y+ <br />
z= <br />
<br />
43. x+y+z=<br />
x+z=<br />
−y+z=<br />
46. − <br />
x− <br />
y+ <br />
z=− <br />
<br />
<br />
x+ <br />
y− <br />
z= <br />
<br />
− <br />
x+ <br />
y+ <br />
z=− <br />
<br />
44. <br />
x− <br />
z=− <br />
<br />
<br />
x+ <br />
y= <br />
<br />
<br />
y− <br />
z= <br />
<br />
EXTENSIONS<br />
<br />
47. x− <br />
+y − <br />
+z − <br />
=<br />
x+y+z=<br />
x+ <br />
+y+z− <br />
=<br />
48. x− <br />
−y+ <br />
+z=−<br />
x+ <br />
+y+ <br />
−z=<br />
x+y− z− <br />
=<br />
49. x− <br />
−y− <br />
+z=−<br />
x+ <br />
+y+ <br />
+z+ <br />
=<br />
x+y+z=<br />
50. x− <br />
+y+ <br />
−z= 51. x− <br />
−y− <br />
+z=−<br />
x+ <br />
−y− <br />
+z= <br />
x+ <br />
+y+ <br />
+z+ <br />
=<br />
x− <br />
+y+ <br />
+z= <br />
x+y+z=<br />
REAL-WORLD APPLICATIONS<br />
<br />
52. <br />
a fle fl<br />
a fl<br />
<br />
53. <br />
. Th<br />
<br />
<br />
h fl<br />
54. <br />
<br />
ft<br />
ft<br />
55. <br />
ft
646 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
56. <br />
<br />
ft<br />
<br />
58. Thm fl<br />
<br />
e fl<br />
<br />
<br />
<br />
y fl<br />
60. <br />
s. Th<br />
<br />
Th<br />
<br />
<br />
57. <br />
. Th<br />
<br />
<br />
<br />
fift<br />
<br />
59. e fl<br />
<br />
<br />
<br />
les. Th<br />
<br />
<br />
<br />
<br />
<br />
61.
SECTION 7.7 SOLVING SYSTEMS WITH INVERSES<br />
647<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the inverse of a matrix.<br />
Solve a system of linear equations using an inverse matrix.<br />
7.7 SOLVING SYSTEMS WITH INVERSES<br />
ffThfi<br />
<br />
<br />
Th<br />
fift<br />
<br />
Finding the Inverse of a Matrix<br />
aa − aa − =a − a =( a ) a = − = <br />
( <br />
) =ThA<br />
A − Th<br />
I n<br />
Th<br />
××<br />
I <br />
=[ ] <br />
<br />
I =[ <br />
<br />
<br />
AI=IA=A<br />
] <br />
<br />
<br />
AA − =I<br />
<br />
A − A =I<br />
<br />
AA − =A − A =A<br />
A − fi×<br />
××<br />
the identity matrix and multiplicative inverse<br />
identity matrixI n<br />
<br />
I <br />
=[ <br />
] I <br />
=[ <br />
<br />
] <br />
× ×<br />
An ×nBn ×nAB=BA=I n<br />
B =A − multiplicative inverse<br />
of a matrixA
648<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 1 Showing That the Identity Matrix Acts as a 1<br />
AAI=IA=A<br />
<br />
A =[ − ] <br />
Solution A<br />
A<br />
AI=[ − ] <br />
[ ] = [ ċ+ċ ċ+ċ <br />
<br />
−ċ+ċ −ċ+ċ ] = [ − ] <br />
AI=[ ] [ − ] = [<br />
ċ+ċ− ċ+ċ <br />
<br />
ċ+ċ− ċ+ċ ]<br />
=[ − ] <br />
How To…<br />
<br />
1. An ×nBn ×nAB<br />
2. AB=n fiBABA=IB =A − A =B − <br />
Example 2<br />
Showing That Matrix A Is the Multiplicative Inverse of Matrix B<br />
<br />
A =[ − − ] B = [ − − ] <br />
Solution ABBA<br />
AB=[ − − ] [ − − ] <br />
<br />
= [ −+ −+<br />
<br />
<br />
<br />
−−− −−− ]<br />
<br />
=[ ] <br />
BA=[ − − ] [ − − ] <br />
= [ −−− −−− <br />
<br />
+− −+− ]<br />
<br />
=[ ] <br />
AB<br />
Try It #1<br />
<br />
A=[ −− ] B= [ − − ] <br />
Finding the Multiplicative Inverse Using Matrix Multiplication<br />
fi<br />
fi
SECTION 7.7 SOLVING SYSTEMS WITH INVERSES<br />
649<br />
Example 3<br />
Finding the Multiplicative Inverse Using Matrix Multiplication<br />
o fi<br />
A =[ − − ] <br />
Solution A<br />
[ − c d ] =[ ] <br />
<br />
− ] [ a b<br />
[ − − ] [ a b<br />
c d ] = [ a −c b −d <br />
a −c b −d ] <br />
fi<br />
<br />
a −c = R <br />
<br />
a −c = R <br />
<br />
−R <br />
+R <br />
→R <br />
c<br />
<br />
a −c =<br />
<br />
+c =−<br />
<br />
c =−<br />
a<br />
<br />
a −−=<br />
<br />
a +=<br />
<br />
a =−<br />
<br />
<br />
b −d = R <br />
<br />
b −d = R <br />
<br />
−R <br />
+R <br />
=R <br />
d<br />
<br />
b −d =<br />
<br />
+d =<br />
<br />
d =<br />
b<br />
<br />
b −=<br />
<br />
b −=<br />
<br />
b =<br />
A − =[ − − ] <br />
Finding the Multiplicative Inverse by Augmenting with the Identity<br />
fiA<br />
IIA − <br />
<br />
A =[ ] <br />
A
650<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
[ ∣ ] <br />
A<br />
1. <br />
2. −<br />
3. −<br />
4. <br />
[ ∣ ] <br />
[ ∣ − ] <br />
[ − ∣ − − ] <br />
5. −<br />
ThA − <br />
<br />
[ − ∣ − − ] <br />
[ ∣ − − ] <br />
A − =[ − − ] <br />
Finding the Multiplicative Inverse of 2 × 2 Matrices Using a Formula<br />
fi×<br />
<br />
A×<br />
A = [ a b<br />
c d ] <br />
A<br />
<br />
A − = <br />
ad−bc [ d −b<br />
−c a ] <br />
ad−bc≠ad−bc=A<br />
Example 4 Using the Formula to Find the Multiplicative Inverse of Matrix A<br />
o fi<br />
A =[ − − ] <br />
Solution <br />
<br />
A − = <br />
<br />
<br />
−−− [ − − ] <br />
<br />
= <br />
−+ [ − − ] <br />
=[ − − ] <br />
Analysis We can check that our formula works by using one of the other methods to calculate the inverse. Let’s augment<br />
A with the identity.<br />
[ − − ∣ ] <br />
Perform row operations with the goal of turning A into the identity.
SECTION 7.7 SOLVING SYSTEMS WITH INVERSES<br />
651<br />
1. Multiply row by −and add to row <br />
[ − ∣ − ] <br />
2. Multiply row by and add to row<br />
[ ∣ − − ] <br />
So, we have verified our original solution.<br />
A − =[ − − ] <br />
Try It #2<br />
o fiA<br />
A − =[ − ] <br />
Example 5<br />
Finding the Inverse of the Matrix, If It Exists<br />
<br />
A = [ ] <br />
Solution <br />
1. <br />
[ ∣ ] <br />
[ ∣ ] <br />
2. −<br />
[ ∣ − ] <br />
3. Th. Th<br />
Finding the Multiplicative Inverse of 3 × 3 Matrices<br />
×fi×<br />
<br />
×<br />
<br />
A =[ <br />
<br />
] <br />
A<br />
<br />
A <br />
<br />
A I =[ <br />
<br />
∣<br />
<br />
<br />
<br />
] A <br />
Aft<br />
ftfi<br />
<br />
How To…<br />
×, fi<br />
1. <br />
2. ft<br />
3. <br />
4. AA − =IA − A =I
652<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
Example 6 Finding the Inverse of a 3 × 3 Matrix<br />
×A, fi<br />
<br />
A =[ <br />
<br />
] <br />
Solution AA<br />
ThA<br />
<br />
[ <br />
<br />
∣<br />
<br />
<br />
<br />
] <br />
<br />
[ <br />
∣<br />
<br />
<br />
<br />
] <br />
<br />
<br />
−<br />
−R <br />
+R <br />
=R <br />
→[ <br />
<br />
∣<br />
<br />
<br />
<br />
] <br />
<br />
<br />
−R <br />
+R <br />
=R <br />
<br />
→[ <br />
∣<br />
<br />
− <br />
<br />
] − <br />
<br />
−<br />
R <br />
↔R <br />
→ [ ∣<br />
<br />
<br />
− <br />
] <br />
<br />
−R <br />
+R <br />
=R <br />
→ [ <br />
−<br />
<br />
∣<br />
<br />
<br />
− <br />
] − <br />
<br />
−R <br />
+R <br />
=R →[ <br />
−<br />
<br />
∣<br />
<br />
<br />
− <br />
<br />
] − −<br />
<br />
− <br />
A − =B<br />
<br />
=[ <br />
− <br />
] − −<br />
Analysis To prove that B = A −1 , let’s multiply the two matrices together to see if the product equals the identity, if<br />
A A − = I and A − A = I.<br />
R <br />
R <br />
<br />
− <br />
]<br />
<br />
[ ] ] <br />
] [ <br />
<br />
] <br />
<br />
<br />
<br />
] <br />
<br />
AA =[ − <br />
<br />
<br />
<br />
− <br />
<br />
<br />
− −<br />
−+−+ ++− ++−<br />
=[ <br />
−+−+<br />
++− ++−<br />
] <br />
<br />
−+−+ ++− ++−<br />
<br />
=[ <br />
<br />
<br />
− <br />
A − A<br />
<br />
=[ <br />
− <br />
<br />
− −<br />
−++ −++ −++<br />
<br />
=[ <br />
<br />
−++ −++ −++<br />
] <br />
<br />
+−+− +−+− +−+−<br />
<br />
=[
SECTION 7.7 SOLVING SYSTEMS WITH INVERSES<br />
653<br />
Try It #3<br />
×<br />
] <br />
− <br />
A=[ <br />
<br />
− −<br />
<br />
−<br />
Solving a System of Linear Equations Using the Inverse of a Matrix<br />
fiX<br />
B<br />
fi<br />
<br />
AX=B<br />
AffiX<br />
BThAX=B<br />
<br />
<br />
a <br />
x +b <br />
y =c <br />
<br />
<br />
a <br />
x +b <br />
y =c <br />
<br />
ffi<br />
A = [ a b ] a <br />
b <br />
<br />
Th<br />
<br />
<br />
<br />
AX=B<br />
X =<br />
[ x <br />
y ] <br />
B =<br />
[ c c ] <br />
[ a b ] a <br />
b <br />
[ x <br />
c ] <br />
y ] = [ c <br />
− =( <br />
) =<br />
ax=bx<br />
a <br />
<br />
ax=b<br />
<br />
<br />
<br />
<br />
<br />
( <br />
a ) ax=( a ) b<br />
a − ax=a − b<br />
a − ax=a − b<br />
x =a − b<br />
x =a − b<br />
Thfffi<br />
<br />
<br />
××
654<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
solving a system of equations using the inverse of a matrix<br />
ffiAXB <br />
<br />
AX=B<br />
A<br />
<br />
<br />
<br />
A − AX=A − B<br />
A − AX=A − B<br />
IX=A − B<br />
X =A − B<br />
Q & A…<br />
If the coefficient matrix does not have an inverse, does that mean the system has no solution?<br />
ffi<br />
fi<br />
Example 7 Solving a 2 × 2 System Using the Inverse of a Matrix<br />
<br />
<br />
x +y =<br />
<br />
x +y =<br />
Solution ffi<br />
A=[ ] X = [ x y] B = [ ] <br />
<br />
[ ] [ x y] = [ ] <br />
A − <br />
<br />
A − = <br />
ad−bc [<br />
d −b<br />
−c a ] <br />
<br />
= <br />
− <br />
[ − − ] <br />
<br />
= <br />
<br />
[ − − ] <br />
<br />
A − =[ − − ] <br />
A − <br />
<br />
[ − <br />
A − AX=A − B<br />
− ] <br />
[ ] [ x y] = <br />
[ − ] [ <br />
<br />
−<br />
<br />
[ ] <br />
[ x y] = [<br />
+− <br />
−+ ]<br />
<br />
] <br />
<br />
Th−<br />
<br />
[ x <br />
] <br />
y ] = [ −
SECTION 7.7 SOLVING SYSTEMS WITH INVERSES<br />
655<br />
Q & A…<br />
Can we solve for X by finding the product B A −1 ?<br />
A − B ≠BA − <br />
<br />
A − AX=A − B<br />
<br />
A − A=A − B<br />
<br />
IX=A − B<br />
<br />
X =A − B<br />
fiA − A − A <br />
B<br />
Example 8 Solving a 3 × 3 System Using the Inverse of a Matrix<br />
<br />
<br />
x +y +z =<br />
<br />
−x −y −z =−<br />
<br />
−x−y −z =−<br />
Solution AX=B<br />
x<br />
[ <br />
<br />
− − <br />
−<br />
] [ y<br />
<br />
] [ = <br />
<br />
<br />
− ] <br />
− − − z −<br />
l fiA<br />
<br />
[<br />
− <br />
− <br />
−∣<br />
<br />
− − − <br />
] <br />
<br />
<br />
<br />
[ <br />
− <br />
− <br />
− <br />
∣<br />
<br />
<br />
] <br />
− − − <br />
<br />
<br />
<br />
[<br />
<br />
<br />
<br />
<br />
<br />
∣<br />
<br />
<br />
] <br />
<br />
− − − <br />
<br />
<br />
−<br />
<br />
[ <br />
<br />
<br />
]<br />
<br />
∣ <br />
<br />
<br />
<br />
− <br />
[ − <br />
]<br />
− <br />
<br />
∣ <br />
<br />
<br />
<br />
[<br />
<br />
<br />
<br />
<br />
<br />
− <br />
<br />
∣<br />
<br />
− <br />
<br />
− <br />
<br />
]
656<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
<br />
− <br />
<br />
<br />
− − <br />
[ ∣ ] <br />
<br />
− − <br />
[ <br />
∣ −<br />
− <br />
]<br />
<br />
] <br />
<br />
− − <br />
A =[ − − −<br />
<br />
<br />
A − A − AX=A − B<br />
<br />
<br />
<br />
[ <br />
Th<br />
] [ <br />
x <br />
] [ ] [ = <br />
<br />
− − <br />
− − <br />
<br />
<br />
− − −<br />
y<br />
<br />
− − − z<br />
] [ <br />
] <br />
<br />
− − <br />
− −<br />
<br />
<br />
<br />
− <br />
−<br />
−+−<br />
A − B =[ <br />
<br />
−−+ <br />
<br />
+−<br />
] [ = <br />
] <br />
<br />
Try It #4<br />
ffi<br />
<br />
x −y +z =<br />
<br />
−x+y −z =<br />
<br />
y−z=−<br />
How To…<br />
<br />
1. ffiAB<br />
2. <br />
3. ffiffi<br />
<br />
Example 9 Using a Calculator to Solve a System of Equations with Matrix Inverses<br />
<br />
<br />
x +y +z =<br />
<br />
x +y +z =−<br />
<br />
x +y +z =−<br />
Solution ffiA<br />
B<br />
<br />
A=[ <br />
<br />
] B= <br />
<br />
[ − <br />
] <br />
<br />
−
SECTION 7.7 SOLVING SYSTEMS WITH INVERSES<br />
657<br />
X<br />
<br />
<br />
A − ×B<br />
<br />
<br />
<br />
− [ − ] <br />
<br />
<br />
The Identity Matrix (http://openstaxcollege.org/l/identmatrix)<br />
Determining Inverse Matrices (http://openstaxcollege.org/l/inversematrix)<br />
Using a Matrix Equation to Solve a System of Equations (http://openstaxcollege.org/l/matrixsystem)
658 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
7.7 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
AB≠BA<br />
<br />
<br />
A − A=AA − <br />
3. ×<br />
<br />
5. <br />
<br />
×<br />
2. ×<br />
<br />
<br />
4. <br />
<br />
ALGEBRAIC<br />
AB<br />
6. A=[ <br />
8. A=[ <br />
<br />
<br />
]<br />
<br />
− ] B= [ ] 7. A= [ ] B= [ − − <br />
<br />
− <br />
] B= [ <br />
<br />
<br />
<br />
10.<br />
<br />
A=[ −<br />
] B= <br />
[ <br />
<br />
−<br />
<br />
− <br />
<br />
<br />
12. A=[ <br />
] B= [ <br />
<br />
− −<br />
− <br />
] <br />
− − <br />
]<br />
9. A= [ − <br />
<br />
] [ 11. A= <br />
<br />
− ] B=[ − − − − ] <br />
, fi<br />
] B= <br />
[ <br />
<br />
−<br />
− −<br />
] <br />
<br />
− <br />
13. [ − ] 14. [ − ] 15. [ − ] 16. [ − − − ] <br />
17. [ ] 18. [ ] 19. [ − ] [ 20. <br />
21. [ <br />
<br />
−<br />
<br />
<br />
] 22. [ <br />
<br />
−<br />
− <br />
− − −<br />
] 23. [ <br />
<br />
−<br />
<br />
− <br />
] 24. [ <br />
] <br />
<br />
<br />
− <br />
<br />
] <br />
<br />
− <br />
− −<br />
<br />
<br />
25. [<br />
<br />
]<br />
26. [ <br />
<br />
<br />
<br />
]
SECTION 7.7 SECTION EXERCISES 659<br />
×<br />
27. x−y=−<br />
x+y=−<br />
28. x+y=−<br />
x−y=<br />
29. x−y=<br />
−x+y=−<br />
30. x−y=−<br />
x+y=<br />
31. −x−y=<br />
x+y=−<br />
32. −x+y= <br />
−x+y= <br />
<br />
33. <br />
x− <br />
y= <br />
<br />
− <br />
x+ <br />
y= <br />
34. <br />
x+ <br />
y=− <br />
<br />
<br />
x− <br />
y=− <br />
<br />
×<br />
35. x−y+z=<br />
x+y=<br />
x−y−z=<br />
36. x+y+z=<br />
x−y+z=−<br />
−x+y+z=<br />
37. x−y−z=<br />
−x+y+z=−<br />
x+y+z=<br />
38. x−y+z=−<br />
x+y−z=<br />
x+y+z=<br />
39. x−y+z=−<br />
x+y−z=<br />
y−z=<br />
40. <br />
x− <br />
y+z=− <br />
<br />
<br />
x−y+ <br />
z=−<br />
<br />
x+y− <br />
z=<br />
41. <br />
x− <br />
y+ <br />
z= <br />
<br />
− <br />
x− <br />
y+ <br />
z= <br />
− <br />
x− <br />
y+ <br />
z=<br />
42. x+y+z=−<br />
x−y+z=<br />
y+z=−<br />
TECHNOLOGY<br />
<br />
43. x−y=−<br />
−x+y=<br />
45. x−y −z=<br />
x+y −z=−<br />
y−z=−<br />
44. − <br />
x− <br />
y=− <br />
<br />
<br />
x+ <br />
y= <br />
<br />
46. x−y+z=−<br />
x−y=−<br />
x+y+z=<br />
EXTENSIONS<br />
<br />
47.<br />
[<br />
51.<br />
<br />
<br />
<br />
]<br />
<br />
<br />
<br />
<br />
<br />
[ ] <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
[<br />
<br />
− <br />
48. <br />
−<br />
[ <br />
<br />
49. <br />
<br />
<br />
<br />
<br />
− <br />
] − <br />
− − <br />
] 50.<br />
<br />
[ ]
660 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
REAL-WORLD APPLICATIONS<br />
Th<br />
<br />
52. <br />
e diff<br />
<br />
<br />
54. o diff<br />
s. Th<br />
<br />
<br />
<br />
<br />
56. <br />
ies. Th<br />
<br />
t $0.75. Th<br />
<br />
58. <br />
<br />
<br />
<br />
60. <br />
. Th<br />
<br />
t. Th<br />
<br />
ft ft <br />
<br />
<br />
53. <br />
<br />
<br />
55. <br />
<br />
<br />
<br />
<br />
<br />
57. <br />
e diff<br />
ts. Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
59. <br />
<br />
<br />
<br />
<br />
61.
SECTION 7.8 SOLVING SYSTEMS WITH CRAMER'S RULE<br />
661<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Evaluate 2 × 2 determinants.<br />
Use Cramer’s Rule to solve a system of equations in two variables.<br />
Evaluate 3 × 3 determinants.<br />
Use Cramer’s Rule to solve a system of three equations in three variables.<br />
Know the properties of determinants.<br />
7.8 SOLVING SYSTEMS WITH CRAMER'S RULE<br />
<br />
<br />
<br />
<br />
Evaluating the Determinant of a 2 × 2 Matrix<br />
<br />
<br />
<br />
<br />
Th<br />
<br />
fi<br />
find the determinant of a 2 × 2 matrix<br />
determinant×<br />
A= [<br />
a b<br />
c d ] <br />
fi<br />
A=∣ a b <br />
c d∣ =ad−cb<br />
ThA<br />
∣ A ∣<br />
Example 1<br />
Finding the Determinant of a 2 × 2 Matrix<br />
<br />
Solution<br />
<br />
<br />
A=[ − ] <br />
<br />
A=∣ − ∣ <br />
=−−<br />
=<br />
Using Cramer’s Rule to Solve a System of Two Equations in Two Variables<br />
fi<br />
<br />
Introduction à l'Analyse des lignes Courbes algébriques<br />
ffifi
662<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
fifi<br />
<br />
<br />
<br />
a <br />
x+b <br />
y=c <br />
<br />
a <br />
x+b <br />
y=c <br />
<br />
x<br />
ffiyffiy<br />
y<br />
b <br />
a <br />
x+b <br />
b <br />
y=b <br />
c <br />
R <br />
b <br />
<br />
<br />
−b <br />
a <br />
x−b <br />
b <br />
y =−b <br />
c <br />
R <br />
−b <br />
<br />
<br />
b <br />
a <br />
x−b <br />
a <br />
x=b <br />
c <br />
−b <br />
c <br />
<br />
x<br />
<br />
<br />
b <br />
a <br />
x−b <br />
a <br />
x=b <br />
c <br />
−b <br />
c <br />
<br />
xb <br />
a <br />
−b <br />
a <br />
=b <br />
c <br />
−b <br />
c <br />
<br />
<br />
yx<br />
x= bc [<br />
<br />
−b <br />
c <br />
c b ] b <br />
a <br />
−b <br />
a <br />
=<br />
c <br />
b <br />
<br />
<br />
[ a b ] <br />
a <br />
b <br />
<br />
<br />
a <br />
a <br />
x+a <br />
b <br />
y=a <br />
c <br />
R <br />
a <br />
<br />
<br />
−a <br />
a <br />
x−a <br />
b <br />
y=−a <br />
c <br />
R <br />
−a <br />
<br />
<br />
a <br />
b <br />
y−a <br />
b <br />
y=a <br />
c <br />
−a <br />
c <br />
<br />
y<br />
<br />
a <br />
b <br />
y−a <br />
b <br />
y=a <br />
c <br />
−a <br />
c <br />
<br />
<br />
ya <br />
b <br />
−a <br />
b <br />
=a <br />
c <br />
−a <br />
c <br />
<br />
<br />
<br />
y= ac <br />
−a <br />
c <br />
<br />
_<br />
a <br />
b <br />
−a <br />
b <br />
=a c <br />
−a <br />
c <br />
[ a c a <br />
b <br />
−a <br />
b <br />
=<br />
a <br />
c ] <br />
<br />
[ a b <br />
a <br />
b <br />
<br />
xyffi<br />
xy<br />
<br />
D <br />
D x<br />
x<br />
x= D _ x D <br />
D y<br />
y<br />
<br />
D y<br />
<br />
<br />
y=_<br />
D <br />
Th<br />
xy
SECTION 7.8 SOLVING SYSTEMS WITH CRAMER'S RULE<br />
663<br />
Cramer’s Rule for 2 × 2 systems<br />
Cramer’s Rule<br />
<br />
<br />
<br />
a <br />
x+b <br />
y=c <br />
<br />
<br />
a <br />
x+b <br />
y=c <br />
<br />
Th<br />
x= D [ c b ] _ x<br />
<br />
D =<br />
c <br />
b <br />
<br />
D<br />
_<br />
<br />
[ a b ] y<br />
[ a c <br />
D≠y=<br />
_ <br />
a <br />
b <br />
D =<br />
a <br />
c ] <br />
_<br />
<br />
[ a b ] D≠<br />
a <br />
b <br />
<br />
xxyy<br />
<br />
Example 2<br />
Using Cramer’s Rule to Solve a 2 × 2 System<br />
×<br />
<br />
x +y =<br />
<br />
x −y =<br />
Solution x<br />
x= D x <br />
∣ D = −∣ _<br />
∣ =<br />
−∣ −− <br />
−− =− <br />
− =<br />
y<br />
y= D y<br />
<br />
∣ D = ∣ _<br />
<br />
∣ =<br />
−∣ − <br />
−− =− <br />
=−<br />
Th−<br />
Try It #1<br />
×<br />
x+y=−<br />
<br />
−x+y=−<br />
Evaluating the Determinant of a 3 × 3 Matrix<br />
×fi×<br />
×fi×<br />
Thdown <br />
up Th<br />
<br />
×<br />
<br />
a <br />
b <br />
c <br />
= a<br />
<br />
[ <br />
b <br />
c<br />
<br />
a <br />
b <br />
c <br />
]
664<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
1. A <br />
<br />
a <br />
b <br />
c <br />
<br />
a<br />
A=∣<br />
<br />
b <br />
c<br />
<br />
a <br />
b <br />
<br />
∣ a <br />
b<br />
∣<br />
a <br />
b <br />
c <br />
a <br />
b <br />
<br />
2. <br />
<br />
3. ftfi<br />
<br />
<br />
a <br />
b <br />
c <br />
<br />
<br />
a<br />
A=∣<br />
<br />
b <br />
c<br />
<br />
a <br />
b <br />
<br />
∣ a <br />
b<br />
∣<br />
a <br />
b <br />
c <br />
a <br />
b <br />
<br />
Th<br />
∣ A ∣=a <br />
b <br />
c <br />
+b <br />
c <br />
a <br />
+c <br />
a <br />
b <br />
−a <br />
b <br />
c <br />
−b <br />
c <br />
a <br />
−c <br />
a <br />
b <br />
<br />
Example 3<br />
Finding the Determinant of a 3 × 3 Matrix<br />
×<br />
] <br />
<br />
<br />
=[ − <br />
<br />
e fi. Th<br />
<br />
<br />
<br />
∣ <br />
∣=∣<br />
− <br />
<br />
<br />
∣ −<br />
∣<br />
<br />
<br />
=−++−−−−<br />
<br />
=+++−−<br />
<br />
=<br />
Try It #2<br />
×<br />
<br />
− <br />
<br />
A=∣<br />
∣<br />
− <br />
Q & A…<br />
Can we use the same method to find the determinant of a larger matrix?<br />
××<br />
ft<br />
Using Cramer’s Rule to Solve a System of Three Equations in Three Variables<br />
fi×<br />
<br />
××<br />
<br />
fio fi
SECTION 7.8 SOLVING SYSTEMS WITH CRAMER'S RULE<br />
665<br />
×<br />
<br />
a <br />
x + b <br />
y + c <br />
z =d <br />
<br />
<br />
a <br />
x + b <br />
y + c <br />
z =d <br />
<br />
<br />
a <br />
x + b <br />
y + c <br />
z =d <br />
<br />
x= D D<br />
_ x<br />
<br />
D y=<br />
y<br />
<br />
_<br />
<br />
D z=D _ z<br />
<br />
D D≠<br />
<br />
a<br />
<br />
b <br />
c <br />
d<br />
a D=∣<br />
<br />
b <br />
c<br />
<br />
b <br />
c <br />
<br />
∣D x<br />
d a <br />
b <br />
c <br />
<br />
=∣<br />
<br />
b <br />
c<br />
<br />
a <br />
d <br />
c <br />
<br />
∣D y<br />
a<br />
d <br />
b <br />
c <br />
<br />
=∣<br />
<br />
d <br />
c<br />
<br />
a <br />
b <br />
d <br />
<br />
∣D z<br />
a<br />
a <br />
d <br />
c <br />
<br />
=∣<br />
<br />
b <br />
d<br />
∣<br />
a <br />
b <br />
d <br />
<br />
D x<br />
x<br />
D y<br />
yD z<br />
<br />
z<br />
Example 4<br />
Solving a 3 × 3 System Using Cramer’s Rule<br />
×<br />
<br />
x +y −z =<br />
<br />
x −y +z =−<br />
<br />
x +y −z =<br />
Solution <br />
<br />
<br />
−<br />
<br />
D=∣ =∣ − <br />
∣D <br />
−<br />
− − <br />
−<br />
x <br />
<br />
−<br />
∣D y<br />
<br />
−<br />
=∣ =∣ − <br />
∣D <br />
<br />
<br />
− −<br />
−<br />
z ∣<br />
<br />
<br />
<br />
<br />
Th−<br />
x= D _ x<br />
<br />
D =− <br />
− =<br />
D y<br />
<br />
y=_<br />
<br />
D =− <br />
− =<br />
z= D _ z<br />
<br />
D = <br />
− =−<br />
Try It #3<br />
×<br />
x−y+z=<br />
x+y+z=<br />
x−y+z=<br />
Example 5 Using Cramer’s Rule to Solve an Inconsistent System<br />
<br />
x −y = <br />
x −y = <br />
Solution y fiDD x<br />
D y<br />
<br />
D=∣ − −∣ =−−−=
666<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
fi<br />
<br />
1. −<br />
2. <br />
<br />
−x +y =−<br />
<br />
x −y =<br />
<br />
=−<br />
=−Th<br />
Figure 1<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
<br />
<br />
<br />
<br />
<br />
y = x<br />
y = x − <br />
x<br />
–<br />
–<br />
–<br />
–<br />
Figure 1<br />
Example 6 Use Cramer’s Rule to Solve a Dependent System<br />
fi<br />
x −y +z = <br />
x +y −z = <br />
x −y +z = <br />
Solution s fit fie fi<br />
<br />
∣<br />
− <br />
<br />
<br />
−<br />
<br />
−<br />
∣<br />
<br />
∣<br />
− <br />
−<br />
+−−+−−−−−−−=<br />
fi<br />
o fi<br />
1. −<br />
<br />
<br />
<br />
−x +y −x =<br />
x −y +z =<br />
=<br />
2. = <br />
<br />
Figure 2
SECTION 7.8 SOLVING SYSTEMS WITH CRAMER'S RULE<br />
667<br />
x − y +z = <br />
x−y+z=<br />
x+y+z=<br />
Understanding Properties of Determinants<br />
Figure 2<br />
Th<br />
<br />
properties of determinants<br />
1. <br />
<br />
2. <br />
3. <br />
4. <br />
5. ThA − A<br />
6. <br />
Example 7<br />
Illustrating Properties of Determinants<br />
<br />
Solution <br />
<br />
] <br />
<br />
<br />
<br />
<br />
A=[ <br />
−<br />
Ae fi<br />
<br />
<br />
<br />
A=[ <br />
<br />
<br />
<br />
<br />
∣<br />
<br />
] <br />
− <br />
<br />
<br />
A=−++−−+<br />
<br />
=−<br />
<br />
A=[ − − ] A=−−−=−=−<br />
<br />
B=[ − − ] B=−−−=−=
668<br />
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
<br />
<br />
<br />
A=∣<br />
<br />
<br />
<br />
∣<br />
] − −<br />
<br />
<br />
A=+−++−−<br />
<br />
=−++−−=<br />
. Th<br />
A=[ ] A=−=<br />
A − A <br />
A=[ ] A=−=−<br />
<br />
A − = [<br />
− <br />
− ] A − =−( − <br />
) − ( <br />
<br />
) =− <br />
<br />
<br />
. Th<br />
A=[ ] A=−=−<br />
<br />
B=[ <br />
] B=−=−<br />
Example 8<br />
Using Cramer’s Rule and Determinant Properties to Solve a System<br />
×<br />
x +y +z = <br />
x +y +z =− <br />
x +y +z = <br />
Solution <br />
<br />
<br />
<br />
<br />
D=∣<br />
<br />
∣<br />
<br />
<br />
fio fi<br />
1. <br />
<br />
−x−y−x=−<br />
<br />
x+y+z=<br />
<br />
=−<br />
<br />
<br />
Solve a System of Two Equations Using Cramer's Rule (http://openstaxcollege.org/l/system2cramer)<br />
Solve a Systems of Three Equations using Cramer's Rule (http://openstaxcollege.org/l/system3cramer)
SECTION 7.8 SECTION EXERCISES 669<br />
7.8 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
<br />
ALGEBRAIC<br />
, fi<br />
2. <br />
<br />
×<br />
<br />
4. Th×A<br />
<br />
<br />
<br />
5. ∣ ∣ ∣ 6. − −∣ ∣ 7. <br />
− ∣ ∣ 8. − − ∣ <br />
<br />
− <br />
9. ∣ −∣ ∣ 10. −∣ ∣ 11. ∣ ∣ 12. − ∣ <br />
13. ∣<br />
− <br />
<br />
− <br />
<br />
∣ 14. ∣ − <br />
− ∣ 15. ∣<br />
−<br />
<br />
<br />
<br />
<br />
∣<br />
−<br />
∣<br />
<br />
− <br />
16. <br />
∣<br />
−<br />
∣<br />
<br />
<br />
<br />
17. <br />
∣<br />
<br />
∣<br />
<br />
− <br />
18. − <br />
∣<br />
− <br />
∣<br />
−<br />
<br />
<br />
19. − <br />
− ∣<br />
− −<br />
∣<br />
<br />
− <br />
20. − <br />
− <br />
∣<br />
−<br />
∣<br />
<br />
−<br />
21. ∣<br />
− −<br />
∣<br />
−<br />
22. −<br />
<br />
<br />
∣<br />
− <br />
∣<br />
<br />
23. <br />
−<br />
− <br />
−∣<br />
<br />
∣<br />
24. <br />
− <br />
<br />
<br />
− ∣<br />
<br />
<br />
25. x−y=−<br />
x+y=<br />
26. x−y=<br />
−x+y=<br />
27. x−y=<br />
−x+y=−<br />
28. x+y=<br />
x−y=<br />
29. x+y=<br />
x−y=−<br />
30. x−y=<br />
−x+y=−<br />
31. x−y=−<br />
x+y=−<br />
32. x−y=<br />
−x+y=<br />
33. x+y=<br />
−x−y=−<br />
34. x−y=−<br />
−x+y=<br />
<br />
35. x+y−z=−<br />
x+y+z=<br />
−x−y+z=−<br />
36. −x+y−z=−<br />
x−y−z=−<br />
x−y+z=<br />
37. x+y−z=−<br />
−x−y+z=<br />
y+z=<br />
38. x−y+z=<br />
x−z=−<br />
x+y−z=−<br />
39. x−y+z=<br />
−x+y=−<br />
x+y+z=<br />
40. x+y−z=<br />
−x−y+z=<br />
x−y+z=<br />
41. x−y+z=<br />
−x+y+z=<br />
x+y−z=−<br />
42. −x−y−z=−<br />
x−y+z=<br />
x−y−z=−
670 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
43. x−y+z=<br />
−x+y−z=−<br />
x+y+z=<br />
44. x−y+z=<br />
−x+y−z=−<br />
x+y−z=−<br />
TECHNOLOGY<br />
<br />
∣ ∣ ∣<br />
<br />
<br />
45. <br />
<br />
<br />
<br />
<br />
46.<br />
∣<br />
47.<br />
<br />
<br />
<br />
<br />
− <br />
<br />
<br />
<br />
− ∣<br />
− <br />
<br />
∣<br />
−<br />
<br />
∣<br />
<br />
48. <br />
<br />
∣<br />
<br />
REAL-WORLD APPLICATIONS<br />
Th<br />
<br />
49. <br />
<br />
50. <br />
<br />
<br />
51. o 106. Th <br />
. Th<br />
<br />
52. o 216. Th <br />
s 112. Th<br />
<br />
Th<br />
<br />
53. <br />
<br />
<br />
<br />
55. <br />
<br />
<br />
<br />
<br />
<br />
57. <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
54. <br />
<br />
<br />
t. Th<br />
<br />
<br />
<br />
56. <br />
<br />
<br />
<br />
<br />
58. <br />
<br />
. Th
SECTION 7.8 SECTION EXERCISES 671<br />
59. <br />
. Th<br />
<br />
<br />
<br />
61. <br />
<br />
<br />
<br />
<br />
63. ies. Th<br />
<br />
<br />
. Th<br />
<br />
<br />
<br />
65. <br />
<br />
<br />
<br />
. Th<br />
<br />
<br />
<br />
60. <br />
<br />
<br />
<br />
<br />
62. . Th<br />
<br />
<br />
et. Th<br />
<br />
<br />
<br />
64. <br />
<br />
<br />
Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
t y<br />
<br />
. ThTable 1<br />
Fat (g) Protein (g) Carbohydrates (g)<br />
<br />
<br />
<br />
Table 1<br />
66. <br />
f mix. Th<br />
<br />
<br />
<br />
67. <br />
<br />
<br />
<br />
68.
672 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
CHAPTER 7 REVIEW<br />
Key Terms<br />
addition method <br />
<br />
<br />
augmented matrix <br />
break-even point fi<br />
coefficient matrix <br />
column <br />
consistent system <br />
<br />
cost function <br />
Cramer’s Rule <br />
dependent system <br />
<br />
determinant <br />
<br />
entry <br />
feasible region <br />
<br />
Gaussian elimination <br />
identity matrix <br />
<br />
inconsistent system <br />
<br />
independent system xy<br />
main diagonal <br />
matrix <br />
multiplicative inverse of a matrix <br />
nonlinear inequality <br />
partial fraction decomposition <br />
diff<br />
partial fractions diff<br />
<br />
profit function fiPx=Rx−Cx<br />
revenue function R=xpx=p=<br />
row <br />
row operations <br />
<br />
row-echelon form ft<br />
<br />
row-equivalent AB<br />
<br />
scalar multiple <br />
solution set <br />
substitution method
CHAPTER 7 REVIEW 673<br />
system of linear equations <br />
system of nonlinear equations <br />
system of nonlinear inequalities <br />
<br />
Key Equations<br />
× I <br />
<br />
=[ ] <br />
<br />
<br />
× I =[ <br />
<br />
] <br />
×<br />
A − = <br />
ad−bc <br />
[ −b<br />
−c a ] ad−bc≠<br />
Key Concepts<br />
7.1 Systems of Linear Equations: Two Variables<br />
<br />
<br />
Th <br />
Example 1<br />
<br />
<br />
<br />
Example 2<br />
<br />
Example 3<br />
<br />
Example 4<br />
ft<br />
Example 5Example 6Example 7<br />
<br />
Example 8<br />
Th<br />
Example 9<br />
<br />
fiExample 10Example 11<br />
7.2 Systems of Linear Equations: Three Variables<br />
xyzExample 1<br />
<br />
Th<br />
Example 2<br />
diff<br />
Example 3<br />
ft<br />
Example 4
674 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
ft<br />
Example 5<br />
<br />
<br />
7.3 Systems of Nonlinear Equations and Inequalities: Two Variables<br />
Th<br />
<br />
Example 1<br />
Th<br />
<br />
Example 2<br />
Thfi<br />
<br />
<br />
Example 3<br />
><<br />
Example 4<br />
<br />
Example 5<br />
7.4 Partial Fractions<br />
Px <br />
Qx <br />
A<br />
a <br />
x+b +<br />
B<br />
<br />
a <br />
x+b <br />
<br />
<br />
<br />
Example 1<br />
Th Px <br />
Qx <br />
Example 2<br />
Th Px <br />
Qx <br />
A x<br />
+<br />
Bx+C<br />
Example <br />
3<br />
ax +bx+c<br />
Px <br />
Qx Qx<br />
<br />
Ax+B<br />
__<br />
ax +bx+c +<br />
A <br />
x+B <br />
<br />
__ <br />
ax +bx+c ++<br />
A n<br />
x+B n<br />
<br />
__ <br />
ax +bx+c n Example 4<br />
<br />
7.5 Matrices and Matrix Operations<br />
<br />
Th×<br />
Example 1<br />
<br />
Example 2Example 3Example 4Example 5<br />
Example 6<br />
ftExample 7<br />
fi<br />
<br />
ThABA<br />
BABExample 8Example 9
CHAPTER 7 REVIEW 675<br />
ftExample 10<br />
ftExample 11<br />
7.6 Solving Systems with Gaussian Elimination<br />
Example 1<br />
Example 2<br />
<br />
Example 3Example 4Example 5<br />
Example 6<br />
<br />
Example 7Example 8<br />
Example 9<br />
Example 10Example 11<br />
7.7 Solving Systems with Inverses<br />
AI=IA=AExample 1<br />
AA − =A − =IExample 2<br />
×Example 3<br />
ThExample 4<br />
Example 5<br />
×<br />
Example 6<br />
AX=BAA − AX=A − BExample 7<br />
Example 8<br />
Example 9<br />
7.8 Solving Systems with Cramer's Rule<br />
Th [ a b<br />
c d ] ad−bcExample 1<br />
x= D D<br />
_ x<br />
<br />
D y=<br />
y<br />
<br />
_<br />
Example 2<br />
D<br />
× <br />
Example 3<br />
<br />
x= D D<br />
_ x<br />
<br />
D y=<br />
y<br />
<br />
_<br />
<br />
D z=D _ z<br />
Example 4<br />
D<br />
<br />
Example 5Example 6<br />
<br />
<br />
○ <br />
<br />
○ <br />
<br />
○ <br />
<br />
○ <br />
○ ThA − A<br />
<br />
○ <br />
Example 7Example 8
676 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
CHAPTER 7 REVIEW EXERCISES<br />
SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES<br />
<br />
1. x−y=<br />
x+y=−−<br />
2. x−y=<br />
−x+y=<br />
<br />
3. x+y=−<br />
x−y=−<br />
4. <br />
x+ <br />
y= <br />
<br />
<br />
x− <br />
y=− <br />
<br />
5. x+y=<br />
x+y=<br />
<br />
6. x+y=−<br />
x+y=<br />
7. x+y=<br />
x+y=<br />
8. x+y=<br />
x−y=<br />
<br />
9. <br />
Cx=x+<br />
Rx=x<br />
10. Cx=x+x<br />
. Th<br />
ft<br />
<br />
<br />
SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES<br />
<br />
11. x−y=<br />
−y+x=<br />
x+z=<br />
12. x+y−z=<br />
x−y+z=<br />
x+y+z=<br />
13. x+y+z=<br />
x+y+z=<br />
x+y=<br />
14. x−y+z=−<br />
x+y+z=−<br />
x+y−z=<br />
17. x−y+z=<br />
x+y−z=<br />
x−y−z=<br />
15. x+y−z=−<br />
x−y+z=<br />
−x+y+z=−<br />
18. x−y−z=<br />
x+y−z=<br />
y−z=<br />
16. x+z=−<br />
x−y=<br />
y−z=−<br />
<br />
19. o 61. Th<br />
<br />
<br />
20. . Th<br />
f $8,070.00. Th
CHAPTER 7 REVIEW 677<br />
SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES<br />
<br />
21. y=x −<br />
y=x−<br />
22. y=x −<br />
y=x+<br />
23. x +y =<br />
y=x−<br />
24. x +y =<br />
y=x +<br />
25. x +y =<br />
y−x =<br />
<br />
26. y>x − 27. <br />
x +y <<br />
<br />
28. x +y +x<<br />
y>−x −<br />
29. x −x+y −x<<br />
y
678 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
56. x−y=−<br />
−x+y=<br />
59. x+y+z=<br />
−x−y−z=−<br />
x+y+z=<br />
57. x−y=<br />
−x+y=<br />
60. −x+y−z=<br />
y+z=−<br />
−x+y+z=<br />
58. −x−y=<br />
−x−y=<br />
SOLVING SYSTEMS WITH INVERSES<br />
, fi<br />
61. [ − − ] [ 62. <br />
<br />
− <br />
<br />
− <br />
] <br />
<br />
−<br />
[ 63. <br />
<br />
− <br />
− − <br />
] [ 64. <br />
<br />
] <br />
, fi<br />
65. x−y=−<br />
−x+y=<br />
66. x−y=−<br />
−x+y=<br />
67. x+y−z=−<br />
x−y−z=−<br />
x+z=−<br />
68. −x−y+z=<br />
−x+y+z=−<br />
−y+z=−<br />
<br />
69. <br />
<br />
<br />
<br />
<br />
70. <br />
ies. Th<br />
<br />
$1. Th<br />
<br />
SOLVING SYSTEMS WITH CRAMER'S RULE<br />
, fi<br />
<br />
−<br />
71. [ ] 72. [ − − <br />
− ] [ 73. <br />
] <br />
√<br />
] [ — 74. <br />
— √ <br />
√ <br />
— <br />
<br />
75. x−y=<br />
−x−y=−<br />
76. x−y=<br />
−x+y=<br />
77. −x+y=<br />
−x+y=<br />
78. x+y+z=<br />
x+y+z=<br />
x−y+z=<br />
79. x−y+z=− <br />
<br />
x−y−z= <br />
<br />
x−y−z= <br />
<br />
80. <br />
x− <br />
y− <br />
z=− <br />
<br />
x− <br />
y− <br />
z=− <br />
<br />
x− <br />
y− <br />
z=−
CHAPTER 7 PRACTICE TEST<br />
679<br />
CHAPTER 7 PRACTICE TEST<br />
<br />
1. −x−y=<br />
x+y=−<br />
<br />
<br />
2. <br />
x− <br />
y=<br />
<br />
x−y=<br />
3. − <br />
x−y=<br />
x+y=<br />
4. x−y=<br />
−x+y=−<br />
5. x−y−z= <br />
x−y+z=− <br />
<br />
x+y−z= <br />
<br />
6. x+z=<br />
x+y+z=<br />
x+y+z=<br />
7. x−y−z=<br />
x+y+z=<br />
x−y−z=<br />
8. y=x +x−<br />
y=x−<br />
9. y +x =<br />
y −x =<br />
<br />
10. y<br />
y
680 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES<br />
<br />
23. x−y=<br />
x−y=<br />
24. x+y+z=−<br />
x−y+z=<br />
x−y−z=<br />
<br />
25. x−y=−<br />
−x+y=<br />
26. <br />
x− <br />
y+ <br />
z=−<br />
<br />
x− <br />
y− <br />
z=<br />
<br />
x− <br />
y− <br />
z=<br />
<br />
27. x−y=<br />
x+y=<br />
28. x+y−z=−<br />
x−y+z=−<br />
x−y+z=−<br />
<br />
29. <br />
<br />
Cx=x +x+Rx=x +x<br />
<br />
fi<br />
fi<br />
30.
8<br />
Analytic Geometry<br />
a<br />
b<br />
Figure 1 (a) Greek philosopher Aristotle (384–322 BCE) (b) German mathematician and astronomer Johannes Kepler (1571–1630)<br />
CHAPTER OUTLINE<br />
8.1 The Ellipse<br />
8.2 The Hyperbola<br />
8.3 The Parabola<br />
8.4 Rotation of Axes<br />
8.5 Conic Sections in Polar Coordinates<br />
Introduction<br />
Th<br />
<br />
d diff<br />
<br />
<br />
<br />
<br />
<br />
<br />
fi<br />
fifi<br />
h fi<br />
681
682<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Write equations of ellipses in standard form.<br />
Graph ellipses centered at the origin.<br />
Graph ellipses not centered at the origin.<br />
Solve applied problems involving ellipses.<br />
8.1 THE ELLIPSE<br />
Figure 1 The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)<br />
<br />
ThFigure 1 <br />
whispering chamber<br />
<br />
<br />
Writing Equations of Ellipses in Standard Form<br />
conicTh<br />
Figure 2<br />
<br />
Figure 2<br />
<br />
Thffi<br />
Th
SECTION 8.1 THE ELLIPSE 683<br />
ellipsex y<br />
o fih fifocusfoci<br />
<br />
<br />
fi<br />
g. ThFigure 3<br />
<br />
Figure 3<br />
Thmajor axis<br />
minoraxisvertexvertices<br />
Thcenter of an ellipseTh<br />
Th<br />
Figure 4<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 4<br />
<br />
Th<br />
xy
684<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Deriving the Equation of an Ellipse Centered at the Origin<br />
−ccTh<br />
xyxyFigure 5<br />
x, y<br />
y<br />
a, <br />
d 1<br />
d 2<br />
c, <br />
c, )<br />
a, <br />
x<br />
Figure 5<br />
a−caa −−c=a +cThc<br />
aa −c . Th<br />
<br />
a+c+a−c=a<br />
xyfi<br />
d <br />
=−cxy<br />
d <br />
=cxy<br />
fid <br />
+d <br />
xy<br />
aad <br />
+d <br />
=a<br />
. Th<br />
d <br />
+d <br />
=√ ——<br />
x−−c +y− +√ — x−c +y− =a<br />
√ — x+c +y +√ — x−c +y =a<br />
<br />
√ — x+c +y =a −√ — x−c +y <br />
<br />
x+c +y = a − √ — x−c +y <br />
x +cx+c +y =a −a √ — x−c +y +x−c +y <br />
<br />
x +cx+c +y =a −a √ — x−c +y +x −cx+c +y <br />
<br />
cx=a −a √ — x−c +y −cx <br />
cx−a =−a √ — x−c +y <br />
cx−a =−a √ — x−c +y <br />
<br />
<br />
[ cx−a ] =a [ √ — x−c +y ] <br />
<br />
c x −a cx+a =a ( x −cx+c +y ) <br />
<br />
c x −a cx+a =a x −a cx+a c +a y <br />
a <br />
a x −c x +a y =a −a c <br />
<br />
x a −c +a y =a a −c <br />
<br />
x b +a y =a b <br />
b =a −c <br />
x b <br />
a b y <br />
+a a b b <br />
=a a b <br />
a b <br />
x <br />
a +y b =<br />
<br />
Th x <br />
_<br />
<br />
a +y b =Thfi<br />
a >bb >a
SECTION 8.1 THE ELLIPSE 685<br />
Writing Equations of Ellipses Centered at the Origin in Standard Form<br />
<br />
<br />
<br />
<br />
Th<br />
<br />
ThThfi<br />
<br />
<br />
<br />
standard forms of the equation of an ellipse with center (0, 0)<br />
Thx<br />
x _<br />
<br />
a +y _<br />
<br />
b<br />
<br />
=<br />
a>b<br />
a<br />
±a<br />
b<br />
±b<br />
±cc =a −b Figure 6a.<br />
Thy<br />
x _<br />
<br />
b +y _<br />
<br />
a<br />
<br />
=<br />
a>b<br />
a<br />
±a<br />
b<br />
±b<br />
±cc =a −b Figure 6b.<br />
c =a −b <br />
fi<br />
<br />
y<br />
y<br />
b<br />
a<br />
c<br />
−a<br />
−c<br />
<br />
<br />
<br />
c<br />
a<br />
x<br />
−b<br />
<br />
<br />
<br />
<br />
<br />
b<br />
x<br />
−c<br />
−b<br />
(a)<br />
Figure 6 (a) Horizontal ellipse with center (0, 0) (b) Vertical ellipse with center (0, 0)<br />
−a<br />
(b)
686<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
How To…<br />
<br />
1. xy<br />
<br />
a. ±a±c<br />
x x _<br />
<br />
a +y _<br />
<br />
b =<br />
b. ±a±c<br />
y x _<br />
<br />
b +y _<br />
<br />
a =<br />
2. c =a −b b <br />
3. a b <br />
Example 1<br />
Writing the Equation of an Ellipse Centered at the Origin in Standard Form<br />
±±<br />
Solution Thxx. Th<br />
<br />
x <br />
a +y _<br />
<br />
b =<br />
Th±a =a =<br />
Th±c =c =<br />
c =a −b b <br />
c =a −b <br />
=−b c a <br />
b = b <br />
a =b =<br />
x _<br />
<br />
+y _<br />
<br />
=<br />
Try It #1<br />
√ — <br />
Q & A…<br />
Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?<br />
<br />
±a±a±c±c<br />
acc =a −b o fib <br />
Writing Equations of Ellipses Not Centered at the Origin<br />
h<br />
khkTh<br />
xx−hyy−k
SECTION 8.1 THE ELLIPSE 687<br />
standard forms of the equation of an ellipse with center (h, k)<br />
Thhkx<br />
_<br />
x −h <br />
a +y _<br />
−k <br />
b =<br />
<br />
a >b<br />
a<br />
h±ak<br />
b<br />
hk ±b<br />
h±ckc =a −b Figure 7a<br />
Thhky<br />
_<br />
x −h <br />
b +y _<br />
−k <br />
a =<br />
<br />
a >b<br />
a<br />
h k±a<br />
b<br />
h±bk<br />
h, k±cc =a −b Figure 7b<br />
h, k<br />
c = a − b . <br />
<br />
h −a, k<br />
h −c <br />
, k<br />
y<br />
h, k + b<br />
h, k<br />
<br />
<br />
<br />
h + c <br />
, k<br />
h + a, k<br />
x<br />
h −b, k<br />
y<br />
h, k + a<br />
<br />
<br />
hk<br />
<br />
<br />
h, k + c<br />
h + b, k<br />
x<br />
h, k −b<br />
hk −a<br />
(a)<br />
(b)<br />
Figure 7 (a) Horizontal ellipse with center (h, k) (b) Vertical ellipse with center (h, k)<br />
hk −c<br />
How To…<br />
<br />
1. xy<br />
a. yx<br />
_<br />
x −h <br />
a +y _<br />
−k <br />
b =<br />
b. xy<br />
_<br />
x −h <br />
b +y _<br />
−k <br />
a =
688<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
2. hk<br />
3. a a<br />
4. c hk<br />
5. b c =a −b <br />
6. hka b <br />
Example 2<br />
Writing the Equation of an Ellipse Centered at a Point Other Than the Origin<br />
−−−−−−<br />
Solution xyTh<br />
<br />
<br />
_<br />
x −h <br />
b +y _<br />
−k <br />
a =<br />
hk. Th−−−<br />
<br />
<br />
hk=( −+− <br />
−+ ) <br />
<br />
=−−<br />
fia Thaafi<br />
y<br />
<br />
a =−−<br />
<br />
a =<br />
<br />
a =<br />
a =<br />
fic Thhk ±chk −c=−−hk +c=−k =−<br />
c<br />
<br />
k +c =<br />
<br />
−+c =<br />
<br />
c =<br />
c =<br />
b c =a −b <br />
<br />
c =a −b <br />
<br />
=−b <br />
<br />
b =<br />
hka b <br />
<br />
_ x+ +_ y+ =<br />
<br />
Try It #2<br />
−−√ — <br />
+√ — <br />
Graphing Ellipses Centered at the Origin<br />
<br />
_<br />
x <br />
_<br />
<br />
a +y b =a >bx _<br />
<br />
_<br />
<br />
b +y a<br />
a >b<br />
=
SECTION 8.1 THE ELLIPSE 689<br />
How To…<br />
<br />
1. <br />
a. _<br />
x <br />
_<br />
<br />
a +y b=a >b<br />
x<br />
±a<br />
±b<br />
±c<br />
b. _<br />
x <br />
_<br />
<br />
b +y a=a >b<br />
y<br />
±a<br />
±b<br />
±c<br />
2. cc =a −b <br />
3. <br />
Example 3<br />
Graphing an Ellipse Centered at the Origin<br />
x _ + <br />
y _<br />
=<br />
Solution >yTh<br />
x _<br />
<br />
b +y _<br />
<br />
a =b =a =<br />
<br />
±a=±√ — =±<br />
±b=±√ — =±<br />
±cc =a −b c<br />
<br />
c =±√ — a −b <br />
<br />
=±√ — −<br />
<br />
=±√ — <br />
<br />
=±<br />
Th±<br />
<br />
Figure 8<br />
y<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
Figure 8
690<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Try It #3<br />
x <br />
+y _<br />
=<br />
<br />
Example 4<br />
Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form<br />
x +y =Th<br />
<br />
Solution <br />
<br />
x +y =<br />
<br />
x <br />
+y _<br />
=<br />
<br />
<br />
_ x <br />
+y _<br />
= <br />
>xTh<br />
_<br />
x <br />
_<br />
<br />
=b =<br />
a +y b =a<br />
<br />
±a=±√ — =±<br />
±b=±√ — =±<br />
±cc =a −b c<br />
<br />
c =±√ — a −b <br />
<br />
=±√ — −<br />
<br />
=±√ — <br />
Th±√ — <br />
<br />
Figure 9<br />
y<br />
<br />
<br />
<br />
<br />
x<br />
−<br />
Figure 9<br />
Try It #4<br />
x +y =Th<br />
<br />
Graphing Ellipses Not Centered at the Origin<br />
fi<br />
hk x _<br />
−h <br />
a +y _<br />
−k <br />
b =a >b<br />
_<br />
x −h <br />
b +y _<br />
−k <br />
a =a >b
SECTION 8.1 THE ELLIPSE 691<br />
How To…<br />
hk<br />
1. <br />
<br />
a. _<br />
x −h <br />
a +y _<br />
−k <br />
b =a >b<br />
hk<br />
x<br />
h±ak<br />
hk ±b<br />
h±ck<br />
b. _<br />
x −h <br />
b +y _<br />
−k <br />
a =a >b<br />
hk<br />
y<br />
hk ±a<br />
h±bk<br />
hk ±c<br />
2. cc =a −b <br />
3. <br />
Example 5 Graphing an Ellipse Centered at (h, k)<br />
_ x+ + y _ − =<br />
<br />
<br />
Solution >y<br />
Th_<br />
x −h <br />
b +y _<br />
−k <br />
a =b =a =<br />
hk=−<br />
hk ±a=−±√ — =−±−−<br />
h±bk=−±√ — =−±−<br />
hk ±cc =a −b c<br />
<br />
c =±√ — a −b <br />
<br />
=±√ — −<br />
<br />
=±√ — <br />
Th−−√ — −+√ — <br />
<br />
y<br />
−<br />
−+√<br />
−<br />
−<br />
<br />
−−√<br />
−<br />
x<br />
Figure 10
692<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Try It #5<br />
_ x− +_ y− =<br />
<br />
<br />
How To…<br />
hk<br />
1. ax +by +cx+dy+e =<br />
2. <br />
<br />
3. ffix y <br />
4. <br />
m <br />
x−h +m <br />
y−k =m <br />
m <br />
m <br />
m <br />
<br />
5. <br />
Example 6<br />
Graphing an Ellipse Centered at ( h, k) by First Writing It in Standard Form<br />
x +y −x +y +=<br />
<br />
Solution <br />
<br />
x +y −x +y +=<br />
<br />
<br />
x −x+y +y=−<br />
ffi<br />
<br />
x −x+y +y=−<br />
<br />
<br />
x −x ++y +y +=−++<br />
<br />
<br />
x− +y+ =<br />
<br />
<br />
_ x− +_ y+ =<br />
<br />
><br />
xTh_<br />
x −h <br />
a +y _<br />
−k <br />
b =a =b =<br />
<br />
hk=−<br />
h±ak=±√ — −=±−−−<br />
hk ±b=−±√ — =−±−<br />
h±ckc =a −b c<br />
<br />
c =±√ — a −b <br />
<br />
=±√ — −<br />
<br />
=±√ — <br />
Th−√ — −+√ — −<br />
<br />
Figure 11
SECTION 8.1 THE ELLIPSE 693<br />
Try It #6<br />
−√5, − +√5, −<br />
<br />
<br />
x<br />
<br />
−<br />
<br />
−<br />
−<br />
<br />
<br />
−<br />
Figure 11<br />
<br />
<br />
x +y −x+y+=<br />
Solving Applied Problems Involving Ellipses<br />
<br />
fl<br />
<br />
<br />
Th<br />
fl Figure 12<br />
<br />
<br />
Example 7<br />
Figure 12 Sound waves are reflected between foci in an elliptical room, called a whispering chamber.<br />
Locating the Foci of a Whispering Chamber<br />
Th<br />
Figure 13<br />
a. <br />
<br />
b. <br />
<br />
<br />
<br />
Figure 13
694<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Solution<br />
<br />
a. <br />
x _<br />
<br />
_<br />
<br />
a +y b=a >ba<br />
b<br />
<br />
aa =a =a =<br />
bb =b =b =<br />
Th_<br />
x <br />
+ _<br />
y <br />
=<br />
b. ±c<br />
c =a −b c<br />
c =a −b <br />
c =− <br />
c =±√ — − <br />
c =±√ — <br />
c ≈± <br />
Th±. Th=<br />
Try It #7<br />
<br />
a. <br />
<br />
b. <br />
<br />
<br />
Conic Sections: The Ellipse (http://openstaxcollege.org/l/conicellipse)<br />
Graph an Ellipse with Center at the Origin (http://openstaxcollege.org/l/grphellorigin)<br />
Graph an Ellipse with Center Not at the Origin (http://openstaxcollege.org/l/grphellnot)
SECTION 8.1 SECTION EXERCISES 695<br />
8.1 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
3. <br />
<br />
5. <br />
<br />
y<br />
4. <br />
<br />
<br />
ALGEBRAIC<br />
<br />
6. x +y= 7. x +y = 8. x −y =<br />
9. x +y = 10. x −x+y −y+=<br />
<br />
<br />
11. _ x + _<br />
y = 12.<br />
<br />
_ x <br />
+ y _<br />
<br />
=<br />
13. x +y =<br />
14. x +y = 15. x− <br />
+y− = 16. x− _<br />
+_ y+ =<br />
<br />
17. _ x+ +_ y− = 18. x− + y− = 19. x −x+y −y+=<br />
20. x −x+y −y+= 21. x −x+y −y+=<br />
22. x +x+y −y+= 23. x +x+y −y+=<br />
24. x +x+y −y+= 25. x +x+y +y+=<br />
26. x +x+y +y+=<br />
, fi<br />
27. _ x+ + _ y+ = 28. _ x+ + _ y− = 29. x +y =<br />
<br />
30. x +y +x+y= 31. x +y +x =<br />
GRAPHICAL<br />
<br />
32. _ x <br />
+y _<br />
<br />
=<br />
33. _ x <br />
+y _<br />
= <br />
34. x +y =<br />
35. x +y = 36. _ x− +_ y− = 37. _ x+ +_ y− =<br />
<br />
38. _ x + <br />
_ y+ = 39. x −x+y −y−=<br />
<br />
40. x −x+y −y+= 41. x +x+y −y+=<br />
42. x +x+y −y−= 43. x +x+y −y+=<br />
44. x +x+y −y+= 45. x +x+y +y+=<br />
<br />
46. x-<br />
y<br />
47. <br />
x-y−
696 CHAPTER 8 ANALYTIC GEOMETRY<br />
48. <br />
x-y<br />
<br />
49. +√ — <br />
50. +√ — 51. −−+√ — <br />
<br />
52.<br />
55.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
x<br />
x<br />
53. y<br />
54.<br />
<br />
56.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
EXTENSIONS<br />
, fi. Th=a ċb ċπ<br />
57. _ x− +_ y− = 58. _ x+ +_ y− = 59. _ x+ +_ y− =<br />
<br />
60. x −x+y −y+= 61. x −x+y −y+=<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
x<br />
REAL-WORLD APPLICATIONS<br />
62. t fi<br />
<br />
64. <br />
e). Th<br />
<br />
<br />
<br />
66. <br />
et. Th<br />
<br />
<br />
<br />
68. <br />
<br />
<br />
<br />
63. t fi<br />
<br />
h<br />
65. . Th<br />
<br />
<br />
<br />
<br />
67.
SECTION 8.2 THE HYPERBOLA 697<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Locate a hyperbola’s vertices and foci.<br />
Write equations of hyperbolas in standard form.<br />
Graph hyperbolas centered at the origin.<br />
Graph hyperbolas not centered at the origin.<br />
Solve applied problems involving hyperbolas.<br />
8.2 THE HYPERBOLA<br />
<br />
Th<br />
<br />
Figure 1<br />
<br />
<br />
<br />
<br />
Figure 1<br />
A shock wave intersecting the ground forms a portion of a conic and results in a sonic boom.<br />
ft<br />
fiflTh<br />
Thfi<br />
<br />
Locating the Vertices and Foci of a Hyperbola<br />
hyperbola<br />
Th<br />
Figure 2<br />
Figure 2 A hyperbola<br />
fi<br />
xyffxy
698<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
fiTh<br />
fidifferencefisum<br />
Thtransverse axis<br />
Th<br />
conjugate axisThcenter of a<br />
hyperbola<br />
asymptotes<br />
Thcentral rectangle<br />
<br />
Figure 3<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
Figure 3<br />
Key features of the hyperbola<br />
<br />
xy<br />
<br />
Deriving the Equation of an Ellipse Centered at the Origin<br />
−ccThxy<br />
ffxyFigure 4<br />
y<br />
xy<br />
d <br />
d <br />
x<br />
–c<br />
c<br />
–a<br />
a<br />
Figure 4<br />
a−caa −−c=a +cThc<br />
ac −a. Th<br />
<br />
a+c−c−a=a
SECTION 8.2 THE HYPERBOLA 699<br />
xyfi<br />
d <br />
=−cxy<br />
<br />
d <br />
=cxy<br />
fid <br />
−d <br />
xyff<br />
aad <br />
−d <br />
=a<br />
Th<br />
<br />
d <br />
−d <br />
=√ ——<br />
x−−c +y− −√ — x−c +y− =a <br />
√ — x +c +y −√ — x −c +y =a<br />
<br />
√ — x +c +y =a +√ — x −c +y <br />
<br />
x +c +y =a +√ — x −c +y <br />
x +cx +c +y =a +a √ — x −c +y +x −c +y <br />
<br />
x +cx +c +y =a +a √ — x −c +y +x −cx +c +y <br />
<br />
cx =a +a √ — x −c +y −cx <br />
cx −a =a √ — x −c +y <br />
cx −a =a √ — x −c +y <br />
cx−a =a ( √ — x−c +y ) <br />
<br />
<br />
<br />
c x −a cx+a =a x −cx+c +y <br />
c x −a cx+a =a x −a cx+a c +a y a <br />
a +c x =a x +a c +a y <br />
c x −a x −a y =a c −a <br />
x c −a −a y =a c −a <br />
x b −a y =a b b =c −a <br />
x b <br />
a b y <br />
−a a b b <br />
=a a b <br />
a b <br />
_ x <br />
_<br />
<br />
a −y b =<br />
Thfi±a±b<br />
standard forms of the equation of a hyperbola with center (0, 0)<br />
Thx<br />
_ x <br />
_<br />
<br />
a −y b =<br />
<br />
a<br />
±a<br />
b<br />
±b<br />
cc =a +b <br />
±c<br />
y=± b __<br />
a x
700<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Figure 5a<br />
Thy<br />
y _<br />
<br />
_<br />
<br />
a −x b =<br />
<br />
a<br />
±a<br />
b<br />
±b<br />
cc =a +b <br />
±c<br />
y =±_<br />
a b x<br />
Figure 5b<br />
c =a +b <br />
<br />
y<br />
y<br />
y = − b<br />
a x<br />
b<br />
y = b<br />
a x<br />
y = − a<br />
b x<br />
c<br />
a<br />
y = a<br />
b x<br />
−c<br />
−a<br />
<br />
a<br />
c<br />
x<br />
−b<br />
<br />
b<br />
x<br />
−b<br />
−c<br />
−a<br />
(a)<br />
Figure 5<br />
(a) Horizontal hyperbola with center (0, 0) (b) Vertical hyperbola with center (0, 0)<br />
(b)<br />
How To…<br />
<br />
1. xya <br />
ffifi<br />
<br />
<br />
a. _<br />
x <br />
_<br />
<br />
a −y b =xTh<br />
±a±c<br />
b. _<br />
y <br />
_<br />
<br />
a −x b =yTh<br />
±a±c<br />
2. aa =√ — a <br />
3. cc =√ — a +b
SECTION 8.2 THE HYPERBOLA 701<br />
Example 1<br />
Locating a Hyperbola’s Vertices and Foci<br />
_<br />
y −x _<br />
=<br />
Solution Th_<br />
y <br />
_<br />
<br />
a −x b =yTh<br />
yo fix =y<br />
<br />
<br />
<br />
<br />
<br />
Th±cc<br />
<br />
=_<br />
y −x _<br />
=_<br />
y − _<br />
=_<br />
y y =<br />
y =±√ — =±<br />
c =√ — a +b =√ — +=√ — =<br />
Th±<br />
Try It #1<br />
_ x − <br />
_<br />
y =<br />
Writing Equations of Hyperbolas in Standard Form<br />
<br />
<br />
fi<br />
Thfi<br />
<br />
Hyperbolas Centered at the Origin<br />
<br />
c =a +b b =c −a Th<br />
<br />
How To…<br />
<br />
1. xy<br />
a. ±a±c<br />
x_<br />
x <br />
_<br />
<br />
a −y b =<br />
b. ±a±c<br />
y_<br />
y <br />
_<br />
<br />
a −x b =<br />
2. b b =c −a <br />
3. a b
702<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Example 2<br />
Finding the Equation of a Hyperbola Centered at (0, 0) Given its Foci and Vertices<br />
±±√ — <br />
Solution ThxTh_<br />
x <br />
_<br />
<br />
a −y b =<br />
Th±a =a =<br />
Th±√ — c =√ — c =<br />
b <br />
<br />
b =c −a <br />
b =− c a <br />
b = <br />
a =b =_<br />
x <br />
_<br />
<br />
a −y b =Th<br />
_<br />
x <br />
−y _<br />
=Figure <br />
6<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
Figure 6<br />
Try It #2<br />
√ — <br />
Hyperbolas Not Centered at the Origin<br />
h<br />
khkTh<br />
xx−hyy−k<br />
standard forms of the equation of a hyperbola with center (h, k)<br />
Thhkx<br />
_<br />
x −h <br />
a −y _<br />
−k <br />
b =<br />
<br />
a<br />
h±ak<br />
b<br />
hk ±b<br />
cc =a +b <br />
h±ck
SECTION 8.2 THE HYPERBOLA 703<br />
ThTh<br />
abTh± b <br />
a hk<br />
point-slope formulay =± b <br />
a x−h+k<br />
Figure 7a<br />
Thhky<br />
y _<br />
−k <br />
a −x _<br />
−h <br />
b =<br />
<br />
a<br />
hk ±a<br />
b<br />
h±bk<br />
cc =a +b <br />
hk ±c<br />
y =± a x−h+kFigure 7b<br />
b<br />
y<br />
y = − b x − h+k<br />
a<br />
h, k + b<br />
y<br />
h, k + c<br />
h, k + a<br />
y = a x − h+k<br />
b<br />
h − b, k<br />
h + b, k<br />
h − c, k<br />
h − a, k<br />
h, k<br />
h + c, k<br />
h + a, k<br />
x<br />
h, k<br />
h, k − a<br />
y = − a x − h+k<br />
b<br />
x<br />
h, k − b<br />
h, k − c<br />
y =<br />
a<br />
b x − h+k<br />
<br />
<br />
Figure 7 (a) Horizontal hyperbola with center (h, k) (b) Vertical hyperbola with center (h, k)<br />
hk<br />
c =a +b <br />
o fi<br />
How To…<br />
hk<br />
1. xy<br />
a. yx<br />
_<br />
x −h <br />
a −y _<br />
−k <br />
b =<br />
b. xy<br />
y _<br />
−k <br />
a −x _<br />
−h <br />
b =<br />
2. hk
704<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
3. a a<br />
4. c hk<br />
5. b b =c −a <br />
6. hka b <br />
Example 3<br />
Finding the Equation of a Hyperbola Centered at (h, k) Given its Foci and Vertices<br />
−−−−<br />
−<br />
Solution yx <br />
<br />
<br />
_<br />
x −h <br />
a −y _<br />
−k <br />
b =<br />
hkTh−−<br />
<br />
<br />
hk=( + <br />
−+− ) =−<br />
fia Thafia fi<br />
x<br />
<br />
a =|−|<br />
<br />
a =<br />
<br />
a =<br />
<br />
a =<br />
fic Thh±ckh−ck=−−h+ck=−<br />
xc−h =<br />
<br />
h +c =<br />
<br />
+c =<br />
<br />
c =<br />
<br />
c =<br />
b b =c −a <br />
<br />
b =c −a <br />
<br />
=−<br />
<br />
=<br />
hka b <br />
<br />
_ x− −_ y+ =<br />
<br />
Try It #3<br />
−−<br />
Graphing Hyperbolas Centered at the Origin<br />
<br />
<br />
_<br />
x <br />
_<br />
y <br />
_<br />
<br />
a −x b =<br />
_<br />
<br />
a −y b =
SECTION 8.2 THE HYPERBOLA 705<br />
How To…<br />
<br />
1. <br />
2. fi<br />
<br />
a. _<br />
x <br />
_<br />
<br />
a −y b =<br />
x<br />
±a<br />
±b<br />
±c<br />
y =±_<br />
a b x<br />
b. _<br />
y <br />
_<br />
<br />
a −x b =<br />
y<br />
±a<br />
±b<br />
±c<br />
y =± a <br />
b x<br />
3. c =±√ — a +b <br />
4. <br />
<br />
Example 4<br />
Graphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form<br />
_ y −_ x =<br />
<br />
<br />
Solution Th_<br />
y <br />
_<br />
<br />
a −x b<br />
y<br />
=Th<br />
Th±a=±√ — =±<br />
Th±b=±√ — =±<br />
Th±cc =±√ — a +b c<br />
Th±<br />
c=±√ — a +b =±√ — +=±√ — =±<br />
Thy =±<br />
__ a b x =± <br />
x =± <br />
x<br />
<br />
<br />
Th<br />
Figure 8
706<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
y<br />
y = − x<br />
<br />
<br />
<br />
y = x<br />
<br />
−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
x<br />
−<br />
<br />
<br />
x<br />
<br />
−<br />
−−<br />
−<br />
<br />
Try It #4<br />
Figure 8<br />
_<br />
x <br />
−y _<br />
= <br />
<br />
<br />
Graphing Hyperbolas Not Centered at the Origin<br />
hk<br />
x _<br />
−h <br />
a −y _<br />
−k <br />
b =<br />
y _<br />
−k <br />
a −x _<br />
−h <br />
b =<br />
<br />
<br />
How To…<br />
hk<br />
1. <br />
<br />
2. fi<br />
<br />
a. _<br />
x −h <br />
a −y _<br />
−k <br />
b =<br />
x<br />
hk<br />
h±ak<br />
hk ±b<br />
h±ck<br />
y =± b <br />
a x−h+k
SECTION 8.2 THE HYPERBOLA 707<br />
b. y −k <br />
a −x −h <br />
b =<br />
y<br />
hk<br />
hk ±a<br />
h±bk<br />
hk ±c<br />
y =±_<br />
a<br />
b x−h+k<br />
3. c =±√ — a +b <br />
4. <br />
<br />
Example 5<br />
Graphing a Hyperbola Centered at (h, k) Given an Equation in General Form<br />
x −y −x −y −=<br />
<br />
Solution <br />
<br />
<br />
x −x−y +y=<br />
ffi<br />
<br />
x −x−y +y=<br />
<br />
<br />
x −x +−y +y +=+−<br />
<br />
<br />
x− −y+ =<br />
<br />
<br />
_ x− − y _ + =<br />
<br />
Th_<br />
x−h <br />
a −y−k _<br />
<br />
b =a =b =a =<br />
b =. Thx<br />
hk=−<br />
h±ak=±−−−−<br />
hk ±b=−±−<br />
h±ckc =±√ — a +b c<br />
c=±√ — +=±√ — =±√ — <br />
Th−√ — −+√ — −<br />
Thy =± b <br />
a x−h+k =± <br />
x−−<br />
<br />
Figure 9
708<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
y<br />
<br />
x<br />
−−<br />
−<br />
−<br />
−<br />
Figure 9<br />
Try It #5<br />
_ y+ −_ x− =<br />
<br />
<br />
Solving Applied Problems Involving Hyperbolas<br />
fi<br />
Thffi<br />
ft<br />
ffi<br />
Figure 10<br />
<br />
Figure 10 Cooling towers at the Drax power station in North Yorkshire, United Kingdom (credit: Les Haines, Flickr)<br />
Thfi<br />
Example 6fi
SECTION 8.2 THE HYPERBOLA 709<br />
Example 6<br />
Solving Applied Problems Involving Hyperbolas<br />
ThFigure 11ThTh<br />
<br />
<br />
<br />
<br />
<br />
Figure 11 Project design for a natural draft cooling tower<br />
<br />
fi<br />
fi<br />
Solution <br />
_<br />
x <br />
_<br />
<br />
a −y b =<br />
t fia b <br />
fia aTh<br />
a =. Tha =a =<br />
b xy<br />
fixy<br />
yx<br />
ThyTh<br />
<br />
<br />
_<br />
x <br />
_<br />
<br />
a −y b =<br />
b =<br />
y <br />
_<br />
_<br />
x <br />
a −<br />
<br />
b <br />
<br />
=<br />
<br />
_<br />
<br />
−<br />
a xy<br />
≈ <br />
Th<br />
_ x <br />
− y y _<br />
<br />
=x _<br />
<br />
−<br />
_<br />
<br />
=
710<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Try It #6<br />
Figure 12<br />
<br />
e fid fi<br />
<br />
<br />
<br />
<br />
Figure 12<br />
<br />
Conic Sections: The Hyperbola Part 1 of 2 (http://openstaxcollege.org/l/hyperbola1)<br />
Conic Sections: The Hyperbola Part 2 of 2 (http://openstaxcollege.org/l/hyperbola2)<br />
Graph a Hyperbola with Center at Origin (http://openstaxcollege.org/l/hyperbolaorigin)<br />
Graph a Hyperbola with Center not at Origin (http://openstaxcollege.org/l/hbnotorigin)
SECTION 8.2 SECTION EXERCISES 711<br />
8.2 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
<br />
3. 4. <br />
<br />
5. <br />
<br />
ALGEBRAIC<br />
<br />
<br />
6. y +x= 7. x <br />
−y <br />
=<br />
8. y +x =x 9. x −y =<br />
10. −x +x+y +y−=<br />
<br />
<br />
11. x <br />
x −y =<br />
12.<br />
<br />
<br />
−y <br />
=<br />
13. y <br />
−x <br />
=<br />
14. y −x =<br />
15. x− <br />
−y − <br />
= 16. y − <br />
−x+ =<br />
17. x− <br />
−y + <br />
= 18. x −x−y −y+=<br />
19. −x −x+y −y+= 20. x −x−y −y+=<br />
21. −x +x+y −y+= 22. −x +x+y −y+=<br />
23. x +x−y −y+= 24. −x +x+y +y+=<br />
25. x +x−y +y−=<br />
, fi<br />
26. y <br />
−x =<br />
27. x− <br />
−y + <br />
=<br />
28. y − <br />
−x+ <br />
= 29. x −x−y +y−=<br />
30. y +y−x +x+=<br />
GRAPHICAL<br />
<br />
31. x <br />
−y <br />
=<br />
33. y <br />
−x <br />
=<br />
32. x <br />
−y <br />
=<br />
34. x −y =<br />
35. y + <br />
−x− <br />
= 36. x− <br />
−y+ <br />
=<br />
37. y − <br />
−x− <br />
= 38. −x −x+y −y−=
712 CHAPTER 8 ANALYTIC GEOMETRY<br />
39. x −x−y −y−= 40. −x +x+y −y+=<br />
41. x +x−y −y−= 42. x +x−y −y−=<br />
43. −x +x+y −y−= 44. x +x−y +y+=<br />
, fi<br />
45. − 46. −<br />
−<br />
47. 48. −√ — <br />
49. +√ — 50. +√ — <br />
, fi<br />
51.<br />
52.<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
x<br />
53.<br />
<br />
y<br />
54.<br />
y<br />
<br />
−<br />
<br />
<br />
−<br />
x<br />
<br />
<br />
<br />
−−<br />
<br />
<br />
55.<br />
y<br />
<br />
<br />
x<br />
––<br />
––<br />
<br />
–
SECTION 8.2 SECTION EXERCISES 713<br />
EXTENSIONS<br />
yx<br />
<br />
56. _ x − <br />
_ y = <br />
57. _ y − <br />
_ x = <br />
58. _ x− − y _ + =<br />
<br />
59. −x −x+y −y−= 60. x −x−y −y+=<br />
REAL-WORLD APPLICATIONS<br />
<br />
<br />
61. Thy=x<br />
y=−x<br />
<br />
62. Thy=x<br />
y=−x<br />
<br />
63. Thy= <br />
x<br />
y=− <br />
x<br />
<br />
64. Thy= <br />
x<br />
y=− <br />
x<br />
<br />
65. Thy= <br />
x<br />
y=− <br />
x<br />
<br />
<br />
x<br />
fl<br />
66. Th<br />
y=x−<br />
<br />
<br />
<br />
y=−x+<br />
68. Th<br />
y=x+<br />
<br />
<br />
y=−x−<br />
70. Th<br />
y=x−<br />
<br />
<br />
y=−x+<br />
67. Th<br />
y=x−<br />
<br />
<br />
y=−x+<br />
69. Th<br />
y= <br />
x−<br />
<br />
<br />
y=− <br />
x+
714<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Graph parabolas with vertices at the origin.<br />
Write equations of parabolas in standard form.<br />
Graph parabolas with vertices not at the origin.<br />
Solve applied problems involving parabolas.<br />
8.3 THE PARABOLA<br />
Figure 1 The Olympic torch concludes its journey around the world when it is used to light the<br />
Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)<br />
Th<br />
flTh<br />
fi<br />
Figure 1<br />
e fl<br />
flTh<br />
<br />
fl<br />
<br />
ffi<br />
Graphing Parabolas with Vertices at the Origin<br />
The Ellipse<br />
. ThparabolaFigure 2<br />
Figure 2 Parabola
SECTION 8.3 THE PARABOLA 715<br />
fi<br />
xyfidirectrixfi<br />
focus<br />
Quadratic Functions<br />
Figure 3<br />
. Th<br />
Thlatus rectumTh<br />
fidP<br />
P<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
Figure 3 Key features of the parabola<br />
<br />
<br />
y<br />
y = −p<br />
p<br />
<br />
d<br />
xy<br />
d<br />
x−p<br />
x<br />
Figure 4<br />
xypy =−pFigure 4<br />
dxyx−pffyd =y +pTh<br />
pxyd<br />
<br />
<br />
d =√ — x− +y−p <br />
=√ — x +y−p <br />
dy<br />
xypxyx−p<br />
<br />
√ — x +y−p =y +p<br />
<br />
<br />
x +y−p =y+p <br />
<br />
<br />
<br />
x +y −py+p =y +py+p <br />
x −py=py<br />
x =py<br />
Thy =pxxx =py<br />
yTh
716<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
standard forms of parabolas with vertex (0, 0)<br />
Table 1Figure 5<br />
Axis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum<br />
x y =px p x=−p p±p<br />
y x =py p y=−p ±pp<br />
Table 1<br />
y<br />
y =px<br />
p > <br />
y =px<br />
p < <br />
y<br />
p|p|<br />
p|p|<br />
<br />
p<br />
p−|p|<br />
x<br />
p<br />
p−|p|<br />
<br />
x<br />
x =−p<br />
(a)<br />
(b)<br />
x = −p<br />
x =py<br />
p > <br />
y<br />
y<br />
<br />
y =−p<br />
x<br />
−|p|, p<br />
p<br />
|p|, p<br />
−|p|, p<br />
p<br />
|p|, p<br />
<br />
(c)<br />
x<br />
y = −p<br />
(d)<br />
x =py<br />
p < <br />
Figure 5 (a) When p > 0 and the axis of symmetry is the x-axis, the parabola opens right. (b) When p < 0 and the axis of symmetry is<br />
the x-axis, the parabola opens left. (c) When p < 0 and the axis of symmetry is the y-axis, the parabola opens up.<br />
(d) When p < 0 and the axis of symmetry is the y-axis, the parabola opens down.<br />
ThFigure 5<br />
<br />
<br />
<br />
Figure 6<br />
y<br />
<br />
y =x<br />
<br />
x<br />
−<br />
x=−<br />
Figure 6
SECTION 8.3 THE PARABOLA 717<br />
How To…<br />
<br />
1. y =pxx =py<br />
2. fi<br />
<br />
a. y =px<br />
xy =<br />
p xpp >p <<br />
<br />
p p<br />
p x =−p<br />
p p±px =p<br />
b. x =py<br />
yx =<br />
p ypp >p <<br />
<br />
p p<br />
p y =−p<br />
p ±pp<br />
3. <br />
Example 1<br />
Graphing a Parabola with Vertex (0, 0) and the x-axis as the Axis of Symmetry<br />
y =x<br />
Solution Thy =pxThx<br />
<br />
=pp =p ><br />
p=<br />
x =−p=−<br />
x x = 6<br />
±<br />
Figure 7<br />
y<br />
x= −<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – <br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
Figure 7<br />
–<br />
x
718<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Try It #1<br />
y =−x<br />
Example 2<br />
Graphing a Parabola with Vertex (0, 0) and the y-axis as the Axis of Symmetry<br />
x =−y<br />
Solution Thx =pyThy<br />
<br />
−=pp =− p <<br />
<br />
p=( − <br />
) <br />
y =−p = <br />
<br />
y = <br />
( ±− <br />
) <br />
<br />
y<br />
y = <br />
<br />
x<br />
−− <br />
− <br />
− <br />
Figure 8<br />
x = −y<br />
Try It #2<br />
x =y<br />
Writing Equations of Parabolas in Standard Form<br />
<br />
<br />
How To…<br />
<br />
1. xy<br />
a. px<br />
y =px<br />
b. py<br />
x =py<br />
2. p<br />
3. <br />
Example 3<br />
Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix<br />
( − <br />
) x = <br />
<br />
Solution Thpy =px<br />
pp =−_ =− <br />
py =px=−x<br />
Thy =−x
SECTION 8.3 THE PARABOLA 719<br />
Try It #3<br />
( <br />
) y=− <br />
<br />
Graphing Parabolas with Vertices Not at the Origin<br />
h<br />
khkTh<br />
xx−hyy−k<br />
hky−k =px−h<br />
xx−h =py−k<br />
yTh<br />
<br />
standard forms of parabolas with vertex (h, k)<br />
Table 2Figure 9hk<br />
Axis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum<br />
y = k y−k =px−h h+pk x=h −p h+pk ±p<br />
x = h x−h =py−k hk +p y=k −p h±pk +p<br />
Table 2<br />
y<br />
y−k =px −h<br />
p > <br />
y<br />
y−k =px −h<br />
p < <br />
h+p k +p<br />
h + p k + p|<br />
y=k<br />
y=k<br />
h, k<br />
h+p k<br />
h+p k −p<br />
h+p k<br />
h + p k −p|<br />
hk<br />
x=h −p<br />
x<br />
x=h −p<br />
x<br />
(a)<br />
(b)<br />
y<br />
x−h =py −h<br />
p > <br />
y<br />
x−h =py −k<br />
p < <br />
x=h<br />
hk +p<br />
y = k − p<br />
x=h<br />
h, k<br />
h−p k +p h−p| k +p h+p| k +p<br />
h+p k +p<br />
hk +p<br />
x<br />
y=k −p h, k<br />
(c)<br />
x<br />
(d)<br />
Figure 9 (a) When p > 0, the parabola opens right. (b) When p < 0, the parabola opens left.<br />
(c) When p > 0, the parabola opens up. (d) When p < 0, the parabola opens down.
720<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
How To…<br />
hk<br />
1. y−k =px−hx−h =py−k<br />
2. fi<br />
<br />
a. y−k =px−h<br />
hkhk<br />
ky =k<br />
p x−hpp ><br />
p <br />
p <<br />
hkp hk +p<br />
kp y =k −p<br />
hkp h±pk +p<br />
3. <br />
Example 4<br />
Graphing a Parabola with Vertex ( h, k) and Axis of Symmetry Parallel to the x-axis<br />
y− =−x+<br />
<br />
Solution Thy−k =px−hTh<br />
x<br />
hk=−<br />
y =k =<br />
−=pp =−p <<br />
h+pk=−+−=−<br />
x =h −p =−−−=<br />
h+pk ±p=−+−±−−−−<br />
<br />
Figure 10<br />
−<br />
y<br />
y− = −x +<br />
−<br />
−<br />
y=<br />
x<br />
−−<br />
x=<br />
Figure 10
SECTION 8.3 THE PARABOLA 721<br />
Try It #4<br />
y+ =x−<br />
<br />
Example 5 Graphing a Parabola from an Equation Given in General Form<br />
x −x −y −=<br />
<br />
Solution Th<br />
x−h =p y−Thy<br />
x<br />
<br />
x −x −y −=<br />
<br />
x −x =y +<br />
<br />
x −x +=y ++<br />
<br />
x− =y +<br />
<br />
x− =y+<br />
<br />
x− =ċċy+<br />
<br />
hk=−<br />
x =h =<br />
p =p ><br />
hk +p=−+=−<br />
y =k −p =−−=−<br />
h±pk +p=±−+−−−<br />
<br />
Figure 11<br />
y x− = y+ <br />
−−<br />
−<br />
−<br />
x<br />
−<br />
y= −<br />
x= <br />
Figure 11<br />
Try It #5<br />
x+ =−y−<br />
<br />
Solving Applied Problems Involving Parabolas<br />
<br />
<br />
fl<br />
flFigure 12Th
722<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
<br />
<br />
<br />
<br />
<br />
Figure 12<br />
<br />
Thffi<br />
d fi<br />
Example 6<br />
Solving Applied Problems Involving Parabolas<br />
fiFigure 13Thfl <br />
<br />
fl<br />
a. <br />
<br />
b. <br />
<br />
<br />
<br />
Solution<br />
<br />
Figure 13 Cross-section of a travel-sized solar fire starter<br />
a. Thx =py<br />
p >0. Th. Thp =<br />
x =py <br />
x =y p<br />
x =y <br />
b. Th <br />
=x<br />
<br />
x =y <br />
=y x<br />
y ≈ y<br />
Th
SECTION 8.3 THE PARABOLA 723<br />
Try It #6<br />
Th<br />
. Thfl <br />
a. <br />
x<br />
<br />
b. o fi<br />
<br />
Conic Sections: The Parabola Part 1 of 2 (http://openstaxcollege.org/l/parabola1)<br />
Conic Sections: The Parabola Part 2 of 2 (http://openstaxcollege.org/l/parabola2)<br />
Parabola with Vertical Axis (http://openstaxcollege.org/l/parabolavertcal)<br />
Parabola with Horizontal Axis (http://openstaxcollege.org/l/parabolahoriz)
724 CHAPTER 8 ANALYTIC GEOMETRY<br />
8.3 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
p<br />
<br />
3. <br />
p<br />
<br />
<br />
5. <br />
<br />
<br />
4. e eff<br />
<br />
p<br />
ALGEBRAIC<br />
<br />
<br />
6. y =−x 7. y=x 8. x −y =<br />
9. y− =x− 10. y +x−y−=<br />
VF<br />
d<br />
11. x=y 12. y= <br />
x 13. y=−x 14. x= <br />
y 15. x=y 16. x= <br />
y<br />
17. x− =y− 18. y− = <br />
x+<br />
19. y− =x+<br />
20. x+ =y+ 21. x+ =y+ 22. y+ =x+<br />
23. y +x−y+= 24. x −x−y+= 25. x −x−y+=<br />
26. y −x+y−= 27. x −x+y−= 28. y −y+x−=<br />
29. y −x−y+= 30. x +x+y−=<br />
GRAPHICAL<br />
<br />
31. x= <br />
y 32. y=x 33. y= <br />
x<br />
34. y=−x 35. y− =− <br />
x+<br />
36. −x+ =y+<br />
37. −y+ =x− 38. y −y−x+= 39. x +x+y+=<br />
40. x +x−y+= 41. y −x+y+= 42. x +x+y+=<br />
43. y +y−x+= 44. −x +x−y−=<br />
, fi<br />
45. y=− 46. x=−<br />
47. x=−√ — <br />
+√ — <br />
49. √ — −√ — x=√ — <br />
−√ — <br />
48. −x=− <br />
( − <br />
) <br />
50. y= <br />
( <br />
)
SECTION 8.3 SECTION EXERCISES 725<br />
<br />
51. 52.<br />
y<br />
<br />
<br />
<br />
−<br />
<br />
<br />
x<br />
53.<br />
<br />
−<br />
y<br />
54.<br />
<br />
<br />
− <br />
<br />
<br />
− <br />
<br />
<br />
−<br />
y<br />
x<br />
x<br />
<br />
<br />
55. y<br />
<br />
<br />
<br />
<br />
x
726 CHAPTER 8 ANALYTIC GEOMETRY<br />
EXTENSIONS<br />
<br />
56. V− 57. V−−−<br />
58. V− 59. V−−−<br />
60. V−( − <br />
) ( − <br />
) <br />
REAL-WORLD APPLICATIONS<br />
61. Th<br />
<br />
<br />
x =y<br />
<br />
63. <br />
<br />
. Th<br />
<br />
12 f<br />
<br />
65. <br />
<br />
<br />
<br />
67. <br />
<br />
<br />
<br />
69. <br />
y=−x +xx<br />
y<br />
<br />
<br />
62. <br />
<br />
<br />
64. <br />
<br />
<br />
66. <br />
<br />
<br />
<br />
68. <br />
<br />
<br />
<br />
20 f<br />
70. <br />
y=−x +x
SECTION 8.4 ROTATION OF AXIS 727<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Identify nondegenerate conic sections given their general form equations.<br />
Use rotation of axes formulas.<br />
Write equations of rotated conics in standard form.<br />
Identify conics without rotating axes.<br />
8.4 ROTATION OF AXIS<br />
<br />
xfiTh<br />
<br />
<br />
<br />
Figure 1<br />
Diagonal Slice Horizontal Slice Deep Vertical Slice Vertical Slice<br />
Ellipse<br />
Circle Hyperbola Parabola<br />
Figure 1 The nondegenerate conic sections<br />
nondegenerate conic sections<br />
degenerate conic sectionsFigure 2<br />
x
728<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Intersecting Lines Single Line Single Point<br />
Figure 2 Degenerate conic sections<br />
Identifying Nondegenerate Conics in General Form<br />
<br />
Th<br />
ffi<br />
<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
ABCffi<br />
<br />
xy<br />
xyB<br />
Conic Sections<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Example<br />
x +y =<br />
x +y =<br />
x −y =<br />
x =yy =x<br />
x +y =<br />
x−y+=<br />
x−x−=<br />
x +y =<br />
x +y =−<br />
Table 1
SECTION 8.4 ROTATION OF AXIS 729<br />
general form of conic sections<br />
nondegenerate conic section<br />
ABC<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
Table 2ffB =ACTh<br />
<br />
Conic Sections<br />
<br />
<br />
<br />
<br />
Example<br />
Ax +Cy +Dx+Ey+F =A ≠CAC><br />
Ax +Cy +Dx+Ey+F =A =C<br />
Ax −Cy +Dx+Ey+F =−Ax +Cy +Dx+Ey+F =<br />
AC<br />
Ax +Dx+Ey+F =Cy +Dx+Ey+F =<br />
Table 2<br />
How To…<br />
<br />
1. Ax +Bxy+Cy +Dx+Ey+F =<br />
2. AC<br />
a. AC<br />
b. AC<br />
c. AC<br />
d. AC<br />
Example 1<br />
Identifying a Conic from Its General Form<br />
<br />
a. x −y +x +y −=<br />
b. y +x +y −=<br />
c. x +y −x −y −=<br />
d. −x −y +x +y +=<br />
Solution<br />
a. <br />
<br />
<br />
<br />
<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
x +xy+−y +x +y +−=<br />
A=C =−ACs. Th<br />
b. <br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
x +xy+y +x +y +−=<br />
A=C =A
730<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
<br />
<br />
<br />
<br />
c. <br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
x +xy+y +−x+−y+−=<br />
A=C =A =C<br />
d. <br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
−x +xy+−y +x +y +=<br />
A=−C =−AC>A ≠C<br />
Try It #1<br />
<br />
a. y −x +x−y−= b. x +y +x+y−=<br />
Finding a New Representation of the Given Equation after Rotating through a Given Angle<br />
xyx<br />
y-xyxy<br />
θxy<br />
xyx′y′fi<br />
x′y′Figure 3<br />
y<br />
y’<br />
x’<br />
θ<br />
x<br />
Figure 3 The graph of the rotated ellipse<br />
<br />
x +y −xy−=<br />
xyx′y′Figure 4<br />
y<br />
y'<br />
x'<br />
θ<br />
θ<br />
θ<br />
θ<br />
x<br />
− θ<br />
θ<br />
Figure 4 The Cartesian plane with x- and y-axes and the resulting x′− and y′−axes formed by a rotation by an angle θ.
SECTION 8.4 ROTATION OF AXIS 731<br />
ThxyijThi′j′ <br />
θangle of rotationFigure 5<br />
<br />
<br />
i′=θi+θ j<br />
j′=−θi+θ j<br />
y'<br />
j'<br />
y<br />
j<br />
x'<br />
θ<br />
i'<br />
θ<br />
− θ<br />
θ<br />
θ<br />
i<br />
x<br />
Figure 5 Relationship between the old and new coordinate planes.<br />
u<br />
<br />
u =x′i′+y′j′<br />
u =x′iθ +jθ+y′−iθ +jθ <br />
u =ix'θ +jx'θ −iy'θ +jy'θ <br />
u =ix'θ −iy'θ +jxθ +jy'θ <br />
u =x'θ −y'θi+x'θ +y'θj <br />
u =x′i′+y′j′xy<br />
x=x′θ −y′θ y =x′θ +y′θ<br />
equations of rotation<br />
xy<br />
θx<br />
x′y′xyx′y′<br />
x=x′θ −y′θ y =x′θ +y′θ<br />
How To…<br />
, fift<br />
1. x yx =x′θ −y′θy =x′θ +y′θ<br />
2. xy<br />
3. x′y′<br />
Example 2 Finding a New Representation of an Equation after Rotating through a Given Angle<br />
x −xy+y −=ftθ =<br />
Solution xyx =x′θ −y′θy =x′θ +y′θ
732<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
θ =<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x =x′−y′<br />
x =x′<br />
( _ <br />
x = x′−y′ _<br />
<br />
√ — <br />
_<br />
√ — ) <br />
√ — ) −y′ ( <br />
y =x′+y′<br />
y =x′<br />
( _ <br />
y = x′+y′ _<br />
<br />
√ — <br />
_<br />
√ — ) <br />
√ — ) +y′ ( <br />
x =x′θ−y′θy =x′θ +y′θx −xy+y −=<br />
<br />
<br />
/<br />
<br />
(<br />
x′−y′ _<br />
<br />
√ — )<br />
− (<br />
x′−y′ _<br />
<br />
√ — )<br />
(<br />
x′+y′ _<br />
<br />
√ — )<br />
+ (<br />
x′+y′ _<br />
<br />
<br />
√ — )<br />
−=<br />
x′−y′x′−y′ __<br />
<br />
/ −x′−y′x′+y′ __<br />
<br />
+ /<br />
x′+y′x′+y′ __<br />
−= <br />
/ <br />
<br />
x′ −x′y′+y′ − x′ −y′ <br />
_<br />
+x′ +x′y′+y′ −=<br />
<br />
<br />
<br />
x′ +y′ − x′ −y′ <br />
_<br />
= <br />
<br />
<br />
( x′ +y′ − x′ −y′ <br />
_<br />
<br />
) = <br />
x′ +y′ −x′ −y′ = <br />
x′ +y′ −x′ +y′ = <br />
<br />
x′y′<br />
_ x′ +<br />
_ y′ =<br />
_ <br />
<br />
<br />
<br />
_ x′ +<br />
y′ _<br />
=<br />
<br />
ThFigure 6<br />
y’<br />
<br />
<br />
<br />
x’<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ=<br />
<br />
<br />
x<br />
<br />
<br />
<br />
Figure 6
SECTION 8.4 ROTATION OF AXIS 733<br />
Writing Equations of Rotated Conics in Standard Form<br />
fi<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
x′y′<br />
x′y′θfi<br />
θ= A −C<br />
B<br />
<br />
<br />
<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
ABCB ≠xy<br />
θθ= A −C<br />
<br />
B<br />
θ>θ θ<br />
θ
734<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
e fi ————<br />
θ = √<br />
<br />
__________ −θ √<br />
− —————<br />
__ <br />
<br />
= √<br />
__ <br />
_<br />
<br />
= − __ <br />
<br />
_<br />
<br />
= √<br />
_____ − <br />
ċ __ = √<br />
__ <br />
= √<br />
__ <br />
<br />
θ =_<br />
<br />
√ — <br />
————<br />
θ = √<br />
<br />
__________ +θ √<br />
+ —————<br />
__ <br />
<br />
= √<br />
__ <br />
_<br />
<br />
= + __ <br />
<br />
_<br />
<br />
= √<br />
_____ + <br />
ċ __ = √<br />
_<br />
<br />
= √ __ <br />
<br />
θ =_<br />
<br />
√ — <br />
θθx =x′θ −y′θy =x′θ +y′θ<br />
<br />
x =x′θ −y′θ<br />
<br />
<br />
<br />
<br />
x =x′<br />
( _ <br />
x = x′−y′ _<br />
<br />
√ — <br />
_<br />
√ — ) <br />
√ — ) −y′ ( <br />
y =x′θ +y′θ<br />
<br />
y =x′<br />
( _ <br />
_<br />
√ — ) <br />
√ — ) +y′ ( <br />
<br />
y = x′+y′ _<br />
<br />
√ — <br />
xy<br />
<br />
<br />
(<br />
x′−y′ _<br />
<br />
√ — )<br />
− (<br />
x′−y′ _<br />
<br />
√ — )<br />
=<br />
√ — )<br />
(<br />
x′+y′ _<br />
<br />
√ — )<br />
+ (<br />
x′+y′ _<br />
<br />
<br />
<br />
<br />
<br />
<br />
( x′−y′x′−y′ __ ) −( x′−y′x′+y′ __ ) +( x′+y′x′+y′ __ ) =<br />
<br />
x′ −x′y′+y′ −x′ +x′y′−y′ +x′ +x′y′+y′ =<br />
x′ −x′y′+y′ −x′ −x′y′+y′ +x′ +x′y′+y′ =<br />
x′ +y′ =<br />
<br />
<br />
x′ + <br />
y′ = <br />
<br />
x′y′<br />
_ x′ + <br />
_ y′ = <br />
Figure 8<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
Figure 8<br />
Try It #2<br />
x −√ — xy+y =x′y′x′y′
SECTION 8.4 ROTATION OF AXIS 735<br />
Example 4<br />
Graphing an Equation That Has No x´y´ Terms<br />
x′y′<br />
<br />
x +xy−y =<br />
Solution e fiθ<br />
<br />
x +xy−y =⇒A =B =C =−<br />
<br />
θ= A −C<br />
B<br />
<br />
<br />
θ= −− <br />
<br />
<br />
θ= <br />
θ= Figure 9<br />
<br />
y<br />
<br />
θ <br />
<br />
θ<br />
<br />
x<br />
<br />
Th<br />
<br />
<br />
<br />
Figure 9<br />
θ= <br />
= _ <br />
+ =h <br />
+=h <br />
=h <br />
<br />
h =<br />
e fiθθ<br />
<br />
————<br />
θ =<br />
√ −θ √<br />
− —————<br />
<br />
= √<br />
_<br />
_<br />
<br />
= − _<br />
<br />
_<br />
<br />
= √ <br />
<br />
ċ = _ <br />
√ — <br />
————<br />
<br />
θ =<br />
√ +θ √<br />
+ —————<br />
<br />
= √<br />
_<br />
_<br />
<br />
= + _<br />
<br />
_<br />
<br />
= √ <br />
ċ = _ <br />
√ — <br />
e fixy<br />
<br />
x =x′θ −y′θ<br />
<br />
<br />
x =x′<br />
( <br />
<br />
_<br />
√ — ) −y′ ( <br />
x = x′−y′ _<br />
<br />
√ — <br />
_ <br />
√ — )
736<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
<br />
<br />
<br />
<br />
y =x′θ +y′θ<br />
y =x′<br />
( <br />
<br />
_<br />
√ — ) +y′ ( <br />
y = x′+y′ _<br />
<br />
√ — <br />
_ <br />
√ — ) <br />
x = x′−y′ _<br />
<br />
√ — y =x′+y′ _<br />
<br />
√ — x +xy−y =<br />
<br />
<br />
<br />
(<br />
x′−y′ _<br />
<br />
√ — )<br />
+ (<br />
x′−y′ _<br />
<br />
√ — )<br />
=<br />
√ — )<br />
(<br />
x′+y′ _<br />
<br />
√ — )<br />
− (<br />
x′+y′ _<br />
<br />
( ) x′−y′ +x′−y′ x′+y′ −x′+y′ =<br />
( ) x′ −x′y′+y′ +x′ +x′y′−y′ −x′ +x′y′+y′ =<br />
<br />
<br />
<br />
<br />
( ) x′ −x′y′+y′ +x′ +x′y′−y′ −x′ −x′y′−y′ = <br />
<br />
( ) x′ −y′ = <br />
x′ −y′ = <br />
<br />
Figure 10_ x′ − <br />
_ y′ =<br />
<br />
y’<br />
<br />
<br />
<br />
<br />
y<br />
_ x′ − <br />
_ y′ =<br />
<br />
x’<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
<br />
Identifying Conics without Rotating Axes<br />
Figure 10<br />
<br />
<br />
<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
<br />
<br />
A′ x′ +B′x′y′+C′y′ +D′x′+E′y′+F′=<br />
B −AC=B′ −A′C′Thft<br />
ThB −ACft
SECTION 8.4 ROTATION OF AXIS 737<br />
using the discriminant to identify a conic<br />
Ax +Bxy+Cy +Dx+Ey+F =<br />
A′x′ +B′x′y′+C′y′ +D′x′+E′y′+F′=B −AC=B′ −A′C′<br />
ThAx +Bxy+Cy +Dx+Ey+F =<br />
<br />
B −AC<br />
<<br />
=<br />
><br />
Example 5 Identifying the Conic without Rotating Axes<br />
<br />
a. x +√ — xy+y −=<br />
b. x +√ — xy+y −=<br />
Solution<br />
a. ABC<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
A B<br />
<br />
x +√ — <br />
xy+ y −=<br />
B −AC=( √ — ) −<br />
=−<br />
=−<br />
=−<<br />
Thx +√ — xy+y −=<br />
b. AB <br />
A B<br />
e fi<br />
C<br />
<br />
<br />
x +√ — <br />
xy+ y −=<br />
<br />
<br />
C<br />
B −AC=( √ — ) −<br />
=−<br />
=−<br />
=−<<br />
Thx +√ — xy+y −=<br />
Try It #3<br />
<br />
a. x −xy+y −=<br />
b. x −xy+y −=<br />
<br />
Introduction to Conic Sections (http://openstaxcollege.org/l/introconic)
738 CHAPTER 8 ANALYTIC GEOMETRY<br />
8.4 SECTION EXERCISES<br />
VERBAL<br />
1. t effxy<br />
<br />
3. <br />
Ax +Bxy+Cy +Dx+Ey+F=<br />
B −AC<br />
2. <br />
Ax +By +Cx+Dy+E=AB=<br />
<br />
4. ax +x+y −=<br />
a<br />
5. Ax +Bxy+Cy +Dx+Ey+F=θ θ= A−C <br />
B <br />
<br />
ALGEBRAIC<br />
<br />
6. x +y +x+y−= 7. x −x+y−=<br />
8. x −y +x−y−= 9. x −y +x−=<br />
10. y −x+y+= 11. x +y −x−y+=<br />
12. x +xy+y −y−= 13. x +xy+y −y−=<br />
14. −x +√ — xy−y += 15. x +√ — xy+y −x−=<br />
16. −x +√ — xy+y −y+= 17. x +√ — xy+y −x+=<br />
, fift<br />
18. x +xy+y −=θ= 19. x −xy+y −=θ=<br />
20. x +xy−=θ= 21. −x +xy+=θ=<br />
22. x +√ — xy+y +y+=θ=<br />
θxy<br />
xy<br />
23. x +√ — xy+y +y−= 24. x +√ — xy+y +y−=<br />
25. x −√ — xy+y +y−= 26. −x −√ — xy−y −x=<br />
27. x +xy+y +x−y+= 28. x +xy+y +x−=<br />
29. x +xy+y −x+= 30. x −√ — xy+y −=<br />
GRAPHICAL<br />
<br />
<br />
31. y=−x θ=− 32. x=y θ= 33. _ x + <br />
_ y =θ=<br />
<br />
34. _<br />
y <br />
+x _<br />
=θ= <br />
35. y −x =θ= 36. y=_ x θ= <br />
<br />
37. x=y− θ= 38. _ x + <br />
_ y =θ=
SECTION 8.4 SECTION EXERCISES 739<br />
x′y′x′y′<br />
39. xy= 40. x +xy+y −=<br />
41. x −xy+y −= 42. x −√ — xy+y −=<br />
43. x +√ — xy+y −= 44. x +√ — xy+y −=<br />
45. x +√ — xy+y −= 46. x +xy+y −x+y=<br />
47. x +xy+y −x+y= 48. x −√ — xy+y −=<br />
49. x −xy+y −√ — x−√ — y=<br />
xyTh<br />
<br />
50. x −√ — xy+y +x−y= 51. x −xy+y +x−y=<br />
52. x −√ — xy+y +x−y= 53. x +√ — xy+y +x−y=<br />
54. x +xy+y +x−= 55. x +xy+y +x−y=<br />
k<br />
56. x +kxy+y +x+y−=<br />
k<br />
58. x +kxy+y −x+y+=<br />
k<br />
57. x +kxy+y +x−y+=<br />
k<br />
59. kx +xy+y −x+y+=<br />
k<br />
60. x +xy+ky +x+y+=<br />
k
740<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Identify a conic in polar form.<br />
Graph the polar equations of conics.<br />
Define conics in terms of a focus and a directrix.<br />
8.5 CONIC SECTIONS IN POLAR COORDINATES<br />
Figure 1 Planets orbiting the sun follow elliptical paths. (credit: NASA Blueshift, Flickr)<br />
<br />
<br />
ft<br />
<br />
<br />
<br />
<br />
<br />
fi<br />
Th<br />
fi <br />
<br />
Identifying a Conic in Polar Form<br />
fi<br />
x =+y Figure 2<br />
Pr, θ<br />
D<br />
<br />
r<br />
θ<br />
F<br />
<br />
x=+y<br />
Figure 2
SECTION 8.5 CONIC SECTIONS IN POLAR COORDINATES 741<br />
The Parabolafififi<br />
fifi<br />
Prθ<br />
FfiDfiefi<br />
eccentricityfi<br />
ThPe = PF <br />
PD fi<br />
PPFP<br />
De<br />
e<br />
≤e <<br />
e =<br />
e ><br />
fifix =±peθ<br />
Thpolar equationrθ<br />
the polar equation for a conic<br />
x =±ppeccentricity<br />
epolar equation<br />
ep<br />
r =_<br />
± e θ<br />
y =±pp<br />
e<br />
ep<br />
r =_<br />
± e θ<br />
How To…<br />
<br />
1. <br />
<br />
2. effi<br />
3. e<br />
4. x =py =pep<br />
xy<br />
Example 1<br />
Identifying a Conic Given the Polar Form<br />
<br />
<br />
a. r =<br />
b. r = <br />
+θ<br />
<br />
c. r = <br />
+θ<br />
<br />
−θ<br />
Solution <br />
Thfi<br />
<br />
c c<br />
a. <br />
<br />
<br />
( <br />
r=<br />
<br />
+θ ċ ) <br />
( <br />
_<br />
) <br />
=<br />
__<br />
=__<br />
<br />
( <br />
) ( <br />
) + ( <br />
) θ + <br />
_<br />
θ
742<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
θy =pe = <br />
<br />
Th<br />
<br />
=ep<br />
<br />
= <br />
p<br />
( <br />
) = ( <br />
<br />
) <br />
p<br />
<br />
=p<br />
e a. The = <br />
c. <br />
r =<br />
_ <br />
−θ ċ<br />
x = <br />
=<br />
( _ <br />
) <br />
( <br />
_<br />
<br />
) <br />
_ <br />
<br />
( _ <br />
) <br />
r = <br />
( _ <br />
) −( <br />
_<br />
<br />
) θ<br />
__<br />
<br />
_ <br />
r=<br />
_ <br />
−θ<br />
y =−pe =<br />
Th<br />
<br />
<br />
=ep<br />
<br />
<br />
=p<br />
<br />
<br />
=p<br />
e =. The =y =− <br />
=−
SECTION 8.5 CONIC SECTIONS IN POLAR COORDINATES 743<br />
Try It #1<br />
<br />
r= <br />
−θ <br />
Graphing the Polar Equations of Conics<br />
<br />
Th<br />
<br />
fiThfi<br />
Th<br />
eThθr<br />
θ π <br />
ππ <br />
<br />
Example 2 Graphing a Parabola in Polar Form<br />
<br />
r =<br />
<br />
+θ <br />
Solution <br />
<br />
<br />
<br />
( _ <br />
r =_<br />
<br />
) <br />
<br />
+θ = __<br />
<br />
( _ <br />
) +( <br />
_<br />
<br />
) θ<br />
<br />
<br />
<br />
r=_<br />
<br />
+θ<br />
e =Thθ<br />
x =p<br />
<br />
<br />
=ep<br />
<br />
<br />
=p<br />
<br />
<br />
=p<br />
Thx = <br />
<br />
Table 1Figure 3<br />
A B C D<br />
θ π <br />
π π<br />
<br />
<br />
<br />
r= <br />
+θ<br />
<br />
<br />
≈<br />
<br />
≈ fi <br />
≈<br />
Table 1<br />
B<br />
F<br />
A<br />
<br />
<br />
D<br />
r<br />
x = <br />
<br />
Figure 3
744<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Analysis We can check our result with a graphing utility. See Figure 4.<br />
<br />
Example 3<br />
Graphing a Hyperbola in Polar Form<br />
Figure 4<br />
<br />
r = <br />
−θ <br />
<br />
Solution <br />
<br />
<br />
<br />
( _ <br />
r =_<br />
<br />
) <br />
<br />
−θ = <br />
( _ <br />
) −( <br />
_<br />
<br />
) θ<br />
__<br />
<br />
<br />
r=_<br />
<br />
−_ θ <br />
e = e >Thθ<br />
<br />
y =−p<br />
<br />
=ep<br />
<br />
=( <br />
) p<br />
<br />
( <br />
) =p<br />
<br />
<br />
=p<br />
Thy =− <br />
<br />
Table 2Figure 5<br />
A B C D<br />
θ π <br />
π π<br />
<br />
<br />
<br />
r= <br />
−θ<br />
<br />
− <br />
=<br />
Table 2<br />
C<br />
D<br />
r<br />
A<br />
<br />
B<br />
Figure 5
SECTION 8.5 CONIC SECTIONS IN POLAR COORDINATES 745<br />
Example 4<br />
Graphing an Ellipse in Polar Form<br />
<br />
r = <br />
−θ <br />
<br />
Solution <br />
<br />
<br />
<br />
r =<br />
<br />
−θ =<br />
( _<br />
<br />
( _<br />
) <br />
<br />
_<br />
) θ<br />
__<br />
) − ( <br />
<br />
<br />
r=_<br />
<br />
− <br />
θ<br />
e = e
746<br />
CHAPTER 8 ANALYTIC GEOMETRY<br />
Analysis<br />
We can check our result with a graphing utility.<br />
See Figure 7.<br />
10<br />
Figure 7 r =<br />
_<br />
graphed on a viewing window<br />
5 − 4 cos θ<br />
of [−3, 12, 1] by [ −4, 4, 1], θ min = 0 and θ max = 2π.<br />
Try It #2<br />
<br />
r= <br />
−θ <br />
<br />
Defining Conics in Terms of a Focus and a Directrix<br />
<br />
<br />
How To…<br />
<br />
1. y<br />
x<br />
2. p <br />
3. ffi<br />
4. p<br />
Example 5<br />
Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix<br />
e =y =−<br />
Solution Thy =−p<br />
y =−−<<br />
ep<br />
<br />
r = <br />
−eθ<br />
e =|−|==p<br />
Th<br />
<br />
<br />
r = <br />
−θ<br />
<br />
<br />
r = <br />
−θ<br />
Example 6 Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and<br />
Directrix<br />
e =_ x =<br />
<br />
Solution x =px =><br />
<br />
ep<br />
<br />
r = <br />
+ e θ<br />
e = <br />
||==p
SECTION 8.5 CONIC SECTIONS IN POLAR COORDINATES 747<br />
Th<br />
<br />
( _ <br />
r = <br />
) <br />
<br />
+_ θ <br />
<br />
_ <br />
r = <br />
<br />
+_ θ <br />
<br />
_ <br />
r = <br />
<br />
( _ <br />
) +_ θ <br />
<br />
_ <br />
r = <br />
<br />
_ +_ θ <br />
<br />
<br />
r =_ <br />
ċ +θ<br />
<br />
r = <br />
+θ<br />
<br />
<br />
Try It #3<br />
e=x=−<br />
Example 7<br />
Converting a Conic in Polar Form to Rectangular Form<br />
<br />
r = <br />
−θ <br />
Solution r =√ — x +y x =rθy =rθ<br />
<br />
<br />
r = <br />
−θ <br />
<br />
<br />
r ċ− θ) = <br />
−θ ċ−θ <br />
r −r θ = <br />
r =+rθ r<br />
r =+r θ) <br />
x +y =+y r =√ — x +y y = r θ<br />
x +y =+y +y <br />
x −y = <br />
Try It #4<br />
<br />
r= <br />
+θ <br />
<br />
Polar Equations of Conic Sections (http://openstaxcollege.org/l/determineconic)<br />
Graphing Polar Equations of Conics - 1 (http://openstaxcollege.org/l/graphconic1)<br />
Graphing Polar Equations of Conics - 2 (http://openstaxcollege.org/l/graphconic2)
748 CHAPTER 8 ANALYTIC GEOMETRY<br />
8.5 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
3. <br />
θ<br />
<br />
5. <br />
<br />
2. <br />
<br />
4. <br />
<br />
<br />
ALGEBRAIC<br />
<br />
<br />
6. r = <br />
−θ <br />
<br />
7. r = <br />
−θ <br />
<br />
8. r = <br />
−θ <br />
<br />
9. r = <br />
+θ <br />
<br />
10. r = <br />
+θ <br />
<br />
11. r = <br />
<br />
+θ <br />
<br />
12. r = <br />
−θ <br />
<br />
13. r = <br />
+θ <br />
14. r−θ= 15. r+θ= 16. r−θ= 17. r+θ=<br />
<br />
<br />
18. r = <br />
+θ <br />
<br />
19. r = <br />
−θ <br />
<br />
20. r = <br />
−θ <br />
<br />
21. r = <br />
+θ <br />
<br />
22. r = 23. r = <br />
+θ −θ <br />
<br />
24. r = <br />
+θ <br />
<br />
25. r = <br />
−θ <br />
θ<br />
26. r+θ= 27. r−θ= 28. r−θ= 29. r = <br />
−+θ <br />
θ<br />
30. r = <br />
+θ <br />
<br />
<br />
<br />
31. r = <br />
+θ <br />
<br />
32. r = <br />
+θ <br />
<br />
33. r = <br />
−θ <br />
<br />
34. r = <br />
+θ <br />
<br />
35. r = <br />
−θ <br />
<br />
36. r = <br />
−θ <br />
<br />
37. r =_<br />
<br />
− θ<br />
<br />
38. r = <br />
+θ <br />
39. r+θ= 40. r−θ= 41. r−θ= 42. r−θ=<br />
fi<br />
<br />
43. x = e = <br />
44. x =−e = 45. y =e =<br />
<br />
46. y =− e = <br />
47. x =e = 48. x =− e =<br />
<br />
49. x =− <br />
e = <br />
<br />
52. x =− e = <br />
<br />
55. x =− e = <br />
<br />
50. y = <br />
e = <br />
<br />
53. x =− e = <br />
<br />
51. y = e = <br />
<br />
54. y =e =<br />
EXTENSIONS<br />
Rotation of Axesxy<br />
rθ<br />
56. xy= 57. x 2 +xy+y 2 = 58. x 2 +xy+y 2 =<br />
59. x 2 +xy+y 2 = 60. xy+y =
CHAPTER 8 REVIEW 749<br />
CHAPTER 8 REVIEW<br />
Key Terms<br />
angle of rotation θ> θ <br />
θ< θ θ= θ =<br />
center of a hyperbola <br />
center of an ellipse <br />
conic section <br />
conjugate axis <br />
degenerate conic sections <br />
<br />
directrix <br />
<br />
eccentricity P F D <br />
e = PF e <br />
PD<br />
ellipse x, y o fi<br />
foci <br />
focus (of a parabola) a fi<br />
focus (of an ellipse) fi<br />
x, y <br />
hyperbola x, y x, y<br />
<br />
latus rectum <br />
<br />
major axis <br />
minor axis <br />
nondegenerate conic section <br />
<br />
parabola x, yfifi<br />
<br />
polar equation rθ<br />
transverse axis <br />
Key Equations<br />
x 2 <br />
a 2+y <br />
b 2= a > b<br />
x 2 <br />
b 2+y <br />
a 2= a > b<br />
h,k (x − h)2<br />
+ (y − k)2<br />
= a > b<br />
a 2 b 2<br />
h,k (x − h)2<br />
+ (y − k)2 = a > b<br />
b 2 a 2<br />
x- x 2 <br />
a 2−y <br />
b 2=<br />
y- y 2 <br />
a 2−x <br />
b 2=
750 CHAPTER 8 ANALYTIC GEOMETRY<br />
h, kx (x − h)2<br />
− (y − k)2<br />
= <br />
a 2 b 2<br />
(h,k), y- (y − k)2<br />
− (x − h)2<br />
= <br />
a 2 b 2<br />
x<br />
y 2 =px<br />
y<br />
x 2 =py<br />
h, kx y−k 2 =p(x − h)<br />
(h,k), y- (x − h) 2 =p(y − k)<br />
<br />
<br />
<br />
<br />
Ax 2 +Bxy+Cy 2 +Dx+Ey+ F =<br />
x =x θ −y θ<br />
y =x θ +y θ<br />
θθ= A − B <br />
Key Concepts<br />
8.1 The Ellipse<br />
x, y <br />
<br />
<br />
Example 1Example 2<br />
<br />
<br />
Example 3Example 4<br />
<br />
Example 5Example 6<br />
<br />
Example 7<br />
8.2 The Hyperbola<br />
x, y ffx, y <br />
<br />
ThExample 1<br />
<br />
Example 2Example 3<br />
<br />
Example 4Example 5<br />
<br />
Example 6
CHAPTER 8 REVIEW 751<br />
8.3 The Parabola<br />
x, y <br />
<br />
Thx<br />
p >p < Example 1<br />
Thy<br />
p >p p < Example 4<br />
Thh, ky<br />
p >p
752 CHAPTER 8 ANALYTIC GEOMETRY<br />
CHAPTER 8 REVIEW EXERCISES<br />
THE ELLIPSE<br />
Th<br />
<br />
1. x <br />
<br />
+y <br />
=<br />
2. (x − <br />
+(y + <br />
=<br />
3. x +y +x −y += 4. x +y −x +y +=<br />
<br />
5. x <br />
<br />
+y <br />
<br />
=<br />
6. (x − <br />
+(y + <br />
=<br />
7. x +y +x +y −= 8. x 2 +y 2 −x +y +=<br />
o fi<br />
9. − 10. −−−<br />
11. <br />
<br />
<br />
<br />
THE HYPERBOLA<br />
Th<br />
<br />
12. x <br />
<br />
−y <br />
=<br />
13. (y + <br />
−(x − <br />
=<br />
14. y 2 −x 2 +y −x += 15. x −y −x −y −=<br />
<br />
16. x <br />
<br />
−y <br />
<br />
=<br />
17. (y − <br />
−(x + <br />
=<br />
18. x −y +x +y −= 19. y −x −y −=<br />
, fi<br />
20. − 21. <br />
THE PARABOLA<br />
Th<br />
<br />
22. y =x 23. (x + = <br />
(y − 24. y −y −x −= 25. x 2 + x − y +=<br />
<br />
26. x +y = 27. (y − = <br />
(x + 28. x −x −y += 29. y +y +x +=
CHAPTER 8 REVIEW 753<br />
<br />
30. − x = 31. ( _ <br />
) y=_ <br />
32. <br />
<br />
<br />
<br />
ROTATION OF AXES<br />
<br />
33. x +xy+y +x −y −= 34. x +xy+y +x −y +=<br />
35. x +xy+y +x −y +=<br />
θ xy<br />
xy<br />
36. x +xy−y −= 37. x −xy+y −=<br />
x'y'x'y'<br />
38. x −xy+y −x −y += 39. x −xy+y −=<br />
40. x +xy−y −x +y +=<br />
CONIC SECTIONS IN POLAR COORDINATES<br />
<br />
<br />
<br />
41. r= <br />
−θ <br />
<br />
42. r= <br />
+θ <br />
<br />
43. r= <br />
+θ <br />
<br />
44. r= <br />
−θ <br />
<br />
<br />
<br />
45. r= <br />
−θ <br />
<br />
46. r= <br />
+θ <br />
<br />
47. r= <br />
+θ <br />
<br />
48. r= <br />
−θ <br />
fi<br />
<br />
49. x = e = 50. y =− e =
754 CHAPTER 8 ANALYTIC GEOMETRY<br />
CHAPTER 8 PRACTICE TEST<br />
<br />
1. x 2<br />
<br />
+y 2<br />
<br />
=<br />
2. y 2 +x 2 −y +x −=<br />
<br />
3. (x − <br />
+(y − <br />
= 4. x 2 +y 2 +x −y −=<br />
5. <br />
<br />
6. <br />
<br />
<br />
<br />
<br />
7. x 2<br />
<br />
−y 2<br />
<br />
=<br />
8. y 2 −x 2 +y +=<br />
<br />
<br />
9. (x − <br />
−(y + <br />
= 10. y 2 −x 2 +y −x −=<br />
11. <br />
<br />
<br />
<br />
12. y 2 +x = 13. x 2 −x − y +=<br />
<br />
14. (x − =−(y + 15. y 2 +x −y +=<br />
16. <br />
y =−<br />
17. <br />
<br />
<br />
<br />
<br />
<br />
θ xy<br />
18. x 2 −xy+y 2 = 19. x 2 +xy+y 2 +x −y =<br />
x'y'x'y'<br />
20. x 2 +√ — xy+y 2 = 21. x 2 +xy+y 2 −x =<br />
<br />
<br />
22. r= <br />
23. r= <br />
−θ<br />
+θ<br />
<br />
<br />
<br />
24. r= <br />
−θ <br />
<br />
25. r= <br />
+θ<br />
26. <br />
e = x =
9<br />
Sequences, Probability and<br />
Counting Theory<br />
Figure 1 (credit: Robert S. Donovan, Flickr.)<br />
CHAPTER OUTLINE<br />
9.1 Sequences and Their Notations<br />
9.2 Arithmetic Sequences<br />
9.3 Geometric Sequences<br />
9.4 Series and Their Notations<br />
9.5 Counting Principles<br />
9.6 Binomial Theorem<br />
9.7 Probability<br />
Introduction<br />
<br />
<br />
Th<br />
fi<br />
<br />
ThftTh<br />
<br />
<br />
<br />
<br />
<br />
755
756<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Write the terms of a sequence defined by an explicit formula.<br />
Write the terms of a sequence defined by a recursive formula.<br />
Use factorial notation.<br />
9.1 SEQUENCES AND THEIR NOTATIONS<br />
Th<br />
Thfi<br />
Table 1<br />
Day <br />
Hits <br />
Table 1<br />
fi<br />
fi<br />
<br />
Writing the Terms of a Sequence Defined by an Explicit Formula<br />
sequence<br />
. Th<br />
<br />
fitermTh<br />
fit fi<br />
fi<br />
ffifi<br />
fi<br />
fi<br />
fififi<br />
finn<br />
n<br />
<br />
↓ ↓ ↓ ↓ ↓ ↓<br />
<br />
n <br />
Thfi = = =Thn<br />
n<br />
abcn. Th<br />
<br />
a n<br />
= n <br />
n<br />
fi<br />
fifi <br />
n<br />
<br />
a <br />
= <br />
<br />
=
SECTION 9.1 SEQUENCES AND THEIR NOTATIONS 757<br />
Th<br />
Th<br />
<br />
Thfifin<br />
Table 2<br />
n n<br />
nth term of the sequence, a n<br />
n<br />
Table 2<br />
<br />
Figure 1Th<br />
<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 1<br />
<br />
<br />
<br />
n <br />
fiinfinite sequence. Thfi<br />
e fi<br />
n <br />
Thfinite sequencefi<br />
sequence<br />
sequencefinite sequence<br />
finThtermsTha<br />
<br />
<br />
a <br />
a <br />
a <br />
a n<br />
<br />
a <br />
fia <br />
a <br />
<br />
Thnth term of the sequenceexplicit<br />
formulafinfi<br />
infinite sequence<br />
n<br />
Q & A…<br />
Does a sequence always have to begin with a 1<br />
?<br />
fia <br />
a
758<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
How To…<br />
e fin<br />
1. n n =o fie fia <br />
<br />
2. o fia <br />
n =<br />
3. fin<br />
Example 1<br />
Writing the Terms of a Sequence Defined by an Explicit Formula<br />
e fit fifia n<br />
=−n +<br />
Solution n =n<br />
n = a <br />
=−+=<br />
n = a <br />
=−+=<br />
n = a <br />
=−+=−<br />
n = a <br />
=−+=−<br />
n = a <br />
=−+=−<br />
The fit fi−−−<br />
Analysis The sequence values can be listed in a table. A table, such as Table 3, is a convenient way to input the function<br />
into a graphing utility.<br />
n <br />
a n<br />
− − −<br />
Table 3<br />
Figure 2<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2<br />
Try It #1<br />
t fi t n<br />
=n−
SECTION 9.1 SEQUENCES AND THEIR NOTATIONS 759<br />
Investigating Alternating Sequences<br />
Th<br />
fi<br />
n<br />
−−<br />
fi<br />
Th<br />
<br />
How To…<br />
e fin<br />
1. nn =fifia <br />
Th<br />
− n <br />
2. o fia <br />
n =<br />
3. fin<br />
Example 2<br />
Writing the Terms of an Alternating Sequence Defined by an Explicit Formula<br />
e fit fi<br />
<br />
Solution n =n =<br />
a n<br />
=_<br />
−n n <br />
n + <br />
<br />
<br />
<br />
<br />
n = a <br />
= − <br />
+ =− <br />
<br />
n = a <br />
= − <br />
+ = <br />
<br />
n = a <br />
= − <br />
+ =− <br />
<br />
n = a <br />
= − <br />
+ = <br />
<br />
<br />
n = a <br />
= − <br />
+ =− <br />
<br />
The fit fi{ − <br />
<br />
− <br />
<br />
− <br />
} <br />
Analysis The graph of this function, shown in Figure 3, looks different from the ones we have seen previously in this<br />
section because the terms of the sequence alternate between positive and negative values.<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
<br />
− <br />
<br />
− <br />
<br />
<br />
<br />
− <br />
Figure 3
760<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Q & A…<br />
In Example 2, does the (−1) to the power of n account for the oscillations of signs?<br />
nn +n −<br />
<br />
Try It #2<br />
t fi<br />
n<br />
a n<br />
=<br />
− n<br />
<br />
Investigating Piecewise Explicit Formulas<br />
Th<br />
fi<br />
ff<br />
How To…<br />
e fin<br />
1. n =<br />
2. o fie fia <br />
n =<br />
3. n =<br />
4. o fia <br />
n =<br />
5. fin<br />
Example 3<br />
Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula<br />
e fi<br />
<br />
n<br />
a n<br />
= n <br />
<br />
<br />
__ n n<br />
Solution n =n =n n<br />
_ n n<br />
<br />
a <br />
= = n <br />
a <br />
= = n <br />
<br />
a <br />
= <br />
= n <br />
<br />
a <br />
= = n <br />
a <br />
= = n <br />
<br />
a <br />
= <br />
= n <br />
<br />
The fi<br />
Analysis Every third point on the graph shown in Figure 4 stands out from the two nearby points. This occurs because<br />
the sequence was defined by a piecewise function.<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
Figure 4
SECTION 9.1 SEQUENCES AND THEIR NOTATIONS 761<br />
Try It #3<br />
<br />
a n<br />
= <br />
n<br />
{ n <br />
<br />
n <br />
n<br />
Finding an Explicit Formula<br />
Thfi<br />
n<br />
fifiTh<br />
fi<br />
<br />
How To…<br />
e fi, fi<br />
1. <br />
2. <br />
3. <br />
4. a n<br />
nn =n =n =<br />
Example 4<br />
Writing an Explicit Formula for the nth Term of a Sequence<br />
n<br />
a.{ − <br />
− <br />
− <br />
} <br />
b.{ − <br />
− <br />
− <br />
− <br />
− <br />
} <br />
c.e e e e e <br />
Solution <br />
a. Th− n Th<br />
n +1. Thn +<br />
<br />
b. Th<br />
a n<br />
= −n n+ n+ <br />
<br />
{ − <br />
− <br />
− <br />
− <br />
− <br />
} <br />
Th<br />
{ − <br />
<br />
− <br />
− <br />
− <br />
− <br />
− <br />
− } Th<br />
n<br />
n + <br />
<br />
a n<br />
=−<br />
<br />
<br />
n+<br />
c. Then = e n +<br />
a n<br />
=e n +<br />
Try It #4<br />
n<br />
−−
762<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Try It #5<br />
n<br />
Try It #6<br />
{ − <br />
n<br />
− <br />
− <br />
− <br />
− <br />
} <br />
{ <br />
e <br />
_<br />
e<br />
e e } <br />
Writing the Terms of a Sequence Defined by a Recursive Formula<br />
<br />
fl<br />
Th<br />
Th<br />
<br />
ff<br />
Th<br />
recursive formula<br />
fi<br />
fia n<br />
<br />
<br />
<br />
a <br />
=<br />
<br />
a n<br />
=a n−<br />
−n ≥<br />
n fie fi<br />
<br />
a <br />
=<br />
<br />
a <br />
=a <br />
−=−=<br />
<br />
a <br />
=a <br />
−=−=<br />
<br />
a <br />
=a <br />
−=−=<br />
e fi<br />
Thfifi<br />
<br />
<br />
a <br />
=<br />
<br />
a <br />
=<br />
<br />
a n<br />
=a n−<br />
+a n−<br />
n ≥<br />
fi<br />
<br />
<br />
a <br />
=a <br />
+a <br />
=+=<br />
recursive formula<br />
recursive formulafi
SECTION 9.1 SEQUENCES AND THEIR NOTATIONS 763<br />
Q & A…<br />
Must the first two terms always be given in a recursive formula?<br />
Thfifi<br />
. The fifi<br />
How To…<br />
e fie fin<br />
1. a <br />
. The fi<br />
2. o fia <br />
a n−<br />
<br />
3. o fia <br />
<br />
4. n<br />
Example 5<br />
Writing the Terms of a Sequence Defined by a Recursive Formula<br />
e fit fifi<br />
<br />
a <br />
=<br />
<br />
a n<br />
=a n−<br />
−n ≥<br />
Solution Thfia n−<br />
<br />
<br />
n = a <br />
=<br />
n = a <br />
=a <br />
−=−=−=<br />
n = a <br />
=a <br />
−=−=−=<br />
n = a <br />
=a <br />
−=−=−=−<br />
n = a <br />
=a <br />
−=−−=−−=−<br />
The fit fiFigure 5.<br />
a n<br />
<br />
− <br />
−<br />
−<br />
−<br />
−<br />
<br />
n<br />
<br />
−<br />
−<br />
Try It #7<br />
Figure 5<br />
e fit fifi<br />
<br />
a <br />
=<br />
<br />
a n<br />
a n−<br />
+n≥<br />
How To…<br />
e fin<br />
1. a <br />
<br />
2. a <br />
<br />
3. o fi<br />
4. n
764<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Example 6<br />
Writing the Terms of a Sequence Defined by a Recursive Formula<br />
e fifi<br />
<br />
a <br />
=<br />
<br />
a <br />
=<br />
<br />
a n<br />
=a n−<br />
+a n−<br />
n ≥<br />
Solution Thfia n−<br />
a n−<br />
<br />
<br />
n = a <br />
=a <br />
+a <br />
=+=<br />
n = a <br />
=a <br />
+a <br />
=+=<br />
n = a <br />
=a <br />
+a <br />
=+=<br />
n = a <br />
=a <br />
+a <br />
=+=<br />
The fi Figure 6<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
n<br />
Figure 6<br />
<br />
Try It #8<br />
e fifi<br />
<br />
a <br />
=<br />
<br />
a <br />
=<br />
<br />
a <br />
=<br />
<br />
a n<br />
= a a n− +a n−<br />
n ≥<br />
n−<br />
Using Factorial Notation<br />
Thn factorialn<br />
n<br />
<br />
=ċċċ=<br />
<br />
=ċċċċ=<br />
a n<br />
=n+<br />
n<br />
<br />
a <br />
=+===<br />
Thnnn−ċ
SECTION 9.1 SEQUENCES AND THEIR NOTATIONS 765<br />
n factorial<br />
n factorialfiThn<br />
nfin<br />
<br />
=<br />
<br />
=<br />
<br />
n=nn−n−⋯n ≥<br />
Thfi=<br />
Q & A…<br />
Can factorials always be found using a calculator?<br />
<br />
<br />
Example 7 Writing the Terms of a Sequence Using Factorials<br />
n<br />
e fit fifia n<br />
=<br />
<br />
n+ <br />
Solution n =n =<br />
n =<br />
<br />
a <br />
=<br />
<br />
+ = <br />
=<br />
<br />
= <br />
<br />
n =<br />
<br />
a <br />
=<br />
<br />
+ = <br />
=<br />
<br />
= <br />
n =<br />
<br />
a <br />
=<br />
<br />
+ = <br />
=<br />
<br />
= <br />
<br />
n =<br />
<br />
a <br />
=<br />
<br />
+ = <br />
=<br />
<br />
<br />
= <br />
n =<br />
<br />
a <br />
=<br />
<br />
+ = <br />
=<br />
<br />
<br />
= <br />
<br />
<br />
The fit fi{ <br />
<br />
<br />
<br />
<br />
<br />
} <br />
Analysis Figure 7 shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the<br />
factorial term in the denominator results in the denominator becoming much larger than the numerator as n increases.<br />
This means the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero.<br />
Try It #9<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 7<br />
t fi a n<br />
= n+ <br />
n<br />
<br />
<br />
<br />
<br />
<br />
n<br />
<br />
Finding Terms in a Sequence (http://openstaxcollege.org/l/findingterms)
766 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
9.1 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
<br />
3. <br />
<br />
<br />
5. <br />
<br />
<br />
2. <br />
4. a n<br />
<br />
<br />
n<br />
<br />
ALGEBRAIC<br />
e fi<br />
6. a n<br />
= n −<br />
<br />
7. a n<br />
=−<br />
<br />
n+ 8. a n =−−n− 9. a n<br />
= n<br />
<br />
n <br />
10. a n<br />
= n+ n 11. a =ċ n −n− 12. a n<br />
=− ċ − n− n<br />
13. a n<br />
=<br />
<br />
n+ <br />
14. a n<br />
=− n + 15. a n<br />
=−( ċ−n− ) <br />
e fi<br />
<br />
16. a n<br />
= −n −n<br />
n− n<br />
18. a n<br />
= n+ n<br />
_ n<br />
n<br />
17. a n<br />
= _ n <br />
n+ n≤<br />
n − n><br />
19. a n<br />
= −ċn− n<br />
ċ− n− n<br />
20. a n<br />
= n − n≤n><br />
_<br />
n − <br />
SECTION 9.1 SECTION EXERCISES 767<br />
<br />
34. −−−−− 35. −−− 36. <br />
37. 38. <br />
<br />
<br />
<br />
<br />
39. 40. ( <br />
) <br />
41. <br />
<br />
e fi<br />
43. a n<br />
= n <br />
n <br />
44. a =ċn <br />
n<br />
ċn <br />
45. a = n<br />
n <br />
n −n− <br />
<br />
42. <br />
<br />
46. a = ċn<br />
n<br />
nn− <br />
GRAPHICAL<br />
e fit fi<br />
47. a n<br />
= −n <br />
n<br />
+n<br />
48. a n<br />
= +n _<br />
<br />
n n<br />
+n n<br />
50. a n<br />
=a n<br />
=a n−<br />
+ 51. a n<br />
= n+ <br />
n− <br />
49. a <br />
=a n<br />
=−a n−<br />
+ <br />
e fit fi<br />
52.<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
53. a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
54. a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
e fit fi<br />
55.<br />
56.<br />
a n a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
<br />
<br />
<br />
<br />
n
768 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
TECHNOLOGY<br />
fi<br />
a <br />
[ENTER]<br />
<br />
[2ND] ANSa n−<br />
[ENTER]<br />
[ENTER]<br />
o fi<br />
57. t fia <br />
= <br />
a n<br />
= <br />
a + <br />
n−<br />
>Frac<br />
<br />
59. t fi<br />
a <br />
=a n<br />
= a n − − +<br />
61. a <br />
=a n<br />
=na n−<br />
58. <br />
a <br />
=a n<br />
=a n−<br />
+<br />
60. <br />
a <br />
=a n<br />
= a + _<br />
n− a n−<br />
<br />
te a fifi<br />
<br />
[2ND] LIST<br />
OPS“seq(”[ENTER]<br />
“Expr:”[X,T, θ, n]n<br />
“Variable:”<br />
“start:”n<br />
“end:”n<br />
[ENTER]<br />
[ENTER] <br />
<br />
<br />
[2ND] LIST<br />
OPS“seq(”[ENTER]<br />
“Expr”, “Variable”,“start”, “end”<br />
<br />
[ENTER] <br />
<br />
fi<br />
<br />
62. t fi<br />
a n<br />
=− <br />
n+ _<br />
<br />
63. <br />
a n<br />
= n −n +n− <br />
n<br />
<br />
64. t fi<br />
65. <br />
a n<br />
= nċ−n− a n<br />
= n +n−<br />
66. a n<br />
= n <br />
n <br />
EXTENSIONS<br />
67. a n<br />
=−−n<br />
a n<br />
=−<br />
68. a n<br />
= n +n+ <br />
n+ <br />
<br />
69. <br />
−−−−<br />
<br />
<br />
71. <br />
70. <br />
a n<br />
= n+ <br />
n− b n<br />
=n +n +n
SECTION 9.2 ARITHMETIC SEQUENCES 769<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the common difference for an arithmetic sequence.<br />
Write terms of an arithmetic sequence.<br />
Use a recursive formula for an arithmetic sequence.<br />
Use an explicit formula for an arithmetic sequence.<br />
9.2 ARITHMETIC SEQUENCES<br />
ftTh<br />
Th<br />
<br />
ft<br />
fiTh<br />
fiThftfift<br />
ftftfi<br />
fi<br />
Finding Common Differences<br />
Tharithmetic sequence<br />
common difference<br />
ff−<br />
− − − − −<br />
<br />
Thff<br />
o fi<br />
+ + + +<br />
<br />
<br />
arithmetic sequence<br />
arithmetic sequenceff<br />
Thcommon differencea <br />
fi<br />
dff<br />
a n<br />
=a <br />
a <br />
+da <br />
+da <br />
+d<br />
Example 1<br />
Finding Common Differences<br />
, fiff<br />
a. b. −<br />
Solution ff<br />
a. Thn diff<br />
−= −= −= −=<br />
b. Thn diff. Thn diff<br />
−−= −= −= −=
770<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Analysis ThFigure 1. We can see from the graphs that, although both<br />
sequences show growth, a is not linear whereas b is linear. Arithmetic sequences have a constant rate of change so their<br />
graphs will always be points on a line.<br />
a n<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−<br />
<br />
n<br />
<br />
−<br />
<br />
<br />
<br />
<br />
n<br />
Figure 1<br />
Q & A…<br />
If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the<br />
common difference?<br />
<br />
o fiff<br />
Try It #1<br />
, fiff<br />
<br />
Try It #2<br />
, fiff<br />
<br />
Writing Terms of Arithmetic Sequences<br />
fifi<br />
ffThfiff<br />
nd<br />
a n<br />
=a <br />
+n−d<br />
How To…<br />
e fiff, fie fi<br />
1. ffe fio fi<br />
2. ffo fi<br />
3. fi<br />
4. <br />
Example 2<br />
Writing Terms of Arithmetic Sequences<br />
e fit fia <br />
=d =−<br />
Solution −fifi<br />
<br />
The fit fi
SECTION 9.2 ARITHMETIC SEQUENCES 771<br />
Analysis As expected, the graph of the sequence consists of points on a line as shown in Figure 2.<br />
a n<br />
<br />
<br />
<br />
<br />
Figure 2<br />
<br />
<br />
<br />
n<br />
Try It #3<br />
t fia <br />
=d=<br />
How To…<br />
y fi, fi<br />
1. a <br />
a n<br />
na n<br />
=a <br />
+n−dd<br />
2. a <br />
nda n<br />
=a <br />
+n−d<br />
Example 3<br />
a <br />
=a <br />
=, fia <br />
<br />
Writing Terms of Arithmetic Sequences<br />
Solution Thffd<br />
<br />
+d+d+d<br />
a <br />
+d =+d<br />
n fiffd<br />
<br />
a n<br />
=a <br />
+n−d<br />
<br />
a <br />
=a <br />
+d<br />
a <br />
=+d a <br />
d<br />
=+d a <br />
<br />
d = n diff<br />
e fiftff<br />
<br />
a <br />
=a <br />
+=<br />
Analysis Notice that the common difference is added to the first term once to find the second term, twice to find the third<br />
term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to<br />
the first term nine times or by using the equation a n<br />
= a 1<br />
+ (n − 1)d.<br />
Try It #4<br />
a <br />
=a <br />
= a <br />
<br />
Using Recursive Formulas for Arithmetic Sequences<br />
fiTh<br />
fi
772<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
ffff<br />
e fi<br />
a n<br />
=a n−<br />
+d n ≥<br />
recursive formula for an arithmetic sequence<br />
Thff<br />
a n<br />
=a n−<br />
+d n ≥<br />
How To…<br />
<br />
1. o fiff<br />
2. ff<br />
Example 4<br />
Writing a Recursive Formula for an Arithmetic Sequence<br />
<br />
−−<br />
Solution Thfi−Thfffi<br />
<br />
d=−−−=<br />
ff<br />
<br />
a <br />
=−<br />
<br />
a n<br />
=a n−<br />
+n ≥<br />
Analysis We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as<br />
shown in Figure 3. The growth pattern of the sequence shows the constant difference of units.<br />
<br />
<br />
<br />
<br />
−<br />
−<br />
a n<br />
Figure 3<br />
<br />
n<br />
Q & A…<br />
Do we have to subtract the first term from the second term to find the common difference?<br />
<br />
fiftf fiff<br />
Try It #5
SECTION 9.2 ARITHMETIC SEQUENCES 773<br />
Using Explicit Formulas for Arithmetic Sequences<br />
<br />
Thff<br />
<br />
a n<br />
=a <br />
+dn−<br />
fiyfffi<br />
<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
Thff−−fiy<br />
−−−=+=fiy<br />
ThFigure4<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
Figure 4<br />
<br />
n<br />
y =mx+ba n<br />
yn<br />
xmb<br />
−<br />
a n<br />
=−n +<br />
fi<br />
a n<br />
=−n−fia n<br />
=−n +<br />
explicit formula for an arithmetic sequence<br />
n<br />
a n<br />
=a <br />
+d n − <br />
How To…<br />
e fi<br />
1. ffa <br />
−a <br />
<br />
2. ffe fia n<br />
=a <br />
+dn−<br />
Example 5 Writing the nth Term Explicit Formula for an Arithmetic Sequence
774<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Solution Thffe fi<br />
<br />
d =a <br />
−a <br />
<br />
<br />
=−<br />
=<br />
Thfffffi<br />
<br />
<br />
a n<br />
=+n−<br />
<br />
a n<br />
=n −<br />
Analysis The graph of this sequence, represented in Figure 5, shows a slope of and a vertical intercept of −.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
Figure 5<br />
<br />
n<br />
Try It #6<br />
<br />
<br />
Finding the Number of Terms in a Finite Arithmetic Sequence<br />
fifi<br />
fffffi<br />
e fi<br />
How To…<br />
e fif a fi, fi<br />
1. ffd<br />
2. ffe fia n<br />
=a <br />
+dn−<br />
3. a n<br />
n<br />
Example 6<br />
Finding the Number of Terms in a Finite Arithmetic Sequence<br />
e fi<br />
−−<br />
Solution Thffe fi<br />
<br />
−=−
SECTION 9.2 ARITHMETIC SEQUENCES 775<br />
Thff−ffn<br />
<br />
<br />
a n<br />
=a <br />
+dn−<br />
<br />
a n<br />
=+−n−<br />
<br />
a n<br />
=−n<br />
−a n<br />
n<br />
<br />
−=−n<br />
<br />
=n<br />
Th<br />
Try It #7<br />
e fi<br />
<br />
Solving Application Problems with Arithmetic Sequences<br />
fta <br />
a <br />
<br />
ff<br />
<br />
a n<br />
=a <br />
+dn<br />
Example 7<br />
Solving Application Problems with Arithmetic Sequences<br />
fi<br />
a. <br />
b. <br />
Solution<br />
<br />
<br />
<br />
a. Thdiff<br />
Anft<br />
<br />
A n<br />
=+n<br />
b. <br />
−=<br />
Try It #8<br />
ftfi<br />
<br />
A <br />
=+=<br />
Th<br />
<br />
ftn<br />
<br />
<br />
Arithmetic Sequences (http://openstaxcollege.org/l/arithmeticseq)
776 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
9.2 SECTION EXERCISES<br />
VERBAL<br />
1. 2. n diff<br />
<br />
3. <br />
<br />
5. <br />
y diff<br />
ALGEBRAIC<br />
4. in diff<br />
<br />
<br />
, fiff<br />
6. 7. { <br />
<br />
} <br />
o fiff<br />
8. 9. <br />
fififi<br />
<br />
10. a <br />
=−d=− 11. a <br />
=d= <br />
<br />
e fit fi<br />
12. a <br />
=a <br />
=− 13. a <br />
=−a <br />
=−<br />
fififi<br />
<br />
14. n diff <br />
t<br />
16. n diff <br />
t<br />
18. n diff <br />
t<br />
15. n diff <br />
t<br />
17. n diff <br />
t<br />
<br />
19. a <br />
<br />
a <br />
=a <br />
=<br />
21. a <br />
<br />
a <br />
=a <br />
=<br />
23. a <br />
<br />
a <br />
=a <br />
=<br />
20. a <br />
<br />
a <br />
=a <br />
=<br />
22. a <br />
<br />
a <br />
=a <br />
=<br />
, fifi<br />
24. a <br />
=a <br />
=−a <br />
25. a <br />
=−a <br />
=−a <br />
<br />
e fit fi<br />
26. a <br />
=a n<br />
=a n−<br />
− 27. a <br />
=−a n<br />
=a n−<br />
−
SECTION 9.2 SECTION EXERCISES 777<br />
<br />
28. a= 29. a= 30. a=−<br />
31. a= 32. a=−− 33. a=<br />
34. a=−−− 35. a={ <br />
<br />
} <br />
36. a= { − <br />
− <br />
− } <br />
37. a={ <br />
− <br />
− } <br />
e fit fi<br />
38. a= 39. a= <br />
40. a= <br />
e fit fi<br />
41. a=−n 42. a= <br />
n− <br />
<br />
<br />
43. a= 44. a= 45. a=−<br />
46. a=−−− 47. a= 48. a=−−−<br />
49. a= 50. a={ <br />
− <br />
− } <br />
52. a= { −− <br />
− <br />
} <br />
51. a= { <br />
<br />
} <br />
, fin fi<br />
53. a=−−− 54. a= 55. a={ <br />
<br />
} <br />
GRAPHICAL<br />
<br />
56.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
− <br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
a n 57.<br />
<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
− <br />
−<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n
778 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
e fi<br />
58. a <br />
=d= 59. a <br />
=a n<br />
=a n−<br />
− 60. a n<br />
=−+n<br />
TECHNOLOGY<br />
a n<br />
=n −<br />
<br />
MODE<br />
<br />
SEQ]<br />
<br />
DOT] <br />
<br />
ENTER<br />
Y=<br />
n n=<br />
unun=n −<br />
un un=<br />
2NDWINDOWTBLSET<br />
<br />
=<br />
<br />
Δ=<br />
<br />
<br />
2NDGRAPHTABLE]<br />
61. <br />
un<br />
62. n=<br />
un<br />
63. WINDOWn=n=<br />
x=x=y=−<br />
y=14. ThGRAPH<br />
<br />
a n<br />
= n +<br />
<br />
<br />
64. <br />
unTABLE] <br />
65. <br />
WINDOW]<br />
<br />
EXTENSIONS<br />
66. <br />
<br />
68. <br />
bbb<br />
70. <br />
<br />
72. <br />
{ _ <br />
_ _ <br />
} <br />
67. <br />
<br />
69. <br />
a−ba+b−a+b,<br />
71. { _ _ _ <br />
<br />
} <br />
<br />
73. <br />
<br />
<br />
74.
SECTION 9.3 GEOMETRIC SEQUENCES 779<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Find the common ratio for a geometric sequence.<br />
List the terms of a geometric sequence.<br />
Use a recursive formula for a geometric sequence.<br />
Use an explicit formula for a geometric sequence.<br />
9.3 GEOMETRIC SEQUENCES<br />
fffl<br />
fi<br />
<br />
ftftft<br />
<br />
<br />
Finding Common Ratios<br />
Thgeometric sequence<br />
common ratioTh<br />
<br />
<br />
<br />
a n<br />
definition of a geometric sequence<br />
geometric sequenceTh<br />
common ratioTh<br />
a <br />
r<br />
<br />
a <br />
a <br />
ra <br />
r a <br />
r <br />
How To…<br />
<br />
1. <br />
2. <br />
Example 1<br />
Finding Common Ratios<br />
, fi<br />
a. b. <br />
Solution <br />
a. _ = <br />
_ = <br />
_ = <br />
_ = <br />
Th. Th<br />
b. _ = _ <br />
= _ _ =_ <br />
<br />
Th
780<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Analysis The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have<br />
the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so<br />
this interpretation holds, but (b) is not.<br />
a n<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
<br />
a<br />
<br />
b<br />
n<br />
Figure 1<br />
Q & A…<br />
If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the<br />
common ratio?<br />
<br />
o fi<br />
Try It #1<br />
, fi<br />
Try It #2<br />
, fi<br />
Writing Terms of Geometric Sequences<br />
<br />
<br />
<br />
fi<br />
fiTh<br />
fifi<br />
a <br />
=−r =fi−ċ−<br />
−ċ−<br />
<br />
a <br />
=−<br />
<br />
a <br />
=−ċ=−<br />
<br />
a <br />
=−ċ=−<br />
<br />
a <br />
=−ċ=−<br />
The fi−−−−<br />
How To…<br />
e fi, fie fi<br />
1. a <br />
o fia <br />
<br />
2. a n<br />
=a <br />
o fia <br />
a <br />
o fia <br />
fi<br />
3.
SECTION 9.3 GEOMETRIC SEQUENCES 781<br />
Example 2<br />
Writing the Terms of a Geometric Sequence<br />
e fia <br />
=r =−<br />
Solution a <br />
−o fia <br />
a <br />
o fia <br />
<br />
<br />
a <br />
=<br />
<br />
a <br />
=−a <br />
=−<br />
<br />
a <br />
=−a <br />
=<br />
<br />
a <br />
=−a <br />
=−<br />
The fi−−<br />
Try It #3<br />
t fia <br />
=r= <br />
<br />
Using Recursive Formulas for Geometric Sequences<br />
fi<br />
. Th<br />
<br />
recursive formula for a geometric sequence<br />
Thrd fia <br />
<br />
a n<br />
=r a n−<br />
n ≥<br />
How To…<br />
e fi<br />
1. <br />
2. <br />
3. <br />
Example 3<br />
Using Recursive Formulas for Geometric Sequences<br />
<br />
<br />
<br />
Solution The fi. The fi<br />
<br />
r =_ = <br />
fia <br />
<br />
<br />
a n<br />
=ra n−<br />
<br />
a n<br />
=a n−<br />
n ≥<br />
<br />
a <br />
=<br />
Analysis The sequence of data points follows an exponential pattern. The common ratio is also the base of an<br />
exponential function as shown in Figure 2.<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2<br />
n
782<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Q & A…<br />
Do we have to divide the second term by the first term to find the common ratio?<br />
<br />
e fiftf fi<br />
Try It #4<br />
<br />
{ _ <br />
_ _ <br />
} <br />
Using Explicit Formulas for Geometric Sequences<br />
<br />
o fi<br />
a n<br />
=a <br />
r n −<br />
Th<br />
<br />
a n<br />
= n −<br />
Th Figure 3<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
<br />
<br />
n<br />
Figure 3<br />
explicit formula for a geometric sequence<br />
n<br />
a n<br />
=a <br />
r n −<br />
Example 4<br />
Writing Terms of Geometric Sequences Using the Explicit Formula<br />
a <br />
=a <br />
=, fia <br />
<br />
Solution Thr<br />
rr r
SECTION 9.3 GEOMETRIC SEQUENCES 783<br />
<br />
a n<br />
=a <br />
r n −<br />
a <br />
=r a <br />
r<br />
=r a <br />
=r <br />
r = <br />
e fi<br />
<br />
<br />
<br />
a <br />
=a <br />
=<br />
=<br />
Analysis The common ratio is multiplied by the first term once to find the second term, twice to find the third term,<br />
three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common<br />
ratio nine times or by multiplying by the common ratio raised to the ninth power.<br />
Try It #5<br />
a <br />
=a <br />
= a <br />
<br />
Example 5<br />
Writing an Explicit Formula for the nth Term of a Geometric Sequence<br />
n<br />
<br />
Solution The fi. The fi<br />
<br />
<br />
=<br />
The fi<br />
<br />
<br />
a n<br />
=a <br />
r n−<br />
a n<br />
=ċ n −<br />
ThFigure4<br />
<br />
<br />
<br />
<br />
<br />
<br />
a n<br />
<br />
n<br />
Figure 4<br />
Try It #6
784<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Solving Application Problems with Geometric Sequences<br />
a <br />
a <br />
<br />
<br />
a n<br />
=a <br />
r n<br />
Example 6<br />
Solving Application Problems with Geometric Sequences<br />
<br />
<br />
a. <br />
b. <br />
Solution<br />
<br />
a. ThTh<br />
<br />
Pnft<br />
<br />
P n<br />
=ċ n<br />
<br />
b. <br />
−=<br />
ftn<br />
<br />
P <br />
=ċ ≈<br />
Th<br />
Try It #7<br />
Th<br />
<br />
a. <br />
b. <br />
<br />
Geometric Sequences (http://openstaxcollege.org/l/geometricseq)<br />
Determine the Type of Sequence (http://openstaxcollege.org/l/sequencetype)<br />
Find the Formula for a Sequence (http://openstaxcollege.org/l/sequenceformula)
SECTION 9.3 SECTION EXERCISES 785<br />
9.3 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
<br />
3. <br />
<br />
5. <br />
y diff<br />
ALGEBRAIC<br />
, fi<br />
4. e diff<br />
<br />
6. 7. −−− 8. −− <br />
− <br />
− <br />
− <br />
<br />
, fi<br />
9. −−−−− 10. 11. − <br />
− <br />
<br />
− <br />
<br />
12. 13. <br />
fififi<br />
14. a <br />
=r= 15. a <br />
=r= <br />
<br />
e fit fi<br />
16. a <br />
=a <br />
= 17. a <br />
=a <br />
=<br />
fififi<br />
18. Th <br />
<br />
19. Th − <br />
<br />
<br />
<br />
20. a n<br />
=−−a <br />
21. a n<br />
={ − <br />
− <br />
<br />
} a <br />
<br />
e fit fi<br />
22. a <br />
=−a n<br />
=− <br />
a n−<br />
23. a <br />
=a n<br />
=a n−<br />
<br />
24. a n<br />
=−− 25. a n<br />
=−−−−<br />
26. a n<br />
= 27. a n<br />
=−−<br />
28. a n<br />
= 29. a n<br />
={ <br />
<br />
<br />
<br />
} <br />
30. a n<br />
={ − <br />
− <br />
<br />
} <br />
31. a = n { <br />
− <br />
<br />
− <br />
} <br />
e fit fi<br />
32. a n<br />
=−ċ n− 33. a n<br />
=ċ( − <br />
<br />
) n−
786 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
<br />
34. a n<br />
=−−−− 35. a n<br />
=<br />
36. a n<br />
=−−−− 37. a n<br />
=−−<br />
38. a n<br />
=−−−− 39. a n<br />
={ −− <br />
− <br />
− } <br />
40. a n<br />
={ <br />
<br />
<br />
} <br />
41. a = n { − <br />
− <br />
} <br />
, fifi<br />
42. a <br />
=a n<br />
=−a n−<br />
a <br />
43. a n<br />
=−( − <br />
) n− a <br />
<br />
, fin fi<br />
44. a n<br />
=−− 45. a n<br />
={ <br />
<br />
<br />
} <br />
GRAPHICAL<br />
<br />
46.<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
−<br />
<br />
<br />
<br />
− <br />
−<br />
−<br />
−<br />
− −<br />
n<br />
47.<br />
<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
− <br />
−<br />
−<br />
−<br />
<br />
e fit fi<br />
48. a <br />
=r= <br />
49. a =a =a 50. a<br />
n n− n<br />
=ċ n−<br />
EXTENSIONS<br />
51. <br />
<br />
52. <br />
<br />
n<br />
53. <br />
bbb<br />
55. <br />
<br />
57. <br />
a n<br />
=−( <br />
) n− <br />
59. <br />
<br />
<br />
<br />
54. <br />
a−ba−ba−b<br />
56. <br />
{ <br />
<br />
<br />
<br />
<br />
} <br />
58. <br />
<br />
<br />
60.
SECTION 9.4 SERIES AND THEIR NOTATIONS 787<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Use summation notation.<br />
Use the formula for the sum of the first n terms of an arithmetic series.<br />
Use the formula for the sum of the first n terms of a geometric series.<br />
Use the formula for the sum of an infinite geometric series.<br />
Solve annuity problems.<br />
9.4 SERIES AND THEIR NOTATIONS<br />
ThTh<br />
<br />
<br />
<br />
Using Summation Notation<br />
fi<br />
Thseries<br />
<br />
<br />
+++++<br />
nth partial sumfifiTh<br />
S n<br />
<br />
<br />
S <br />
=<br />
<br />
S <br />
=+=<br />
<br />
S <br />
=++=<br />
<br />
S <br />
=+++=<br />
Summation notationft<br />
fi<br />
fi. <br />
index of summationThlower<br />
limit of summationfiTh<br />
upper limit of summation<br />
→<br />
→<br />
k ← k<br />
←<br />
<br />
∑<br />
k =<br />
fia k<br />
=kk =<br />
k =k<br />
<br />
a <br />
==<br />
<br />
a <br />
==<br />
<br />
a <br />
==<br />
<br />
a <br />
==<br />
<br />
a <br />
==<br />
n fi<br />
<br />
<br />
∑<br />
k =++++=<br />
k =
788<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
summation notation<br />
The finseriessummation notation<br />
n<br />
∑<br />
a k<br />
<br />
k =<br />
a k<br />
k =k =n<br />
kindex of summationlower limit of summationnupper limit of summation<br />
Q & A…<br />
Does the lower limit of summation have to be 1?<br />
Th<br />
<br />
How To…<br />
<br />
1. <br />
2. <br />
3. k<br />
4. o fi<br />
Example 1<br />
∑ k<br />
<br />
<br />
k =<br />
Using Summation Notation<br />
Solution fi<br />
k k =k =fik =<br />
k o fi<br />
<br />
<br />
<br />
<br />
∑<br />
k <br />
= + + + + <br />
k =<br />
=++++<br />
=<br />
Try It #1<br />
<br />
∑ k−<br />
k=<br />
Using the Formula for Arithmetic Series<br />
<br />
ff dTh<br />
arithmetic sequencefin<br />
<br />
<br />
S n<br />
=a <br />
+a <br />
+d+a <br />
+d++a n<br />
−d+a n<br />
<br />
<br />
<br />
S n<br />
=a n<br />
+a n<br />
−d+a n<br />
−d++a <br />
+d+a <br />
<br />
fi n<br />
e fin
SECTION 9.4 SERIES AND THEIR NOTATIONS 789<br />
<br />
S n<br />
=a <br />
+a <br />
+d+a <br />
+d++a n<br />
−d+a n<br />
<br />
+S n<br />
=a n<br />
+a n<br />
−d+a n<br />
−d++a <br />
+d+a <br />
<br />
S n<br />
=a <br />
+a n<br />
+a <br />
+a n<br />
++a <br />
+a n<br />
<br />
n<br />
<br />
S n<br />
=na <br />
+a n<br />
<br />
o fie fin<br />
<br />
S n<br />
= n a <br />
+a n <br />
formula for the sum of the first n terms of an arithmetic series<br />
arithmetic seriesThfin<br />
<br />
S n<br />
= n a <br />
+a n <br />
How To…<br />
, fie fin<br />
1. a <br />
a n<br />
<br />
2. n<br />
3. a <br />
a n<br />
nS n<br />
= n a <br />
+a<br />
_ n <br />
<br />
<br />
4. o fiS n<br />
<br />
Example 2<br />
Finding the First n Terms of an Arithmetic Series<br />
<br />
a. +++++++++<br />
b. ++++−<br />
<br />
c. ∑ k−<br />
Solution<br />
k=<br />
a. a <br />
=a n<br />
=<br />
o fin =<br />
a <br />
a n<br />
n<br />
<br />
<br />
<br />
<br />
S n<br />
= n a <br />
+a n <br />
S <br />
= + <br />
=<br />
b. a <br />
=a n<br />
=−<br />
o fin<br />
<br />
a n<br />
=a <br />
+n−d<br />
<br />
−=+n−−<br />
<br />
−=n−−<br />
<br />
=n −<br />
<br />
=n<br />
a <br />
a n<br />
n<br />
S n<br />
= n a <br />
+a n <br />
S <br />
= − <br />
=−
790<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
<br />
<br />
c. a <br />
k =<br />
a k<br />
=k −<br />
a <br />
=−=−<br />
n =o fia <br />
k =<br />
<br />
a k<br />
=k −<br />
<br />
a <br />
=−=<br />
a <br />
a n<br />
n<br />
<br />
S n<br />
= na +a _ n<br />
<br />
<br />
<br />
Try It #2<br />
o fi<br />
S <br />
=__<br />
−+ <br />
=<br />
++++++++++<br />
Try It #3<br />
o fi<br />
Try It #4<br />
o fi<br />
++++<br />
<br />
∑<br />
k=<br />
−k<br />
Example 3<br />
Solving Application Problems with Arithmetic Series<br />
ft<br />
ft<br />
Solution Tha <br />
= <br />
d = <br />
<br />
ftn =S <br />
fia <br />
<br />
<br />
<br />
a n<br />
=a <br />
+dn−<br />
<br />
<br />
<br />
<br />
<br />
a <br />
= <br />
+ <br />
−= <br />
<br />
S n<br />
= na +a _ n <br />
( <br />
+ <br />
) <br />
<br />
=<br />
__<br />
<br />
=<br />
Try It #5<br />
ft
SECTION 9.4 SERIES AND THEIR NOTATIONS 791<br />
Using the Formula for Geometric Series<br />
<br />
geometric series<br />
re fin<br />
<br />
S n<br />
=a <br />
+ra <br />
+r a <br />
++r n a <br />
<br />
fin<br />
r<br />
<br />
rS n<br />
=ra <br />
+r a <br />
+r a <br />
++r n a <br />
<br />
<br />
S n<br />
=a <br />
+ra <br />
+r a <br />
++r n a <br />
<br />
<br />
−rS n<br />
=−ra <br />
+r a <br />
+r a <br />
++r n a <br />
<br />
<br />
−rS n<br />
=a <br />
−r n a <br />
fi<br />
S n<br />
−r<br />
<br />
S n<br />
= a −rn <br />
−r r ≠<br />
formula for the sum of the first n terms of a geometric series<br />
geometric seriesThfin<br />
<br />
S n<br />
= a −r n <br />
<br />
−r r ≠<br />
How To…<br />
, fie fin <br />
1. a <br />
rn<br />
2. a <br />
rnS n<br />
= a −r n <br />
−r <br />
3. o fiS n<br />
<br />
Example 4<br />
Finding the First n Terms of a Geometric Series<br />
o fi<br />
a. S <br />
+−++<br />
<br />
b. ∑ ċ <br />
k <br />
Solution<br />
k=<br />
a. a <br />
=n =<br />
n fire fi<br />
r= − <br />
=− <br />
<br />
a <br />
rn<br />
S n<br />
= a −rn <br />
_ −r<br />
<br />
( − ( − <br />
S <br />
= <br />
−( − <br />
) <br />
) ) <br />
__<br />
≈
792<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
<br />
b. a <br />
k =<br />
a <br />
=ċ =<br />
r =. Thn =<br />
a <br />
rn<br />
S n<br />
= a −r n <br />
<br />
−r<br />
<br />
S <br />
= − <br />
− =<br />
Try It #6<br />
o fi<br />
S <br />
+++<br />
Try It #7<br />
o fi<br />
<br />
∑<br />
k=<br />
k<br />
Example 5<br />
Solving an Application Problem with a Geometric Series<br />
<br />
<br />
Solution Tha <br />
=n =r =<br />
a <br />
rno fi<br />
S n<br />
= a −rn <br />
−r<br />
<br />
S <br />
= − <br />
<br />
− ≈<br />
<br />
Try It #8<br />
<br />
<br />
Using the Formula for the Sum of an Infinite Geometric Series<br />
Thfi<br />
fifininfinite seriesfi<br />
fi++++<br />
∞<br />
Th∑ kfi<br />
k =<br />
Th<br />
fifidiverges<br />
Determining Whether the Sum of an Infinite Geometric Series is Defined<br />
fififiTh<br />
<br />
+++++
SECTION 9.4 SERIES AND THEIR NOTATIONS 793<br />
Thr =nr n <br />
ff<br />
fiThfi−
794<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
fir = <br />
r n n<br />
<br />
( _ <br />
) = <br />
<br />
<br />
( _ <br />
) = <br />
<br />
<br />
( _ <br />
) = <br />
Thr n n<br />
<br />
<br />
( _ <br />
) <br />
=<br />
<br />
<br />
( _ <br />
) <br />
= <br />
<br />
<br />
<br />
( _ <br />
) <br />
= <br />
<br />
<br />
nr n nr n r n <br />
−r n a <br />
Th<br />
fi<br />
formula for the sum of an infinite geometric series<br />
Thfi−
SECTION 9.4 SERIES AND THEIR NOTATIONS 795<br />
<br />
<br />
a <br />
=r =− o <br />
fi<br />
a <br />
S =_<br />
<br />
−r <br />
<br />
S =__<br />
=<br />
−( − <br />
) <br />
d. Thr > 1. Th<br />
Example 8<br />
Finding an Equivalent Fraction for a Repeating Decimal<br />
_ <br />
Solution _ =<br />
<br />
_ =+++<br />
fi<br />
<br />
<br />
_ =+ + <br />
<br />
fi<br />
fi<br />
<br />
<br />
Try It #12<br />
<br />
Try It #13<br />
<br />
Try It #14<br />
<br />
a <br />
S =_<br />
<br />
−r =<br />
<br />
− = <br />
= <br />
<br />
+ <br />
+ <br />
+<br />
∞<br />
∑<br />
k =<br />
k +<br />
∞<br />
∑<br />
k =<br />
( − <br />
) k <br />
Solving Annuity Problems<br />
<br />
annuity<br />
<br />
<br />
fi<br />
<br />
<br />
a <br />
=<br />
r =100.5% = <br />
<br />
n n<br />
n =a <br />
=r =n =<br />
<br />
<br />
S <br />
= − <br />
<br />
− ≈<br />
<br />
=
796<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
How To…<br />
, fi<br />
1. a <br />
<br />
2. n<br />
3. r<br />
a. <br />
b. o fir<br />
4. a <br />
rne fin<br />
S n<br />
= a rn <br />
_ r<br />
5. o fiS n<br />
ftn<br />
Example 9<br />
Solving an Annuity Problem<br />
Th<br />
ft<br />
<br />
Solution Tha <br />
=<br />
n =firfi<br />
<br />
<br />
r =+ <br />
=<br />
a <br />
=r =n =fin<br />
o fi<br />
<br />
S <br />
= − <br />
<br />
− ≈<br />
ft<br />
Try It #15<br />
Th <br />
<br />
<br />
<br />
Arithmetic Series (http://openstaxcollege.org/l/arithmeticser)<br />
Geometric Series (http://openstaxcollege.org/l/geometricser)<br />
Summation Notation (http://openstaxcollege.org/l/sumnotation)
SECTION 9.4 SECTION EXERCISES 797<br />
9.4 SECTION EXERCISES<br />
VERBAL<br />
1. n 2. e diff<br />
<br />
3. 4. <br />
ies diff n<br />
5. <br />
ALGEBRAIC<br />
<br />
6. Thm +mm=m= 7. Thn=n=n<br />
8. Thk−k=−k= 9. Ther 4 fi<br />
<br />
<br />
10. +++++++++ 11. ++++<br />
12. <br />
++ <br />
+++<br />
e fin<br />
13. <br />
++ <br />
++ <br />
14. ++++ 15. ++++<br />
<br />
<br />
16. +++++++ 17. ++++<br />
18. − <br />
+ <br />
− <br />
++ <br />
<br />
n<br />
<br />
19. +++ _ + _ <br />
<br />
20. ∑ ċ <br />
n− <br />
n=<br />
<br />
21. ∑ ċ a− <br />
a=<br />
fi<br />
<br />
22. ++++ 23. ++++ 24. ∑ m− <br />
∞<br />
25. ∑ −( − <br />
) k− <br />
k=<br />
∞<br />
m=
798 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
GRAPHICAL<br />
<br />
ft<br />
<br />
26. <br />
<br />
∞<br />
∑ <br />
k =<br />
( <br />
) k <br />
27. <br />
<br />
28. 29. S n<br />
<br />
<br />
<br />
NUMERIC<br />
, fi<br />
<br />
30. ∑ a <br />
a=<br />
<br />
31. ∑ nn− <br />
n=<br />
<br />
32. ∑ k <br />
k=<br />
<br />
33. ∑ k <br />
e fino fi<br />
34. −+−++++ 35. + <br />
++ <br />
++ <br />
+<br />
<br />
36. −++++ 37. ∑ ( k <br />
− <br />
) <br />
finfi<br />
38. S <br />
−−−− 39. S <br />
−+−<br />
k =<br />
k=<br />
<br />
40. ∑ k− <br />
k=<br />
<br />
41. ∑ −ċ( <br />
n= ) n− <br />
, fifi<br />
42. +++ <br />
<br />
∞<br />
45. ∑ ċ n− <br />
n=<br />
43. −− <br />
− <br />
− <br />
<br />
k=<br />
<br />
∞<br />
44. ∑ ċ( <br />
) k− <br />
<br />
<br />
46. 47. <br />
<br />
<br />
48. <br />
<br />
49. <br />
<br />
EXTENSIONS<br />
50. Th−k k=x<br />
x<br />
52. n<br />
n<br />
∑<br />
k=<br />
k−<br />
51. a k<br />
<br />
<br />
∑<br />
k=<br />
a k<br />
=<br />
53. <br />
−−−−−
SECTION 9.4 SECTION EXERCISES 799<br />
54. _ <br />
n. Th<br />
<br />
<br />
56. <br />
<br />
. Th<br />
<br />
ff<br />
<br />
fter fi<br />
<br />
55. Th s fi<br />
<br />
<br />
57. <br />
<br />
<br />
ff<br />
<br />
REAL-WORLD APPLICATIONS<br />
58. <br />
<br />
<br />
<br />
ft<br />
60. <br />
<br />
ft<br />
59. g 6 f<br />
<br />
<br />
ft<br />
61. <br />
<br />
<br />
<br />
<br />
<br />
62. <br />
. Th
800<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Solve counting problems using the Addition Principle.<br />
Solve counting problems using the Multiplication Principle.<br />
Solve counting problems using permutations involving n distinct objects.<br />
Solve counting problems using combinations.<br />
Find the number of subsets of a given set.<br />
Solve counting problems using permutations involving n non-distinct objects.<br />
9.5 COUNTING PRINCIPLES<br />
<br />
<br />
Th<br />
Th<br />
ff<br />
Th<br />
<br />
<br />
Using the Addition Principle<br />
ThffTh<br />
ThAddition Principle<br />
fi<br />
Figure 1<br />
Figure 1<br />
the Addition Principle<br />
Addition Principlem<br />
ne fim +n<br />
Example 1<br />
Using the Addition Principle<br />
Th
SECTION 9.5 COUNTING PRINCIPLES 801<br />
Solution fi<br />
<br />
+ + + + + + <br />
↓ ↓ ↓ ↓ ↓ ↓ ↓<br />
+ + + + + + <br />
Th<br />
Try It #1<br />
<br />
<br />
Using the Multiplication Principle<br />
ThMultiplication Principle<br />
n a fi<br />
Figure 2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2<br />
Th<br />
1. <br />
2. <br />
3. , fi<br />
4. , fi<br />
5. <br />
6. <br />
7. <br />
8. <br />
9. , fi<br />
10. , fi<br />
11. <br />
12. <br />
o fi<br />
<br />
××<br />
× × =
802<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
the Multiplication Principle<br />
Multiplication Principlemn<br />
ftfim ×nTh<br />
Fundamental Counting Principle<br />
Example 2<br />
Using the Multiplication Principle<br />
<br />
fifi<br />
fi<br />
Solution fififi<br />
<br />
××<br />
× × =<br />
Thfi<br />
Try It #2<br />
ffTh<br />
<br />
<br />
Finding the Number of Permutations of n Distinct Objects<br />
Th<br />
<br />
permutation<br />
Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle<br />
ftTh<br />
fi<br />
<br />
<br />
× ×<br />
The fie fi<br />
× ×<br />
fte fin fi<br />
× ×<br />
ftfi<br />
e fi<br />
× × =<br />
Th<br />
How To…<br />
n<br />
1. e fi<br />
2. <br />
3. e fi<br />
4.
SECTION 9.5 COUNTING PRINCIPLES 803<br />
Example 3<br />
Finding the Number of Permutations Using the Multiplication Principle<br />
<br />
a. <br />
b. <br />
<br />
c. <br />
Solution<br />
a. <br />
<br />
× ×<br />
<br />
Thfifi<br />
e fi<br />
× × =<br />
o fi<br />
b. <br />
<br />
× ×<br />
<br />
fifiTh<br />
<br />
× × =<br />
o fis fi<br />
c. <br />
× × × × × × × ×<br />
Thfi<br />
<br />
× × × × × × × × =<br />
Th<br />
Analysis Note that in part c, we found there were ways for people to line up. The number of permutations of n<br />
distinct objects can always be found by n!.<br />
Try It #3<br />
fi <br />
<br />
Try It #4<br />
fi <br />
<br />
Try It #5<br />
fi
804<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Finding the Number of Permutations of n Distinct Objects Using a Formula<br />
<br />
<br />
nr<br />
P nr n<br />
P r<br />
<br />
Pn, rfinnn−r<br />
n−r<br />
Th<br />
fi<br />
Th<br />
××=<br />
<br />
= <br />
==<br />
Thffi<br />
. Th<br />
n<br />
Pn, r=<br />
<br />
n− <br />
n<br />
n−nnn<br />
n <br />
n<br />
formula for permutations of n distinct objects<br />
nr<br />
n<br />
Pn, r=<br />
<br />
n−r <br />
How To…<br />
<br />
1. n<br />
2. r<br />
3. nr<br />
4. <br />
Example 4<br />
Finding the Number of Permutations Using the Formula<br />
<br />
<br />
Solution n =r =<br />
n<br />
<br />
Pn, r=<br />
<br />
n−r <br />
<br />
<br />
P=<br />
<br />
− = <br />
=<br />
Th<br />
Analysis We can also use a calculator to find permutations. For this problem, we would enter 12, press the n<br />
P r<br />
function],<br />
enter 9], and then press the equal sign. The n<br />
P r<br />
function] may be located under the MATH] menu with probability<br />
commands.<br />
Q & A…<br />
Could we have solved Example 4 using the Multiplication Principle?<br />
ċċċċċċċċo fi
SECTION 9.5 COUNTING PRINCIPLES 805<br />
Try It #6<br />
fi<br />
<br />
Try It #7<br />
fi<br />
<br />
Find the Number of Combinations Using the Formula<br />
Th<br />
<br />
combinations. rn<br />
C nrCn, r n<br />
C r<br />
<br />
<br />
n<br />
C n, r=<br />
<br />
rn−r <br />
<br />
<br />
fi<br />
<br />
Th==Th <br />
<br />
<br />
ThnotTh<br />
<br />
formula for combinations of n distinct objects<br />
nr<br />
n<br />
C n, r=<br />
<br />
rn−r <br />
How To…<br />
<br />
1. n<br />
2. r<br />
3. nr<br />
4. <br />
Example 5<br />
Finding the Number of Combinations Using the Formula<br />
ffs fi<br />
a <br />
b. <br />
Solution<br />
<br />
<br />
a. <br />
<br />
C=<br />
<br />
− =<br />
b. <br />
<br />
C=<br />
<br />
− =
806<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Analysis We can also use a graphing calculator to find combinations. Enter 5, then press n<br />
C r<br />
, enter 3, and then press<br />
the equal sign. The n<br />
C r<br />
function may be located under the MATH menu with probability commands.<br />
Q & A…<br />
Is it a coincidence that parts ( a) and ( b) in Example 5 have the same answers?<br />
rn notn−r. ThCn, r=Cn, n −r<br />
Try It #8<br />
ffs 10 fle 3 fl<br />
Finding the Number of Subsets of a Set<br />
r<br />
ff<br />
ff<br />
ThC=<br />
ThC=<br />
<br />
C+C+C+C+C+C=<br />
Th. Th <br />
n <br />
n<br />
<br />
<br />
formula for the number of subsets of a set<br />
n n <br />
Example 6<br />
Finding the Number of Subsets of a Set<br />
ffff<br />
<br />
Solution n =<br />
<br />
n = <br />
<br />
Th<br />
Try It #9<br />
=<br />
diff<br />
<br />
Finding the Number of Permutations of n Non-Distinct Objects<br />
<br />
<br />
<br />
Th
SECTION 9.5 COUNTING PRINCIPLES 807<br />
n<br />
<br />
r <br />
r <br />
r k<br />
<br />
<br />
fi<br />
Th<br />
<br />
=<br />
Th<br />
formula for finding the number of permutations of n non-distinct objects<br />
nr <br />
r <br />
r <br />
r k<br />
<br />
<br />
n<br />
<br />
<br />
r <br />
r <br />
r k<br />
<br />
Example 7<br />
Finding the Number of Permutations of n Non-Distinct Objects<br />
<br />
Solution Thn =r <br />
=r <br />
=<br />
Th<br />
<br />
=<br />
Try It #10<br />
<br />
<br />
Combinations (http://openstaxcollege.org/l/combinations)<br />
Permutations (http://openstaxcollege.org/l/permutations)
808 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
9.5 SECTION EXERCISES<br />
VERBAL<br />
nAmB<br />
AB<br />
1. <br />
AB<br />
<br />
3. <br />
<br />
<br />
<br />
<br />
2. <br />
AB<br />
4. nts diff<br />
r<br />
n<br />
5. r<br />
n<br />
r <br />
<br />
<br />
NUMERIC<br />
Th<br />
<br />
6. A=−−−<br />
<br />
A<br />
8. <br />
<br />
10. <br />
<br />
12. A<br />
B<br />
A=bcd B=aeiou<br />
14. <br />
<br />
7. B=−−−−<br />
<br />
A<br />
9. <br />
<br />
<br />
11. <br />
<br />
13. <br />
<br />
<br />
15. P 16. P 17. P 18. P 19. P<br />
20. C 21. C 22. C 23. C 24. C<br />
, fi<br />
25. 26. abcz<br />
27. 28. Th<br />
<br />
29. Th<br />
<br />
, fi<br />
30. Th 31. Th<br />
32. Th 33. Th<br />
<br />
34. Th
SECTION 9.5 SECTION EXERCISES 809<br />
EXTENSIONS<br />
35. ThS<br />
<br />
SHint<br />
<br />
t 0.)<br />
36. Th<br />
n<br />
<br />
n<br />
<br />
37. Cn, rPn,r 38. A<br />
A<br />
39. <br />
<br />
<br />
REAL-WORLD APPLICATIONS<br />
40. <br />
<br />
<br />
a. <br />
<br />
b. <br />
<br />
c. <br />
<br />
42. <br />
<br />
<br />
y diff<br />
<br />
44. <br />
<br />
<br />
<br />
<br />
46. <br />
<br />
<br />
48. <br />
<br />
<br />
<br />
<br />
<br />
50. <br />
<br />
diff<br />
52. ff<br />
<br />
<br />
<br />
<br />
<br />
54. <br />
<br />
<br />
41. ffs 6 diff<br />
d 8 diff<br />
3 p<br />
<br />
<br />
43. ff<br />
<br />
<br />
y diff<br />
<br />
45. s in 12 diff<br />
<br />
<br />
47. <br />
<br />
49. <br />
s—diff<br />
<br />
<br />
51. ff<br />
<br />
<br />
<br />
<br />
53.
810<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Apply the Binomial Theorem.<br />
9.6 BINOMIAL THEOREM<br />
<br />
<br />
o fix+y n n<br />
Identifying Binomial Coefficients<br />
Counting Principlesfix+y n <br />
fiffi( _ n r<br />
)<br />
Cn, r<br />
( _ n n<br />
r<br />
)=Cn, r=<br />
<br />
rn−r <br />
Th( _ n r<br />
)binomial coefficientffi( <br />
<br />
) =C=<br />
binomial coefficients<br />
nrn ≥rbinomial coefficient<br />
( _ n n<br />
r<br />
)=Cn, r=<br />
<br />
rn−r <br />
Q & A…<br />
Is a binomial coefficient always a whole number?<br />
ffi<br />
<br />
Example 1<br />
ffi<br />
Finding Binomial Coefficients<br />
a. ( <br />
) b ( <br />
) c. ( <br />
Solution<br />
) <br />
ffi n<br />
C r<br />
<br />
a. ( <br />
) =<br />
<br />
− =ċċ <br />
=<br />
( _ n n<br />
r<br />
)=Cn, r= <br />
rn−r <br />
<br />
b. ( <br />
c. ( <br />
) =<br />
<br />
− =ċċ <br />
=<br />
) =<br />
<br />
− =ċċ <br />
=<br />
Analysis <br />
<br />
<br />
( _ n r<br />
)=( <br />
n_<br />
n − r )
SECTION 9.6 BINOMIAL THEOREM 811<br />
Try It #1<br />
ffi<br />
a. ( <br />
) b. ( <br />
) <br />
Using the Binomial Theorem<br />
x+y n binomial expansion<br />
ffix+y x+yfiftTh<br />
fifi<br />
<br />
<br />
x+y =x +xy+y <br />
<br />
<br />
x+y =x +x y +xy +y <br />
x+y =x +x y +x y +xy +y <br />
xy<br />
. Thn<br />
ffiffi<br />
Thffi<br />
( n _<br />
) ( n _<br />
) ( n _<br />
) ( n _<br />
n )<br />
ThBinomial Theorem<br />
n<br />
<br />
x+y =∑<br />
n ( n _ k ) xn−k y k <br />
<br />
k =<br />
n_<br />
=x n +( n _<br />
) xn− y +( n _<br />
) xn− y ++( n − ) xy n − +y n<br />
ffix +y<br />
<br />
<br />
x+y =x +y<br />
<br />
x+y =x +xy+y <br />
<br />
<br />
x+y =x +x y +xy +y <br />
x+y =x +x y +x y +xy +y <br />
x+y <br />
Pascal’s Triangle<br />
Exponent<br />
x+y = x+y<br />
<br />
Pattern<br />
+ <br />
# of Terms<br />
<br />
x+y = x +xy + y <br />
<br />
+ <br />
<br />
x+y = x +x y + xy + y <br />
<br />
+ <br />
<br />
x+y = x +x y + x y + xy + y <br />
<br />
+ <br />
<br />
n<br />
n+ <br />
n+ <br />
+<br />
+<br />
+<br />
+<br />
+<br />
xy xy xy xy xy<br />
x <br />
y <br />
Figure 1
812<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Figure1<br />
Thn +x+y n <br />
Thn<br />
Thxn<br />
Thyn<br />
Th <br />
x+y n =+=<br />
x<br />
x xx =<br />
y =yy <br />
x x yx y x y xy y <br />
Th<br />
x+y =x +x y +x y +x y +xy +y <br />
ffiThffiffi<br />
Figure2<br />
+ = <br />
<br />
Pascal’s Triangle<br />
<br />
<br />
<br />
<br />
<br />
<br />
+ = <br />
Figure 2<br />
fl<br />
+finfl<br />
<br />
ffi<br />
<br />
<br />
<br />
→ x+y =<br />
→ x+y =x +y<br />
→ x+y =x +xy+y <br />
→ x+y =x +x y +xy +y <br />
→ x+y =x +x y +x y +xy +y <br />
→ x+y =x +x y +x y +x y +xy +y <br />
the Binomial Theorem<br />
Binomial Theorem<br />
n<br />
<br />
x+y =∑<br />
n ( n k ) x n −k y k <br />
k =<br />
<br />
n_<br />
=x n +( n _<br />
) x n − y +( n _<br />
) x n − y ++( n − ) xy n − +y n
SECTION 9.6 BINOMIAL THEOREM 813<br />
How To…<br />
<br />
1. n<br />
2. k =k =nl Th<br />
3. <br />
Example 2<br />
Expanding a Binomial<br />
<br />
a x+y <br />
b (x −y <br />
Solution<br />
a. n =k =k =<br />
<br />
<br />
<br />
<br />
x+y =( _<br />
) x y +( _<br />
) x y +( _<br />
) x y +( _<br />
) x y +( _<br />
) x y +( _<br />
) x y <br />
x+y =x +x y +x y +x y +xy +y <br />
b. n =k =k =x<br />
x−yy<br />
x −y =( _<br />
) x −y +( _<br />
) x −y +( _<br />
) x −y +( _<br />
) x −y +( _<br />
) x −y <br />
x −y =x −x y +x y −xy +y <br />
Analysis Notice the alternating signs in part b. This happens because (−y) raised to odd powers is negative, but (−y)<br />
raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.<br />
Try It #2<br />
<br />
a. x−y b. xy <br />
Using the Binomial Theorem to Find a Single Term<br />
x+y <br />
<br />
o fifi<br />
ffix+y <br />
x+y =x +( _<br />
) x y +( _<br />
) x y +( _<br />
) x y +( _<br />
) xy +y <br />
Th( _<br />
) x y. Th( _<br />
) x y <br />
( n _<br />
r )x n−r y r<br />
the (r + 1)th term of a binomial expansion<br />
r+x+y n <br />
( n _<br />
r )x n −r y r
814<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
How To…<br />
fi<br />
1. n<br />
2. r+<br />
3. r<br />
4. rr+<br />
Example 3<br />
Writing a Given Term of a Binomial Expansion<br />
x+y <br />
Solution r +=r =<br />
Try It #3<br />
( n _<br />
r )x n−r y r<br />
( _<br />
) x− y =x y <br />
x −y <br />
<br />
The Binomial Theorem (http://openstaxcollege.org/l/binomialtheorem)<br />
Binomial Theorem Example (http://openstaxcollege.org/l/btexample)
SECTION 9.6 SECTION EXERCISES 815<br />
9.6 SECTION EXERCISES<br />
VERBAL<br />
1. <br />
<br />
2. <br />
<br />
<br />
3. l Th 4. <br />
Th<br />
ALGEBRAIC<br />
ffi<br />
5. ( _<br />
) <br />
6. ( _<br />
) <br />
7. ( _<br />
) <br />
8. ( _<br />
) <br />
9. ( _<br />
) <br />
10. ( _<br />
) <br />
11. ( _<br />
) <br />
12. ( _<br />
) <br />
l Th<br />
13. a−b 14. a+ 15. a+b 16. x+y 17. x+y <br />
18. x−y 19. x−y 20. ( _ x<br />
+y ) 21. x − +y − 22. √ — x −√ — y <br />
l The fi<br />
23. a+b 24. x− 25. a−b 26. x−y 27. a+b <br />
28. a+b 29. x −√ — y <br />
, fi<br />
30. Thx−y 31. Thx−y <br />
32. Thx−y 33. Th+y <br />
34. Tha+b 35. Th x−y <br />
36. Thx− 37. Tha−b <br />
38. Th( x − <br />
) <br />
<br />
<br />
39. Th( y _<br />
+_ x<br />
) <br />
GRAPHICAL<br />
Thf x=x+ Thfi<br />
<br />
40. f <br />
xf <br />
x <br />
<br />
42. f <br />
xf <br />
x<br />
<br />
41. f <br />
xf <br />
x<br />
<br />
43. f <br />
xf <br />
x<br />
<br />
44. f <br />
xf <br />
x<br />
t fi
816 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
EXTENSIONS<br />
45. x+y n <br />
( n _<br />
k ) an−k b k k<br />
n ( n _<br />
<br />
k ) = ( _<br />
) <br />
46. a+b n a n−k b k <br />
<br />
47. x+b <br />
bk<br />
49. <br />
l Th<br />
a. x −x+<br />
b. √ — a +√ — a − <br />
c. x +y −z <br />
d. x −√ — y <br />
48. ( <br />
n_<br />
k − 1 ) + ( n _<br />
<br />
k ) <br />
( n _<br />
k ) <br />
pp≥<br />
p=pp−
SECTION 9.7 PROBABILITY 817<br />
LEARNING OBJECTIVES<br />
In this section, you will:<br />
Construct probability models.<br />
Compute probabilities of equally likely outcomes.<br />
Compute probabilities of the union of two events.<br />
Use the complement rule to find probabilities.<br />
Compute probability using counting theory.<br />
9.7 PROBABILITY<br />
Figure 1 An example of a “spaghetti model,” which can be used to predict possible paths of a tropical storm. [33]<br />
<br />
Figure 1Th<br />
Th<br />
<br />
Constructing Probability Models<br />
experiment<br />
ThoutcomesTh<br />
sample spaceTh<br />
event<br />
ThprobabilityThpfi<br />
≤p ≤. probability model<br />
<br />
ffleTable 1<br />
Outcome<br />
ffle<br />
ffle<br />
Table 1<br />
Probability<br />
<br />
<br />
Th<br />
The fi
818<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
How To…<br />
<br />
1. <br />
2. <br />
3. <br />
Example 1<br />
Constructing a Probability Model<br />
<br />
Solution Th<br />
. Th<br />
<br />
Th<br />
<br />
<br />
Outcome <br />
Probability<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 2<br />
Q & A…<br />
Do probabilities always have to be expressed as fractions?<br />
<br />
<br />
Try It #1<br />
<br />
Computing Probabilities of Equally Likely Outcomes<br />
SS<br />
fi<br />
S<br />
fiTh<br />
S <br />
= <br />
<br />
computing the probability of an event with equally likely outcomes<br />
ThES<br />
PE= E <br />
S = n(E) ____<br />
n(S) <br />
ES≤PE≤<br />
Example 2<br />
Computing the Probability of an Event with Equally Likely Outcomes<br />
<br />
Solution ThTh<br />
o fi<br />
PE= <br />
=
SECTION 9.7 PROBABILITY 819<br />
Try It #2<br />
<br />
Computing the Probability of the Union of Two Events<br />
ftfi<br />
fi<br />
Thunion of two eventsEFE ∪F<br />
<br />
PE∪F=PE+PF−PE∩F<br />
Figure 2o fib<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 2<br />
Th <br />
= <br />
Th<br />
bb <br />
= <br />
<br />
bfi<br />
bb<br />
<br />
+ <br />
− <br />
= <br />
<br />
Thb <br />
<br />
probability of the union of two events<br />
ThEF E ∪FE<br />
FEEF<br />
E ∩F<br />
PE∪F=PE+PF−PE∩F<br />
Example 3<br />
Computing the Probability of the Union of Two Events<br />
<br />
Solution <br />
<br />
Th<br />
<br />
<br />
Th<br />
<br />
PH= <br />
P= <br />
PH∩= <br />
<br />
PE∪F=PE+PF−PE∩F<br />
Th <br />
<br />
= <br />
+ <br />
− <br />
<br />
=
820<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Try It #3<br />
<br />
Computing the Probability of Mutually Exclusive Events<br />
Figure 2<br />
dThd<br />
mutually exclusive events<br />
<br />
PE ∪F=PE+PF<br />
EFTh<br />
<br />
= <br />
d <br />
fi<br />
d<br />
PE∪F=PE+PF<br />
Thd <br />
<br />
= <br />
+ <br />
<br />
=<br />
<br />
probability of the union of mutually exclusive events<br />
ThE F <br />
PE∪ F=PE+PF<br />
How To…<br />
<br />
1. e fi<br />
2. e fi<br />
3. <br />
4. <br />
5. <br />
Example 4<br />
Computing the Probability of the Union of Mutually Exclusive Events<br />
<br />
Solution Th<br />
Th <br />
<br />
<br />
<br />
<br />
+ <br />
= <br />
<br />
Try It #4
SECTION 9.7 PROBABILITY<br />
821<br />
<br />
<br />
Thcomplement of an eventEE′<br />
E<br />
WW<br />
fi<br />
<br />
PE′=−PE<br />
Th. Th<br />
<br />
<br />
<br />
<br />
− = <br />
the complement rule<br />
Thcomplement of an event<br />
PE′=−PE<br />
Example 5<br />
Using the Complement Rule to Calculate Probabilities<br />
<br />
a. <br />
b. <br />
Solution ThfiTh<br />
fi<br />
×<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 3<br />
a. <br />
. Th<br />
<br />
= <br />
<br />
b. <br />
fi<br />
<br />
PE′=−PE<br />
=−<br />
<br />
<br />
=<br />
<br />
<br />
Try It #5
822<br />
CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Computing Probability Using Counting Theory<br />
<br />
fi<br />
Th<br />
<br />
fi<br />
<br />
Th<br />
CTh<br />
C. Th<br />
<br />
____<br />
<br />
<br />
=C _<br />
<br />
C <br />
<br />
<br />
Example 6<br />
Computing Probability Using Counting Theory<br />
= <br />
<br />
= <br />
<br />
a. <br />
b. <br />
c. <br />
Solution<br />
<br />
<br />
a. <br />
ThCThC<br />
<br />
C <br />
C =<br />
<br />
=<br />
<br />
<br />
b. <br />
ThCTh<br />
C<br />
ThCċC<br />
<br />
CC <br />
C =ċ <br />
=<br />
<br />
<br />
c. ft <br />
<br />
<br />
ThC<br />
C<br />
<br />
<br />
C <br />
C =<br />
<br />
<br />
<br />
<br />
ThCċC<br />
<br />
CC <br />
C =ċ <br />
=
SECTION 9.7 PROBABILITY 823<br />
<br />
o fi<br />
<br />
<br />
<br />
<br />
+<br />
<br />
= <br />
<br />
o fi<br />
<br />
− <br />
= <br />
<br />
<br />
Try It #6<br />
<br />
<br />
a. <br />
b. <br />
c. <br />
<br />
Introduction to Probability (http://openstaxcollege.org/l/introprob)<br />
Determining Probability (http://openstaxcollege.org/l/determineprob)
824 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
9.7 SECTION EXERCISES<br />
VERBAL<br />
1. 2. <br />
<br />
<br />
3. 4. e diff<br />
<br />
<br />
5. union of two sets <br />
<br />
union<br />
of two events<br />
diff<br />
NUMERIC<br />
Figure 3o fi<br />
6. <br />
<br />
<br />
7. <br />
<br />
<br />
8. <br />
9. <br />
<br />
<br />
<br />
<br />
10. <br />
11. <br />
12. <br />
Figure 3<br />
13. <br />
<br />
14. 15. <br />
16. 17. <br />
<br />
18. 19. <br />
20. 21. <br />
<br />
22. 23. <br />
24. <br />
<br />
25. <br />
<br />
<br />
<br />
26. 27. 28. 29. <br />
30. 31. 32. <br />
<br />
33. <br />
<br />
34.
SECTION 9.7 SECTION EXERCISES 825<br />
35. <br />
<br />
37. <br />
<br />
39. <br />
<br />
36. <br />
n 9.<br />
38. <br />
40. <br />
<br />
41. 42. <br />
<br />
<br />
<br />
43. 44. <br />
45. 46. <br />
<br />
<br />
47. 48. <br />
49. 50. <br />
EXTENSIONS<br />
<br />
ft<br />
<br />
<br />
51. <br />
<br />
52. <br />
<br />
53. <br />
5 <br />
54. <br />
55. <br />
3 w<br />
<br />
REAL-WORLD APPLICATIONS<br />
<br />
<br />
56. <br />
<br />
<br />
58. <br />
<br />
<br />
57. et fi<br />
<br />
<br />
59. et fi<br />
<br />
<br />
60. e in fi
826 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
CHAPTER 9 REVIEW<br />
Key Terms<br />
Addition Principle mn<br />
m +n<br />
annuity <br />
arithmetic sequence e diff<br />
arithmetic series <br />
binomial coefficient rn<br />
Cnr( _<br />
n r )<br />
binomial expansion x+y n <br />
Binomial Theorem <br />
combination <br />
common difference e diff<br />
common ratio <br />
complement of an event E<br />
diverge <br />
event <br />
experiment <br />
explicit formula <br />
finite sequence n<br />
n<br />
Fundamental Counting Principle mnft <br />
m ×n<br />
geometric sequence <br />
geometric series <br />
index of summation <br />
<br />
infinite sequence <br />
infinite series <br />
lower limit of summation <br />
Multiplication Principle mnft <br />
m ×n<br />
mutually exclusive events <br />
n factorial n<br />
nth partial sum n<br />
nth term of a sequence <br />
outcomes <br />
permutation <br />
probability <br />
probability model
CHAPTER 9 REVIEW 827<br />
recursive formula <br />
sample space <br />
sequence <br />
series <br />
summation notation <br />
<br />
term <br />
union of two events <br />
upper limit of summation <br />
Key Equations<br />
<br />
<br />
<br />
n<br />
n<br />
n<br />
n<br />
e fin<br />
e fin<br />
fi−
828 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Key Concepts<br />
9.1 Sequences and Their Notations<br />
<br />
Example 1Example 2<br />
Example 3<br />
nExample 4<br />
<br />
<br />
Example 5Example 6<br />
<br />
ThnnExample 7<br />
9.2 Arithmetic Sequences<br />
e diff<br />
Thn diff<br />
Thdiff<br />
Example 1<br />
Thdiff<br />
Example 2Example 3<br />
diffda n<br />
=a n−<br />
+dn ≥<br />
Example 4<br />
<br />
diffda n<br />
=a <br />
+dn−Example 5<br />
Example 6<br />
a n<br />
=a <br />
+dnExample 7<br />
9.3 Geometric Sequences<br />
<br />
Th<br />
ThExample 1<br />
Th <br />
Example 2Example 4<br />
ra n<br />
=ra n−<br />
n ≥<br />
Example 3<br />
ra n<br />
=a <br />
r n − Example 5<br />
a n<br />
=a <br />
r n Example 6<br />
9.4 Series and Their Notations<br />
Th<br />
<br />
Example 1<br />
Th<br />
Th nExample 2Example 3<br />
Th<br />
Th nExample 4Example 5<br />
Th −
CHAPTER 9 REVIEW 829<br />
Example 6Example 7Example 8<br />
ts. Th<br />
Example 9<br />
9.5 Counting Principles<br />
mn <br />
m +nExample 1<br />
mnft <br />
m ×nExample 2<br />
n<br />
nrPn, r<br />
Pn, rExample 3<br />
Example 4<br />
<br />
nrC n, r<br />
Example 5.<br />
n n Example 6<br />
<br />
Example 7<br />
9.6 Binomial Theorem<br />
r ) C n, rExample 1<br />
Thl ThExample 2<br />
Example 3<br />
( n _<br />
9.7 Probability<br />
<br />
<br />
ThExample 1<br />
<br />
<br />
Example 2<br />
<br />
Example 3<br />
<br />
Example 4<br />
The diff<br />
Example 5<br />
<br />
Example 6
830 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
CHAPTER 9 REVIEW EXERCISES<br />
SEQUENCES AND THEIR NOTATION<br />
1. <br />
a <br />
=a n<br />
=a n− <br />
+n<br />
3. <br />
a n<br />
= n +<br />
<br />
2. <br />
− <br />
<br />
4. <br />
n<br />
a n<br />
=<br />
<br />
nn+ <br />
ARITHMETIC SEQUENCES<br />
5. _ _ _ _ <br />
<br />
n diff<br />
7. a <br />
=<br />
n diffd=− <br />
fi<br />
9. <br />
−−<br />
11. <br />
_ _ _ _ <br />
<br />
6. <br />
n diff<br />
8. a <br />
=<br />
a <br />
=− <br />
10. <br />
−_ −−_ <br />
<br />
12. <br />
<br />
GEOMETRIC SEQUENCES<br />
13. <br />
<br />
15. a <br />
=<br />
a <br />
= t fi<br />
17. t fi<br />
a <br />
=a n<br />
=ċa n−<br />
<br />
19. <br />
−_ −_<br />
− _ <br />
− _ <br />
<br />
14. <br />
<br />
16. a <br />
=−<br />
r=_ <br />
<br />
18. <br />
_ _ _<br />
<br />
20. <br />
−−_ −_ − <br />
<br />
<br />
SERIES AND THEIR NOTATION<br />
21. <br />
_ m+m=m=<br />
<br />
23. n<br />
<br />
<br />
22. <br />
<br />
24. <br />
n diff. Th
CHAPTER 9 REVIEW 831<br />
25. n 26. Th <br />
S <br />
<br />
Table 1<br />
<br />
<br />
Year Membership Fees<br />
<br />
<br />
<br />
<br />
<br />
<br />
Table 1<br />
27. <br />
∞<br />
∑<br />
k−<br />
ċ ( − <br />
) k= <br />
29. <br />
<br />
<br />
ft<br />
28. _ <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
30. Th<br />
<br />
<br />
<br />
<br />
<br />
<br />
COUNTING PRINCIPLES<br />
31. <br />
−−<br />
<br />
33. <br />
<br />
32. <br />
<br />
<br />
34. <br />
<br />
<br />
<br />
35. P 36. <br />
<br />
<br />
37. C 38. ff<br />
<br />
<br />
ff<br />
39. <br />
<br />
41. <br />
<br />
40. <br />
<br />
<br />
<br />
<br />
<br />
42.
832 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
BINOMIAL THEOREM<br />
43. ( _ <br />
) <br />
45. l Th <br />
a+b <br />
44. l Th( x+_ y ) <br />
46. a −b <br />
<br />
PROBABILITY<br />
<br />
47. 48. <br />
49. 50. <br />
<br />
51. <br />
<br />
52. <br />
53. <br />
<br />
<br />
<br />
<br />
54. <br />
<br />
56. <br />
<br />
55. <br />
<br />
57.
CHAPTER 9 PRACTICE TEST<br />
833<br />
CHAPTER 9 PRACTICE TEST<br />
1. <br />
a=−a n<br />
= +an <br />
<br />
3. <br />
n diff<br />
5. <br />
−− _ −− _ <br />
<br />
<br />
7. −−− _ − _ <br />
<br />
<br />
9. <br />
− _ <br />
_ − _ <br />
<br />
11. <br />
k − <br />
kk=−k=<br />
13. <br />
<br />
∑ −ċ− <br />
k− <br />
k=<br />
15. <br />
. Th<br />
<br />
<br />
<br />
<br />
17. <br />
m 5 diffs, 3 diff<br />
<br />
<br />
<br />
19. <br />
<br />
<br />
21. <br />
<br />
<br />
2. <br />
a n<br />
= n −n− n <br />
<br />
4. a <br />
=−<br />
diffd=−<br />
_ <br />
<br />
6. <br />
<br />
8. <br />
−−−−<br />
10. <br />
−_ _ −<br />
_ <br />
12. <br />
<br />
n. Th<br />
<br />
14. <br />
∞<br />
∑<br />
k=<br />
<br />
_ ċ ( − _<br />
) k−<br />
16. <br />
<br />
<br />
<br />
<br />
18. <br />
<br />
<br />
<br />
20. <br />
<br />
<br />
22. l Th( _ x−_ y ) <br />
23. ( x − <br />
)
834 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY<br />
Figure 1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Figure 1<br />
24. <br />
<br />
e t <br />
25. <br />
<br />
26. 27. <br />
<br />
28. <br />
<br />
29. <br />
y fl
Try It Answers<br />
Chapter 1<br />
Section 1.1<br />
1. a. b. __ c.− <br />
__ <br />
2. a. b. _ <br />
c.− 3. a.<br />
b. c. d.<br />
e. 4. a.<br />
b. c. d.<br />
e.<br />
5. <br />
a. − <br />
N W I Q Q' 6. a. b. c.<br />
d. e.<br />
b. × × ×<br />
c. √ — × × × ×<br />
d. √ — ×<br />
e. ×<br />
7. a.<br />
<br />
b. <br />
c. d. <br />
<br />
<br />
e.<br />
<br />
8. Constants Variables 9. a. b. c.<br />
d.<br />
a. πrr + h π r, h<br />
10. a. b. c. <br />
b. L + W L, W<br />
π<br />
d. e.<br />
c. y +y 4 y 11. <br />
12. a.−y−z−y + zb. −c. pq−p+q d. r−s+<br />
t<br />
13. A=P+rt<br />
Section 1.2<br />
1. a.k b.( y ) c.t 2. a.s b.− c.ef <br />
3. a.y) b.t c.−g 4. a. b. c. d.<br />
<br />
<br />
5. a. <br />
−t b. <br />
f c. <br />
k 6. a.t − = <br />
t b. <br />
<br />
<br />
7. a.g h b. t c.−y d.<br />
<br />
a b e. r <br />
s <br />
8. a. b <br />
c b. u c.− <br />
w d.q <br />
p e. <br />
c d <br />
9. a. v <br />
u b. <br />
x c. e _<br />
<br />
f<br />
_<br />
s<br />
d.r<br />
e. f. h <br />
10. a.× <br />
b.× c.× d. × − e. × − <br />
11. a. b.− c. −<br />
d. 12. a.−× b.× − c. × <br />
d. −× e. ≈ × 13. × <br />
× × <br />
Section 1.3<br />
1. a. b. c. d. 2. ∣x ∣ ∣ y ∣ √ — yz <br />
xy? Th<br />
3. ∣ x ∣ 4. x√— <br />
y <br />
y <br />
5. b √ — ab 6. √ — 7. 8. √ — 9. −√ — <br />
10. a.− b. c. √ 11. ( √ — ) = = 12. x y 13. x <br />
Section 1.4<br />
1. Th−x <br />
− 2. x +x −x− 3. −x −x +x − <br />
4. x −x −x + x+ 5. x + x−<br />
6. x − x+ 7. x − 8. x + xy−x −y+<br />
Section 1.5<br />
1. b −ax+ 2. x−x− 3. a.x+x+<br />
b. x−x+ 4. x− 5. y+y−<br />
6. a+ba −ab+b 7. x−x +x+<br />
8. a− − a−<br />
Section 1.6<br />
<br />
1. <br />
x+ 2. x+x+ <br />
<br />
x+x+<br />
3. <br />
x−<br />
4. <br />
x+x− <br />
5. x −y xy <br />
Chapter 2<br />
Section 2.1<br />
2. x 3. √ — =√ — <br />
y<br />
– –<br />
<br />
<br />
<br />
<br />
<br />
– – – –<br />
–<br />
–<br />
y<br />
<br />
x<br />
4. ( − <br />
) <br />
<br />
1. <br />
x y = 1__<br />
y<br />
x + 2<br />
2 (x, y)<br />
− y = − +=<br />
−<br />
<br />
− y = − += ( − <br />
<br />
)<br />
<br />
<br />
y = +=<br />
<br />
− <br />
y = += ( – – – – – <br />
–<br />
)<br />
–<br />
y = += <br />
x<br />
A-1
A-2<br />
TRY IT ANSWERS<br />
Section 2.2<br />
1. x=− 2. x=− 3. x= 4. x=<br />
<br />
5. x = − <br />
− <br />
− <br />
<br />
6. x = <br />
7. m = − 8. y =x − 9. x + y=<br />
<br />
10. y=<br />
11. y<br />
<br />
<br />
Section 2.3<br />
12. y =x + <br />
1. 2. C =x + 3. <br />
4. L=W= 5. 250 ft <br />
Section 2.4<br />
1. √ — −=+i √ — 2.<br />
3. −i−+i=−i<br />
4. <br />
−i<br />
5. +i<br />
6. −−i 7. −<br />
Section 2.5<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
–<br />
–<br />
–<br />
i<br />
<br />
1. x−x+=x=x=− 2. x−x+=<br />
x=x=− 3. x+x−=x=−x=<br />
4. x+x+=x=− <br />
x=− <br />
<br />
5. x=x=−<br />
x=− 6. x=±√ — 7. x=±√ — 8. x=− <br />
<br />
x= 9. <br />
<br />
Section 2.6<br />
1. <br />
2. 3. − 4. x=x= __ x=− <br />
<br />
5. x=− 6. x=−<br />
<br />
− 7. x=−x= 8. x=−−ii<br />
<br />
9. x=x= 10. x=−<br />
Section 2.7<br />
1. − 2. −∞−∪∞ 3. x< 4. x≥ −<br />
5. ∞ 6. [ − <br />
8. ( − <br />
<br />
∞ ) 7. < x ≤<br />
) 9. ∣ x−∣≤<br />
y<br />
10. k≤k≥ <br />
<br />
<br />
<br />
<br />
−∞∪∞<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – –<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
– – – – – –<br />
–<br />
<br />
<br />
<br />
x<br />
x<br />
r<br />
Chapter 3<br />
Section 3.1<br />
1. a. b. <br />
<br />
2. w =f d 3. 4. g= 5. m=<br />
6. y=f x= ___ √ <br />
<br />
— x 7. g= 8. x=x=<br />
<br />
9. a.<br />
b.<br />
c.<br />
10. a. <br />
b. . Th<br />
e 100 diff<br />
fi<br />
11. <br />
12. <br />
Section 3.2<br />
1. − 2. −∞∞ 3. ( −∞ __<br />
<br />
) ∪ ( __<br />
∞ ) <br />
5. a.–<br />
4. [ − <br />
__ ∞ ) <br />
–<br />
b. x |x≤––≤x< c. –∞–∪–<br />
6. = =<br />
7. −∞−∞<br />
8.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
Section 3.3<br />
<br />
1. ___________<br />
− <br />
= _____ <br />
= 2. __<br />
<br />
3. a + 4. Th−<br />
−80). Th<br />
−∞−∪∞−<br />
− <br />
<br />
– –<br />
– –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f x<br />
Section 3.4<br />
<br />
<br />
<br />
<br />
<br />
−<br />
1. a. fgx=fxgx=x−x −= x −x −x+<br />
f−gx=fx−gx=x−−x −=x−x <br />
b.<br />
x<br />
x
TRY IT ANSWERS A-3<br />
2. aGr<br />
r<br />
GaF<br />
3. fg =f=g f=g = 4. g f=g =<br />
5. a. b. 6. −∪∞<br />
7. gx=√ — <br />
+ x hx=____<br />
<br />
− x f= h ∘ g<br />
Section 3.5<br />
1. bt=ht+=−t +t+<br />
2.<br />
y<br />
3.<br />
5. a.<br />
– – – –<br />
– – – – –<br />
– – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
hx<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
gx<br />
fx<br />
<br />
<br />
<br />
<br />
x<br />
hxx – + <br />
<br />
<br />
<br />
<br />
x<br />
<br />
x<br />
Thf x<br />
g x. Th<br />
<br />
ft. Thft<br />
<br />
<br />
b.<br />
<br />
4. gx=<br />
____ <br />
x– +<br />
– – –<br />
y<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
6. a.gx=−f x b. hx=f −x<br />
x − x − <br />
g(x) − − − − h(x) <br />
7.<br />
– – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
y fx = x <br />
hx = f− x= − x <br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
gx = −f x = −x <br />
<br />
<br />
hx=f−x<br />
fx<br />
8. <br />
9. x <br />
g(x) <br />
10. g(x) = x−<br />
11. gx=f ( __<br />
x ) <br />
gx= √ <br />
x <br />
Section 3.6<br />
1. p| p−|≤<br />
2. f x= −| x+|+ 3. x= −x = <br />
Section 3.7<br />
1. h= 2. 3. 4. Th<br />
f − −∞−f − ∞<br />
5. a.f=<br />
b. f − =<br />
<br />
<br />
x<br />
<br />
6. a. b. 7. x=y+ 8. f − x=−x <br />
f∞f − −∞<br />
9. y<br />
y=x<br />
<br />
<br />
<br />
<br />
<br />
fx <br />
<br />
<br />
<br />
– – – –<br />
–<br />
–<br />
Chapter 4<br />
Section 4.1<br />
f – x<br />
<br />
1. m= − − = <br />
− =− __<br />
m<<br />
2. m= − <br />
<br />
− = <br />
= <br />
3. y=−x+ 4. H(x)=x+<br />
5. <br />
7.<br />
– – – –<br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y = x<br />
y<br />
y = x + <br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
y = x<br />
6. <br />
−−−<br />
8. <br />
9. a. f x=x<br />
b. gx=−<br />
__ x<br />
10. y=−<br />
__ x+<br />
Section 4.2<br />
1. Cx=x+25,000; Thy<br />
<br />
2. a. b. 3. <br />
Section 4.3<br />
1. 2. <br />
Chapter 5<br />
Section 5.1<br />
1. Th−<br />
hx= −<br />
__ <br />
x+ +h−<br />
h−≈<br />
2. gx=x −x+gx=x− +
A-4<br />
TRY IT ANSWERS<br />
3. Ths. Thfx≥ <br />
[ <br />
∞ ) 4. yx<br />
5. <br />
<br />
Section 5.2<br />
1. fxfx=x <br />
Th<br />
2. x fx<br />
x→±∞fx→−∞<br />
3. Ths 6. Thm i−x <br />
Th − 4. x→∞fx→−∞<br />
x→−∞fx→−∞<br />
5. Th<br />
x x<br />
fxx<br />
fx<br />
6. yx−<br />
7. Thx<br />
8. Th<br />
x<br />
<br />
9. x<br />
−y<br />
<br />
Section 5.3<br />
1. yx−<br />
2. Th−−<br />
<br />
3.<br />
<br />
– – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
4. f<br />
f<br />
f<br />
<br />
x=x=<br />
x<br />
5. fx=−<br />
<br />
x− x+ x− 6. Th<br />
−<br />
<br />
Section 5.4<br />
<br />
1. x −x+− <br />
x+ 2.<br />
3. x −x+<br />
Section 5.5<br />
x −x +x−+<br />
<br />
x+ <br />
1. f −=− 2. −−<br />
3. 4. − <br />
<br />
5. f x=−<br />
<br />
x +<br />
<br />
x −x+ 6. <br />
<br />
<br />
7. <br />
Section 5.6<br />
1. x→±∞f x→x→<br />
f x→∞e nxy<br />
2.<br />
y<br />
Th<br />
hift<br />
<br />
<br />
<br />
<br />
x→fx→∞<br />
<br />
<br />
x→±∞fx→−<br />
3. __<br />
<br />
– – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
y = –<br />
x = <br />
x<br />
Th<br />
<br />
fx=______<br />
<br />
x− −<br />
4. x=x=<br />
5. x=x=<br />
x= 6. x=x=–<br />
y= 7. <br />
<br />
<br />
f x=______<br />
<br />
x− −<br />
<br />
8.<br />
= <br />
___________<br />
−x− x− <br />
_______________<br />
= −x −x+ <br />
x−x− <br />
______________<br />
= −x +x− <br />
x −x+ <br />
–– – –<br />
Section 5.7<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x →±∞f x→−<br />
y =–<br />
<br />
<br />
x =x →<br />
f x→∞<br />
x<br />
<br />
<br />
y ( − ) <br />
y=_<br />
<br />
x=<br />
x=y( _<br />
) <br />
x–<br />
x –<br />
<br />
x<br />
<br />
<br />
<br />
1. f − fx=f ( − x+ _____<br />
) = ( x+ _____<br />
) −=x−+=x<br />
ff − x=fx−=__________<br />
x−+ <br />
=x __<br />
=x<br />
2. f − x= x − 3. f − x= √ — x −<br />
4. f − x=______<br />
x − <br />
x 5. f− x=______<br />
x+ <br />
x− <br />
Section 5.8<br />
1. <br />
____ <br />
2. __<br />
3. x=
TRY IT ANSWERS A-5<br />
Chapter 6<br />
Section 6.1<br />
1. gx= x jx= −x <br />
2. 3. <br />
<br />
4. Nt= t<br />
5. f x= x 6. f x=√ — √ — x <br />
x <br />
7. y≈· x 8. 9. <br />
10. e − ≈ 11. 12. <br />
× − <br />
<br />
<br />
<br />
Section 6.2<br />
1.<br />
2.<br />
fx<br />
fx = 4 x<br />
<br />
<br />
<br />
<br />
– <br />
– – – – – –<br />
<br />
–<br />
<br />
<br />
<br />
– <br />
<br />
fx<br />
– – – – – –<br />
–<br />
<br />
Th−∞∞<br />
∞<br />
y=<br />
x<br />
f(x) = 2 x – + 3<br />
<br />
Th−∞∞<br />
∞<br />
y=<br />
<br />
y = 3<br />
<br />
x<br />
3. x≈−<br />
4. <br />
( _<br />
<br />
) =− 5. = 6. ≈<br />
7. The diff 8. <br />
<br />
<br />
Section 6.4<br />
1. ∞ 2. (∞)<br />
3.<br />
fx<br />
4.<br />
5.<br />
– –<br />
<br />
<br />
<br />
<br />
–––<br />
– – –<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
f (x) = (x) <br />
<br />
x = 0<br />
x= − x= <br />
<br />
<br />
f (x) = (x + )<br />
<br />
y= x<br />
x<br />
– – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
−<br />
−<br />
<br />
<br />
y= x<br />
<br />
<br />
x<br />
x<br />
Th∞<br />
−∞∞<br />
x=<br />
Th−∞<br />
−∞∞<br />
x=–<br />
y fx= x+ Th∞<br />
x= <br />
−∞∞<br />
x=<br />
4.<br />
<br />
<br />
<br />
<br />
– <br />
fx<br />
– – – – – –<br />
–<br />
fx= ( x<br />
<br />
y = 0<br />
<br />
<br />
Th−∞∞<br />
∞<br />
y=<br />
x<br />
6.<br />
x= <br />
y<br />
<br />
<br />
y= x<br />
Th<br />
∞<br />
<br />
fx= x<br />
−∞∞<br />
<br />
x <br />
<br />
x=<br />
5.<br />
g(x) = 1.25 –x <br />
<br />
<br />
<br />
– <br />
––– – – –<br />
gx<br />
<br />
<br />
y = 0<br />
x<br />
<br />
<br />
Th−∞∞<br />
∞<br />
y=<br />
6. fx=−<br />
__ e x −−∞∞<br />
<br />
−∞y=<br />
Section 6.3<br />
1. a. <br />
= =<br />
b. <br />
= = 2. a. =<br />
<br />
= b. = <br />
=<br />
c. − =_ <br />
( _<br />
) =−<br />
3. <br />
=_ √ — = _<br />
=<br />
<br />
7.<br />
8.<br />
y<br />
x = 2<br />
<br />
<br />
<br />
<br />
–––<br />
– – –<br />
<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
y<br />
– – – – –<br />
<br />
–<br />
–<br />
x<br />
Th∞<br />
−∞∞<br />
x=<br />
x<br />
Th−∞<br />
−∞∞<br />
x=<br />
9. x≈ 10. x=<br />
11. fx=x+−
A-6<br />
TRY IT ANSWERS<br />
Section 6.5<br />
1. b<br />
+ b<br />
+ b<br />
+ b<br />
k= b<br />
+ b<br />
k<br />
2. <br />
x+− <br />
x−− <br />
x− 3. x<br />
4. −x 5. <br />
6. x+y−z<br />
7. <br />
__ x<br />
8. __<br />
x−+x+−x+−x−<br />
9. ( ⋅ ____<br />
⋅ ) ( __<br />
) <br />
10. ( x− √ — ___________ x<br />
x− ) <br />
11. ________<br />
x x+ <br />
x+ ( _______<br />
x x+ <br />
x+ ) <br />
12. Th 13. ____<br />
<br />
<br />
14. _____<br />
<br />
≈ _____<br />
=<br />
Section 6.6<br />
1. x= − 2. x= − 3. x=__<br />
<br />
4. 5. x=<br />
6. t = ( __<br />
<br />
) ( __<br />
<br />
) <br />
<br />
( _<br />
) <br />
<br />
7. t=<br />
( _ <br />
√ — ) =− __<br />
8. x= 9. x=e<br />
<br />
10. x=e − 11. x≈ 12. x= x=−<br />
13. t=×<br />
______ ≈<br />
Section 6.7<br />
1. f t=A <br />
e −t 2. <br />
3. f (t) = A <br />
e ( ____<br />
<br />
)t 4. 5. <br />
6. y=e x 7. y=e x <br />
Section 6.8<br />
1. a. Tht fi<br />
y= x b. <br />
ft<br />
2. a. Tht fi<br />
y=+x b. <br />
<br />
3. a. Tht fi<br />
_____________________<br />
<br />
y= <br />
−x<br />
+e<br />
b. <br />
c. <br />
<br />
Chapter 7<br />
Section 7.1<br />
1. <br />
2. Th<br />
−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
− <br />
– ––<br />
– – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
– −<br />
3. −− 4. −− 5. − 6. <br />
7. Th<br />
fixx + <br />
8. <br />
Section 7.2<br />
1. − 2. 3. fi<br />
xx−−x +<br />
Section 7.3<br />
1. ( − _<br />
<br />
4.<br />
_<br />
) 2. − 3. −−−−<br />
y<br />
<br />
+ <br />
<br />
<br />
−− <br />
<br />
– ––<br />
– – <br />
–<br />
<br />
–<br />
− <br />
–<br />
Section 7.4<br />
<br />
1. _<br />
<br />
x− − _ <br />
x− 2. _ <br />
x− − _ <br />
x− <br />
x−<br />
4. _<br />
<br />
x −x+ + __ x+ <br />
x −x+ <br />
Section 7.5<br />
1. A+B= [<br />
<br />
<br />
<br />
−<br />
2. B=[ − − − −<br />
] <br />
Section 7.6<br />
]<br />
+ [<br />
<br />
<br />
<br />
− <br />
− <br />
]<br />
= [<br />
<br />
3. _ <br />
x− +x− _<br />
<br />
x + <br />
] = [<br />
<br />
+ +−<br />
+<br />
<br />
+<br />
+− −+<br />
1. [ − ∣ ] 2. x−y+ z= 3. <br />
x−y+z=<br />
y+z=−<br />
4. [ − <br />
<br />
∣<br />
<br />
5. <br />
]<br />
<br />
6. <br />
Section 7.7<br />
1. AB=[ <br />
<br />
<br />
<br />
− −<br />
] [ − − ] = [ <br />
BA=[ − − <br />
<br />
2. A =[ − <br />
<br />
<br />
<br />
− <br />
<br />
<br />
<br />
<br />
<br />
− 0 ]<br />
<br />
−+ −+<br />
<br />
<br />
−1−+− −−+−] <br />
= [ <br />
<br />
] <br />
] [ <br />
− −<br />
] = [ −+−− <br />
−+−− <br />
= [ <br />
<br />
] <br />
]<br />
3. A− = [<br />
<br />
<br />
+− <br />
<br />
− <br />
−<br />
+− ] <br />
]<br />
4. X= [<br />
<br />
<br />
]
TRY IT ANSWERS A-7<br />
Section 7.8<br />
1. − 2. − 3. ( −_ _ <br />
) <br />
Chapter 8<br />
Section 8.1<br />
1. x +_<br />
y <br />
= 2. x− _<br />
<br />
+y− _<br />
<br />
=<br />
y<br />
3. <br />
<br />
±<br />
<br />
<br />
± −<br />
<br />
±√ — <br />
– ––<br />
– – <br />
–<br />
–<br />
– −<br />
–<br />
4. _<br />
x <br />
+y _<br />
<br />
=<br />
<br />
±<br />
±<br />
±√ — <br />
5. <br />
−<br />
( −√ — )<br />
( +√ — )<br />
<br />
6. x− <br />
+y+ <br />
=−−<br />
−−( −−√ — )<br />
( −+√ — x<br />
) 7. a. <br />
+ y <br />
= b. Th<br />
<br />
<br />
Section 8.2<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
− <br />
– ––<br />
– – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
– −<br />
–<br />
−<br />
<br />
+ <br />
<br />
<br />
−−<br />
<br />
<br />
– ––<br />
– – <br />
–<br />
<br />
–<br />
− <br />
–<br />
1. ±( ±√ — ) 2. y −x <br />
=<br />
3. y− <br />
+x− <br />
=<br />
4. ±<br />
±<br />
±<br />
y<br />
<br />
y=± _ x <br />
−<br />
−<br />
y<br />
−<br />
<br />
<br />
x<br />
5. −−<br />
−−−( −−√ — ) <br />
( −+√ — )y=± x −−<br />
<br />
<br />
−−<br />
−<br />
y<br />
−<br />
−<br />
6. Th<br />
_<br />
x <br />
− y _<br />
<br />
= _ x <br />
y<br />
− _<br />
<br />
=<br />
Section 8.3<br />
1. −<br />
x= <br />
<br />
−±<br />
2. <br />
y= −<br />
<br />
±<br />
3. x =y<br />
4. −<br />
y= −<br />
−<br />
x=<br />
<br />
−<br />
5. −<br />
x= −<br />
−−<br />
−<br />
y=<br />
<br />
−−<br />
−−<br />
−−<br />
−<br />
x=−<br />
−<br />
6. a. y =x b. Th<br />
Section 8.4<br />
1. a. b. 2. x' _<br />
<br />
+y' _<br />
<br />
=<br />
3. a. b. <br />
x<br />
y =−x<br />
−<br />
y<br />
−<br />
−−<br />
<br />
x=<br />
y y + =−x −<br />
y=−<br />
<br />
y y + =−x −<br />
y=−<br />
−<br />
x=<br />
y<br />
<br />
−<br />
−<br />
x<br />
y=<br />
x<br />
x<br />
x + =−y −<br />
x
A-8<br />
TRY IT ANSWERS<br />
Section 8.5<br />
Section 9.7<br />
b. x +x y+xy +y 3. x y <br />
1. e= <br />
x=−<br />
1.<br />
Outcome<br />
y<br />
<br />
2.<br />
3. r=<br />
<br />
<br />
<br />
−θ<br />
4. −x+ x −y = <br />
− −<br />
<br />
−<br />
−<br />
<br />
x<br />
<br />
<br />
<br />
−<br />
<br />
Chapter 9<br />
Section 9.1<br />
1. The fit fi 2. The fit fi<br />
{ −−_ −_ <br />
} 3. The fi<br />
4. a n<br />
=− n+ n 5. a n<br />
=− _ n n<br />
6. a n<br />
=e n− 7. 8. { _ _ <br />
} <br />
9. The fit fi{ _ <br />
} <br />
Section 9.2<br />
1. Th. Thff−<br />
2. Th−≠−<br />
3. 4. a <br />
= 5. a <br />
=a n<br />
=a n − 1<br />
+<br />
n≥ 6. a n<br />
=−n 7. Th<br />
8. ThT n<br />
=+n<br />
<br />
Section 9.3<br />
1. Th_ ≠_ <br />
<br />
2. Th. Th_ <br />
3. { _<br />
_<br />
} 4. a <br />
=a n<br />
=_ a n − <br />
n≥<br />
5. a <br />
= 6. a n<br />
=−− n − 7. a. P n<br />
=3 ∙ 1a n<br />
b. Th<br />
Section 9.4<br />
1. 2. 3. 4. − 5. <br />
6. ≈ 7. 8. 9. Th<br />
fi 10. Thfi<br />
fi 11. Thfifi<br />
12. 13. Th 14. −<br />
_ <br />
<br />
15. <br />
Section 9.5<br />
1. 2. Th 3. <br />
4. 5. 6. P= 7. P=<br />
8. C= 9. 10. <br />
Section 9.6<br />
1. a. b. 2. a. x −x y+x y −x y +xy −y <br />
Probability<br />
2.<br />
_ <br />
3.<br />
_ 4.<br />
<br />
5. _ 6. a. _<br />
b.<br />
<br />
_ <br />
<br />
<br />
_<br />
c. _
Odd Answers<br />
CHAPTER 1<br />
Section 1.1<br />
1. . Th<br />
<br />
3. Th<br />
<br />
d diffff <br />
<br />
5. − 7. − 9. −<br />
11. 13. − 15. 17. 19. 21. <br />
23. − 25. 27. 29. − 31. − 33. ±<br />
35. 37. 39. −y− 41. −b+<br />
43. z− 45. y+ 47. −b+ 49. _ x <br />
51. x 53. −+ <br />
55. <br />
<br />
57. g+−= 59. <br />
61. 63. 65. 67. <br />
Section 1.2<br />
1. <br />
<br />
×× ×<br />
3. <br />
5. 7. 9. <br />
11. 13. 15. <br />
<br />
17. 19. <br />
21. ×− 23. <br />
25. a 27. b c 29. ab d 31. m 33. q <br />
p <br />
35. y <br />
x 37. 39. a 41. c y<br />
<br />
b 43. <br />
z <br />
45. 47 × 49. <br />
51. m 53. a <br />
55. n <br />
a c 57. <br />
a b c <br />
59. <br />
Section 1.3<br />
1. <br />
t. Th<br />
3. Th<br />
5. 7. 9. 11. √ — <br />
13. _____ √— 15. 17. √— 19. √ — 21. √ — <br />
23. √ — 25. <br />
27. √— 29. __________ −+√— <br />
31. √ <br />
— 33. √ — 35. x 37. √ — p<br />
39. m √ — m 41. b√ — a 43. x <br />
45. y √ — <br />
47. ______ √— d<br />
d 49. +√ — ____________ x √— <br />
51. −w√ — w <br />
−x<br />
53. √— x−√ — x 55. n √ — 57. m √— m 59. <br />
d <br />
61. √ <br />
— ______ x 63. z √ <br />
— 65. 67 _________ −√— − <br />
69. ______ √— mnc a cmn 71. x+√ — ___________ √— 73. ____ √— <br />
Section 1.4<br />
1. Th<br />
<br />
Th<br />
<br />
3. <br />
5. 7. 9. 11. x +x+<br />
13. w +w+ 15. b − b +b −b+<br />
17. x −x− 19. b −b +<br />
21. v −v+ 23. n −n +n−<br />
25. y −y+ 27. p +p+<br />
29. y −y+ 31. c − 33. n −<br />
35. −m + 37. q − 39. t +t −t −t+<br />
41. y −y −y+ 43. p −p −p+ <br />
45. a −b 47. t −tu+u 49. t +x +t−tx−x<br />
51. r +rd−d 53. x −x−m <br />
55. t −t +t+ 57. a +a c−ac −c <br />
Section 1.5<br />
1. Th<br />
x <br />
−y <br />
x −y =x+yx−y 3. <br />
x<br />
<br />
5. m 7. m 9. y 11. a−a+<br />
13. n − n+ 15. p+p−<br />
17. h+h− 19. d−d−<br />
21. t+t − 23. x+x−<br />
25. p+p− 27. d+d−<br />
29. b+cb−c 31. n+ 33. y+ <br />
35. p− 37. x+x −x+<br />
39. a+a −a+ 41. x−x +x+<br />
43. r+sr −rs+s 45. c+ − −c−<br />
47. x+ x+ 49. z− − z−<br />
51. x−x+ 53. x+x−<br />
55. x+ x− 57. z +a z+az−a<br />
<br />
59. <br />
<br />
x+x−x+ <br />
Section 1.6<br />
1. <br />
<br />
3. <br />
<br />
<br />
B-1
B-2<br />
ODD ANSWERS<br />
5. y+ y+ 7. b+ 9. x+ x+ 11. a+ a− <br />
13. n− n− 15. c− c+ 17. 19. d − d − <br />
21. t+ t+ 23. x− x+ 25. p+ p+ 27. d+ d+ <br />
29. b+ <br />
b− 31. y− 33. x+y<br />
35. a−<br />
<br />
y+ xy<br />
a −a− <br />
37. y −y+ <br />
y −y− 39. z +z+<br />
41. x+xy+y<br />
<br />
z −z− x+xy+y+ <br />
43. b+a<br />
ab 45. +ab 47. a−b 49. c +c− b<br />
c +c+ <br />
51. x+ <br />
x− 53. x+ x− 55. <br />
y+ 57. <br />
Chapter 1 Review Exercises<br />
1. − 3. 5. y= 7. m 9. <br />
11. 13. 15. a 17. x <br />
y <br />
19. a<br />
21. × 23. 25. √ — 27. √— <br />
29. √— 31. √— 33. − 35. x +x +<br />
37. x −x+ 39. k −k− 41. x +x +x+<br />
43. a +ab−b 45. p 47. a <br />
49. a−a+ 51. x+ 53. h−k <br />
55. p+p −p+ 57. q−pq +pq+p <br />
59. p+ −p− 61. x+ <br />
x− 63. <br />
<br />
67. x+y<br />
69. <br />
xy<br />
<br />
Chapter 1 Practice Test<br />
1. 3. x= 5. 7. <br />
9. 11. x 13. 15. √— x 17. √— <br />
19. q −q −q 21. n −n +n−<br />
23. x+x− 25. c−c +c+<br />
27. z− z− 29. a+b <br />
b<br />
CHAPTER 2<br />
Section 2.1<br />
65. m+ <br />
m− <br />
1. <br />
xy<br />
3. y<br />
y 5. x<br />
y 7. x<br />
y− 9. x<br />
y( <br />
) 11. y=−x 13. y=−x<br />
<br />
15. y=x− <br />
17. d=√— 19. d=√ — =<br />
21. d≈ 23. ( − <br />
) 25. − 27. 29. y=<br />
31. <br />
y<br />
<br />
<br />
<br />
− <br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
35.<br />
x − <br />
y <br />
y<br />
<br />
<br />
<br />
<br />
<br />
− <br />
<br />
<br />
x<br />
– – – – – – <br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
39.<br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x<br />
<br />
<br />
−<br />
x<br />
33. −<br />
37. x − <br />
y <br />
<br />
<br />
<br />
<br />
<br />
− <br />
– – – – – –<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
41.<br />
– – – – –<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
43. d= 45. d= 47. − 49. x=y=−<br />
51. x=y= 53. x=−y=<br />
55. −= 57. 59. <br />
<br />
<br />
61. 63. 54 ft<br />
Section 2.2<br />
1. 3. Th<br />
x 5. <br />
<br />
7. x=<br />
9. x=<br />
11. x= 13. x= 15. x=−<br />
<br />
17. x≠−x=− 19. x≠x=<br />
21. x≠x=−<br />
<br />
23. y=− ___ <br />
x+ <br />
<br />
25. y=− ___ <br />
x+ <br />
27. y= <br />
x+ <br />
29. y=−x− 31. y= 33. y=− 35. x+y=<br />
37. y<br />
39. y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
x<br />
<br />
<br />
x
ODD ANSWERS B-3<br />
41. m=− <br />
43. m= <br />
45. m <br />
=− <br />
m =<br />
<br />
47. y=x−yy <br />
=−y <br />
=−<br />
49. y=−x+y y <br />
=−y <br />
=<br />
51. y=− A <br />
B x+C 53. Th−<br />
B<br />
Th−− <br />
Th<br />
3. Th− <br />
<br />
55. 30 ft 57. 59. <br />
Section 2.3<br />
1. <br />
<br />
3. −x 5. v+<br />
7. 9. +m 11. <br />
13. +P 15. 17. −x 19. <br />
21. <br />
23. 25. B=+x<br />
27. 29. R= 31. r= <br />
<br />
33. W= P−L <br />
=− <br />
=<br />
pq<br />
35. f= <br />
p+q = <br />
+ = <br />
37. m=− <br />
<br />
39. h=<br />
A 41. = =160 ft<br />
b <br />
+b <br />
43. 45. A= 47. 49. h= V <br />
πr <br />
51. r= √ V 53. C=π<br />
πh<br />
Section 2.4<br />
1. <br />
3. ii−<br />
5. −+i 7. +i 9. − <br />
+ <br />
i<br />
11.<br />
i<br />
13.<br />
i<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
r<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
15. −i 17. −+i 19. −i 21. +i<br />
23. −+i 25. −−i 27. 29. − <br />
i<br />
31. −i 33. <br />
+ <br />
i 35. i 37. +i√—<br />
<br />
39. 41. − 43. i 45. ( √— <br />
+ <br />
i ) =−<br />
47. i 49. 51. −i 53. −i 55. <br />
− <br />
i<br />
Section 2.5<br />
1. <br />
3. <br />
a ∙ b=<br />
ff<br />
a=b=<br />
r<br />
5. xx =<br />
ax+b =d<br />
7. x=x= 9. x=− <br />
x=− 11. x=x=−<br />
<br />
13. x=− <br />
x= <br />
15. x=− 17. x=x=− <br />
<br />
19. x=−x= 21. x=x=− 23. x=x=−<br />
25. x=−x= 27. x=±√ — 29. z= <br />
z=− <br />
<br />
31. x= ±√— <br />
33. 35. <br />
<br />
37. 39. x= −±√— <br />
41. x= ±√— 43. x=−±√— <br />
45. x≈x≈ 47. x≈−x≈<br />
49. ax +bx+c=<br />
<br />
x + b <br />
a x=−c <br />
a <br />
x + a b x+b <br />
<br />
a =−c <br />
a + b <br />
a <br />
<br />
<br />
( x+ b <br />
a ) = b −ac<br />
a <br />
<br />
=±√ <br />
x+ b <br />
a b−ac<br />
a <br />
<br />
<br />
x= −b±√— b −ac <br />
<br />
a<br />
51. xx+= 53. x=<br />
55. Th<br />
x−x −x+=300. Thx<br />
57. <br />
Section 2.6<br />
1. <br />
<br />
<br />
3. <br />
−<br />
. Th<br />
<br />
− 5. <br />
<br />
<br />
7. x= 9. x= 11. x=x= <br />
13. x=−− 15. y= <br />
− <br />
<br />
17. m=−<br />
19. x= <br />
± i 21. x= 23. t= 25. x=<br />
<br />
27. x=− 29. x=− <br />
31. x=− <br />
<br />
<br />
33. x=− 35. x=− 37. x=−−<br />
39. x=− 41. x= 43. <br />
45. x=−− 47. 49. <br />
Section 2.7<br />
1. <br />
<br />
3. −∞∞
B-4<br />
ODD ANSWERS<br />
5. x=<br />
<br />
y = x−e left<br />
y<br />
7. ( −∞ __ <br />
] 9. [ − ____ <br />
∞ ) 11. −∞ 13. ( −∞− ___ <br />
] <br />
15. −∞∞ 17. ( −∞− <br />
) ∪∞<br />
19. −∞−∪+∞ 21. 23. −<br />
25. 27. −<br />
29. x>−x>− <br />
x>−−+∞<br />
31. x
ODD ANSWERS B-5<br />
Chapter 2 Practice Test<br />
1. y= <br />
x+<br />
x <br />
y <br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
3. −<br />
– – – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
−<br />
5. −∞ 7. x=− 9. x≠−x=− <br />
<br />
11. x= ±√— 13. − 15. y=− <br />
x− <br />
<br />
17. y= <br />
x− 19. i 21. <br />
− <br />
i 23. x=− <br />
<br />
25. x= <br />
±√— 27. 29. x= <br />
−<br />
CHAPTER 3<br />
Section 3.1<br />
1. <br />
<br />
3. <br />
<br />
<br />
5. <br />
<br />
<br />
<br />
7. 9. 11. 13. <br />
15. 17. 19. <br />
21. 23. 25. <br />
27. f−=−f=−f−a=−a−−fa=−a+<br />
fa+h=a+h− 29. f−=√ — +f=<br />
f−a=√ — +a +−fa=−√ — −a −fa+h=<br />
√ — −a−h + 31. f−=f=−<br />
f−a=∣ −a−∣−∣ −a+∣−fa=−∣ a−∣+∣ a+∣<br />
fa+h=∣ a+h−∣−∣ a+h+∣<br />
33. gx−ga _<br />
x−a<br />
=x + a+ x ≠ a 35. a. f −= b. x=<br />
37. a. f = b. x=− 39. a. r=−<br />
__ t<br />
b. f−= c. t= 41. 43. <br />
45. 47. 49. <br />
51. 53. a. f = b. f x=−x=−<br />
55. 57. <br />
59. 61. 63. <br />
65. 67. f x=x=<br />
69. f−=f−=f=f=f=<br />
71. f−=f−=f=f=f=<br />
x<br />
73. f−=<br />
__ f−= __<br />
f=f=f= 75. <br />
77. Th 79. Th<br />
<br />
−<br />
y<br />
y<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
– <br />
–<br />
–<br />
–<br />
–<br />
81. Th<br />
−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
– –<br />
– <br />
– <br />
– <br />
– <br />
– <br />
<br />
85. Th<br />
−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
x<br />
x<br />
x<br />
<br />
<br />
<br />
<br />
<br />
–<br />
<br />
–<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
83. Th<br />
0, 10<br />
y<br />
<br />
<br />
<br />
<br />
<br />
–<br />
x<br />
<br />
87. Th<br />
−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – <br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
89. a. g=b. Th<br />
f 100 s 91. a. <br />
fts 200 ft.<br />
b. Thfts 350 ft.<br />
Section 3.2<br />
1. Th<br />
<br />
3. Thxfx= √ — x<br />
<br />
−∞∞<br />
x<br />
fx= √ — x <br />
∞ 5. <br />
<br />
xy<br />
<br />
−∞∞ <br />
7. −∞∞ 9. −∞<br />
11. −∞∞ 13. −∞∞ 15. ( −∞−<br />
<br />
) ∪ ( −<br />
<br />
∞ ) <br />
17. −∞−∪−∪∞ 19. −∞−∪−∪∞<br />
x<br />
x
B-6<br />
ODD ANSWERS<br />
21. −∞ 23. ∞ 25. −∞−∪−∪∞<br />
27. 29. −<br />
31. − 33. −∞∞<br />
35. [ −−<br />
<br />
] ∪[ <br />
] [ −−<br />
<br />
] ∪[ <br />
] <br />
37. −∞∞<br />
39. −∞∞ 41. −∞∞<br />
y<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
43. −∞∞<br />
y<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
45. −∞∞<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
47. f−=f−=f−=f=<br />
49. f−=−f=f=f=<br />
51. f −=−f=f=f=<br />
53. −∞∪∞<br />
55.<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
x<br />
<br />
x<br />
<br />
Th Th<br />
<br />
<br />
<br />
57. 59. fx= <br />
√ — x−<br />
61. a. Th b. Th<br />
c. Th<br />
Section 3.3<br />
1. <br />
3. Th<br />
<br />
5. b+ 7. 9. x+h<br />
<br />
11. −<br />
_________ <br />
+h 13. h +h+ 15. x+h−<br />
17. 19. −∞−∪∞<br />
<br />
− 21. −∞∪<br />
∪∞<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
x<br />
23. −<br />
25. <br />
−−<br />
27. a.− b. −<br />
29. − 31. 33. ≈− 35. <br />
−−∞∞<br />
37. −−−−<br />
−∞ 39. −<br />
−−−−∞<br />
−−−−∞<br />
41. 43. b= 45. ≈<br />
47. ≈−<br />
Section 3.4<br />
1. g<br />
f<br />
g<br />
3. f x= x+gx= x−<br />
Thf (gx=f x−=x−+=x <br />
g ( f x=g x+=x+−=x f∘g=g∘f<br />
5. f+gx=x+−∞∞<br />
f−gx=x +x−−∞∞<br />
fgx=−x −x +x +x−∞∞<br />
( f _<br />
g ) x=x +x <br />
−x −∞− √— ∪−√ — √ — ∪√ — ∞<br />
7. f+gx= x +x + <br />
−∞∪∞<br />
x<br />
f−gx= x +x − <br />
−∞∪∞<br />
x<br />
fgx=x+−∞∪∞<br />
( f _<br />
g ) x=x +x −∞∪∞<br />
9. f+gx=x +√ — x−∞<br />
f−gx=x −√ — x−∞<br />
fgx=x √ — x−∞<br />
( f _<br />
g ) x= x ∞ 11. a.fg=<br />
√ — x−<br />
b.fgx=x −x+ c.gfx=x −<br />
d.g ∘ gx=x− e.f ∘ f−=<br />
13. fgx=√ — x ++gfx=x+√ — x +<br />
√<br />
15. fgx= — <br />
x+ √<br />
x<br />
; gfx= <br />
— x +<br />
x<br />
<br />
17. fgx=<br />
__ x x≠gfx=x−x≠<br />
<br />
<br />
19. fghx= <br />
x+ +<br />
<br />
21. a.g∘f x=−<br />
<br />
√ — −x b. ( −∞ __<br />
) <br />
23. a. ∪∞x=−b.∞c.∞ 25. ∞<br />
27. fx=x gx=x−<br />
29. fx= x ; gx=x+<br />
31. fx= √ — <br />
x ; gx= <br />
x− <br />
33. fx= √ — x ; gx= x− <br />
x+ <br />
35. fx=√ — x ; gx=x+
ODD ANSWERS B-7<br />
37. fx= √ — x gx=x−<br />
<br />
39. fx=x gx= <br />
x− <br />
41. fx=√ — x ; gx= x− x+ <br />
43. 45. 47. 49. 51. 53. <br />
55. 57. 59. 61. 63. 65. <br />
67. 69. 71. 73. fg=gf=−<br />
75. fg= <br />
gf= 77. fgx=x +x+<br />
79. g∘gx=x+ 81. f∘gx=g∘f x=<br />
83. −∞∞ 85. 87. f∘g =g∘f =<br />
89. f∘g =g∘f = 91. <br />
93. At=π ( √ — t+) A=π ( √ — ) =π<br />
95. A=π<br />
97. a.NTt=t +t− b. ≈<br />
Section 3.5<br />
1. <br />
<br />
3. <br />
<br />
<br />
5. f<br />
−xxfx<br />
f−x= fx<br />
<br />
f−x= −fx<br />
<br />
7. gx= ∣ x−∣−<br />
<br />
9. gx=<br />
<br />
x++ 11. Thfx+<br />
f<br />
13. Thfx− <br />
f 15. Thfx+<br />
f 17. Thfx−<br />
f 19. Th<br />
fx+− hift<br />
f 21. −∞−<br />
−∞ 23. ∞<br />
25. y 27.<br />
y<br />
– – – – – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– <br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
h<br />
<br />
x<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
f<br />
<br />
x<br />
29. y 31. gx= fx−<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
k<br />
<br />
<br />
<br />
hx= fx+<br />
33. fx= ∣ x−∣−<br />
35. fx= √ — x+−<br />
37. fx= x− <br />
39. fx= ∣ x+∣−<br />
x 41. fx= − √ — x<br />
43. fx= −x+ +<br />
45. fx= √ — −x +<br />
47. 49. <br />
51. 53. Th<br />
g <br />
x<br />
f 55. Thg<br />
<br />
f<br />
57. Thg <br />
<br />
f 59. Thg<br />
f 61. Thg<br />
y<br />
f 63. gx= ∣ −x ∣<br />
<br />
65. gx=<br />
<br />
x+ − 67. gx=<br />
<br />
x− +<br />
69. hift<br />
<br />
<br />
hift<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
73. <br />
<br />
<br />
<br />
y<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
g<br />
<br />
<br />
m<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
71. <br />
<br />
hift<br />
<br />
<br />
hift<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
75. Th<br />
<br />
hift<br />
<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
h<br />
<br />
<br />
<br />
<br />
p<br />
<br />
x<br />
x
B-8<br />
ODD ANSWERS<br />
77. Th<br />
fx= √ — x hift<br />
<br />
<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
Section 3.6<br />
79.<br />
– – – – – – –<br />
81.<br />
– – – – – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
g<br />
<br />
g<br />
<br />
<br />
1. <br />
∣ A ∣ =B<br />
A<br />
B<br />
A<br />
−B 3. <br />
x<br />
<br />
x 5. Thx<br />
∣ x−∣= 7. ∣ x−∣≥<br />
9. Thx 11. −<br />
13. −− 15. −<br />
17.<br />
y 19.<br />
y<br />
21.<br />
–<br />
–<br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
23.<br />
– – – –<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
x<br />
25.<br />
– – – – –<br />
29.<br />
– – – –<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
33. −<br />
y<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
f<br />
<br />
<br />
<br />
x<br />
x<br />
35.<br />
<br />
<br />
27.<br />
– – – – –<br />
31.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
– – – – – –<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
x<br />
<br />
37. Tha<br />
yTh<br />
yx=<br />
39. ∣ p− ∣ ≤ 41. ∣ x−∣≤<br />
SECTION 3.7<br />
1. <br />
<br />
y<br />
<br />
y<br />
3. f x= x <br />
5. y=f − x<br />
7. f − x=x− 9. f − x=−x 11. f − x=−<br />
_ x<br />
x− <br />
13. f x−∞f − x=√ — x −<br />
15. f x∞f − x=√ — x+<br />
17. fgx=xgfx=x 19. <br />
21. 23. 25. 27. <br />
29. y<br />
31. 33. <br />
35. − 37. 39. <br />
<br />
41. <br />
<br />
f<br />
<br />
– x <br />
f<br />
<br />
f −1 (x) <br />
–– – –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x
ODD ANSWERS B-9<br />
43. f − x=+ x <br />
– – – –<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
f<br />
<br />
f –<br />
Chapter 3 Review Exercises<br />
<br />
<br />
<br />
45. f − x = x −<br />
<br />
47. td=_ d t= <br />
Th<br />
<br />
<br />
x<br />
<br />
1. 3. 5. f−=−f=−<br />
f−a=−a −a; −fa=a −a;<br />
fa +h=−a −ah −h +a +h<br />
7. 9. 11. <br />
13.<br />
y<br />
15. 17. −<br />
<br />
<br />
19. −+a−a <br />
−+a<br />
<br />
23.<br />
– – – –<br />
– – – –<br />
<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
=−a +a≠<br />
21. −∞−∪(−∪(∞<br />
25. <br />
27. ∞<br />
−∞<br />
29. −−∞−∞<br />
31. −−<br />
33. <br />
35. f∘g x=−x, g∘f x=−−x<br />
37. f∘g x= √ <br />
x<br />
+ g∘f x=_<br />
<br />
√ — x+ <br />
<br />
<br />
<br />
x 39. f∘g x=<br />
+ =_<br />
+x<br />
<br />
<br />
+x ( −∞− <br />
) ∪ ( − <br />
) ∪(∞)<br />
+<br />
x<br />
41. f∘g x= <br />
√ — x ∞)<br />
43. gx= x − <br />
x + <br />
fx=√ — x <br />
45.<br />
y<br />
47.<br />
y<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
49.<br />
53.<br />
– – – –<br />
– – – – –<br />
67.<br />
– – –<br />
75. <br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
y<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
–<br />
–<br />
<br />
–<br />
–<br />
–<br />
–<br />
Chapter 3 Practice Test<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
51.<br />
– – –<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
55. fx=∣ x −∣<br />
57. 59. <br />
61. <br />
63. fx= <br />
∣ x +∣+ <br />
65. fx=−∣ x−∣+<br />
69. f − x= x − <br />
71. f − x=√ — x −<br />
73. Th<br />
y<br />
<br />
<br />
<br />
<br />
<br />
– – – – –<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
1. 3. − 5. Th<br />
<br />
7. a −a 9. −a+b+b≠a 11. √ — <br />
13.<br />
15. 17. <br />
y<br />
19. f − x=<br />
x+ <br />
<br />
<br />
<br />
<br />
21. −∞−∞<br />
<br />
23. − 25. f=<br />
– – – – –<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
CHAPTER 4<br />
Section 4.1<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
27. fx= {<br />
|x| x ≤<br />
x > <br />
29. x = 31. <br />
33. f − x=− x − <br />
−x _<br />
<br />
1. <br />
3. dt = −t<br />
5. Thaa <br />
ya<br />
xa. Th<br />
<br />
x<br />
x
B-10<br />
ODD ANSWERS<br />
7. 9. 11. 13. 15. <br />
17. 19. 21. <br />
23. 25. 27. − 29.<br />
_ x − <br />
<br />
31. y = x− 33. y = −<br />
<br />
x+ 35. y = −x−<br />
<br />
37. 39. 41. yx<br />
43.<br />
<br />
y−x( )<br />
47. m=−m=−<br />
49. m=−m=<br />
45. yx−<br />
51. m=− m=−<br />
53. y =x− 55. y=−<br />
__ t+ 57. 59. y=−<br />
__ x+<br />
61. y =x− 63. y =− 65. 67. 69. <br />
71.<br />
– – – – –<br />
75.<br />
– – – – –<br />
79.<br />
– – – – –<br />
83.<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
x<br />
73.<br />
– – – – –<br />
77.<br />
– – – – –<br />
81.<br />
– – – – –<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
85. y = <br />
87. x = −<br />
89. gx = −x+<br />
91. fx = x−<br />
93. gx = −<br />
__ <br />
x+<br />
95. fx = x−<br />
97. fx = −x+<br />
x<br />
x<br />
x<br />
99.<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
– –<br />
–<br />
–<br />
–<br />
<br />
<br />
<br />
x<br />
x<br />
101. aa = b = <br />
b.qp = p−<br />
103. y 105. x = −<br />
__ 107. x = a<br />
<br />
<br />
109. y=____<br />
d<br />
x−____<br />
ad<br />
<br />
c−a c−a<br />
<br />
111. y=x−<br />
113. x< <br />
x> <br />
115. <br />
<br />
117. Th<br />
ts. Th<br />
<br />
119. Th−<br />
<br />
121. <br />
Section 4.2<br />
1. <br />
3. <br />
5. <br />
7. 9. 11. <br />
13. Pt=t + 15. −<br />
<br />
17. ft<br />
19. Wt=t+ 21. −15, 0) Th<br />
x<br />
<br />
23. <br />
25. Ct=−t 27. <br />
<br />
<br />
29. 31. y =−t +<br />
33. fi 35. y =t −<br />
37. fi<br />
39. 41. 43. <br />
45. a. b. c. d.<br />
e. Pt=+t f. 47. a. Cx=x +<br />
b. The fl<br />
c. 49. a. Pt=t +<br />
b. 51. a. Rt= −t +<br />
b. c. <br />
53. 55. <br />
57.
ODD ANSWERS B-11<br />
Section 4.3<br />
1. ft<br />
3. <br />
5. Th<br />
<br />
7. <br />
9. <br />
11. <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
13. 15. 17. <br />
<br />
19.<br />
21.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
23. <br />
25. y=x+r= 27. y=−x+<br />
r=− 29. y=−x+r=−<br />
31. y=x−r= 33. −−−<br />
−(−− 35. <br />
fi<br />
37. y=x+ 39. y=x−<br />
41. y=−x+<br />
Chapter 4 Review Exercises<br />
1. 3. 5. y=−x+ 7. <br />
9. y=x− 11. 13. 15. −−<br />
17. m−m=− 19. y=−x+<br />
21.<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
– – – – – – –<br />
<br />
–<br />
–<br />
–<br />
–<br />
–<br />
35. 37. <br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Population<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Year<br />
Chapter 4 Practice Test<br />
23. <br />
25. <br />
27. y=−x+<br />
29. a. b. <br />
c. Pt= t+<br />
x 31. 33. y=<br />
1. 3. 5. y=−x− 7. y=−x−<br />
9. 11. 13. −− <br />
15. y=−x+<br />
17. =−<br />
y=<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
– – – –<br />
<br />
–<br />
–<br />
–<br />
x<br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
39. <br />
41. y=−x+<br />
r=−<br />
43. <br />
45. <br />
19. 21. <br />
23. y=x+<br />
25. a. b.<br />
<br />
<br />
c.y=−t+<br />
27. 29. <br />
y<br />
31. y=x+<br />
<br />
<br />
33. r=<br />
<br />
<br />
<br />
<br />
<br />
<br />
x
B-12<br />
ODD ANSWERS<br />
CHAPTER 5<br />
Section 5.1<br />
1. <br />
3. a=<br />
5. <br />
7. gx=x+ − −−<br />
9. fx=( x+ <br />
) − <br />
( − <br />
− <br />
) <br />
11. kx= x− −−<br />
13. fx= ( x− <br />
) − <br />
( <br />
− <br />
) <br />
15. − <br />
<br />
x= <br />
<br />
17. − <br />
− <br />
x=− <br />
<br />
19. − <br />
−x=−<br />
21. −∞∞∞ 23. −∞∞<br />
−∞ 25. −∞∞−∞<br />
27. fx=x +x+ 29. fx=x −x+<br />
31. fx=−<br />
__ <br />
x +<br />
__ <br />
x+ __<br />
33. fx=x −x+<br />
35. −) 37. ( <br />
x= <br />
− <br />
) <br />
+√ — − √ — x=<br />
<br />
<br />
<br />
y<br />
( +√— <br />
)( − √— <br />
)<br />
<br />
<br />
<br />
<br />
<br />
− −<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
39. ( <br />
− ) <br />
x= <br />
<br />
( +√— <br />
)( − √— <br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
x<br />
)<br />
x<br />
<br />
<br />
<br />
<br />
<br />
− −<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
41. fx=x +x+<br />
43. fx=−x −x−<br />
y<br />
45. fx=−<br />
__ <br />
x − x+<br />
47. fx=x +x+<br />
49. fx=−x +x <br />
51. Th<br />
<br />
h. Th<br />
<br />
53. Thhift<br />
<br />
55. Th<br />
<br />
57. −∞∞<br />
−∞<br />
59. −∞∞ ∞ 61. fx=x +<br />
63. fx=−x − 65. fx=x +x− 67. <br />
69. 71. Th<br />
<br />
73. 75. <br />
<br />
x<br />
Section 5.2<br />
1. Th <br />
. Th<br />
3. x<br />
fxx<br />
fx 5. Th<br />
7. 9. <br />
11. 13. − 15. <br />
− 17. x→∞fx→∞x→−∞fx→∞<br />
19. x→−∞fx→−∞x→∞fx→−∞<br />
21. x→−∞fx→−∞x→∞fx→−∞<br />
23. x→∞fx→∞x→−∞fx→−∞<br />
25. yt−<br />
27. y−x−<br />
29. yx−<br />
31. 33. 35. 37. 39. <br />
41. <br />
43. <br />
45. <br />
47. x→−∞fx→∞<br />
x→∞fx→∞<br />
x<br />
fx<br />
<br />
<br />
<br />
−<br />
<br />
− <br />
51. y<br />
x<br />
x→−∞fx→∞<br />
x→∞fx→∞<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
x<br />
x<br />
49. x→−∞fx→∞<br />
x→∞fx→−∞<br />
x<br />
<br />
<br />
−<br />
−<br />
fx<br />
−<br />
−<br />
<br />
<br />
53. y<br />
x<br />
x→−∞fx→−∞<br />
x→∞fx→∞<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
55. y 57. y−<br />
x− x−<br />
x→−∞fx→−∞ x→−∞fx→∞<br />
x→∞fx→∞<br />
x→∞fx→∞<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
x<br />
x
ODD ANSWERS B-13<br />
59. yx<br />
−<br />
x→−∞fx→−∞<br />
x→∞fx→∞<br />
61. fx=x −<br />
63. f x=x −x +x<br />
65. f x=x +<br />
67. Vm=m +m +m+<br />
69. Vx=x −x +x<br />
Section 5.3<br />
<br />
<br />
<br />
<br />
<br />
y<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
1. x<br />
x<br />
fx= 3. a b <br />
<br />
a b 5. Th<br />
7. −− 9. −<br />
11. − 13. −<br />
15. −− 17. −( <br />
_ )<br />
19. − 21. ( √ — )( −√ — )<br />
23. −−<br />
25. f =−f = <br />
27. f =f =− <br />
29. f =f =− <br />
31. − <br />
<br />
33. −<br />
35. − <br />
37. <br />
<br />
− <br />
39. <br />
41. <br />
<br />
<br />
<br />
43. x1, 0<br />
−4, 0<br />
y<br />
(0, 4); ax→−∞g x→<br />
−∞x→∞g x→∞<br />
<br />
<br />
<br />
<br />
<br />
gx<br />
−−−−−<br />
−<br />
<br />
−<br />
47. x0, 0−2, 0<br />
<br />
y(0, 0); ax→−∞<br />
n x→∞x→∞<br />
n x→−∞<br />
45. x3, 0<br />
2, 0<br />
y<br />
(0, −x→−∞k x→<br />
−∞x→∞k x→∞<br />
kx<br />
x<br />
<br />
<br />
−−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
nx<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
x<br />
x<br />
x<br />
49. f x=− <br />
x−x+x+ 51. f x= <br />
x+ x−<br />
53. −− 55. −<br />
57. f x=− <br />
x+x−x−<br />
59. f x= <br />
x− x− x+ 61. f x=−x− x− <br />
63. f x=−x+x+x−<br />
65. f x=− <br />
x− x−x+<br />
67. −−−<br />
69. : −− 71. −<br />
73. f x=x− x+ 75. f x=x −x +x<br />
77. f x=x −x +x+<br />
79. f x= π x +x +x+<br />
Section 5.4<br />
1. Th<br />
<br />
3. x++<br />
<br />
x− x+<br />
5. x+x+ 7. x−<br />
<br />
x− 9. x−+<br />
<br />
x− <br />
x+<br />
<br />
11. x−+<br />
<br />
x+ x−<br />
13. x −x+x −x+<br />
<br />
15. x +x++<br />
<br />
x− 17. x −x+− <br />
x+ <br />
19. x −x+− <br />
x+ 21. x +x+<br />
23. x −x+− x+ 25. x −x+<br />
<br />
27. x +x+ <br />
x− 29. x −x+ 31. x −x +<br />
33. x −x +x− 35. x −x +x−<br />
37. x −x+ 39. x−x − 41. <br />
x−x +x +x+ 43. <br />
45. x−x +x+ 47. x−x +x+<br />
49. x +x+− 51. <br />
x +x+ 53. x −x +<br />
x−− 55. x −x +x −x +x −x+<br />
<br />
57. x −x +x−+<br />
<br />
x+ 59. ++i<br />
<br />
x−i <br />
61. +−i<br />
x+i <br />
63. x +ix−+ −i<br />
x−i 65. x + 67. x+<br />
69. x+ 71. x− 73. x −<br />
Section 5.5<br />
1. Th<br />
3. <br />
5. <br />
<br />
7. − 9. 11. <br />
13. − 15. − 17. − 19. −<br />
<br />
21. − <br />
√— − √ — 23. −− 25. −−<br />
<br />
27. −− <br />
29. <br />
+√— <br />
−√— <br />
31. <br />
33. −− 35. <br />
− <br />
−<br />
37. −−√ — −√ — 39. − <br />
− <br />
41. +i−i<br />
43. − <br />
+i−i<br />
45. − <br />
<br />
+i−i
B-14<br />
ODD ANSWERS<br />
47. <br />
<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
51. <br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
x<br />
x<br />
<br />
55. <br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
x<br />
49. <br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− <br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
53. <br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
57. ± <br />
±±± <br />
<br />
59. ±± <br />
± <br />
± <br />
<br />
61. <br />
− <br />
<br />
63. <br />
− <br />
65. <br />
<br />
67. fx= <br />
x +x −x−<br />
69. fx=−<br />
__ x −x <br />
71. 73. <br />
75. 77. <br />
79. <br />
Section 5.6<br />
1. Th<br />
3. Th<br />
5. es. Th<br />
<br />
<br />
7. x=− 9. x=−−<br />
11. x=− <br />
y=<br />
x=− 13. <br />
<br />
x=−y=<br />
x=− 15. x=−<br />
y=x=−<br />
17. x=−y=<br />
x=−<br />
19. x= <br />
y=− <br />
<br />
x= 21. <br />
<br />
23. xy( <br />
) <br />
25. x→− <br />
+ f x→−∞x→− <br />
− f x→∞<br />
x→±∞f x→ <br />
x<br />
x<br />
27. x→ + f x→−∞ x→ − f x→∞;<br />
x→±∞f x→− 29. <br />
x→− <br />
+ f x→∞ x→− <br />
− f x→−∞ x→− <br />
− f x→∞<br />
x→− <br />
+ f x→−∞x→±∞f x→ <br />
<br />
31. y=x+ 33. y=x<br />
35. <br />
x=<br />
y= y<br />
37. x=<br />
y=<br />
y<br />
<br />
y=<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
x=<br />
39. <br />
x=−<br />
y=( <br />
) ( − <br />
) px<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
x= −<br />
<br />
43. <br />
x=− <br />
<br />
y=<br />
( − <br />
) ( ) fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− −<br />
−<br />
x= −<br />
−<br />
−<br />
−<br />
−<br />
y= <br />
x<br />
x<br />
x= <br />
y= <br />
x<br />
<br />
47. <br />
x=<br />
y=x+−( <br />
) <br />
( <br />
) hx<br />
<br />
<br />
<br />
<br />
<br />
y=x+<br />
x=<br />
−−−−−<br />
<br />
−<br />
−<br />
−<br />
−<br />
−<br />
x<br />
<br />
<br />
<br />
<br />
<br />
−−−− −<br />
y= −<br />
−<br />
−<br />
−<br />
−<br />
<br />
x=<br />
41. x=<br />
y=<br />
sx<br />
x=<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
y= <br />
<br />
45. x=−<br />
y=<br />
− <br />
ax<br />
<br />
<br />
x=− <br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
x<br />
x<br />
y= <br />
x<br />
<br />
49. x=−<br />
y=<br />
−( − <br />
) wx<br />
x=<br />
<br />
x= −<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
y=<br />
x
ODD ANSWERS B-15<br />
51. fx=________<br />
x − x− x − 53. +x− fx=x <br />
x +x+ <br />
55. fx= − x+ <br />
⋅x x−<br />
57. fx= <br />
x+ x − x− <br />
59. fx=− x− x − 61. fx= <br />
∙ +x− x x− <br />
x−<br />
63. fx=−____________<br />
<br />
x+x− <br />
65. x=y=<br />
x <br />
y − −<br />
x <br />
y <br />
67. x=−y=<br />
x − − − − −<br />
y − −<br />
x <br />
y <br />
69. x=−y=<br />
x −0 −0 −0 − −<br />
y <br />
x <br />
y <br />
71. ( <br />
∞ )<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
–– – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
x<br />
73. −∞∪∞<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
–– – – – –<br />
–<br />
–<br />
–<br />
–<br />
<br />
75. () 77. () 79. (−) <br />
+t<br />
81. Ct=<br />
+t<br />
83. ft 85. <br />
87. <br />
Section 5.7<br />
x<br />
<br />
1. diffict oxy<br />
3. <br />
5. f − x=√ — x + 7. f − x=√ — x+ −<br />
9. f − x=√ — −x 11. f − x= ± √ x− <br />
√<br />
—<br />
13. f − x= x− <br />
15. f− x= −x <br />
<br />
17. f − x=<br />
______ −x <br />
∞ 19. f− x=<br />
__________ x− + <br />
√<br />
21. f − x=−x 23. f − x=<br />
______ x+ <br />
x<br />
—<br />
27. f − x=<br />
______ x− x+ 29. f− x=√ — x+−<br />
25. f − x=<br />
______ x+ <br />
−x<br />
31. f − x=√ — x−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
35. f − x= √ <br />
— x−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
− −−−− − −<br />
−<br />
−<br />
−<br />
−<br />
−<br />
39. <br />
<br />
<br />
<br />
<br />
<br />
y<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
43. −∪ ∞<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
47. −<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−− –<br />
−<br />
−<br />
−<br />
−<br />
−<br />
51. f − x= <br />
x−b <br />
a<br />
55. f − x= cx−b<br />
a−x <br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
x<br />
x<br />
x<br />
33. f − x=√ — x −<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
37. f − x=√ — x+ −<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
41. −∪∞<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
45. −∞−ċ−<br />
f x<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
49. −−−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−− –<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
53. f − x= √ x −b <br />
a<br />
57. th= √ −h <br />
<br />
59. r A= √ A <br />
π , ≈ 61. lT= ( __<br />
π) T ≈ 3.26 ft<br />
63. rA= √ A+π<br />
−2, 3.99 ft 65. rV= √ V ≈ 5.64 ft<br />
π<br />
π<br />
x<br />
x<br />
x<br />
x<br />
x
B-16<br />
ODD ANSWERS<br />
Section 5.8<br />
1. Th<br />
3. 5. y=x <br />
7. y=x 9. y=x 11. y= <br />
x 13. y= <br />
x <br />
15. y= 17. y=xzw 19.<br />
√ <br />
— y=x√ — z<br />
x<br />
21. y= xz<br />
xz<br />
w<br />
23. y=<br />
25. y=<br />
√ — wt <br />
27. y= 29. y= 31. y= 33. y=<br />
35. y= 37. y= 39. y= <br />
41. y= <br />
x y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
45. y=<br />
__ <br />
x <br />
<br />
<br />
<br />
<br />
<br />
y<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Chapter 5 Review Exercises<br />
1. f x = x − −<br />
−<br />
−−<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
x<br />
x<br />
x<br />
43. y=<br />
__ <br />
√— x <br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
47. ≈ <br />
49. ≈<br />
51. <br />
53. <br />
55. ≈<br />
57. ≈<br />
59. ≈<br />
3. f x= <br />
x+ +<br />
5. <br />
<br />
<br />
7. <br />
<br />
9. <br />
<br />
11. x→−∞f x→−∞<br />
x→∞f x→∞<br />
13. −− <br />
−<br />
15. 17. <br />
<br />
19. x +<br />
<br />
21. x −x+− x+ <br />
23. x −x−x+x −x−<br />
25. { −−<br />
<br />
} 27. { <br />
29. <br />
} <br />
x<br />
31. −( −<br />
<br />
) <br />
x=y=<br />
y = <br />
<br />
<br />
<br />
<br />
<br />
y<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
x = <br />
x<br />
33. −<br />
( <br />
) x=−<br />
y=<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
x = −<br />
y = <br />
x<br />
<br />
x = <br />
35. y=x− 37. f − x=√ — x + 39. f − x=√ — x+−<br />
41. f − x= x+ −<br />
x≥−<br />
<br />
43. y= 45. y=<br />
47. ≈ <br />
Chapter 5 Practice Test<br />
1. − 3. x→−∞<br />
f x→∞ x→∞f x→∞ 5. f x=x− <br />
7. <br />
<br />
9. −<br />
<br />
<br />
<br />
11. x +x +x++ x− 13. { −− <br />
} <br />
15. −−<br />
<br />
<br />
17. f x=− <br />
x− x−x+ 19. <br />
21. −( <br />
) 23. f − x=x− +x≥<br />
x=−y=<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
−<br />
−<br />
−<br />
x = −<br />
−<br />
CHAPTER 6<br />
Section 6.1<br />
<br />
x = <br />
y = <br />
x<br />
25. f − x= x+ <br />
x− <br />
27. y=<br />
1. <br />
<br />
3. <br />
<br />
t. Th<br />
<br />
nominal 5. <br />
7. <br />
<br />
9. <br />
11. ft<br />
13.
ODD ANSWERS B-17<br />
t infl<br />
<br />
<br />
15. <br />
17. <br />
19. f x= x 21. f x=( <br />
__ <br />
) − <br />
( <br />
__ <br />
) x ≈ x <br />
23. 25. 27. 29. <br />
31. 33. P=At⋅( + n) r 35. <br />
37. 39. <br />
41. <br />
43. 45. f −=− 47. f −≈−<br />
49. f ≈ 51. y=⋅ x 53. y≈⋅ x<br />
55. y≈⋅ x<br />
57. = At−a a( + ___ r<br />
) −a<br />
a = <br />
a <br />
a[ ( + ___ r<br />
) −] <br />
= <br />
a<br />
<br />
=( + r <br />
) − <br />
In=( + n) r −<br />
59. ff x=a⋅( b) x <br />
b> 1. Thn><br />
fx=a⋅( b) x =ab − x =ae n − x =ae −n x =ae −nx <br />
61. 63. 65. <br />
67. <br />
Section 6.2<br />
1. <br />
x. Th<br />
<br />
<br />
3. gx= −x y<br />
<br />
5. gx) =− x +y<br />
<br />
7. gx=( <br />
) x y<br />
<br />
9. y−<br />
y<br />
<br />
<br />
<br />
<br />
gx=− x<br />
x<br />
−−−−− <br />
−<br />
−<br />
−<br />
−<br />
−<br />
g−x=− −x<br />
11.<br />
<br />
<br />
<br />
<br />
<br />
y<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
gx= x<br />
hx= x<br />
f x= x<br />
<br />
13. 15. 17. 19. 21. <br />
23. y<br />
25. y<br />
− −fx= − <br />
−<br />
−<br />
−<br />
fx=− x + <br />
<br />
<br />
<br />
<br />
fx= x<br />
<br />
<br />
<br />
−f x= x − <br />
<br />
<br />
x<br />
−−−−− <br />
−<br />
−−−−− <br />
−<br />
x<br />
−<br />
−<br />
−<br />
−<br />
x<br />
27. <br />
hx=<br />
<br />
<br />
−−−−−<br />
hx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
29. x→∞fx→−∞<br />
x→−∞f x→−1<br />
31. x → ∞fx→<br />
x→ −∞fx→∞<br />
33. fx= x −<br />
35. fx= x − <br />
37. fx= −x<br />
39. y=− x + <br />
41. y=− x +<br />
43. g≈<br />
45. h−=− 47. x≈− 49. x≈−<br />
51. Thgx) =( b) x y<br />
fx=b x b><br />
fx=b ( x <br />
b) x yf −x<br />
53. Thgxhx<br />
fxn<br />
b>fx=b ( x <br />
b ) b x <br />
n<br />
fx−n<br />
Section 6.3<br />
1. <br />
b <br />
b. Th<br />
yb y b<br />
x<br />
d txby<br />
3. <br />
<br />
b y =x<br />
x 5. Th<br />
b <br />
e e<br />
(x)<br />
(x)<br />
7. a c =b 9. x y = 11. b =a 13. a =<br />
15. e n =w 17. c<br />
k=d 19. <br />
y=x<br />
21. n<br />
= 23. y<br />
( <br />
) =x 25. h=k<br />
27. x= <br />
29. x= 31. x= 33. x= <br />
<br />
35. x=e 37. 39. 41. 43. <br />
<br />
45. 47. − 49. − 51. 53. <br />
55. ≈ 57. ≈ 59. <br />
x=x=<br />
fx=x). Thn<br />
n=<br />
n =n<br />
Thx=f x=x<br />
61. x<br />
x=x=e <br />
x=e . Th <br />
x=e = 63. = e <br />
<br />
<br />
65.
B-18<br />
ODD ANSWERS<br />
Section 6.4<br />
1. <br />
y=xab<br />
ba<br />
3. hift<br />
y<br />
ff 5. <br />
<br />
<br />
7. ( −∞, <br />
) −∞∞<br />
9. ( − <br />
∞ ) −∞∞ <br />
11. ∞x=<br />
13. ( −<br />
<br />
∞ ) x=− <br />
<br />
15. −∞x=−<br />
17. ( <br />
∞ ) x= <br />
<br />
x→( <br />
) + fx→−∞s x → ∞fx → ∞<br />
19. −∞x=−<br />
x→−+fx→−∞x→∞fx→∞<br />
21. ∞−∞∞x=<br />
x( <br />
) y<br />
23. −∞, 0); ra−∞∞<br />
x=x−e y<br />
25. ∞−∞∞x=<br />
xe y<br />
27. 29. 31. 33. <br />
35. y 37.<br />
<br />
<br />
<br />
<br />
<br />
− −<br />
−<br />
−<br />
−<br />
−<br />
41.<br />
fx= x<br />
<br />
gx= x <br />
<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
45.<br />
gx<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
x<br />
<br />
<br />
<br />
x<br />
x<br />
y<br />
39. <br />
<br />
<br />
<br />
<br />
f x= e x<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
43.<br />
x<br />
<br />
gx= x<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− −− −<br />
−<br />
−<br />
−<br />
−<br />
47. f x= <br />
−x−<br />
49. f x= <br />
x+<br />
51. x= <br />
53. x≈ <br />
55. x≈−<br />
<br />
x<br />
57. Thfx= xgx=− x<br />
<br />
b≠ b<br />
x=−_<br />
<br />
b x<br />
59. <br />
x+ <br />
x− ><br />
fx= x+ <br />
x− ><br />
x−∞−<br />
∞Th−∞−∪∞<br />
Section 6.5<br />
1. <br />
<br />
_<br />
. Th b<br />
x n = n x b<br />
3. b<br />
+ b<br />
+ b<br />
x+ b<br />
y<br />
5. b<br />
− b<br />
7. −k 9. xy<br />
11. b<br />
13. b<br />
15. x+y−z<br />
17. x− y <br />
19.<br />
23. ( xz <br />
√ — y<br />
)<br />
<br />
x+ <br />
y <br />
25. = <br />
a <br />
21. x <br />
27. <br />
= 29. <br />
b<br />
<br />
( <br />
) =a−b 31. <br />
b b−<br />
33. ≈ 35. ≈ 37. ≈−<br />
39. x=<br />
<br />
x+− <br />
x−= <br />
( <br />
_____ x+ x− ) =<br />
x<br />
=<br />
_____ x+ <br />
x− <br />
=<br />
_____ x+ <br />
x− −<br />
=<br />
_____ x+ <br />
x− −x− _______<br />
<br />
x− <br />
=<br />
_____________<br />
x+−x+ <br />
x− <br />
=<br />
_____ x− <br />
x− <br />
x=<br />
<br />
+− <br />
−=<br />
<br />
− <br />
x=<br />
41. bnn 1. Th<br />
b<br />
n= n n <br />
n<br />
b = <br />
n<br />
b <br />
Section 6.6<br />
1. <br />
<br />
<br />
<br />
3. Th<br />
<br />
<br />
<br />
. Th<br />
<br />
5. x=− <br />
7. n=− 9. b= <br />
<br />
11. x= 13. 15. p=( <br />
) − <br />
(<br />
17. k=−<br />
<br />
) − <br />
19. x= <br />
<br />
21. x=
ODD ANSWERS B-19<br />
( <br />
) −<br />
23. x=<br />
<br />
25. 27. x=<br />
<br />
29. − = <br />
31. n= 33. k= <br />
35. x=−e <br />
<br />
37. n= 39. 41. <br />
43. x=±<br />
45. x= 47. x= 49. x= <br />
<br />
51. x=<br />
53. x= e ≈<br />
y<br />
y<br />
<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
55. x=−<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−−− –<br />
−<br />
−<br />
−<br />
59. <br />
<br />
<br />
<br />
y<br />
−−−− −<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
63. x= <br />
≈<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
x<br />
<br />
65. <br />
y<br />
x<br />
x<br />
<br />
<br />
<br />
<br />
<br />
−− −<br />
−<br />
−<br />
−<br />
−<br />
<br />
57. x= e+ <br />
≈<br />
y<br />
<br />
− −<br />
−<br />
−<br />
−<br />
−<br />
−<br />
x<br />
<br />
61. x= <br />
≈<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−− −<br />
−<br />
−<br />
−<br />
−<br />
x<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
fx=e x<br />
<br />
<br />
− x<br />
−<br />
x<br />
67. <br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Section 6.7<br />
x<br />
69. ≈ 71. ≈<br />
73. ≈<br />
75. ≈−<br />
77. <br />
79. t=( ( y <br />
_<br />
A) <br />
k<br />
)<br />
k)<br />
81. t=( ( − T−T<br />
s<br />
<br />
T <br />
−T s)<br />
_ <br />
1. <br />
s. Th<br />
<br />
3. <br />
<br />
Th<br />
<br />
5. <br />
<br />
<br />
<br />
t diff<br />
<br />
<br />
orders of magnitude<br />
7. f ≈<br />
9. 11. f x= x<br />
13. <br />
15. <br />
y<br />
fx<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
17. <br />
<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−−−− −<br />
−<br />
<br />
<br />
<br />
<br />
Pt<br />
19. 21. <br />
23. <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
t<br />
x
B-20<br />
ODD ANSWERS<br />
25. M= <br />
( S <br />
S ) <br />
<br />
M= ( S <br />
S ) <br />
M __ =<br />
( S <br />
S ) <br />
S <br />
M __ =S<br />
27. y=b x <br />
bb≠ 1. Th<br />
y=b x <br />
y=xb<br />
e y =e xb<br />
y=e xb<br />
29. A=e −t A≈ 31. <br />
33. f t=e −t ; <br />
35. r≈−<br />
37. f t=e t ftP ≈<br />
39. f t=e t <br />
41. 43. Tt=e −t +t<br />
45. 113 47. <br />
x=x≈<br />
49. ≈ 51. N≈ 53. <br />
Section 6.8<br />
1. <br />
<br />
<br />
3. <br />
t fi<br />
<br />
<br />
. Th<br />
<br />
<br />
<br />
5. y<br />
<br />
7. 9. 11. P =<br />
13. p≈ 15. y 17. <br />
19. <br />
21. y<br />
31.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
x<br />
23. <br />
25. <br />
27. f x= x<br />
29. y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
33. f x=e −x<br />
35. f x=x≈<br />
37. y=+x<br />
39. y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
x<br />
41. y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
51. y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
53. y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
43. f ≈<br />
45. f x=x≈<br />
<br />
47. f x= <br />
+e −x<br />
49. <br />
<br />
<br />
<br />
55. f x=x≈ 57. f x= x <br />
g x=<br />
y =x<br />
_ 59. f − x= a−( c x<br />
−) <br />
<br />
<br />
b<br />
Chapter 6 Review Exercises<br />
1. <br />
3. y = x 5. 7. <br />
<br />
y<br />
9. <br />
<br />
<br />
y<br />
11. gx= −x y<br />
13. x =<br />
15. a<br />
b= − 17. x = 19. =−<br />
<br />
x<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−−−−−<br />
<br />
−<br />
x
ODD ANSWERS B-21<br />
21. e − =−<br />
23.<br />
gx<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
x<br />
25. x>−<br />
x =−<br />
x→− + <br />
f x→−∞<br />
x→∞f x→∞<br />
27. <br />
xy<br />
29. ( xy) z <br />
31. y<br />
<br />
33. +b+ b+−b− <br />
35. <br />
(<br />
v w <br />
) <br />
√ u<br />
37. x = 39. x =− 41. 43. <br />
<br />
45. x = 47. a=e − 49. x =± <br />
__ <br />
<br />
51. 53. f − x= √ — x − <br />
55. f t= t f ≈g 57. <br />
59. <br />
61. <br />
y<br />
63. y = x ; y =e –x<br />
65. <br />
67. <br />
y =−x<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Chapter 6 Practice Test<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
<br />
−<br />
x<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
1. 3. <br />
5. y 7. a = 9. x = <br />
<br />
y<br />
11. ≈−<br />
fx= −x f−x= −x<br />
13. x<<br />
x = <br />
x→ − f x→−∞<br />
x→−∞f x→∞<br />
15. t<br />
<br />
17. y+z+ x− <br />
_______<br />
<br />
+<br />
19. x =__<br />
≈ 21. a=<br />
________ + <br />
23. 25. x = 27. x =±<br />
____ √— <br />
<br />
29. f t=e −t ; <br />
31. Tt=e −t +T≈°F<br />
x<br />
x<br />
33. <br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
CHAPTER 7<br />
Section 7.1<br />
x<br />
35. <br />
y = x<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
37. <br />
<br />
y = <br />
<br />
+e −x<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
1. <br />
3. <br />
<br />
fi 5. <br />
xy<br />
7. 9. 11. − 13. −<br />
<br />
15. ( −<br />
<br />
) 17. 19. ( <br />
<br />
) <br />
21. − 23. ( − __ <br />
___ ) 25. <br />
27. ( − __ <br />
33. ( <br />
__ <br />
) <br />
<br />
29. ( x x+ ) 31. −<br />
) 35. ( <br />
) 37. xx−<br />
39. ( − __ <br />
__ <br />
) 41. <br />
43. <br />
45. <br />
47. − 49. − 51. ( A+B<br />
<br />
A−B ) <br />
<br />
53. ( −<br />
<br />
A−B A A−B ) 55. ( _<br />
EC−BF<br />
AE−BD DC−AF _<br />
BD−AE )<br />
x<br />
x
B-22<br />
ODD ANSWERS<br />
57. Thn a pfi 59. <br />
61. Th 63. <br />
65. 67. <br />
69. <br />
71. 73. <br />
75. <br />
77. <br />
<br />
Section 7.2<br />
1. <br />
3. . Th <br />
<br />
<br />
x+y−z=−x−y+z=−x+y+z=<br />
5. <br />
<br />
<br />
7. 9. <br />
11. − 13. ( − ____ <br />
____ <br />
____ <br />
) 15. ( <br />
) <br />
17. − 19. ( x−x _______<br />
+x ______<br />
) 21. ( − ___ <br />
___ <br />
− ) <br />
23. 25. 27. ( __ <br />
− __ <br />
− __ <br />
) <br />
29. 31. − 33. <br />
35. −−− 37. 39. ( <br />
<br />
<br />
) <br />
41. ( <br />
<br />
<br />
) 43. 45. <br />
47. ( <br />
<br />
<br />
) 49. − 51. <br />
53. 55. <br />
<br />
57. Th <br />
59. <br />
61. Th<br />
63. <br />
<br />
65. <br />
<br />
67. <br />
69. e 19.3%, fi<br />
<br />
Section 7.3<br />
1. <br />
s. A<br />
f a pd a cir<br />
<br />
3. . Th<br />
e a fsider a sys<br />
<br />
<br />
<br />
5. <br />
CxRx<br />
Cx
ODD ANSWERS B-23<br />
5. x=−B<br />
A= <br />
x=− <br />
B−<br />
<br />
7. <br />
x+ − 9. <br />
x− x+ + <br />
x+ <br />
11. <br />
<br />
x− + 13. <br />
x− x+ + <br />
x− <br />
15 <br />
<br />
x+ + <br />
x− 17. <br />
x+ + <br />
x− <br />
<br />
19. <br />
x− − 21. <br />
x− x− + <br />
x− <br />
23. − <br />
<br />
x+ + <br />
x+ 25. − <br />
x− − <br />
x− <br />
27. <br />
x − <br />
x+ + <br />
x+ <br />
29. <br />
x + <br />
<br />
x − <br />
x+ + <br />
x+ <br />
31. x+<br />
<br />
x +x+ + 33. −x x+ x +x+ + <br />
x− <br />
35. x−<br />
<br />
x +x+ + 37. <br />
x+ x +x+ + <br />
x− <br />
39. <br />
<br />
x −x+ + 41. − <br />
x+ x +x+ + <br />
x− <br />
43. <br />
x + <br />
x+ −___________<br />
x <br />
x −x+ 45. x+ <br />
x + + x+ <br />
x + <br />
47. x+ <br />
x+ + x+ <br />
x+ 49. <br />
x +x+ − x <br />
x +x+ <br />
51. <br />
x − x <br />
x + + −x <br />
x + <br />
53. − <br />
x − <br />
<br />
x + <br />
x− − <br />
x− <br />
<br />
55. <br />
x+ − <br />
x+ + <br />
x+ <br />
57. <br />
<br />
x− − <br />
x+ + <br />
x+ − <br />
x− <br />
59. − <br />
x − <br />
x+ + <br />
x+ + <br />
x− <br />
Section 7.5<br />
1. <br />
f diff<br />
s a 2×<br />
s a 2× [ ] + [ ] <br />
3. Am × nB<br />
n × m 5. <br />
AB A B<br />
AB BA <br />
B A BATh<br />
<br />
<br />
<br />
7. <br />
[ 9. ] [ − <br />
] <br />
11. <br />
<br />
<br />
13. <br />
[ <br />
15. ] [ − − − − <br />
<br />
<br />
− − − − ] <br />
] 19. [ ] <br />
<br />
− <br />
23. <br />
[ − − ] <br />
<br />
17.<br />
<br />
[ <br />
<br />
<br />
<br />
21. [ <br />
<br />
− − ] <br />
− <br />
25. <br />
−<br />
<br />
− −<br />
27. <br />
[ − <br />
−<br />
<br />
−<br />
29. ] [ <br />
]<br />
<br />
− <br />
31. [ − ] <br />
33. <br />
35. [ <br />
− ] 37. [ <br />
− ] <br />
39. [ ] 41. [ − −− ] <br />
43. [ − <br />
− − <br />
− − −<br />
] 45. <br />
[ − <br />
]<br />
− <br />
<br />
− −<br />
47. <br />
[ − <br />
49. ] [ − <br />
− ] <br />
<br />
− <br />
51. <br />
[ −<br />
<br />
<br />
53. ] [ <br />
]<br />
− <br />
<br />
<br />
<br />
55. <br />
[ <br />
57. ] [ ]<br />
<br />
<br />
59. B n =<br />
[<br />
{ <br />
<br />
n<br />
<br />
] <br />
<br />
<br />
[ ] n<br />
<br />
Section 7.6<br />
1. <br />
d a v<br />
<br />
3. <br />
n a m<br />
ThR <br />
=R <br />
−R <br />
<br />
R <br />
=R <br />
−R <br />
. Th<br />
5. <br />
<br />
7. [ − <br />
] 9. <br />
[ <br />
<br />
<br />
] <br />
−<br />
11. −x+y=<br />
x−y=<br />
15. x+y−z=<br />
y+z=<br />
x+y − z=−<br />
23. 25. ( <br />
<br />
) <br />
31. ( ____ <br />
− <br />
13. x+y=<br />
−x−y+z=<br />
x+y+z=<br />
17. <br />
19. −−<br />
21. <br />
27. ( x <br />
x+ ) 29. <br />
___ <br />
) 33. − 35. ( <br />
<br />
<br />
<br />
)
B-24<br />
ODD ANSWERS<br />
37. ( ___ <br />
___ <br />
− ___ <br />
) 39. ( xy <br />
− x − <br />
y )<br />
41. ( x−x __ <br />
− ) 43. − 45. −<br />
47. 49. ( −z+<br />
__ z−__<br />
z ) <br />
51. 53. <br />
55. 57. <br />
59. <br />
61. <br />
Section 7.7<br />
1. A − AA − =I<br />
A − A − =I<br />
r a 2 × 3. adbc<br />
ad−bc=<br />
5. [ ] . Th<br />
<br />
<br />
A − = <br />
− −<br />
[ <br />
] <br />
− ] = [ <br />
7. AB=BA= [ ] = 9. AB=BA= [ ] =<br />
<br />
<br />
11. <br />
AB=BA=[ = 13. ] <br />
<br />
<br />
[ − ] <br />
15. <br />
[ − ] 17. Th 19. <br />
[ − ] <br />
21. −<br />
<br />
<br />
[ <br />
−<br />
− <br />
<br />
] 23. <br />
− [ <br />
−<br />
<br />
<br />
−<br />
]<br />
− −<br />
−<br />
25. [ <br />
<br />
− − <br />
27. − 29. <br />
]<br />
−<br />
31. ( <br />
− <br />
) 33. ( − __ <br />
− ___ <br />
) 35. ( <br />
<br />
) <br />
<br />
37. − 39. ( −<br />
__ <br />
41. ( _<br />
<br />
−__<br />
−__<br />
<br />
<br />
) <br />
− ___<br />
− _ <br />
) 43. ( − <br />
<br />
) <br />
− −<br />
45. (<br />
<br />
− <br />
) 47. <br />
<br />
[ <br />
<br />
−<br />
<br />
− <br />
<br />
<br />
− <br />
−<br />
49. <br />
<br />
[ <br />
<br />
− <br />
<br />
− <br />
<br />
] <br />
− − −<br />
<br />
<br />
<br />
<br />
51. [ <br />
<br />
<br />
] 53. <br />
<br />
<br />
<br />
<br />
<br />
− − − − −<br />
55. <br />
57. <br />
59. <br />
61. <br />
] <br />
Section 7.8<br />
1. <br />
<br />
3. Th 5. −<br />
7. 9. − 11. 13. − 15. 17. −<br />
19. 21. 23. − 25. 27. ( <br />
29. 31. ( −− __ <br />
)<br />
) <br />
33. 35. <br />
37. − 39. ( <br />
) 41. <br />
43. 45. 47. 49. <br />
51. 53. <br />
55. 57. <br />
59. <br />
61. 63. <br />
<br />
65. 67. <br />
<br />
Chapter 7 Review Exercises<br />
1. 3. − 5. − 7. <br />
9. 11. − 13. <br />
15. −− 17. ( xx ___ <br />
x ____ <br />
) 19. <br />
21. − 23. 25. <br />
27. y<br />
29. y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
x<br />
<br />
<br />
<br />
<br />
<br />
−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
<br />
31. − <br />
x+ + 33. <br />
x+ x+ − x+ <br />
<br />
35. <br />
x− + −x+ <br />
x +x+ 37. x− <br />
x − + x+ <br />
x − <br />
<br />
39. [ − − − ] 41. <br />
43. <br />
]<br />
47. [<br />
<br />
<br />
<br />
−<br />
45. <br />
[ −<br />
<br />
<br />
<br />
− <br />
− <br />
−<br />
<br />
49. <br />
51. x−z=<br />
y+z=− <br />
−<br />
<br />
<br />
53. <br />
[ −<br />
<br />
<br />
<br />
55. ] [ − <br />
<br />
<br />
]<br />
− <br />
−<br />
57. 59. <br />
61. <br />
[ ] 63. 65. −<br />
67. − 69. <br />
71. 73. 75. ( __ <br />
) 77. xx+ 79. ( − <br />
]<br />
<br />
x<br />
<br />
)
ODD ANSWERS B-25<br />
Chapter 7 Practice Test<br />
1. 3. 5. ( <br />
<br />
<br />
) <br />
7. ( xx ____ <br />
−x ____ <br />
) <br />
9. −√ — −√ — −√ — √ — √ — −√ — √ — √ — <br />
11.<br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
x<br />
<br />
13. <br />
x+ − x+ <br />
x+ <br />
15. [ − ] <br />
−<br />
17. [ − ] <br />
19. − <br />
<br />
<br />
− <br />
21. <br />
[ − <br />
−<br />
<br />
<br />
−<br />
] 23. x<br />
− <br />
25. 27. ( ____ <br />
) <br />
CHAPTER 8<br />
Section 8.1<br />
29. <br />
1. <br />
s a co<br />
3. e a cir 5. <br />
xy<br />
7. x <br />
+y = 9. x <br />
( + y <br />
( =<br />
<br />
) <br />
) <br />
11. x <br />
+y =−<br />
−√ — −√ — <br />
y<br />
13. x <br />
+<br />
<br />
<br />
( =−<br />
<br />
) <br />
( <br />
) ( − <br />
) ( √— <br />
) <br />
( − √— <br />
) 15. x− <br />
+y − <br />
=<br />
−−<br />
+√ — −√ — 17. x + <br />
+y − <br />
=<br />
−−<br />
−−−+√ — −−√ — <br />
19. x − <br />
+y − <br />
=−<br />
+√ — <br />
−√ — 21. x − <br />
( √ — ) + y − <br />
( √ — ) =<br />
+√ — −√ — <br />
+√ — −√ — −<br />
23. _______ x + <br />
+y _______ − <br />
=−<br />
−−−+√ — <br />
−−√ — <br />
25. _______ x + <br />
+y _______ + <br />
=−<br />
−−−−−−<br />
−+√ — −−−√ — − <br />
27. −−+√ — −−−√ — 29. <br />
31. −−−<br />
33. <br />
−−<br />
√ — −√ — <br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
37. −<br />
−−−<br />
−<br />
s a cir. Th<br />
<br />
<br />
y<br />
<br />
<br />
<br />
<br />
−−−− <br />
−<br />
−<br />
−<br />
−<br />
x<br />
x<br />
35. ( __ <br />
( − __ <br />
) ( <br />
( √— <br />
) <br />
<br />
) ( − __ <br />
) <br />
<br />
) <br />
) ( − √— <br />
<br />
<br />
<br />
y<br />
−−.−<br />
−<br />
<br />
−.<br />
−<br />
39. <br />
−−<br />
+√ — −√ — <br />
y<br />
<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
41. − 43. −<br />
−−−− −−−−<br />
−+√ — −−√ — −+√ — −−√ — <br />
y<br />
y<br />
<br />
<br />
<br />
<br />
−−−− <br />
−<br />
−<br />
−<br />
45. −−<br />
−−−−<br />
−−−−<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
y<br />
x<br />
x<br />
<br />
<br />
<br />
−−.− <br />
−<br />
−.<br />
−<br />
47. x <br />
+y <br />
=<br />
49. x − <br />
+y − <br />
=<br />
51. x + <br />
+y − <br />
=<br />
53. x <br />
+y <br />
=<br />
55. x + <br />
+y − <br />
=<br />
57. =π 59. =√ — π<br />
x<br />
x<br />
x
B-26<br />
ODD ANSWERS<br />
61. =π 63. x <br />
y <br />
h + <br />
<br />
=<br />
<br />
h<br />
65. x <br />
+ y <br />
=<br />
67. <br />
Section 8.2<br />
1. ts in a pe diff<br />
s a p<br />
3. Th<br />
5. Th<br />
7. x <br />
<br />
−y =<br />
9. x <br />
<br />
−y = <br />
11. x <br />
<br />
−y =−<br />
√ — −√ — y= <br />
xy=− <br />
x<br />
13. y <br />
<br />
<br />
−x =−√—<br />
−√ — y= <br />
xy=− <br />
x<br />
15. x − <br />
−y − <br />
=−<br />
−y= <br />
x −+y=− x −+<br />
<br />
17. x − <br />
−y + <br />
=−−−<br />
+√ — −−√ — −y=x−y=−x−<br />
19. x + <br />
−y − <br />
=−<br />
−+√ — −−√ — y=x+y=−x<br />
33.<br />
−<br />
35.<br />
−<br />
−<br />
−<br />
y<br />
<br />
<br />
<br />
−<br />
−<br />
<br />
− − − −<br />
− −<br />
37.<br />
−<br />
<br />
−<br />
−<br />
<br />
−<br />
−<br />
<br />
<br />
<br />
−<br />
−<br />
−<br />
−<br />
y<br />
y<br />
<br />
x<br />
<br />
−<br />
−<br />
<br />
<br />
−<br />
−<br />
<br />
<br />
x<br />
<br />
x<br />
<br />
<br />
−<br />
21. y − <br />
−x − <br />
=+√— <br />
−√ — y= <br />
x −+y=− x −+<br />
<br />
23. _______ y + <br />
−x _______ + <br />
=−−−<br />
−−+√ — −−−√ — <br />
y<br />
39.<br />
−− <br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−−<br />
−<br />
<br />
−<br />
x<br />
y= ___ <br />
x+−y=− ___ <br />
x+−<br />
25. x + <br />
−y − <br />
=−<br />
−+√ — −−√ — y= x ++<br />
<br />
y=− __ x ++ <br />
27. y= __ <br />
x−−y=− __ x −−<br />
<br />
29. y= <br />
x −+y=− <br />
x−+<br />
31.<br />
y<br />
− <br />
−<br />
− − − −<br />
<br />
<br />
−<br />
−<br />
<br />
x<br />
<br />
41.<br />
y<br />
43.<br />
<br />
<br />
−− −<br />
x<br />
− − <br />
− −<br />
−− − −<br />
45. x <br />
−y <br />
=<br />
47. x − <br />
−y − <br />
=<br />
−<br />
<br />
<br />
<br />
−<br />
−<br />
−<br />
y<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
−<br />
<br />
−<br />
49. x − <br />
−y − <br />
= 51. y <br />
−x <br />
=<br />
53. y <br />
−x + <br />
= 55. x + <br />
−y + <br />
=
ODD ANSWERS B-27<br />
57. yx=√ — x + 59. yx=+√ — x +x+ 31. y<br />
33.<br />
y<br />
yx=−√ — x +<br />
yx=−√ — x = −<br />
x +x+<br />
<br />
<br />
<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
<br />
<br />
<br />
−−−−<br />
−<br />
<br />
<br />
<br />
−<br />
x<br />
<br />
<br />
− − − − <br />
−<br />
x<br />
x<br />
−−−−−<br />
−<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
y = −<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
35.<br />
61. x <br />
−y <br />
=<br />
63. x <br />
−y <br />
=<br />
y 37.<br />
y<br />
<br />
<br />
y<br />
y<br />
x− x = <br />
<br />
<br />
x<br />
<br />
− − − − <br />
<br />
<br />
<br />
<br />
<br />
−<br />
<br />
<br />
<br />
x<br />
x<br />
−<br />
− − − − − − − <br />
x<br />
−<br />
− − − <br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
39.<br />
y 41.<br />
y<br />
−<br />
65. x <br />
− y <br />
=<br />
67. x <br />
<br />
− <br />
− y <br />
=<br />
y= <br />
x<br />
<br />
− − − − x= −<br />
y<br />
69. x − <br />
−y <br />
=<br />
−<br />
x<br />
−−<br />
− − <br />
<br />
−<br />
− −<br />
<br />
x<br />
−<br />
−<br />
−−−−− <br />
−<br />
−<br />
−<br />
−<br />
−<br />
43. y<br />
45. x =−y<br />
Section 8.3<br />
<br />
47. y − =√ — x −<br />
1. <br />
49. y +√ — =−√ — x −√ — <br />
<br />
m a fixd a fix<br />
x= <br />
51. x =y<br />
3. Th<br />
<br />
53. y −<br />
5. Th<br />
= x +<br />
x<br />
<br />
7. y=x 9. y − =x −<br />
− 55. y −√ — <br />
11. y = <br />
xVF ( <br />
) dx=− −<br />
=√ — x +√ — <br />
<br />
<br />
−<br />
57. y =−x<br />
13. x =− <br />
yVF ( − −<br />
) dy= <br />
<br />
<br />
59. y + =x +<br />
15. y = <br />
xVF ( <br />
) dx=− −<br />
61. <br />
<br />
<br />
63. 65. <br />
17. x − =y −VFdy=<br />
67. x =−y − 69. <br />
19. y− =x+V−F( − <br />
) dx=− <br />
<br />
Section 8.4<br />
21. x + =y +V−−F−dy=−<br />
23. y − =−x +V−F−dx=<br />
1. xyes a r<br />
25. x − = <br />
y +V−F ( − <br />
) dy=− <br />
<br />
3. Ths a h 5. <br />
xy<br />
27. x − =−y −VF( <br />
) dy= <br />
<br />
7. AB= 9. AB=−<<br />
29. y − = <br />
x−VF ( <br />
) dx= 11. AB=> 13. B −AC=<br />
<br />
15. B −AC= 17. B −AC=−<<br />
19. x′ +y′ −= 21. x′ +x′y′−y′ +=<br />
23. θ=x′ −y′ +√ — x′+y′−=<br />
25. θ=x′ +y′ +x′−√ — y′−=<br />
27. θ≈x′ +x′−y′+=<br />
29. θ=x′ −y′ −√ — x′+√ — y′+=
B-28<br />
ODD ANSWERS<br />
31. ____ √— <br />
x′+y′ = __ x′−y′ <br />
<br />
y<br />
−<br />
−<br />
<br />
<br />
−<br />
−<br />
35. ________ x′+y′ <br />
−x′−y′ ________ <br />
=<br />
y<br />
−<br />
39.<br />
43.<br />
−<br />
−<br />
<br />
<br />
−<br />
−<br />
θ = <br />
−<br />
−<br />
−−<br />
<br />
<br />
<br />
<br />
<br />
<br />
−<br />
−<br />
<br />
y<br />
<br />
x<br />
<br />
<br />
−−−−<br />
−<br />
<br />
−<br />
−<br />
47. y<br />
θ = <br />
<br />
−<br />
−<br />
−<br />
−<br />
51. θ=<br />
y<br />
y'<br />
<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
55. θ≈<br />
y<br />
y'<br />
<br />
<br />
<br />
<br />
x'<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
<br />
x'<br />
<br />
<br />
x<br />
x<br />
x<br />
x<br />
x<br />
x<br />
33. x′−y′ <br />
+x′+y′ <br />
=<br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
37. √— <br />
41.<br />
49.<br />
y<br />
x′− <br />
y′=<br />
( __ <br />
____ x′+√— <br />
y′− ) <br />
y<br />
<br />
<br />
−<br />
−<br />
−<br />
θ = <br />
−<br />
−<br />
θ = y<br />
45.<br />
θ = <br />
−<br />
−<br />
−<br />
<br />
<br />
−<br />
−<br />
y<br />
<br />
−<br />
53. θ=<br />
y<br />
y'<br />
<br />
<br />
−<br />
−<br />
<br />
<br />
<br />
y<br />
<br />
θ = <br />
<br />
x<br />
<br />
<br />
<br />
<br />
x'<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
57. −√ —
ODD ANSWERS B-29<br />
Chapter 8 Review Exercises<br />
1. x <br />
+y =−<br />
−√ — −√ — 3. x + <br />
+y − <br />
=<br />
−−−−−−−+√ — <br />
−−√ — <br />
5. <br />
−−<br />
√ — −√ — <br />
<br />
<br />
<br />
−−−− <br />
−<br />
−<br />
−<br />
y<br />
x<br />
7. −−<br />
−−−−<br />
−−−−+√ — <br />
−−−√ — <br />
y<br />
<br />
<br />
<br />
−−−− <br />
−<br />
−<br />
−<br />
9. x <br />
+y = 11. <br />
<br />
13. y + <br />
−x − <br />
=−<br />
−−+√ — −−√ — <br />
15. x − <br />
− y + <br />
( √ — ) =−−<br />
−−−−<br />
17. y<br />
<br />
−<br />
<br />
−−<br />
<br />
<br />
<br />
<br />
<br />
−<br />
−−−− <br />
−<br />
−<br />
−<br />
−<br />
<br />
−−<br />
x<br />
19.<br />
<br />
−<br />
<br />
<br />
<br />
<br />
y<br />
<br />
<br />
−−−− <br />
−<br />
−<br />
−<br />
−<br />
<br />
−<br />
x<br />
<br />
<br />
x<br />
21. x − <br />
−y − <br />
= 23. x + = y −<br />
<br />
−( − <br />
) y= <br />
<br />
25. x + =y +−−( −− <br />
) <br />
y=− <br />
<br />
27. <br />
29. <br />
y<br />
<br />
y<br />
x= <br />
x = − <br />
<br />
<br />
<br />
−−−−−<br />
−<br />
<br />
−<br />
−<br />
−<br />
x<br />
− −<br />
<br />
<br />
31. x − = ( <br />
) y −<br />
33. B −AC= 35. B −AC=−<<br />
37. θ=x′ +y′ −=<br />
−<br />
<br />
<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
x<br />
39. θ= 41. e=<br />
y<br />
<br />
−<br />
43. e= <br />
<br />
<br />
<br />
45.<br />
47.<br />
y<br />
y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x<br />
−−−− <br />
−<br />
y= − −<br />
− − − −<br />
− <br />
−<br />
−<br />
<br />
49. r= <br />
+θ −<br />
Chapter 8 Practice Test<br />
<br />
<br />
<br />
1. x <br />
+y =−<br />
−√ — −√ — <br />
3. <br />
−−<br />
+√ — −√ — <br />
y<br />
<br />
<br />
<br />
−−− <br />
−<br />
−<br />
−<br />
x<br />
5. x − <br />
+y − <br />
=<br />
7. x <br />
<br />
−y =<br />
−<br />
√ — −√ — <br />
y=± <br />
x<br />
9. −−−−+√ — −<br />
−√ — −y=± x −−<br />
<br />
y<br />
−− −<br />
<br />
<br />
x<br />
−<br />
<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
−<br />
11. y − <br />
−x − <br />
= 13. x − = y +<br />
<br />
−( − ___ <br />
) y=− <br />
<br />
15.<br />
y<br />
17. <br />
<br />
19. θ≈<br />
−<br />
−<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
−<br />
−<br />
<br />
−−−− −<br />
<br />
−<br />
−<br />
−<br />
<br />
<br />
−<br />
−<br />
x<br />
<br />
x=−<br />
<br />
x<br />
x
B-30<br />
ODD ANSWERS<br />
21. x′ −x′+y′=<br />
23. e= <br />
47.<br />
a n<br />
49.<br />
a n<br />
y<br />
<br />
⎛<br />
<br />
<br />
<br />
⎝ ⎞ <br />
<br />
<br />
⎭<br />
<br />
<br />
<br />
<br />
<br />
<br />
45. − <br />
<br />
55. Th<br />
57. Th<br />
x<br />
<br />
<br />
−−−−−− <br />
−<br />
<br />
<br />
<br />
−<br />
n<br />
−−−−−− −−−−−− <br />
−<br />
−<br />
−<br />
<br />
−<br />
−<br />
−<br />
n<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
25. <br />
y<br />
−<br />
−<br />
<br />
<br />
y= <br />
51. a n<br />
53. a n<br />
= n− <br />
<br />
⎛ ⎝ ⎞ <br />
<br />
55. a<br />
<br />
<br />
=a n<br />
=a n−<br />
−<br />
⎭<br />
<br />
x<br />
<br />
− − − − <br />
<br />
57. t fi<br />
<br />
− <br />
<br />
<br />
n <br />
<br />
<br />
<br />
<br />
−<br />
−−−−−<br />
<br />
−<br />
− <br />
59. t fi<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
<br />
61. a <br />
=<br />
−<br />
63. <br />
65. <br />
67. a n<br />
=−s a t<br />
CHAPTER 9<br />
−=−−nnld a n<br />
−=−−nn=<br />
Section 9.1<br />
1. <br />
en a fini <br />
y a f<br />
a n<br />
=−<br />
69. a <br />
=a <br />
=a n<br />
=a n−<br />
−a n−<br />
71. n + <br />
n − =n +n <br />
+nn − <br />
<br />
n − <br />
y a f =nn +n +=n +n +n<br />
3. <br />
<br />
5. f a p<br />
<br />
Section 9.2<br />
1. <br />
efi y a co 3. <br />
<br />
<br />
e diff<br />
s a co<br />
13 ∙ 12 ∙ 11 ∙ 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1.<br />
diff 5. <br />
7. −− <br />
−− 9. e a co. The diff<br />
<br />
<br />
<br />
<br />
<br />
11. −−<br />
<br />
13. <br />
<br />
<br />
15. r a s 7. Th<br />
<br />
− <br />
− 17. <br />
<br />
<br />
<br />
<br />
<br />
diff 9. Th<br />
<br />
−≠− 11. <br />
<br />
<br />
13. −−−−<br />
19. −−−−−−−−<br />
15. a<br />
21. a n<br />
=n + 23. a n<br />
= <br />
n n− <br />
n 25. a n<br />
=( − <br />
) − <br />
= 17. a <br />
= 19. a <br />
= 21. a <br />
=<br />
n 23. a <br />
= 25. a <br />
=− 27. −−−−−<br />
27. t fi−− 29. t fi<br />
29. a <br />
=a n<br />
=a n−<br />
+n≥ 31. a <br />
=a n<br />
=a n−<br />
+n≥<br />
−− <br />
<br />
31. <br />
<br />
<br />
<br />
<br />
<br />
<br />
33. a <br />
=a n<br />
=a n−<br />
+n≥ 35. a <br />
= <br />
a =a + <br />
n n−<br />
n≥<br />
33. <br />
<br />
35. a=−a =a +n<br />
37. a <br />
= <br />
a =a − <br />
n n−<br />
n≥ 39. a=a =a +a =<br />
n n− <br />
n n−<br />
41. t fi 43. a n<br />
=+n<br />
37. a <br />
=a n<br />
=a n−<br />
+ 39. 41. <br />
45. a n<br />
=−+n 47. a n<br />
=n 49. a n<br />
=+n<br />
43. <br />
<br />
<br />
<br />
51. a n<br />
= <br />
n− <br />
<br />
53. Th
ODD ANSWERS B-31<br />
59.<br />
<br />
<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
−<br />
65.<br />
a n<br />
<br />
−<br />
n<br />
−<br />
−<br />
−<br />
<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− <br />
−<br />
n<br />
61. <br />
63.<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
−−−−−− <br />
−<br />
67. <br />
a n<br />
=n<br />
a n<br />
=+n<br />
69. a <br />
=−a+b<br />
71. Th <br />
a <br />
=− 73. <br />
<br />
a <br />
=<br />
a n<br />
=a n−<br />
−−−a <br />
=−<br />
Section 9.3<br />
1. <br />
3. m in a s<br />
<br />
5. <br />
e a co<br />
<br />
<br />
er difff a<br />
<br />
<br />
7. Th− 9. Th<br />
Th 11. Thic. Th<br />
− 13. Thic. Th<br />
<br />
15. <br />
<br />
<br />
17. <br />
19. a <br />
=− <br />
21. a <br />
=− 23. <br />
<br />
25. a=−a n<br />
= <br />
a 27. a=a =−a n− n n−<br />
29. a <br />
= <br />
a = n <br />
a 31. a= <br />
n− <br />
a =−a n n−<br />
33. −− <br />
<br />
35. a n<br />
= n−<br />
37. a n<br />
= 0.8 ∙ (− n− 39. a n<br />
=−( __ <br />
) n− <br />
41. a n<br />
= 3 ∙ ( − __ <br />
<br />
) n− 43. a <br />
= <br />
<br />
45. Th<br />
47. Tht a g<br />
49.<br />
a n<br />
<br />
<br />
<br />
<br />
<br />
−<br />
−<br />
<br />
<br />
<br />
<br />
<br />
n<br />
51. <br />
a <br />
=a n<br />
=a n−<br />
<br />
a <br />
=a n<br />
=a n−<br />
53. a <br />
=b<br />
55. Th<br />
a <br />
≈<br />
n<br />
57. a <br />
=− <br />
<br />
59. h a de<br />
a n<br />
= 400 ∙ 0.5 n− <br />
a <br />
=<br />
Section 9.4<br />
1. n nf a<br />
3. s in a<br />
5. s a s<br />
n a co<br />
<br />
<br />
<br />
7. ∑ n 9. ∑ 11. ∑ k+ 13. S <br />
= <br />
n=<br />
k=<br />
k=<br />
<br />
<br />
15. S <br />
= 17. ∑ 8 ∙ 0.5 k− <br />
( −( <br />
19. S <br />
= <br />
− <br />
<br />
<br />
) ) <br />
k=<br />
<br />
= <br />
≈<br />
21. S <br />
= − <br />
− = <br />
≈<br />
<br />
23. Th S= <br />
− <br />
25. Th S= − <br />
−( − <br />
) <br />
27. y<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
29. S: Th<br />
S n<br />
<br />
g 1. Th<br />
∞<br />
∑ ( <br />
k= ) k <br />
<br />
x<br />
<br />
<br />
S= <br />
−( <br />
) =<br />
31. 33. 35. S <br />
= <br />
37. S <br />
= <br />
<br />
39. S <br />
= 41. S <br />
=− <br />
43. S=− <br />
<br />
45. S= 47. 49. <br />
51. a k<br />
=−k 53. 55. r= <br />
<br />
57. 59. 61. <br />
Section 9.5<br />
1. Thm + nAB<br />
3. Th<br />
. Th <br />
<br />
. Th<br />
lem. Th<br />
5. <br />
n<br />
Cn, r=<br />
<br />
7. += 9. ++=<br />
n−rr<br />
11. = 13. = 15. P=<br />
17. P= 19. P= 21. C=<br />
23. C= 25. = 27. =<br />
29. = 31. = <br />
33. 35. <br />
<br />
37. r=r=r=<br />
Cnr=Pnr=r=r=Cnr=Pnr=n
B-32<br />
ODD ANSWERS<br />
39. ×= <br />
41. −+−= 43. ××= 33.<br />
<br />
45. ××= 47. P=<br />
1 2 3 4 5 6<br />
49. C×C×C= 51. =<br />
53. 1<br />
<br />
<br />
=<br />
<br />
2<br />
<br />
<br />
Section 9.6<br />
3<br />
<br />
<br />
1. <br />
Cnr (<br />
n r n<br />
)<br />
=Cnr= <br />
rn−r 4<br />
<br />
<br />
n<br />
3. Thl Th x +y =∑ ( n <br />
k= k )<br />
x n−k y k 5<br />
<br />
<br />
<br />
5. 7. 9. 11. <br />
6<br />
<br />
<br />
13. a −a b +ab −b 15. a +a b +ab +b <br />
17. x +x y +x y +x y +xy +y 35. <br />
37. 39. <br />
41. <br />
43. <br />
<br />
19. x −x y +x y −x y +xy −y <br />
21. <br />
<br />
x + <br />
x y + <br />
x y + <br />
xy + <br />
y 23. 45. <br />
a +a b +a b 47. C <br />
C = <br />
49. CC <br />
<br />
C = <br />
<br />
25. a −a b +a b <br />
51. CC <br />
<br />
C ≈ 53. CC <br />
<br />
C ≈<br />
27. a +a b +a b <br />
29. x −x √ — y +x y 31. −x y <br />
55. +−=<br />
33. y 35. x y <br />
57. CC <br />
<br />
37. a b 39. y x <br />
<br />
C =<br />
59.<br />
41. f <br />
x=x +x 43. f <br />
x=x +x +x +x<br />
CC <br />
<br />
C =<br />
y<br />
y<br />
f x<br />
f x<br />
<br />
<br />
<br />
<br />
Chapter 9 Review Exercises<br />
<br />
<br />
<br />
<br />
1. 3. <br />
<br />
<br />
<br />
<br />
5. Thic. Thn diffd= <br />
<br />
<br />
.<br />
x<br />
<br />
7. −− 9. a<br />
−−−−− <br />
<br />
=−a n<br />
=a n−<br />
+<br />
−<br />
<br />
−<br />
x<br />
11. a<br />
−<br />
−−−−−−−−− <br />
n<br />
= <br />
−<br />
n+ 13. r= 15. <br />
<br />
−<br />
−<br />
17. 19. a<br />
−<br />
−<br />
n<br />
=− <br />
∙ ( <br />
) n− <br />
−<br />
<br />
−<br />
21. ∑ ( <br />
45. x y −<br />
m= m+ ) 23. S = 25. S ≈<br />
<br />
<br />
−<br />
−<br />
27. S= <br />
47. k −<br />
−<br />
29. 31. 33. =<br />
35. P= 37. C=<br />
49. Thx +y −z <br />
l Th 39. =× 41. = <br />
43. <br />
<br />
45. a<br />
Section 9.7<br />
+a b+a b <br />
47.<br />
1 2 3 4 5 6<br />
1. <br />
<br />
1 <br />
3. <br />
2 <br />
5. Thunion of two events<br />
3 <br />
<br />
4 <br />
<br />
5 <br />
ABAB<br />
ABh. The diff<br />
6 <br />
<br />
7. <br />
<br />
49. <br />
9. 51. <br />
53. <br />
55. −C <br />
C ≈<br />
<br />
11. <br />
13. <br />
15. <br />
17. <br />
19. <br />
<br />
21. <br />
23. <br />
25. <br />
27. <br />
29. 57. CC <br />
<br />
<br />
31. <br />
C ≈
ODD ANSWERS B-33<br />
Chapter 9 Practice Test<br />
1. −−− 3. Thic. Th<br />
n diffd=. 5. a <br />
=−a n<br />
=a n−<br />
− <br />
<br />
a <br />
=− 7. Thic. Th<br />
<br />
r= <br />
<br />
. 9. a=a =− n <br />
∙ a n−<br />
11. ∑ ( − <br />
k=− k ) <br />
13. S <br />
=− 15. <br />
17. ××××=<br />
19. C= 21.<br />
<br />
= 23. x <br />
25.<br />
<br />
27. <br />
29. CC <br />
<br />
C ≈
Index<br />
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r Th <br />
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C-1
C-2<br />
INDEX<br />
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fi <br />
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Q<br />
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INDEX C-3<br />
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hift <br />
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W<br />
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X<br />
x <br />
x <br />
x <br />
Y<br />
y <br />
y <br />
y <br />
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Z