College Algebra, 2015a

College Algebra 

SENIOR CONTRIBUTING AUTHOR 

JAY ABRAMSON, ARIZONA STATE UNIVERSITY

About Our Team 

Senior Contributing Author 

Jay Abramson has been teaching College Algebra for 33 years, the last 14 at Arizona State University, where he is a principal 

lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid 

and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing 

author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught 

at Texas State Technical College and Amarillo College. He received Teacher of the Year awards at both institutions. 

Contributing Authors 

Valeree Falduto, Palm Beach State College 

Rachael Gross, Towson University 

David Lippman, Pierce College 

Melonie Rasmussen, Pierce College 

Rick Norwood, East Tennessee State University 

Nicholas Belloit, Florida State College Jacksonville 

Jean-Marie Magnier, Springfield Technical Community College 

Harold Whipple 

Christina Fernandez 

The following faculty contributed to the development of OpenStax 

Precalculus, the text from which this product was updated and derived. 

Honorable Mention 

Nina Alketa, Cecil College 

Kiran Bhutani, Catholic University of America 

Brandie Biddy, Cecil College 

Lisa Blank, Lyme Central School 

Bryan Blount, Kentucky Wesleyan College 

Jessica Bolz, The Bryn Mawr School 

Sheri Boyd, Rollins College 

Sarah Brewer, Alabama School of Math and Science 

Charles Buckley, St. Gregory's University 

Kenneth Crane, Texarkana College 

Rachel Cywinski, Alamo Colleges 

Nathan Czuba 

Srabasti Dutta, Ashford University 

Kristy Erickson, Cecil College 

Nicole Fernandez, Georgetown University / Kent State University 

David French, Tidewater Community College 

Douglas Furman, SUNY Ulster 

Erinn Izzo, Nicaragua Christian Academy 

John Jaffe 

Jerry Jared, Blue Ridge School 

Stan Kopec, Mount Wachusett Community College 

Kathy Kovacs 

Sara Lenhart, Christopher Newport University 

Joanne Manville, Bunker Hill Community College 

Karla McCavit, Albion College 

Cynthia McGinnis, Northwest Florida State College 

Lana Neal, University of Texas at Austin 

Steven Purtee, Valencia College 

Alice Ramos, Bethel College 

Nick Reynolds, Montgomery Community College 

Amanda Ross, A. A. Ross Consulting and Research, LLC 

Erica Rutter, Arizona State University 

Sutandra Sarkar, Georgia State University 

Willy Schild, Wentworth Institute of Technology 

Todd Stephen, Cleveland State University 

Scott Sykes, University of West Georgia 

Linda Tansil, Southeast Missouri State University 

John Thomas, College of Lake County 

Diane Valade, Piedmont Virginia Community College 

Reviewers 

Phil Clark, Scottsdale Community College 

Michael Cohen, Hofstra University 

Matthew Goodell, SUNY Ulster 

Lance Hemlow, Raritan Valley Community College 

Dongrin Kim, Arizona State University 

Cynthia Landrigan, Erie Community College 

Wendy Lightheart, Lane Community College 

Carl Penziul, Tompkins-Cortland Community College 

Sandra Nite, Texas A&M University 

Eugenia Peterson, Richard J. Daley College 

Rhonda Porter, Albany State University 

Michael Price, University of Oregon 

William Radulovich, Florida State College Jacksonville 

Camelia Salajean, City Colleges of Chicago 

Katy Shields, Oakland Community College 

Nathan Schrenk, ECPI University 

Pablo Suarez, Delaware State University 

Allen Wolmer, Atlanta Jewish Academy 

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Brief Contents 

1 Prerequisites 1 

2 

3 

4 

5 

6 

7 

8 

9 

Equations and Inequalities 73 

Functions 159 

Linear Functions 279 

Polynomial and Rational Functions 343 

Exponential and Logarithmic Functions 463 

Systems of Equations and Inequalities 575 

Analytic Geometry 681 

Sequences, Probability and Counting Theory 755 

v

Contents 

Preface 

xi 

1 Prerequisites 1 

1.1 Real Numbers: Algebra Essentials 2 

1.2 Exponents and Scientific Notation 17 

1.3 Radicals and Rational Expressions 31 

1.4 Polynomials 41 

1.5 Factoring Polynomials 49 

1.6 Rational Expressions 58 

Chapter 1 Review 66 

Chapter 1 Review Exercises 70 

Chapter 1 Practice Test 72 

2 

3 

Equations and Inequalities 73 

2.1 The Rectangular Coordinate Systems and Graphs 74 

2.2 Linear Equations in One Variable 87 

2.3 Models and Applications 102 

2.4 Complex Numbers 111 

2.5 Quadratic Equations 119 

2.6 Other Types of Equations 131 

2.7 Linear Inequalities and Absolute Value Inequalities 142 




Functions 159 

3.1 Functions and Function Notation 160 

3.2 Domain and Range 180 

3.3 Rates of Change and Behavior of Graphs 196 

3.4 Composition of Functions 209 

3.5 Transformation of Functions 222 

3.6 Absolute Value Functions 247 

3.7 Inverse Functions 254 




vii

4 

Linear Functions 279 

4.1 Linear Functions 280 

4.2 Modeling with Linear Functions 309 

4.3 Fitting Linear Models to Data 322 




5 

Polynomial and Rational Functions 343 

5.1 Quadratic Functions 344 

5.2 Power Functions and Polynomial Functions 360 

5.3 Graphs of Polynomial Functions 375 

5.4 Dividing Polynomials 393 

5.5 Zeros of Polynomial Functions 402 

5.6 Rational Functions 414 

5.7 Inverses and Radical Functions 435 

5.8 Modeling Using Variation 446 




6 

Exponential and Logarithmic Functions 463 

6.1 Exponential Functions 464 

6.2 Graphs of Exponential Functions 479 

6.3 Logarithmic Functions 491 

6.4 Graphs of Logarithmic Functions 499 

6.5 Logarithmic Properties 516 

6.6 Exponential and Logarithmic Equations 526 

6.7 Exponential and Logarithmic Models 537 

6.8 Fitting Exponential Models to Data 552 




viii

7 

8 

9 

Systems of Equations and Inequalities 575 

7.1 Systems of Linear Equations: Two Variables 576 

7.2 Systems of Linear Equations: Three Variables 592 

7.3 Systems of Nonlinear Equations and Inequalities: Two Variables 603 

7.4 Partial Fractions 613 

7.5 Matrices and Matrix Operations 623 

7.6 Solving Systems with Gaussian Elimination 634 

7.7 Solving Systems with Inverses 647 

7.8 Solving Systems with Cramer's Rule 661 




Analytic Geometry 681 

8.1 The Ellipse 682 

8.2 The Hyperbola 697 

8.3 The Parabola 714 

8.4 Rotation of Axis 727 

8.5 Conic Sections in Polar Coordinates 740 




Sequences, Probability and Counting Theory 755 

9.1 Sequences and Their Notations 756 

9.2 Arithmetic Sequences 769 

9.3 Geometric Sequences 779 

9.4 Series and Their Notations 787 

9.5 Counting Principles 800 

9.6 Binomial Theorem 810 

9.7 Probability 817 




Try It Answer Section A-1 

Odd Answer Section B-1 

Index C-1 

ix

Preface 

College AlgebraTh 

 

About OpenStax 

fifi 

 

 

Th 

 

 

About OpenStax’s Resources 

Customization 


 

 

 

 

 

 

 

 

 

Errata 

 

 

 

fi 

 

Format 

 

xi

About College Algebra 


Th 

 

 

Coverage and Scope 

 

Thfifl 

 

Th 

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Development Overview 

College Algebra 

 

 

ft 

 

Th 

Th 

 

 

Accuracy of the Content 

 

 

 

Th 

 

 

 

s fi 

xii

Pedagogical Foundations and Features 

Learning Objectives 

 

Th 

 

Narrative Text 

fi 

Th 

 

fifi 

fi 

fi 

Examples 

 

Thff 

 

Th 

Th 

fifl 

 

Figures 

College Algebrafi 

fi 

 

 

 

Supporting Features 

 

How To 

 

Try It 

 

 

Q & A... 

 

ThTh 

 

 

fiCollege Algebra 

xiii

Section Exercises 

 

 

 

Verbal 

Algebraic 

Graphical 

Numeric 

Technology 

 

ExtensionsTh 

 

Real-World Applications fi 

 

Chapter Review Features 

 

 

Key Termsfi 

Key Equations 

Key Concepts 

 

Chapter Review Exercises 

Practice Test 

 

 

Answer Key 

 

Additional Resources 

Student and Instructor Resources 

 

 

 

Partner Resources 

 

Th 

 

Online Homework 

XYZ Homework 

 

 

 

 

WebAssignfi 

fi 

 

 

 

 

 

xiv

Prerequisites 

1 

Figure 1 Credit: Andreas Kambanls 

CHAPTER OUTLINE 

1.1 Real Numbers: Algebra Essentials 

1.2 Exponents and Scientific Notation 

1.3 Radicals and Rational Expressions 

1.4 Polynomials 

1.5 Factoring Polynomials 

1.6 Rational Expressions 

Introduction 

 

Th 

 

 

ff 

 

 

Th 

1

2 

CHAPTER 1 PREREQUISITES 

LEARNING OBJECTIVES 

In this section students will: 

Classify a real number as a natural, whole, integer, rational, or irrational number. 

Perform calculations using order of operations. 

Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity. 

Evaluate algebraic expressions. 

Simplify algebraic expressions. 

1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 

ft 

Th 

 

fi 

 

ThThfi 

 

fl 

 

fift 

 

 

Th 

 

 

ff 

 

Classifying a Real Number 

Thnatural numbers 

fiTh 

counting numbers 

Thwhole numbers 

 

Thintegers−−− 

 

 

 

 

−−− 

Thrational numbers{ _ m n 

∣ 

mnn ≠}fi 

 

 

 

 

 

1. 

= 

2. 

==_

SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 3 

Example 1 

Writing Integers as Rational Numbers 

 

a. b. c. − 

Solution 

a. = 

b. = 

c. −=− 

 

Try It #1 

 

a. b. c. − 

Example 2 

Identifying Rational Numbers 

 

a. − 

b. 

c. 

 

 

Solution 

a. − 

=−_ 

b. 

c. 

Try It # 

 

= 

 

= 

 

a. 

 

b. 

 

 

c. − 

Irrational Numbers 

 

 

 

 

irrational 

Thirrational numbers 

 

 

h | h 

Example 3 

Differentiating Rational and Irrational Numbers 

 

 

a. √ — b. 

 

√ — d. 

 

 

 

Solution 

a. √ — √ — =5. Th√ —

4 


b. 

 

 

 

 

_ 

 

 

= 

 

= 

=_ 

c. √ — . Th√ — 

d. 

 

 

 

 

 

= 

 

= 

= 

 

_ 

 

 

 

 

e. 

Th 

 

Try It #3 

 

 

a. 

 

b. √— c. d. 

 

 

e. √ — 

Real Numbers 

nnTh 

real numbers 

 

+− 

Th 

fi 

 

ThTh 

real number lineFigure 2 

Example 4 

Classifying Real Numbers 

Figure The real number line 

ft 

 

Solution 

− − − − 

a. − 

 

b. √ — c. −√ — d. −π e. 

a. − 

 

b. √ — 

c. −√ — =−√ — =− 

d. −π 

e.


Try It #4 

ft 

 

a. √ — 

b. − c. d. − √— 

e. 

 

 

Sets of Numbers as Subsets 

 

Th 

Figure 3 

N 

W 

I 

Q 

Q 

 

N 

 

W 

−−− 

I 

n≠ 

m 

n 

Q 

Figure Sets of numbers 

sets of numbers 

Thnatural numbers 

Thwhole numbers 

Thintegers−−− 

Thrational numbers{ m n ∣ 

mn ≠} 

Thirrational numbers 

hh 

Example 5 

Differentiating the Sets of Numbers 

NWIQ 

 

a. √ — b. _ 

 

c. √ — d. − e. 

Solution 

N W I Q Q' 

a. √ — = × × × × 

b. 

 

=_ 

× 

c. √ — × 

d. − × × 

e. ×

6 


Try It #5 

NWIQ 

Q 

a. − 

 

b. c. √ — d. √ — e. 

Performing Calculations Using the Order of Operations 

=∙ = 

exponential notationa n a 

n 

n 

a n =a ∙ a ∙ a ∙ … ∙ a 

a n naabasenexponent 

 

+6 ∙ 

− 

 

order of operations. Th 

 

 

 

 

Thft 

d fi 

 

+6 ∙ 

− 

ThTh 

 

+6 ∙ 

− 

+6 ∙ 

− 

 

+6 ∙ 

− 

+− 

 

+− 

− 

 

Th+6 ∙ 

− = 

 

fi 

 

order of operations 

fi 

PEMDAS 

P 

E 

MD 

AS


How To… 

 

1. 

2. 

3. 

4. 

Example 6 

Using the Order of Operations 

 

a. (3 ∙ 2) −+ b. − 

− √— − c. −∣−∣+− 

−3 ∙ 2 

d. 

 

2 ∙ 5− e. 7(5 ∙ 3)−−− + 

Solution 

a. 3 ∙ 2 −+= − 

=− 

 

=− 

 

= 

 

b. − 

− √— −= − − √— 

= − 

− 

 

= − − 

 

 

= 

− 

 

=− 

 

= 

 

fi 

 

c. −−+−=−−+ 

=−+ 

 

=−+ 

 

=+ 

 

= 

 

d. −3 ∙ 2 

2 ∙ 5− =−3 ∙ 2 

2 ∙ 5− 

= − 

− 

= 

 

 

 

= 

 

 

 

e 5 ∙ 3−−− +=−− + 

=−−+ 

=−−+ 

=++ 

 

=

8 


Try It #6 

 

a. √ — − +− b. + 

7 ∙ 5−8 ∙ 4 

− 

c. −+√— + 

 

d. · − + · 

e. − −−− 

Using Properties of Real Numbers 

 

ff 

s fi. Th 

Commutative Properties 

commutative property of additionff 

a+b =b +a 

 

−+= +−= 

commutative property of multiplication 

ff 

a ∙ b = b ∙ a 

 

−∙ (−= −∙ (−= 

− 

−÷≠÷ 

Associative Properties 

associative property of multiplication 

 

a(bc) = (ab)c 

 

∙∙ 5= ∙∙= 

associative property of additionffff 

a + (b + c) = (a + b) + c 

Th 

+−+= +−+= 

 

−−−− 

÷÷÷÷ 

−−− 

÷÷ 

≠− 

≠ 

 

Distributive Property 

distributive property 

 

a ∙ b + c = a ∙ b + a ∙ c


Th 

 

∙+−=4 ∙+4 ∙ (− 

 

=+− 

 

= 

− 

 

Th 

 

+3 ∙ 5+) ∙ (+ 

+∙ 

≠ 

 

a − b = a + −b 

ff−+ff+ 

+ 

+−∙+ 

− 

−+=+−∙+ 

=+−∙ 5+−∙ 

=+− 

= 

Th 

 

 

−+=+−− 

=+− 

= 

Identity Properties 

Thidentity property of addition 

 

a + = a 

Thidentity property of multiplication 

 

a ∙ = a 

−+=−∙=Th 

 

Inverse Properties 

inverse property of additiona 

−a 

a + −a = 

a =−−+= 

inverse property of multiplicationfi 

Tha 

a 

a ∙ a = 

a =− 

 

a −

10 


a ∙ a = ( − 

properties of real numbers 

Thabc 

) ∙ ( − 

) = 

Addition 

Multiplication 

Commutative Property a + b = b + a a ∙ b = b ∙ a 

Associative Property a+b+c=a+b+c abc=abc 

Distributive Property 

a ∙ (b+c=a ∙ b + a ∙ c 

Identity Property 

Inverse Property 

Example 7 

Using Properties of Real Numbers 

Th 

 

a 

a + = a 

 

−a 

 

a+−a= 

Th 

 

a 

a ∙ = a 

a 

 

a 

a ∙ ( a )= 

 

a. ∙ 6+3 ∙ 4 b. ++− c. −+ d. 

 

∙ ( ∙ ) e. ∙ [0.75+− 

Solution 

a. ∙ 6+3 ∙ 4=3 ∙ (6+ 

= 3 ∙ 10 

= 

 

b. ++−=++− 

=+ 

 

= 

 

c. −+=+−+− 

=+− 

 

=− 

 

d. 

__ ∙ __ 

( ∙ __ ) = __ ∙ __ 

( ∙ __ ) 

= __ 

( ∙ __ ) ∙ __ 

= 1 ∙ __ 

 

 

 

= __ 

 

e. 100 ∙ [0.75+−=100 ∙ 0.75+100 ∙ (− 

=+− 

 

=− 

 

Try It #7 

 

a. ( − 

) ·[ · ( − ) ] b. ·+ c. −− d. 

+ [ 

+ ( − 

) ] 

e. ċ−+ċ


Evaluating Algebraic Expressions 

 

x + 

πr √ — m n x +constant 

xvariable 

algebraic expression 

 

 

 

− =−) ∙ (−) ∙ (−) ∙ (−) ∙ (− 

x =x ∙ x ∙ x ∙ x ∙ x 

2 ∙ 7 =2 ∙ 7) ∙ (2 ∙ 7) ∙ (2 ∙ 7 

yz =yz) ∙ (yz) ∙ (yz 

 

 

ff 

 

 

 

 

Example 8 

Describing Algebraic Expressions 

 

Solution 

a. x + b. 

πr c. 

 

√ — m n 

Constants 

Variables 

a. x + x 

b. 

πr 

 

 

π 

r 

c. √ — m n m, n 

Try It #8 

 

a. πrr+h b. L+W c. y +y 

Example 9 

Evaluating an Algebraic Expression at Different Values 

x −x 

a. x = b. x = c. x = 

 

d. x =− 

Solution 

a. x x −=− 

=− 

=− 

b. x x −=− 

=− 

=−

12 


Try It #9 

c. 

x 

x −= ( ) − 

=− 

=− 

d. −x x −=−− 

=−− 

=− 

−yy 

a. y = b. y = c. y = 

 

d. y =− 

Example 10 

Evaluating Algebraic Expressions 

 

a. x +x =− b. 

t t = c. 

t− 

d. a +ab+ba =b =− e. √ — m n m =n = 

Solution 

a. −x x +=−+ 

= 

b. t 

t 

t − = 

− 

 

= 

 

− 

= 

 

πr r = 

c. r 

 

πr = 

π 

= 

π 

= 

π 

d. a−b a + ab+ b =+−+− 

=−− 

=− 

e. mn √ — m n =√ — 

= √ — 

=√ — 

= 

Try It #10 

 

y + a. _ 

y − y = b. −tt =− c. 

πr r = 

d. p q p =−q = e. m−n−n−mm = 

n =


Formulas 

equationTh 

ThTh 

 

x +=x =x 

+= 

formulaft 

fift 

fiAr 

A =πr rAπr 

Example 11 

Using a Formula 

rhS 

S =πrr+hFigure 4 

π 

r 

h 

Figure Right circular cylinder 

Solution πrr + hr =h = 

S =πr+ 

=π+ 

=π 

=π 

Thπ 

Try It #11 

LWTh 

A =L+W+−L ∙ Figure 5 

 

 

 

 

 

Figure

14 


Simplifying Algebraic Expressions 

 

 

 

Example 12 

Simplifying Algebraic Expressions 

 

a. x −y +x −y − b. r −−r+ c. ( t − 

) −( 

t +s ) 

d. mn−m +mn+n 

Solution 

a. x −y +x −y −=x +x −y −y − 

=x −y − 

b. r −−r+=r −+r + 

=r +r −+ 

=r − 

 

c. t −( t − 

) −( 

t + ) =t − 

s − t −s 

 

=t − 

t − s −s 

 

 

= 

t − 

s 

 

d. mn−m +mn+n =mn+mn−m +n 

=mn−m +n 

Try It #12 

 

a. 

y − ( 

y +z ) b. 

t −− + c. pq−+q−p d. r −s+r+−s 

t 

Example 13 

Simplifying a Formula 

LWPP =L +W +L +W 

Solution 

 

P =L +W +L +W 

P =L +L +W +W 

P =L +W 

P =L+W 

Try It #13 

PrtA 

A =P +Prt 

 

Simplify an Expression (http://openstaxcollege.org/l/simexpress) 

Evaluate an Expression1 (http://openstaxcollege.org/l/ordofoper1) 

Evaluate an Expression2 (http://openstaxcollege.org/l/ordofoper2)

SECTION 1.1 SECTION EXERCISES 

15 

1.1 SECTION EXERCISES 

VERBAL 

1. √ — 

t fi 

 

2. 

 

 

3. 

 

 

NUMERIC 

 

4. +·− 5. ÷−÷ 6. +− 7. −·÷− 

8. −+· 9. − 10. +−÷ 11. ÷÷+ 

12. + ÷ 13. −·+ 14. +·÷ 15. ++− 

16. −÷ 17. ·÷− 18. −+· 19. +·− 

20. ÷+· 21. + 22. ÷· 23. ÷ − 

24. −·− 25. ·−− 26. −· 

 

27. −÷ 

ALGEBRAIC 

 

28. x += 29. y +=y 30. a +−a =− 31. z −z+= 

32. y− =− 33. −x +=− 34. +−b =b 35. c −= 

36. −x = 37. 

w − = 

 

38. x +x− 39. y − y − 40. a 

−a ÷ 

41. b −b+ 

42. l ÷l ·− 43. z −+z · 44. ·+x ÷− 45. y +− 

46. ( 

t − ) 47. +b −·b 48. y −+y 49. ( 

) ·x 

50. −m+− 51. x +x+−x +x 52. −x

16 


REAL-WORLD APPLICATIONS 

 

 

53. 

 

 

54. 

 

55. 

es. Th 

π 

 

 

56. 

 

 

 

 

 

ThTh 

 

 

57. 58. g 

 

59. 

 

 

−x = 

x 

TECHNOLOGY 

x 

60. −x = 

 

61. − x −= 

EXTENSIONS 

62. 

 

64. 

Th 

 

66. 

√ — −++ 

63. 

Th 

 

65. 

√ — −−− 

67. Th 

 

68. 

+x −

SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 17 


In this section students will: 

Use the product rule of exponents. 

Use the quotient rule of exponents. 

Use the power rule of exponents. 

Use the zero exponent rule of exponents. 

Use the negative rule of exponents. 

Find the power of a product and a quotient. 

Simplify exponential expressions. 

Use scientific notation. 

1.2 EXPONENTS AND SCIENTIFIC NOTATION 

 

fi 

 

 

Th 

o fil fi 

····ENTERTh1.304596316E13 

ThE13 

· fifi 

 

Using the Product Rule of Exponents 

x ∙ x xff 

 

 

 

 

x ·x =x· x · x · x · x · x · x 

 

 

 

= x · x · x · x · x · x · x 

 

=x 

Thx ∙ x =x + =x 

 

Th 

product rule of exponents 

 

a m ·a n =a m+ n 

 

 

· = + = 

fi 

Th∙ 

ff 

the product rule of exponents 

amn 

 

a m ·a n =a m+n

18 


Example 1 

Using the Product Rule 

 

a. t ∙ t b. − ∙ (− c. x ∙ x ∙ x 

Solution 

a. t ∙ t =t + =t 

b. − ∙ (−=− ∙ (− =− + =− 

c. x ∙ x ∙ x 

fi 

e fi 

 

x ·x ·x =x ·x ·x =x + ·x =x ·x =x + =x 

 

Try It #1 

 

x ·x ·x =x ++ =x 

 

a. k ·k b. ( _ 

y ) ·( _ y ) c.t ·t ·t 

Using the Quotient Rule of Exponents 

quotient rule of exponents 

ff ym 

y n m >n 

y _ 

y 

 

y y · y · y · y · y · y · y ·y 

 

y=y· y· y · y · y 

·y 

 

 

 

 

= y· y· y· y· y· y · y · y ·y 

 

y· y· y· y· y 

 

y· y · y ·y 

= 

=y 

ff 

 

am 

 

a n=am−n 

 

 

 

y =y 

y =y− 

m >n m −n 

Th 

 

the quotient rule of exponents 

amnm > n 

 

am 

 

a n=am−n 

Example 2 

Using the Quotient Rule 

 

a. − 

− b. t 

t c. ( z√ — ) 

z√ —


Solution 

a. − =− 

− =−− 

b. t 

t =t − =t 

( 

c. z√ — ) 

z√ — = ( z√ — ) − =( z√ — ) 

Try It #2 

 

a. s 

s b. − 

− c. ef _ 

ef 

Using the Power Rule of Exponents 

 

x Th 

. Th 

c. − =− ∙ =− 

 

x =x ·x · x 

 

 

=( x·x )·( x·x )·( x·x ) 

 

= x · x · x · x · x · x 

 

=x 

Thx =x ∙ =x 

 

 

a m n =a m∙ n 

ff 

 

 

Product Rule 

Power Rule 

∙ 5 = + = = 3 ∙ 4 = 

x ∙ x = x + = x x = x 5 ∙ 2 = x 

a ∙a = a + = a a = a 7 ∙ 1 = a 

the power rule of exponents 

amn 

 

a m n =a m∙n 

Example 3 Using the Power Rule 

 

a. x b. t c. − 

Solution 

a. x =x ∙ =x 

b. t =t ∙ =t

20 


Try It #3 

 

a. y b. t c. −g 

Using the Zero Exponent Rule of Exponents 

m >nffm −n 

m =n 

 

 

t t 

 

t = t = 

 

 

t =t 

t 

=t− =Th 

 

 

a = 

Th Th 

fi 

the zero exponent rule of exponents 

a 

 

a = 

Example 4 

Using the Zero Exponent Rule 

 

a. c 

c b. j k 

−x x 

c. 

_ 

j k∙j k d. 

r _ 

r 

Solution 

a. c 

 

c =c− 

=c 

 

 

 

 

b. −x 

x =−3 ∙ x 

x 

=−∙x − 

=−∙x 

=−∙ 

=− 

j k 

c. 

_ 

j k 

j k∙j k = 

j k + 

_ 

 

 

= j k 

j k 

=j k − 

=j k 

 

= 

d. r − 

r =r 

=r 

= 5 ∙ 1 

=


Try It #4 

 

a. t 

t b. 

de _ 

de c. ·w 

w w d. t ·t 

t ·t 

Using the Negative Rule of Exponents 

m >n 

h 

hm

22 


Try It #5 

 

a. −t 

−t b. f 

f ⋅f c. k 

k 

Example 6 

Using the Product and Quotient Rules 

 

a. b ∙ b − b. −x ∙ (−x − c. −z 

−z 

Solution 

a. b ∙ b − =b − =b − = 

b 

b. −x ∙ (−x − =−x − =−x = 

c. −z 

−z = −z =−z − = 

−z =−z− −z 

Try It #6 

 

a. t − ⋅t b. 

 

Finding the Power of a Product 

power of a product rule of exponents 

 

pq 

 

 

pq =pq·pq·pq 

 

= p · q · p · q · p · q 

 

 

=p· p · p · q · q ·q 

 

=p ·q 

(pq =p ·q 

the power of a product rule of exponents 

a n 

 

ab n =a n b n 

Example 7 Using the Power of a Product Rule 

 

 

 

a. ab b. t c. −w d. 

−z e. e− f 

Solution fi 

a. ab =a ∙ (b =a 1 ∙ 3 ∙ b 2 ∙ 3 =a b 

b. t = ∙ (t = t =t 

c. −w =− ∙ (w =−8 ∙ w 3 ∙ 3 =−w 

 

d. 

−z = 

 

− ∙ (z = 

 

z 

e. e − f =e − ∙ (f =e −2 ∙ 7 ∙ f 2 ∙ 7 =e − f = f 

e


Try It #7 

 

 

 

a. g h b. t c. −y d. 

a b e. r s − 

Finding the Power of a Quotient 

 

 

 

 

e − f = f 

e 

ff 

 

e − f =( f 

e ) 

 

= f 

e 

 

 

e − f =( f 

e ) 

 

 

 

 

 

= f 

e 

f· 

= 

e · 

= f 

e 

the power of a quotient rule of exponents 

abn 

 

( a b ) n = an 

 

b n 

Example 8 

Using the Power of a Quotient Rule 

 

 

a. ( 

z ) b. ( 

p 

q ) 

c. ( − 

Solution 

a. ( 

z ) 

= 

z = 

z ∙= 

z 

b. ( 

p 

q ) 

= p 

 

 

q =p∙ q ∙=p q 

c. ( − 

t ) = 

− 

t =− 

t ∙=− 

t =− 

t 

t ) d. j k − e. m − n − 

d. j k − =( j 

k )= j 

 

k =j∙ k ∙=j 

k 

 

e. m − n − =( 

m n 

) =( 

m n ) = 

m n = 

 

m ∙ ∙ n ∙= 

 

m n

24 


Try It #8 

 

 

 

c ) 

a. ( b 

b. ( 

u ) c. ( − 

Simplifying Exponential Expressions 

w ) d. p − q e. c − d − 

 

. Th 

Example 9 

Simplifying Exponential Expressions 

 

a. m n − b. ∙ 17 − ∙ 17 − c. ( u− v 

 

v ) − d. −a b − a − b 

e. ( x √ — 

) ( x √ — − 

) 

f. w 

w − 

Solution 

a. m n − = m n − 

= m ∙ n −∙ 

=m n − 

= m n 

b. ∙ 17 − ∙ 17 − = −− Th 

= − 

 

= 

 

 

Th 

c. ( u− v 

 

v 

) − = u− v 

v − 

Th 

 

= u− v v − 

Th 

=u − v −− Th 

=u − v 

 

= v 

u 

Th 

d. −a b − a − b =−2 ∙ 5 ∙ a ∙ a − ∙ b − ∙ b 

=−0 ∙ a − ∙ b −+ Th 

=−ab 

e. ( x √ — ) ( x √ — ) − =( x √ — ) − Th 

=( x √ — ) 

= Th 

 

 

 

 

f. w ∙ (w 

w − = ∙ (w − 

 

= w ∙ 

w −∙ 

= w 

w − 

Th 

Th 

 

= w −− Th 

= w


Try It #9 

 

a. uv − − b. x ·x − ·x c. ( e f − 

Using Scientific Notation 

 

f − ) 

d. r − s r s − e. ( 

 

) tw− − ( 

 

) tw− f. h k 

 

h − k 

× 

 

ff 

 

scientific notation 

fi 

fi 

n 

nn 

fi 

 

ft 

 

ftTh 

ft. Th 

× 

 

 

 

ThTh 

 

× − 

scientific notation 

fia × n ≤a

26 


c. 

← 

× 

d. 

→ 

× − m 

e. 

→ 

× − 

Analysis Observe that, if the given number is greater than , as in examples a–c, the exponent of is positive; and if 

the number is less than , as in examples d–e, the exponent is negative. 

Try It #10 

fi 

a. 

b. 

c. 

d. 

e. 

Converting from Scientific to Standard Notation 

scientific notationn 

nn nn 

n 

Example 11 

Converting Scientific Notation to Standard Notation 

fi 

Solution 

a. × b. −× c × − d. −× − 

a. × 

 

→ 

 

b. −× 

− 

→ 

 

c. × − 

 

← 

 

d. −× − 

− 

←


Try It #11 

fi 

a. × b. −× c. −× − d. × − 

Using Scientific Notation in Applications 

fi 

1 

Th 

× 1 × Th 

×× ×× ≈× 

fi 

fifi 

× ×× =× Thfi 

×. Th 

× =×× =×× =× 

Example 12 

Using Scientific Notation 

fi 

a. × − × 

b. × ÷−× 

c. × × 

d. × ÷× 

e. × −× × 

Solution 

a. × − × =× − × 

 

= 

=× fi 

b. × ÷−× =( 

− ) ( 

) 

 

 

≈− − 

=−× − fi 

c. × × =× × 

 

= 

=× fi 

d. × ÷× = ( ) ( 

) 

 

 

= 

=× fi 

e. × −× × =×−× × × 

 

≈− 

 

=−×

28 


Try It #12 

fi 

a. −× × − 

b. × ÷× 

c. × × 

d. × ÷−× 

e. −× × −× 

Example 13 

Applying Scientific Notation to Solve Problems 

Th 

fififi 

fi 

Solution Th=× 

Th≈× 

o fi 

 

 

 

× ÷× = ( 

 

) 

) × ( 

≈× 

=× 

Th× 

Try It #13 

 

0.000008 

 

fi 

Exponential Notation (http://openstaxcollege.org/l/exponnot) 

Properties of Exponents (http://openstaxcollege.org/l/exponprops) 

Zero Exponent (http://openstaxcollege.org/l/zeroexponent) 

Simplify Exponent Expressions (http://openstaxcollege.org/l/exponexpres) 

Quotient Rule for Exponents (http://openstaxcollege.org/l/quotofexpon) 

Scientific Notation (http://openstaxcollege.org/l/scientificnota) 

Converting to Decimal Notation (http://openstaxcollege.org/l/decimalnota)


29 


VERBAL 

1. 2. 

3. 4. 

NUMERIC 

 

5. 6. − 7. · 8. ÷ 

9. − 10. − 11. ÷ 12. · − 

13. 14. − ÷ 

 

 

15. · ÷ − 16. 

17. · 18. ÷ − 

19. − · − 20. ÷ 

fi 

21. 22. 

fi 

23. × 24. × − 

ALGEBRAIC 

 

25. a a 

a 26. mn 

m − 27. b c x 28. ( − y ) 

− 

29. ab ÷d − 30. w x − 31. m 

n 

32. y− y 

33. p− q 

p q − 34. l ×w 35. y ÷x 36. ( a 

) 

37. m ÷ m 

( 

38. √ — x ) y − 39. 

a − 

41. b − c 42. x y ÷y 43. z − y 

 

40. ma 

 

m a 


44. 

× 

 

46. Th 

 

 

45. . A 

× −3 

 

47.

30 


48. Th 

× 

 

49. × − 

 

50. Th 

 

 

TECHNOLOGY 

 

m 

51. ( 

− ) 

EXTENSIONS 

52. ÷ x 

 

− 

53. ( 

a ) ( a ) 55. m n 

a c −· a− n − 

m c 

56. ( 

x y 

x y −· y − 

x ) − 

57. ( ab c − 

b − ) 

54. − ÷ 

( x 

y ) − 

 

 

58. 

 

 

Th× 

 

59. 

 

. Th 

× −

SECTION 1.3 RADICALS AND RATIONAL EXPONENTS 31 


In this section, you will: 

Evaluate square roots. 

Use the product rule to simplify square roots. 

Use the quotient rule to simplify square roots. 

Add and subtract square roots. 

Rationalize denominators. 

Use rational roots. 

1.3 RADICALS AND RATIONAL EXPONENTS 

 

fi 

Figure 1n Th 

 

c 

Figure 1 

a 2 +b 2 =c 2 

+ =c 2 

=c 2 

fi 

fifi 

Evaluating Square Roots 

= 

Th 

 

aa 

aTh 

Thprincipal square rootaTh 

 

Tha√ — a Th radical 

radicand radical expression 

 

√ — 

 

 

 

 

principal square root 

principal square rootaa 

radical expression radicalradicand√ — a

32 


Q & A… 

Does √ — 25 = ±5? 

− 

Th√ — = 

Example 1 Evaluating Square Roots 

 

a. √ — b. √ — √ c. √ — + d. √ — −√ — 

Solution 

a. √ — = = 

b. √ — √ =√ — = = = 

c. √ — += √ — = = 

d. √ — −√ — =−=− = = 

Q & A… 

For √ — 25 + 144 , can we find the square roots before adding? 

√ — +√ — =+=Th√ — +=Th 

e fi 

Try It #1 

a. √ — b. √ — √ — c. √ — − d. √ — +√ — 

Using the Product Rule to Simplify Square Roots 

Th 

Thfi 

product rulefor simplifying square roots 

√ — √ — ∙ √ — 

 

the product rule for simplifying square roots 

ababab 

 

√ — ab =√ — a · √ — b 

How To… 

 

1. 

2. 

3. 

Example 2 

Using the Product Rule to Simplify Square Roots 

 

a. √ — b. √ — a b 

Solution 

a. √ — 100 ∙ 3 

√ — ∙ √ — 

√ —


b. √ — a b ∙ 2a 

√ — a b ∙ √ — a 

a b √ — a 

 

Try It #2 

√ — x y z 

How To… 

 

1. 

2. 

Example 3 

Using the Product Rule to Simplify the Product of Multiple Square Roots 

√ — ∙ √ — 

Solution 

√ — 2 ∙ 3 

√ — 

 

 

Try It #3 

√ — x ·√ — x x > 

Using the Quotient Rule to Simplify Square Roots 

 

quotient rule for simplifying square roots 

 

 

√ 

 

√— 

√ — 

the quotient rule for simplifying square roots 

Th_ 

a a b b ≠ 

b 

 

 

√ __ a 

b = ____ a √— 

√ — b 

How To… 

 

1. 

2. 

Example 4 

Using the Quotient Rule to Simplify Square Roots 

 

 

√ 

 

Solution 

 

√— √ — 

 

√—

34 


Try It #4 

 

√ ____ x 

y . 

Example 5 

Using the Quotient Rule to Simplify an Expression with Two Square Roots 

√— x y _________ 

√ — x y 

Solution 

√ x y 

x y 

 

√ — x 

x 

Try It #5 

________ 

√— a b √ — a b . 

Adding and Subtracting Square Roots 

 

√ — √ 

— √ 

— ft 

Th√ — 

√ — √ 

— +√ 

— =√ 

— +√ 

— =√ 

— 

How To… 

 

1. 

2. 

Example 6 

Adding Square Roots 

√ — +√ — 

Solution 

√ — √ — √ — √ — √ — 

√ — √ — 

√ — +√ — =√ — 

Try It #6 

√ — +√ — 

Example 7 

Subtracting Square Roots 

√ — a b c−√ — a b c


Solution 

 

 

√ — a b =√ — √ — √ — √ — a √ — a √ — b √ — c 

 

=∣a ∣b √ — ac 

 

=∣ a ∣b √ — ac 

 

√ — a b c =√ — √ — √ — a √ — a √ — b √ — c 

 

=∣ a ∣b √ — ac 

 

=∣ a ∣b √ — ac 

 

 

∣ a ∣b √ — ac −∣ a ∣b √ — ac =∣ a ∣b √ — ac 

Try It #7 

√ — x −√ — x . 

Rationalizing Denominators 

 

rationalizing the 

denominator. 

 

 

 

 

b √ — c ____ 

√— c 

√ — c 

 

 

a +b √ — c a −b √ — c . 

How To… 

 

1. 

2. 

Example 8 

Rationalizing a Denominator Containing a Single Term 

√— 

√ — 

Solution 

√ — √— 

√ — 

√— 

√ — · √— √ — 

√— 

√—

36 


Try It #8 

√— 

√ — 

How To… 

 

1. 

2. 

3. 

4. 

Example 9 

Rationalizing a Denominator Containing Two Terms 

 

 

+√ — 

Solution 

 

+√ — −√ — −√— 

−√ — 

 

 

 

+√ — ∙ −√— −√ — 

 

−√— − 

√ — − 

Try It #9 

 

 

+√ — 

Using Rational Roots 

 

 

Th 

 

Understanding nth Roots 

a 3 =fi = 

 

na na−− 

− =−anprincipal nth roota 

ana 

Thna n √ — a n 

n index 

principal nth root 

a nprincipal nth roota n √ — a 

ana. Thindexn


Example 10 

Simplifying n th Roots 

 

— 

a. √ 

— − 

b.√ 

— √ 

— c. − 

√ 

x d. √ 

— − √ 

— 

Solution 

a. √ 

— −=−− =− 

b. √ 

— = = 

c. − √ 

— x 

√ 

— 

−x 

d. √ 

— − √ 

— 

√ 

— 

 

 

 

 

Try It #10 

 

a.√ 

− b. √ 

 

 

√ 

 

c. √ 

+ √ 

 

Using Rational Exponents 

 

Thna 

 

a n 

= √ n — a 

 

nTh 

 

 

a m n =( n √ — a ) m = n √ — a m 

 

rational exponents 

nTh 

 

 

a m n 

=( √ n — a ) m = √ n — a m 

How To… 

 

1. 

2. 

3. 

Example 11 

Writing Rational Exponents as Radicals 

 

Solution 

 

 

=( √ 

— ) = √ 

—

38 


√ 

— = =fifi 

o fit fi 

Try It #11 

 

 

=( √ 

— ) = = 

Example 12 

Writing Radicals as Rational Exponents 

 

 

 

√ 

— a 

Solution 

 

 

 

a 

 

 

√ 

— a =a − 

 

Try It #12 

x √ — y 

Example 13 

Simplifying Rational Exponents 

 

a. ( x 

Solution 

a. x 

x 

) ( x 

) b. ( 

 

x 

 

) − 

 

+ 

 

x 

 

 

b. ( ) 

 

√ 

 

√— √ — 

 

 

 

 

 

 

Try It #13 

x 

( x 

) 

 

Radicals (http://openstaxcollege.org/l/introradical) 

Rational Exponents (http://openstaxcollege.org/l/rationexpon) 

Simplify Radicals (http://openstaxcollege.org/l/simpradical) 

Rationalize Denominator (http://openstaxcollege.org/l/rationdenom)


39 


VERBAL 

1. 

 

 

3. 

 

2. 

 

4. 

 

NUMERIC 

 

5. √ — 6. √ — √ 7. √ — + 8. √ — −√ — 

9. √ — 10. √ — 11. √ — 12. 

 

√ 

 

13. 

√ 

14. √— 15. √ — +√ — 

 

16. √ 

 

17. 

√ — 18. √— 19. √ — −√ — 20. √ — +√ — 

21. √ — 22. 

 

√ 23. ( √ — )( √ — ) 24. √ — −√ — 

25. 

 

√ 

 

 

26. 

√ 

 

 

27. 

√ 

 

28. 

 

+√ — 

 

29. 

−√ — 

30. √ 

 

— 

31. √ 

— + √ 

— 

— 

√ 

32. − 

 

33. √ 

— 

34. √ 

— −+ √ 

— 

√ 

— 

ALGEBRAIC 

 

35. √ — x 36. √ — y 37. √ — p 38. p q 

 

 

39. m 

√ — 40. √ — m +√ — 41. √ — ab −b√ — a 42. √— n 

√ — n 

43. √ x x 44. √— z +√ — z 45. √ — y 46. √ — bc 

47. 

 

√ d 

48. q √ — p 

√ — 

49. 

 

−√ — x 

50. √ d 

 

 

 

51. w 

√ — 

−w 

√ — 52. √ — x +√ — x 53. √— x +√ — 

54. √— k 

55. √ — n 56. √ q 

q 

57. √ m 

m 

58. √— c −√ — c

40 


59. 

√ 

d 

60. √ 

 

— x + √ 

— x 

 

— 

61. x 

√ 

x 

62. √ 

 

— y 

63. √ 

— z − √ 

— −z 

64. √ 

— c 


65. 

 

 

Th . Th 

 

 

Th 

 

√ — + 

 

66. − √— 

√ — t s t 

ft 

 

EXTENSIONS 

 

67. √— −√ — 

−√ — 

− 

 

 

68. − 

 

 

 

 

69. √— mn 

a √ — c− · a− n − 

√ — m c 

 

 

70. a 

a −√ — c 

71. x√— y +√ — y 

√ — y 

( 72. x √— √ — b )( √— b √ — x ) 

 

 

73. √ √ 

— + √ 

— 

√ — +√ —

SECTION 1.4 POLYNOMIALS 41 



Identify the degree and leading coefficient of polynomials. 

Add and subtract polynomials. 

Multiply polynomials. 

Use FOIL to multiply binomials. 

Perform operations with polynomials of several variables. 

1.4 POLYNOMIALS 

Th 

fi 

Figure 1 

 

 

 

 

x 

x 

t fi 

 

 

 

Figure 1 

A =s 

=x 

=x 

Thn fi 

 

 

 

A =_ bh 

=_ 

x( _ 

) 

=_ x 

t fi 

 

A =lw 

 

= x · 

 

= x 

Th 

x +_ x −x ft x +_ x ft 

 

 

Identifying the Degree and Leading Coefficient of Polynomials 

Thpolynomialff 

 

πcoefficientffi 

a i 

x i πwterm of a polynomial 

constant

42 


x monomial. 

x −binomial−x +x −trinomial 

fidegree 

Thleading termfiThffi 

leading coefficient 

 

 

 

a n 

x n ++a x +a x +a 

 

polynomials 

polynomial 

a n 

x n ++a x 

+a x +a 

a i 

coefficientTha 

constant 

a i 

x i term of a polynomialTh 

degreeThleading termffi 

leading coefficient 

How To… 

ffi 

1. x 

2. xo fi 

3. ffi 

Example 1 

Identifying the Degree and Leading Coefficient of a Polynomial 

ffi 

a. +x −x b. t −t +t c. p −p − 

Solution 

a. ThxTh−x 

Th − 

b. tt 

Th 

c. p−p 

Th − 

Try It #1 

 

x −x +x − 

Adding and Subtracting Polynomials 

 

x −x x xx


How To… 

 

1. 

2. 

Example 2 

Adding Polynomials 

 

x +x −+x +x −x + 

Solution 

x +x +x +x −x+−+ 

x +x +x − 

Analysis We can check our answers to these types of problems usinggraphing calculator. To check, graph the problem 

as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to 

compare the graphs. Using different windows can make the expressions seem equivalent when they are not. 

Try It #2 

 

x +x −x ++x −x − 

Example 3 

Subtracting Polynomials 

 

x −x +x +−x −x +x + 

Solution 

x −x +−x +x +x −x+− 

x −x +x +x − 

Analysis Note that finding the difference between two polynomials is the same as adding the opposite of the second 

polynomial to the first. 

Try It #3 

 

−x −x +x −−x −x −x + 

Multiplying Polynomials 

 

fi 

 

 

 

Multiplying Polynomials Using the Distributive Property 

Th 

x +x + 

fi

44 


How To… 

 

1. e fi 

2. 

3. 

Example 4 

Multiplying Polynomials Using the Distributive Property 

 

x +x −x + 

Solution 

x x −x ++x x + 

x −x +x +x −x + 

x +−x +x +x −x+ 

x +x +x + 

Analysis We can usetable to keep track of our work, as shown in Table 1. Write one polynomial across the top and 

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 

all of the terms together, combine like terms, and simplify. 

x −x + 

x x −x x 

+ x −x 

Table 1 

Try It #4 

 

x + − + 

Using FOIL to Multiply Binomials 

o fi 

foil 

 

 

 

Thfi 

 

How To… 

 

1. e fi 

2. 

3. 

4. 

5. 

6.


Example 5 

Using FOIL to Multiply Binomials 

o fi 

x −x + 

Solution 

 

 

 

e fi 

x − x + xx = x 

 

 

 

 

x − x + x = x 

 

 

 

 

x − x + −x = −x 

 

 

 

 

x − x + − = − 

x +x −x − 

x +x −x− 

x −x − 

Try It #5 

o fi 

x+x − 

Perfect Square Trinomials 

perfect square 

trinomialfi 

 

 

 

x+ =x +x + 

 

x− =x −x + 

 

x − =x −x + 

fifi 

Th 

e fi 

perfect square trinomials 

fi 

 

 

x+a =x+ax+a=x +ax+a

46 


How To… 

 

1. e fi 

2. 

3. 

4. 

Example 6 

x − 

Expanding Perfect Squares 

Solution fi 

 

x −x+− 

 

x −x + 

Try It #6 

x − 

Difference of Squares 

difference of squares 

x+x− 

 

 

x+x−=x −x +x − 

 

=x − 

Thff 

 

 

x+x−=x − 

 

x+x−=x − 

 

x +x −=x − 

 

e fi 

Q & A… 

Is therespecial form for the sum of squares? 

Thff 

Th 

difference of squares 

 

e fi 

 

a+ba−b=a −b 

How To… 

n, fiff 

1. e fi 

2. 

3. e fi


Example 7 

x +x − 

Multiplying Binomials Resulting in a Difference of Squares 

Solution fix =x = 

e fio fix − 

Try It #7 

x +x − 

Performing Operations with Polynomials of Several Variables 

 

 

 

a+ba −b −c 

aa −b −c+ba −b − c 

a −ab−ac+ab−b −bc 

a +−ab+ab−ac−b −bc 

a +ab−ac−bc−b 

Example 8 Multiplying Polynomials Containing Several Variables 

x+x −y + 

Solution 

xx −y ++x −y + 

x −xy+x +x −y + 

x −xy+x +x−y + 

x −xy+x −y + 

Try It #8 

x −x +y − 

 

Adding and Subtracting Polynomials (http://openstaxcollege.org/l/addsubpoly) 

Multiplying Polynomials (http://openstaxcollege.org/l/multiplpoly) 

Special Products of Polynomials (http://openstaxcollege.org/l/specialpolyprod)

48 



VERBAL 

1. t: Th 

 

 

 

3. 

 

 

 

2. 

 

s a diff 

 

4. 

 

 

ALGEBRAIC 

 

5. x −x + 6. m +m −m + 7. −a +b 8. p −p m +m 

9. x +x + 10. y −y +y − 

, fiff 

11. x +x−x − 12. z +z −+−z +z + 

13. w +w +−w −w + 14. a +a −a −+−a −a +a + 

15. b −b +b −b +−b +b +b 16. p −+p −p + 

, fi 

17. x +x − 18. c +cc −c 19. b −b − 20. d −d + 

21. v −v − 22. t +t−t + 23. n −n + 

 

24. x + 25. y − 26. −x 27. p + 

28. m − 29. y − 30. b + 

 

31. c +c − 32. a −a + 33. n −n + 34. b +b − 

35. +m−m 36. p +p − 37. q −q + 

 

38. x +x +x − 39. t +t −t − 40. x −x −x + 

41. y −y −y − 42. k −k +k − 43. p +p −p − 

44. m −m −m + 45. a +ba −b 46. x −yx −y 

47. t −u 48. m +n −m + 49. t −xt −x + 

50. b −a +ab +b 51. r −dr +d 52. x +yx −xy+y 


53. 

. Th 

x +x − x 

 

 

EXTENSIONS 

 

54. 

. Th 

x + . Thx + 

x 

 

 

55. t − t +−t +t + 56. b +b −b − 

57. a +ac +c a −c

SECTION 1.5 FACTORING POLYNOMIALS 49 



Factor the greatest common factor of a polynomial. 

Factor a trinomial. 

Factor by grouping. 

Factor a perfect square trinomial. 

Factor a difference of squares. 

Factor the sum and difference of cubes. 

Factor expressions using fractional or negative exponents. 

1.5 FACTORING POLYNOMIALS 

fiTh 

Figure 1 

 

 

 

 

x 

 

Figure 1 

Th 

 

A =lw 

 

=x·x 

 

=x 

ThTh 

A = = = Th 

x −A =lw=x −=x − 

+x −=x 

Th −x Th 

xx − fi 

 

 

Factoring the Greatest Common Factor of a Polynomial 

x 

greatest common factor (GCF) 

 

Thxxx 

xx 

fi 

ffi

50 


greatest common factor 

greatest common factor 

 

How To… 

 

1. ffi 

2. 

3. o fi 

4. 

5. 

Example 1 

x y +x y +xy 

Factoring the Greatest Common Factor 

Solution fiThThx x xx 

x n y y 

yyo fixy 

 

xyx y =x y xyxy=x y xy=xy 

 

xyx y +xy+ 

Analysis After factoring, we can check our work by multiplying. Use the distributive property to confirm that 

xyx y + xy + = x y + x y + xy. 

Try It #1 

xb −a+b −a 

Factoring a Trinomial with Leading Coefficient 1 

 

Thx +x + 

x+x+ 

x +bx+ficb 

x +x + 

. Thx+x+ 

factoring a trinomial with leading coefficient 1 

x +bx+cx+px+qpq= cp +q =b 

Q & A… 

Can every trinomial be factored asproduct of binomials? 

. Th


How To… 

x +bx+c 

1. c 

2. pqcb 

3. x+px+q 

Example 2 Factoring a Trinomial with Leading Coefficient 1 

x +x − 

Solution ffib =c =−fi 

−Table 1e fi 

Factors of −15 

− 

− 

− 

− 

Sum of Factors 

− 

 

− 

 

fipq−x−x+ 

Analysis We can check our work by multiplying. Use FOIL to confirm that x − x + = x + x − . 

Table 1 

Q & A… 

Does the order of the factors matter? 

 

Try It #2 

x −x + 

Factoring by Grouping 

ffi 

factor by grouping 

Thx +x +x +x+ 

x +x +x + 

xx++x+x+o fi 

factor by grouping 

ax +bx+c ac 

bx 

 

How To… 

ax +bx+c 

1. ac 

2. pqacb 

3. ax +px+qx+c 

4. ax +px 

5. qx+c 

6.

52 


Example 3 

x +x − 

Factoring a Trinomial by Grouping 

Solution a =b =c =−ac=−fi 

−Table 2e fi 

p =−q = 

Factors of −30 

− 

− 

− 

− 

− 

− 

Table 2 

Sum of Factors 

− 

 

− 

 

− 

 

x −x +x − ax +px+qx+c 

xx −+x − 

x −x+ 

Analysis We can check our work by multiplying. Use FOIL to confirm that x − x + = x + x − . 

Try It #3 

a. x +x + b. x +x − 

Factoring a Perfect Square Trinomial 

 

fi 

 

 

a +ab+b =a + b 

 

 

 

a −ab+b =a − b 

 

perfect square trinomials 

 

 

a +ab+b =a+b 

How To… 

 

1. fie fi 

2. fiab 

3. a + b


Example 4 

x +x + 

Factoring a Perfect Square Trinomial 

Solution x x =x = Th 

x. Thx=x 

Thx + 

Try It #4 

x −x + 

Factoring a Difference of Squares 

ffff 

 

 

 

a −b =a + ba − b 

ff 

differences of squares 

A diff 

 

a −b =a + ba−b 

How To… 

ff 

1. fie fi 

2. a+ba−b 

Example 5 

x − 

Factoring a Difference of Squares 

Solution x x =x = Th 

ffx +x − 

Try It #5 

y − 

Q & A… 

Is thereaformula to factor the sum of squares? 

 

Factoring the Sum and Difference of Cubes 

ff 

 

 

a +b =a + ba −ab+b 

ff 

 

a −b =a − ba +ab+b

54 


ffThfi 

SOAP 

 

x − =x−x +x + 

Thfisamex − Thxopposite 

x − always positive 

sum and difference of cubes 

 

 

e diff 

 

a +b =a + ba −ab+b 

a −b =a − ba +ab+b 

How To… 

ff 

1. fie fia +b a −b 

2. a+ba −ab+b ff 

a−ba +ab+b 

Example 6 

x + 

Factoring a Sum of Cubes 

Solution x =x+x −x + 

Analysis After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be 

factored further. However, the trinomial portion cannot be factored, so we do not need to check. 

Try It #6 

a +b 

Example 7 

x − 

Factoring a Difference of Cubes 

Solution x x =x = ff 

x −x +x + 

Analysis Just as with the sum of cubes, we will not be able to further factor the trinomial portion. 

Try It #7 

e diffx −


Factoring Expressions with Fractional or Negative Exponents 

 

 

Thx +x 

x x ( +x ) 

Example 8 

Factoring an Expression with Fractional or Negative Exponents 

xx+ − +x+ 

 

Solution x+ − 

x+ − x +x+ 

x+ − x +x + 

 

 

x+ − x + 

Try It #8 

a − +aa − − 

 

Identify GCF (http://openstaxcollege.org/l/findgcftofact) 

Factor Trinomials when a Equals 1 (http://openstaxcollege.org/l/facttrinom1) 

Factor Trinomials when a is not equal to 1 (http://openstaxcollege.org/l/facttrinom2) 

Factor Sum or Difference of Cubes (http://openstaxcollege.org/l/sumdifcube)

56 



VERBAL 

1. 

 

2. 

r a diff 

 

3. 

ALGEBRAIC 

, fi 

4. x +xy−xy 5. mb −m ba +ma 

6. x y −x y +xy 7. p m −p m +m 

8. j k −j k +j k 9. y −y +y −y 

 

10. x +x − 11. a +a − 12. c +c + 13. n −n − 

14. w −w + 15. p −p − 

 

16. x +x − 17. h −h − 18. b −b − 19. −d + 

20. v −v + 21. t +t − 22. n −n − 23. x − 

24. y − 25. p − 26. m − 27. d − 

28. x − 29. b −c 30. a −a + 31. n +n + 

32. x −x + 33. y +y + 34. m −m + 35. p −p + 

36. q +q + 

 

37. x + 38. y − 39. a + 40. b −d 

41. x − 42. q + 43. r +s 

44. xx − − +x − 

46. tt + +t + 

48. yy − −y − 

 

 

45. cc + − −c + 

47. xx + − +x + 

49. zz − − +z − − 

 

50. dd + − +d +


57 


 

fiThfi 

Thx +x − fi 

Th 

l × w=x +x− 

51. 

 

53. 

in. Th 

x − 

 

52. k. Th 

x +x + 

 

 

 

flThfi 

. The flx −x +y 

x 

−x+ 

 

 

54. e fl 

EXTENSIONS 

 

55. x −x + 56. y − 57. z −a 

58. xx + − +x + 

59. x +x −x − −

58 




Simplify rational expressions. 

Multiply rational expressions. 

Divide rational expressions. 

Add and subtract rational expressions. 

Simplify complex rational expressions. 

1. 6 RATIONAL EXPRESSIONS 

fiTh 

x+x 

 

 

+x 

x 

 

 

 

Simplifying Rational Expressions 

Thrational expression 

 

fi 

 

x +x + 

 

x +x + 

 

x+ 

 

 

x+x+ 

Thx+ 

How To… 

 

1. 

2. 

x + x + 

Example 1 

Simplifying Rational Expressions 

x 

− 

x +x + 

x+x− 

Solution 

 

 

x+x+ 

 

 

x − x + 

x+ 

Analysis We can cancel the common factor because any expression divided by itself is equal to . 

Q & A… 

Can the x 2 term be cancelled in Example 1? 

. Thx

SECTION 1.6 RATIONAL EXPRESSIONS 59 

Try It #1 

x − x − 

Multiplying Rational Expressions 

 

fifi 

 

ft 

How To… 

 

1. 

2. 

3. 

4. 

Example 2 

Multiplying Rational Expressions 

 

Solution 

x +x − x + ∙ x − 

x + 

 

x+x− 

x+ ∙ 

x − 

x+ 

 

 

 

 

x+x−x 

− x+x+ 

x+x−x − x+ x+ 

 

 

x−x 

− x+ 

Try It #2 

 

x +x + 

 

x +x + · x +x + x +x + 

Dividing Rational Expressions 

 

fi 

x ÷x 

 

x ∙ 

x 

 

 

x · 

x = 

x

60 


How To… 

 

1. e fi 

2. 

3. 

4. 

5. 

Example 3 

Dividing Rational Expressions 

 

Solution 

 

x +x − x − ÷ 

x − 

x +x + 

x +x − 

x − +x + ∙x x − 

 

 

 

 

x −x+ 

 

x+x− ∙x+x+ 

 

x+x− 

x −x+x+x+ 

 

x+x−x+x− 

x −x+x+x+ 

 

x+x−x+x− 

x −x 

+ 

x − x − 

 

 

 

Try It #3 

 

x 

− −x − 

x +x − ÷x 

x +x − 

Adding and Subtracting Rational Expressions 

 

o fi 

 

 

 

+ 

= + 

 

= 

 

 

= 

 

 

 

Thleast common denominatorTh 

fi 

x+x+x+x+ 

x+x+x+ 

fi 

x+x+ x + 

x+x+ x + 

x + 

x +


How To… 

 

1. 

2. 

3. 

4. 

5. 

Example 4 

 

 

Adding Rational Expressions 

x + 

y 

Solution fixy 

xy 

 

 

x · y 

y + 

y · 

x x 

y 

xy +x 

xy 

o fi 

x +y 

 

xy 

 

Analysis Multiplying by y _ 

y 

or 

_ x x 

does not change the value of the original expression because any number divided by 

itself is , and multiplying an expression by gives the original expression. 

Example 5 Subtracting Rational Expressions 

 

Solution 

 

 

 

x 

+x + − 

x − 

 

 

 

 

 

 

 

 

 

x+ − 

 

x+x− 

 

 

x+ ∙x 

− x − − 

 

x+x− ∙ x + x + 

x− 

− 

x+ x+ 

 

x− x+ x− 

x −−x 

+ x+ x− 

x − 

 

x+ x− 

x− 

 

x+ 

x− 

 

 

 

 

 

 

Q & A… 

Do we have to use the LCD to add or subtract rational expressions?

62 


Try It #4 

 

 

x + − 

x − 

 

Simplifying Complex Rational Expressions 

 

 

Th 

a 

fi 

 

b +c 

a + bc 

 

 

b 

a 

∙ b 

+bc 

 

ab 

+bc 

How To… 

 

1. 

2. 

3. 

4. 

5. 

6. 

Example 6 

Simplifying Complex Rational Expressions 

y + 

x 

 

x y 

 

Solution 

 

 

 

y ∙ x x + 

x 

xy 

x + 

x 

xy+ x 

 

x 

x 

 

 

 

xy+ x 

__ x 

y 

 

 

xy+ 

x 

÷ x y 

xy+ 

x 

∙ y 

x 

 

 

yxy+ x


Try It #5 

__ x y −y __ 

x y 

 

Q & A… 

Can a complex rational expression always be simplified? 

fi 

 

Simplify Rational Expressions (http://openstaxcollege.org/l/simpratexpress) 

Multiply and Divide Rational Expressions (http://openstaxcollege.org/l/multdivratex) 

Add and Subtract Rational Expressions (http://openstaxcollege.org/l/addsubratex) 

Simplify a Complex Fraction (http://openstaxcollege.org/l/complexfract)

64 



VERBAL 

1. 

 

3. 

 

 

2. 

 

ALGEBRAIC 

 

4. 

x − x −x + 

5. y +y + 

y +y + 

6. −a + a 

a − 7. +b + b 

b + 

8. 

m − m − 

9. +x − x x +x − 

10. x +x − 

 

x +x + 

11. a +a + a +a − 

12. c +c − 

c −c + 

13. −n − n 

 

n −n − 

 

14. x −x − x +x − · x +x − 

x − 15. c +c − 

 

c +c + · c −c + 

 

c −c + 

16. d +d − 

 

d +d + · d +d − 

d +d − 

18. b +b + 

b − · b +b − 

b −b − 

17. h −h − 

h −h + · h −h + 

h −h − 

19. +d + d 

d − · d −d + 

d − 

20. x −x − 

 

x −x − · x −x − 

x +x + 21. 

t − 

t +t + · t +t − 

t −t + 

22. n −n − 

n +n − · n −n + 

 

n −n + 

23. 

x − 

 

 

x +x + · x +x + 

x +x + 

 

24. y −y − +y − 

y −y − ÷y y +y − 

25. p +p − 

−p + 

p +p + ÷p 

p +p − 

q − 

26. −q − 

q +q + ÷q q +q − 

27. +d − d 

+d − 

d −d + ÷d 

 

d −d + 

28. x +x − 

+x + 

x −x − ÷x 

 

x +x + 

30. a −a + 

 

a +a − ÷ 

a − 

 

a +a + 

29. 

b − 

 

−b + 

b −b − ÷b 

b −b − 

31. +y + y 

+y + 

y +y − ÷y 

 

y −y + 

32. x +x − 

+x − 

x −x + ÷x 

 

x −x +


65 

 

33. 

x + y 

34. q − 

p 

 

35. 

a + + 

a − 

36. c + −c − 

39. z 

z + +z + 

z − 

37. y + y − +y − y + 

p 

40. p + −p + 

p 

38. x − x + −x + x + 

x 

41. x + + y 

y + 

 

42. 

y − 

x 

 

 

y 

 

43. _ + 

__ 

a b 

b 

44. _ x 

− p 

p 

 

45. _ 

a 

+_ b 

 

b 

a 

46. 

 

x + + 

x − 

 

x − 

 

x + 

 

47. _ a 

b −b 

a 

 

a +b 

 

ab 

48. 

x 

+x 

49. 

c 

_ x c + +c − c + 

 

c + 

 

c + 

 

50. x _y − y _ 

x 

 

x _ 

y 

+ y _ 

x 


51. Th 

x −x −7 ft Th 

x −x +1 ft 

x −x − 

 

x −x + 

=x −x − 

52. Thx −625 ft A p 

x −x +25 ft 

 

 

53. 

x +x +81 ft 

x −81 ft 

 

 

EXTENSIONS 

 

54. x +x − x −x − · x −x − +x + 

x −x − ÷x 

x +x + 55. y −y + 

 

y +y − ∙ −y − y 

 

y −y − 

 

y − 

 

56. a + a − +a − a + 

 

 

a + 

a 

57. x +x + +x + 

x +x − ÷x 

+x − 

x −x − ÷x x +x −

66 


CHAPTER 1 REVIEW 

Key Terms 

algebraic expression 

associative property of addition 

a +b+c=a+b+c 

associative property of multiplication 

, a·b·c=a·b·c 

base 

binomial 

coefficient a i 

a n 

x n ++a 

x +a 

x +a 

commutative property of addition 

a +b =b +a 

commutative property of multiplication 

a·b =b ·a 

constant 

degree 

difference of squares 

 

distributive property 

a·b+c=a·b + a·c 

equation 

exponent 

 

exponential notation 

factor by grouping ax +bx+cx 

 

formula 

greatest common factor 

identity property of addition 

, a +=a 

identity property of multiplication 

, a·=a 

index n 

integers −−− 

inverse property of addition a 

−a, a +−a= 

inverse property of multiplication a 

a 

, a· a = 

irrational numbers 

 

leading coefficient ffi 

leading term 

least common denominator

CHAPTER 1 REVIEW 67 

monomial 

natural numbers 

order of operations 

perfect square trinomial 

polynomial 

principal nth root a tna 

principal square root f a naa 

radical 

radical expression 

radicand 

rational expression 

rational numbers __ 

m n 

mnn ≠ 

 

real number line 

 

real numbers 

scientific notation a × n ≤∣a ∣< 

n 

term of a polynomial a i 

x i a n 

x n ++a 

x +a 

x +a 

trinomial 

variable 

whole numbers 

Key Equations 

Rules of Exponents 

s a a 

 

a m ∙ a n =a m+n 

 

 

 

 

 

 

 

 

 

 

 

ff 

am 

=a 

a n m−n 

a m n =a m∙n 

a = 

a −n = 

 

a n 

a ∙ b n =a n ∙ b n 

( a __ 

b ) n = an 

b n 

x+a =x+ax+a=x +ax+a 

a+ba−b=a −b 

a −b =a+ba − b 

a +ab+b =a+b 

a +b =a+ba −ab+b 

a −b =a−ba +ab+b

68 


Key Concepts 

1.1 Real Numbers: Algebra Essentials 

Example 1Example 2 

Example 3 

ThExample 4 

Example 5 

ThExample 6 

Th 

Th 

Example 7 

 

Example 8Th 

Example 9Example 10Example 12. 

Th 


1.2 Exponents and Scientific Notation 

Example 1 

Example 2 

Example 3 

fiExample 4 

fiExample 5Example 6 

ThExample 7 

ThExample 8 

ThExample 9 

fiExample 10 

Example 11 

fiExample 12 

Example 13 

1.3 Radicals and Rational Expressions 

Thas a 

Example 1 

ababab 


ab __ a b ab 

Example 4 Example 5 

Example 6 

Example 7 

 

 


Thna ia tna. Th 

Example 10 

 


ThExample 13


1.4 Polynomials 

. Th 

. Th 

ffiffiExample 1 

Example 2 Example 3. 

e fi 

. ThExample 4 

Example 5 


Example 8 

1.5 Factoring Polynomials 

The fi 

Example 1 

ffiy fi 

Example 2 

Example 3 

 


Th Example 6Example 7 

Example 8 

1.6 Rational Expressions 

 

Example 1 

 

Example 2 

Example 3 

s fiExample 4 

Example 5 

. Th 

Example 6

70 


CHAPTER 1 REVIEW EXERCISES 

REAL NUMBERS: ALGEBRA ESSENTIALS 

 

1. −· − 2. ÷·+÷ 3. · +÷ 

 

4. x +=− 5. y + = 

 

6. y +÷·+ 7. m+−m 

 

 

8. 9. 10. 

 

11. √ — 

EXPONENTS AND SCIENTIFIC NOTATION 

 

12. · 13. 

 

 

14. ( a 

b 

) 

15. a ·a 

a − 

16. xy 

y · 

x 

17. x y − 

− 

− 

y ) 

x 18. ( x 

19. ( a 

b 

) ab − − 

20. 

× − 

21. 

RADICALS AND RATIONAL EXPRESSIONS 

 

22. √ — 23. √ — 24. √ — 25. √ — 

26. √ — 27. 

√ 

 

28. √ 

 

29. √ 

 

 

30. 

+√ — 31. √— +√ — 32. √ — −√ — 33. √ 

— − 

34. √ — 

 

√ — −


POLYNOMIALS 

 

35. x +x −+x −x + 36. y +−y −y − 

37. x +x −+x −x + 38. a +a +−a −a + 

39. k +k − 40. h +h − 

41. x +x + 42. m −m +m − 

43. a +ba −b 44. x +yx −y 

FACTORING POLYNOMIALS 

 

45. p +pq −p q 46. x y +xy −xy 47. a b +a b −a 

 

48. x −x − 49. a +a − 50. d −d − 

51. x +x + 52. y −y + 53. h −hk +k 

54. x − 55. p + 56. x − 

57. q −p 58. xx − − +x − 

 

59. pp + −p + 

 

60. r r − − −r − 

 

RATIONAL EXPRESSIONS 

 

61. x −x − x −x + 

62. 

y − 

 

y −y + 

63. a −a − a −a − · a −a − 

a −a − 

64. d − 

d − · d − d − 

65. m +m + 

+m − 

m −m − ÷m 

m −m − 

66. d −d − 

+d + 

d −d + ÷d 

d +d − 

67. x + 

y 

 

 

68. 

a +a + − 

a − 

69. 

 

 

d + 

c 

 

c +d 

d 

70. 

x − 

y 

 

 

x

72 


CHAPTER 1 PRACTICE TEST 

 

 

1. − 2. √ — 

 

3. x +−= 4. y+ −= 

5. × 6. 

 

 

7. −·+· + 8. x +−x + 9. · − 

10. ( 

) 

11. x 

x 

12. y y − 

13. √ — 14. √ — 15. 

√ x 

 

16. √— b 

+√ — b 17. √— +√ — −√ — 18. √ 

— − 

 

√ 

— 

19. q +q −−q +q − 20. p +p ++p − 21. n −n −n + 

22. a −ba +b 

 

23. x − 24. y +y + 25. c − 

 

26. xx − − +x − 

 

27. z +z + z − · z −z + 

z − 28. x 

y + 

x 29. a 

b −b 

a 

 

a −b 

 

a

Equations and Inequalities 

2 

Figure 1 


2.1 The Rectangular Coordinate Systems and Graphs 

2.2 Linear Equations in One Variable 

2.3 Models and Applications 

2.4 Complex Numbers 

2.5 Quadratic Equations 

2.6 Other Types of Equations 

2.7 Linear Inequalities and Absolute Value Inequalities 

Introduction 

 

 

 

A x 

y 

 

B x 

y 

 

73

74 

CHAPTER 2 EQUATIONS AND INEQUALITIES 


In this section you will: 

Plot ordered pairs in a Cartesian coordinate system. 

Graph equations by plotting points. 

Graph equations with a graphing utility. 

Find x-intercepts and y-intercepts. 

Use the distance formula. 

Use the midpoint formula. 

2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 

y 

 

 

 

 

 

 

 

 

 

x 

Figure 1 

 

Figure 1 

 

 

Plotting Ordered Pairs in the Cartesian Coordinate System 

 

fl 

fl 

 

 

 

 

Cartesian coordinate system 

x-axis y-axis 

Th 

x- y- 

quadrantFigure 2.

SECTION 2.1 THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 75 

y 

 

 

x 

 

 

Figure 2 

Thorigin 

x- 

y- x- y-Th 

fiFigure 3 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 3 

x-coordinate 

y-coordinateordered pair 

x,y 

x- y-− 

Figure 4 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

– 

x 

Figure 4 

x- 

y- x- y-

76 


Cartesian coordinate system 

 

x 

y 

fix,y, x 

y 

Example 1 

Plotting Points in a Rectangular Coordinate System 

−− 

Solution −Th x-−ftTh 

y- y 

x- 

y- y 

−Th x-Th 

x- y- y Figure 5 

y 

– 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

– 

 

 

x 

Figure 5 

Analysis Note that when either coordinate is zero, the point must be on an axis. If the x-coordinate is zero, the point 

is on the y-axis. If the y-coordinate is zero, the point is on the x-axis. 

Graphing Equations by Plotting Points 

x y 

equation in two variablesgraph in two variables 

 

y =x − x 

y x- y-Table 1 

x −y 

x y = 2x − 1 (x, y) 

− y=−−=− −− 

− y=−−=− −− 

− y=−−=− −− 

y=−=− − 

y=−= 

y=−= 

y=−= 

Table 1


Th 

Figure 6. Th 

y 

 

 

 

 

 

 

 

– – – – – 

– 

– 

–– – 

– 

–– – 

– 

–– – 

 

– 

Figure 6 

x- 

x 

 

Th 

 

 

 

 

 

x 

How To… 

 

1. x 

 

2. x-fi x- 

 

3. x- y-ff 

4. 

5. 

Example 2 Graphing an Equation in Two Variables by Plotting Points 

y =−x+ 

Solution Table 2 x y 

x y = − x + 2 (x, y) 

− y=−−+= − 

− y=−−+= − 

− y=−−+= − 

y=−+= 

y=−+= 

y=−+=− − 

y=−+=− − 

Table 2

78 


Figure 7 

– 

 

 

 

– 

 

– 

 

 

– – – – – – 

– 

– 

– 

– 

y 

 

 

 

– 

– 

x 

Try It #1 

Figure 7 

y= 

x+ 

Graphing Equations with a Graphing Utility 

Th 

y = Th 

 

 

y =x −Figure 8aFigure 8b 

Th 

−≤ x ≤−≤ y ≤Figure 8c 

Plot1 Plot2 Plot3 

\Y 1 

= 2X–20 

\Y 2 

= 

\Y 3 

= 

\Y 4 

= 

\Y 5 

= 

\Y 6 

= 

\Y 7 

= 

WINDOW 

Xmin = −10 

Xmax = 10 

Xscl = 1 

Ymin = −10 

Ymax = 10 

Yscl = 1 

Xres = 1 

 

Figure 8 (a) Enter the equation. (b) This is the graph in the original window. (c) These are the original settings. 

x- y- 

x- y-Figure 9aFigure 9b 

WINDOW 

Xmin = −5 

Xmax = 15 

Xscl = 1 

Ymin = −30 

Ymax = 10 

Yscl = 1 

Xres = 1 

 

Figure 9 (a) This screen shows the new window settings. (b) We can clearly view the intercepts in the new window. 

y 

x 

 

x 

Example 3 

Using a Graphing Utility to Graph an Equation 

y =− 

x − 

 

Solution y = x- 

y-Figure 10 

– – – – 

– 

y 

 

 

 

– 

– 

Figure 10 

y=− x − 

 

 

 

 

x


Finding x-intercepts and y-intercepts 

interceptsTh x-intercept 

x- y-Th y-intercept 

y- x- 

x- y x y- x 

ys fiy =x − 

o fi x-y = 

 

y =x − 

 

=x − 

 

=x 

 

 

=x 

( 

) 

 

x 

o fi y-x = 

 

y =x − 

 

y =− 

 

y =− 

− y 

fiFigure 11 

 

given an equation, find the intercepts. 

– – – – 

Figure 11 

xy =x 

yx =y 

– 

 

 

 

 

– 

– 

– 

– 

y 

 

y=x − 

 

 

 

 

x 

Example 4 Finding the Intercepts of the Given Equation 

y =−x −. Th 

Solution y =o fi x- 

 

y =−x − 

 

=−x − 

 

=−x 

 

− 

=x 

 

( − 

) 

x− 

x =o fi y- 

 

y =−x − 

 

y =−− 

 

y =− 

− y−

80 


Figure 12 

Try It #2 

Figure 12 

y=− 

x+ 

Using the Distance Formula 

− 

 

– – – – 

– 

 

 

 

– 

– 

– 

– 

– 

– 

y 

Thdistance formulafi 

ThTha +b =c a b 

c Figure 13 

y 

 

 

 

– 

 

x 

 

 

 

 

 

 

x 

y 

 

 

x 

y 

 

d = c 

y 

− y 

= b 

x 

y 

 

x 

− x 

= a 

 

x 

Figure 13 

Th|x 

−x 

||y 

−y 

| d a b c 

 

|−|=Th|x 

−x 

||y 

−y 

| 

o ficn Th 

 

c =a +b →c =√ — a +b 

 

d =x 

−x 

+y 

−y 

→d =√ —— 

x 

−x 

+y 

−y 

 

fi 

the distance formula 

x 

y 

x 

y 

 

d=√ —— 

x 

−x 

+y 

−y 

 

Example 5 Finding the Distance between Two Points 

−− 

Solution fiFigure 14


y 

 

 

 

 

– – – – – 

– 

−− – 

– 

– 

 

 

 

− 

x 

Figure 14 

Th d 

d =√ —— 

x 

−x 

+y 

−y 

 

d =√ —— 

−− +−− 

 

=√ — + 

 

= √ 

— + 

 

=√ — 

Try It #3 

 

Example 6 

Finding the Distance Between Two Locations 

 

 

Figure 1 

d fi 

Solution Thfi 

 

fiTh 

ft 

Figure 15 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 

Figure 15

82 


 

 

 

 

Th 

 

 

 

Table 3 

From/To 

 

 

 

 

 

Number of Feet Driven 

 

 

 

 

 

Table 3 

ThTh 

fi 

 

 

 

 

d =√ — − +− 

=√ — + 

=√ — 

≈ 

Th 

 

fl 

fl 

 

Using the Midpoint Formula 

fiTh 

midpoint formulax 

y 

 

x 

y 

o fiM 

M=( x +x y +y ) 

Figure 16 

 

y 

 

x +x y +y 

 

 

Figure 16 

x


Example 7 Finding the Midpoint of the Line Segment 

− 

Solution o fi 

 

 

( x +x 

 

y +y 

 

−+ ) 

) = ( + 

= ( ) 

Try It #4 

−−− 

Example 8 

Finding the Center of a Circle 

Th−−− 

Solution ThTh 

 

 

 

( x +x 

 

y +y ) 

( −+ −− 

 

) = ( 

− ) =− 

 

Plotting Points on the Coordinate Plane (http://openstaxcollege.org/l/coordplotpnts) 

Find x- and y-intercepts Based on the Graph of a Line (http://openstaxcollege.org/l/xyintsgraph)

84 CHAPTER 2 EQUATIONS AND INEQUALITIES 


VERBAL 

1. 

 

 

3. y- 

 

2. x- 

y- 

4. 

d=√ —— 

x 

−x 

+y 

−y 

 

 

 

ALGEBRAIC 

fi x- y- 

 

5. y =−x + 6. y =x − 7. x −y = 8. x −=y 

9. x +y = 10. x − 

= y + 

 

y x 

11. x +y = 12. x −y = 13. x =−y 14. x −y = 

15. y +=x 16. x +y = 

fi 

 

17. −− 18. − 19. 20. − 

21. 

 

fi 

22. −− 23. −− 24. −−−− 25. − 

26. −− 

GRAPHICAL 

 

27. 28. y- 

x- 

29. x- 

y- 

 

 

30. −− 31. − 

y 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x

SECTION 2.1 SECTION EXERCISES 85 

32. −−−− 33. 

y 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

A 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

B 

 

 

 

C 

 

x 

34. 

a. b. c. d. − e. − 

 

35. y = x + 36. y =−x + 37. y =x + 

 

NUMERIC 

fi x- y- 

 

38. x −y = 39. x −y = 40. y −=x 41. y =−x + 42. y = x − 

e fi 

y 

43. 

 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

44. 

 

45. − 

46. 

 

47. 

TECHNOLOGY 

Y= 

ft2 nd CALC1:valueENTER 

x= x y x 

x =s fi y- 

48. 

=−x + 49. 

= x− 50. =x + 

 

Y= 

ft2 nd CALC2:zeroENTER 

left bound? x-ENTER 

right bound? x-ENTERguess? 

x-ENTER 

x- y-fi x-


 

fi x-ft 

x-, 

fi 

51. 

=−x + 52. 

=x− 53. 

= x + 

EXTENSIONS 

54. 

 

 

 

 

56. AB−C 

D− 

_ AB 

_ CD 

58. 

 

 

55. 

 

 

57. ft 

 

 

59. 

 

– 

– – – – – 

–– 

– 

y 

 

 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

– 

x 


60. Th 

 

 

 

62. 

. Th 

 

 

 

 

 

61. − 

 

 

 

63. 

 

 

 

 

 

64. 

 

y x

SECTION 2.2 LINEAR EQUATIONS IN ONE VARIABLE 87 



Solve equations in one variable algebraically. 

Solve a rational equation. 

Find a linear equation. 

Given the equations of two lines, determine whether their graphs are parallel or perpendicular. 

Write the equation of a line parallel or perpendicular to a given line. 

2.2 LINEAR EQUATIONS IN ONE VARIABLE 

 

 

 

 

 

Figure 1 

y 

 

 

 

 

 

 

 

 

 

Figure 1 

x 

Solving Linear Equations in One Variable 

linear equationTh 

ax+b = 

 

identity equation 

 

x =x +x 

solution set 

x 

conditional equation 

x +=x − 

 

x +=x − 

 

x =− 

 

x =− 

Th− 

 

inconsistent equationx −=x− 

 

 

x −=x − 

x −−x =x −−x x 

−≠−

88 


−≠−. Th 

 

 

linear equation in one variable 

 

ax+b = 

a b a≠ 

How To… 

 

Th 

x = x . Th 

1. 

 

2. ab+c=ab+ac 

3. 

4. ffifi 

ffi 

Example 1 

Solving an Equation in One Variable 

x += 

Solution Thax+b = 

 

 

x += 

x = 

x = 

 

Th 

Try It #1 

x+=− 

Example 2 

Solving an Equation Algebraically When the Variable Appears on Both Sides 

x−+=−x+ 

Solution 

 

x−+=−x+ 

x −+=−x − 

x =−−x 

x =− x- 

 

x =− 

 

 

x =− 

 

Analysis This problem requires the distributive property to be applied twice, and then the properties of algebra are used 

to reach the final line, x = − _ 

 

.


Try It #2 

−x−+x=−x 

Solving a Rational Equation 

ft 

rational equation 

 

 

 

 

x + 

x − 

x − 

x +x − 

 

 

 

 

 

 

Example 3 

Solving a Rational Equation 

 

x − 

 

x = 

 

Solution xxThxxx 

 

x 

 

x( 

x − 

x ) = ( 

) x 

 

x( 

x ) −x( 

x ) =( 

) x 

 

 

x( 

x ) − x( 

x ) =( 

 

) x 

−=x 

 

−=x 

 

=x 

 

 

 

=x 

 

=x 

fi 

x+ 

xx − 

x −xx−x− 

x−Th x 

fi x x−ff 

 

Thxx−Th 

 

 

xx−=xx− 

xx−

90 


x x +x 

x +x =xx+ x xx+ 

 

 

 

a 

b =c d 

fi 

 

 

a 

b =c 

d a 

b =c 

d 

adbc,ad=bc 

 

rational equations 

rational equation 

 

How To… 

 

1. 

2. 

3. 

4. ft 

5. 

6. 

Example 4 

Solving a Rational Equation without Factoring 

 

 

x − 

= 

x 

Solution xxThfi 

x 

x 

 

x ( 

x − 

) = ( 

x ) x 

 

x( 

x ) − x( 

) 

 

=( 

x ) 

x x 

−x = 

 

−x = 

 

−x = 

 

x =− 

 

− 

Th−−− 

 

Try It #3 

 

x = 

− 

 

x


Example 5 

Solving a Rational Equation by Factoring the Denominator 

x = 

 

− 

 

x 

Solution fiThx=ċ 

x =ċċxThxx = 

x 

 

 

 

 

Th 

 

 

− 

x ) x 

=x − 

=x 

x ( x ) = ( 

 

=x 

Try It #4 

− 

x + 

 

x =− 

 

Example 6 Solving Rational Equations with a Binomial in the Denominator 

 

 

a. 

x− = 

x b. x 

x− = 

x− − 

 

c. 

x 

x− = 

x− − 

 

Solution 

a. Thxx −n. Thxx− 

 

 

 

 

x − = 

x 

x =x− 

 

x =x − 

 

−x =− 

 

x = 

Th. Th 

b. Thx−x− 

 

x 

x−( 

x − ) = 

( 

x − − ) x − 

 

x−x 

x − 

x− = x − − x− 

 

 

x =−x− 

 

x =−x + 

 

x =−x 

 

x = 

 

Th . Th 

 

x =

92 


 

 

 

 

 

c. Thx−xx− 

x−( 

x 

x − ) = 

( 

x − − 

) x − 

Th. Th 

x =−x− 

x =−x 

x = 

x = 

Try It #5 

− 

 

x+ = 

x+ 

Example 7 Solving a Rational Equation with Factored Denominators and Stating Excluded Values 

 

ft 

x + − 

x − = x 

 

x − 

Solution x −ff 

x−x+Thx−x+ 

 

 

 

x−x+( 

x + − 

x − ) = ( x 

x−x+ ) 

x −x+ 

 

x−−x+=x 

x −−x −=x 

 

−−x = 

 

−=x 

Th−. Th− 

Try It #6 

 

 

x− + 

x+ = 

x −x− 

Finding a Linear Equation 

y =mx+b 

m =b =y 

The Slope of a Line 

slope y x 

 

 

 

m = y −y 

x −x 

 

ft 

Figure 2Thm =− 

m =m =


y 

y = −x − 

 

 

 

 

 

 

––– – – 

– 

– 

– 

– 

– 

– 

– 

 

y = x + 

 

 

 

y = x + 

 

 

x 

Figure 2 

the slope of a line 

m y xx 

y 

x 

y 

 

 

 

m = y −y 

x −x 

 

Example 8 

Finding the Slope of a Line Given Two Points 

−− 

Solution y- x- 

 

 

 

Th− 

 

m = −− 

−− 

= 

− 

=− 

 

Analysis It does not matter which point is called x 

, y 

or x 

, y 

. As long as we are consistent with the order of the y 

terms and the order of the x terms in the numerator and denominator, the calculation will yield the same result. 

Try It #7 

− 

Example 9 

Identifying the Slope and y-intercept of a Line Given an Equation 

y- y =− x − 

 

Solution y =mx+b m =− y-b =− 

 

Analysis The y-intercept is the point at which the line crosses the y-axis. On the y-axis, x = . We can always identify 

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. 

The Point-Slope Formula 

n fi 

 

y −y 

=mx−x 

 

Thftfi 

ft

94 


the point-slope formula 

 

 

y −y 

=mx−x 

 

Example 10 

Finding the Equation of a Line Given the Slope and One Point 

m =−fi 

 

Solution − m x 

y 

 

 

y −y 

=mx−x 

 

 

y −=−x− 

 

y −=−x + 

 

y =−x + 

Analysis Note that any point on the line can be used to find the equation. If done correctly, the same final equation will 

be obtained. 

Try It #8 

m= 

Example 11 

Finding the Equation of a Line Passing Through Two Given Points 

−fi 

 

Solution 

 

 

m = −− 

− 

= − 

− 

 

= 

 

 

x y 

 

 

y −= 

x− 

y −= 

x − 

 

 

y = x − 

 

y = x − 

 

Analysis To prove that either point can be used, let us use the second point , − and see if we get the same equation. 

 

 

y −−= 

x− 

y += 

x 

 

y = x − 

 

We see that the same line will be obtained using either point. This makes sense because we used both points to calculate 

the slope.


Standard Form of a Line 

 

 

Ax+By= 

AB C Th x- y- 

 

Example 12 

Finding the Equation of a Line and Writing It in Standard Form 

m =−( 

− ) 

Solution 

 

y −−=−( x − 

) 

 

y +=−x + 

 

 

 

 

 

 

Th 

y+=( −x + 

) 

y +=−x + 

x +y =− 

Try It #9 

m =− 

( 

) 

Vertical and Horizontal Lines 

Th 

. Th 

 

x =c 

c Thfi y- 

x-c 

fi−−−− 

−l fi 

− 

 

m = 

−−− = 

fi 

x-fi 

x =−Figure 3 

Th 

 

y =c 

c Th x- y- 

c 

fi−−−− 

−e fi

96 


 

m = −−− 

−− 

 

= 

 

 

= 

x 

y 

y- 

 

y −−=x− 

 

y += 

 

y =− 

Thy =− y-Figure 3 

x= − 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

y= − 

x 

Example 13 

Figure 3 The line x = −3 is a vertical line. The line y = −2 is a horizontal line. 

Finding the Equation of a Line Passing Through the Given Points 

− 

Solution x-. Thx = 

Try It #10 

− 

Determining Whether Graphs of Lines are Parallel or Perpendicular 

ff y- 

Figure 4m = 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

y 

 

y = x − 

y = x + 

y = x + 

x 

 

Figure 4 Parallel lines 

ff y- 

Th


−m 

ċm 

=− 

Figure 5− 

 

m 

ċm 

=− 

 

ċ( − 

) =− 

y 

y = x − 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

y = − 

 

x − 

 

Example 14 

Figure 5 Perpendicular lines 

Graphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither 

y =−x + 

x −y = 

Solution The fi 

 

 

y =−x + 

 

 

 

 

 

Figure 6 

y = − 

 

x + 

 

– – – – 

x −y = 

y =− x + 

 

−y =−x + 

– 

y = x − 

 

 

 

 

 

 

– 

– 

– 

– 

– 

y 

 

y = 

 

x − 

 

 

 

 

 

x 

Figure 6 

 

 

 

m 

=− 

 

m 

= 

 

 

m 

ċm 

=( − 

) ( 

) =− 

Thfi

98 


Try It #11 

y−x=y=x+ 

Writing the Equations of Lines Parallel or Perpendicular to a Given Line 

fi 

fi 

ftr fi 

How To… 

 

1. . Th 

2. 

3. 

Example 15 

Writing the Equation of a Line Parallel to a Given Line Passing Through a Given Point 

x +y = 

Solution o fi 

 

x +y = 

 

y =x + 

 

y =− 

x + 

 

Th m =− 

y- 

 

Th y- 

. Th 

 

 

 

Th y =− 

y −=− 

x− 

y −=− x + 

 

y =− x + 

 

x +Figure 7 Figure 7 

y = − 

 

x + 

 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

y 

y = − 

 

x + 

 

 

x 

Try It #12 

x=+y−−


Example 16 

Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point 

x −y +=− 

Solution The fi 

 

x −y += 

 

−y =−x − 

 

m = 

y = 

x + 

 

Th 

− 

 

 

 

 

 

y −=− 

x−− 

y −=− 

x − 

 

y =− 

x − 

+ 

 

y =− 

x − 

 

 

 

 

 

findperpline)



VERBAL 

1. 

 

3. 

y =x + 

 

 

5. 

 

x − = 

x + 

x =x =− 

 

 

2. 

 

 

4. 

 

ALGEBRAIC 

x 

6. x +=x − 7. x−= 8. x +−=x + 

9. −x +=x − 10. 

− 

x= 

 

11. x 

− 

=x + 

 

12. 

x + 

= 

13. x −+x =x + 14. x 

 

− 

15. x + 

−x − 

= 

=x 

+ 

 

x x- 

16. x − 

= 

 

x 

19. 

x − += 

x − 

 

17. − 

x+ =x+ 

x+ 

20. 

x+ + 

x− = 

− 

x −x− 

18. 

x − = 

x − + 

 

x −x − 

21. 

x = 

+ 

x 

fifi 

 

22. 

 

23. − 

 

24. x− 25. y− 

26. −− 27. 

28. y =x + 

 

29. y =x − 

− 

, fi 

30. −− 31. 

32. Th 

 

34. Th 

 

33. Th 

− 

35. −−


GRAPHICAL 

 

 

x + 

38. y = 

 

36. y =x + 

37. 

y=− x − 

 

NUMERIC 

x −y = 

y −x = 

y =x + 

, fi 

39. x = 

y =− 

40. 41. −− 42. − 43. −− 

44. −− 

fi 

 

45. − 

− 

TECHNOLOGY 

46. 

−− 

 

Y1yminymax 

yyminymax 

47. x−y= 48. x −y = −y 

49. 

x = 

EXTENSIONS 

50. 

y −y 

=mx −x 

x 

x 

yy 

m 

52. 

 

x −y = 

54. 

 

51. 

Ax +By =C y 

ABCx. Th 

 

53. 

 

 

 

− 


55. Th 

 

 

s 2.5 ft, fin 

 

 

56. fi x 

y 

p =x +y y p = 

x = 

Th 

y =+x x 

57. 58. 

 

59.

102 




Set up a linear equation to solve a real-world application. 

Use a formula to solve a real-world application. 

2.3 MODELS AND APPLICATIONS 

Figure 1 Credit: Kevin Dooley 

fifi 

fi 

 

 

 

x 

Th 

 

Setting up a Linear Equation to Solve a Real-World Application 

fifi 

fiTh 

 

ThxTh 

 

 

fi 

o fiC 

 

C =x + 

 

Table 1 

Verbal 

a 

 

a 

a 

Thab 

Tha 

 

Th 

b c 

Table 1 

Translation to Math Operations 

xx +a 

x 

xx +a 

xx −a 

ax−b 

x 

 

x +a =x 

xx−b=c

SECTION 2.3 MODELS AND APPLICATIONS 103 

How To… 

o fi 

1. 

2. 

3. , fie fi 

4. 

5. 

Example 1 

Modeling a Linear Equation to Solve an Unknown Number Problem 

 

 

Solution x fiThfi 

x +. Th 

 

x +x+= 

x += 

 

x = 

 

x = 

 

x +=+ 

 

= 

Th 

Try It #1 

 

 

Example 2 Setting Up a Linear Equation to Solve a Real-World Application 

Thffff A 

B 

a. ff 

b. ff 

c. ff 

d. 

Solution 

a. Th A A =x +Thx 

B 

xBB =x + 

b. 

 

A =+ 

 

=+ 

 

= 

 

B =+ 

 

=+ 

 

= 

B ffff 

A

104 


c. 

 

A =+ 

 

=+ 

 

= 

 

B =+ 

 

=+ 

 

= 

A ff 

B 

d. 

xyxy 

x 

 

x +=x + 

 

x = 

 

x = 

x- 

 

+= 

 

+= 

ThFigure 2. 

 

y 

 

 

B=x+ 

 

 

 

A=x+ 

 

 

 

Figure 2 

x 

 

Try It #2 

 

s. Th 

 

Using a Formula to Solve a Real-World Application 

Thfi 

 

 

areaA =LWperimeter 

P =L +WV =LWHfi 

 

Example 3 

Solving an Application Using a Formula


Solution Thd =rt 

 

 

 

 

r 

 

d. A tTable 2ft 

 

 

d r t 

To Work d r 

 

To Home d r− 

 

Table 2 

d =r ( 

) 

 

 

d =r−( 

) 

r 

 

r( 

) =r−( 

) 

 

 

r = 

r − 

 

 

 

r − 

r =− 

 

 

 

 

− 

r =− 

 

r =− 

− 

r = 

 

d 

 

 

Th 

d =( 

) 

= 

Analysis Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the 

LCD to solve for r. 

 

r( 

) =r−( 

) 

 

r( 

) =r−( 

) 

 

r =r− 

 

r =r − 

 

−r=− 

 

r = 

Try It #3 

er 3.6 h t 

er 4 h t

106 


Example 4 

Solving a Perimeter Problem 

ThftTh 

 

Solution ThP =L +W 

L =W + 

ftFigure 3 

W 

L = W + 

Figure 3 

 

 

P =L +W 

 

=W++W 

 

=W ++W 

 

=W + 

 

=W 

 

=W 

 

+=L 

 

=L 

ThL = W =ft 

Try It #4 

f a r 

Example 5 

Solving an Area Problem 

ThTh 

 

Solution ThA =LW 

Th 

 

ft 

L =W + 

d fi 

 

P =L +W 

 

=W++W 

 

=W ++W 

 

=W + 

 

=W 

 

=W 

 

+=L 

 

=L


e fiL =W = 

 

A =LW 

 

A = 

 

= 

Th 

Try It #5 

 

 

s fiy ft 

Example 6 Solving a Volume Problem 

 

 

Solution ThV =LWH 

L =WH =. Th 

 

V =LWH 

 

=WW 

 

=W 

 

=W 

 

=W 

ThL =W =H = 

Analysis 

Note that the square root of W 2 would result in a positive and a negative value. However, because we are 

describing width, we can use only the positive result.



VERBAL 

1. o fi 

 

3. 

x 

 

2. 

n n +=n 

4. 

n 

n 

5. v 

 


fi 

 

6. 

 

 

 

 

8. y fi 

. Thy fi 

 

y fi 

7. 

 

 

ffff 

y y 

 

9. 

m 

11. 

 

10. 

m 

12. 

 

: ff 

Th 

Th 

P 

 

13. 14. 

 

15. 

 

 

16.


: 

Th 

 

17. x 

 

 

19. es fl 

 

 

21. ft 

 

. Th 

 

18. 

 

 

20. 

ft 

 

 

22. 

 

 

 

23. 

 

 

 

 

ff 

 

24. 

 

26. 

 

25. 

 

27. 

 

fi 

 

28. A=P+rt P 

rt 

P 

A= 

30. F =maFm 

a 

f 12 N i 

29. ThF= mv 

R Fv 

mRRm= 

v=F= 

 

31. Sum = 

−r 

r 

ft 

 

32. WP=L+W 33. 

W 

 

34. f 

p + 

q = f 

35. f 

p =q = 

36. m 

y =mx +b 

37. 

m 

b =


38. ThA = 

hb +b 

 

 

h =b 

=b 

= 

39. hA = 

hb +b 

40. 

A =b 

= 

b 

= 

42. d =rt 

 

55 mi/h f 

44. 

3 h if o 

 

46. hA = 

bh 

48. ThV =πr h 

π 

r 

 

50. 

 

π 

52. 

 

π 

41. 

Th 

 

P =L +W 

43. 

 

 

45. A = bh 

 

 

 

47. 

 

 

49. hV =πr h 

51. rV =πr h 

53. Th 

C =πr 

r =r π 

 

54. π 

π

SECTION 2.4 COMPLEX NUMBERS 111 



Add and subtract complex numbers. 

Multiply and divide complex numbers. 

Simplify powers of i 

2.4 COMPLEX NUMBERS 

Figure 1 

Th 

 

fiTh 

 

 

fift 

Thfi Thfi 

 

t fid fi 

Expressing Square Roots of Negative Numbers as Multiples of i 

fifi 

Thff 

. Th i fi− 

 

√ — −=i 

 

 

i =( √ — −) =− 

i− 

 

√ — −=√ — ċ− 

 

=√ — √ — − 

 

=i 

i−i 

. 

a +bia b +i 

+i√ — 

+i 

 

 

ff

112 


imaginary and complex numbers 

complex numbera +bi 

a 

b 

b =a +bia = b 

imaginary number 

How To… 

 

1. √ — −a √ — a √ — − 

2. √ — −i 

3. √ — a ċ i 

Example 1 

√ — − 

Solution 

 

+i 

Try It #1 

√ — − 

Expressing an Imaginary Number in Standard Form 

Plotting a Complex Number on the Complex Plane 

√ — −=√ — √ — − 

=i 

 

 

complex plane 

 

a,b,a b 

−+iTh− 

−−+iFigure 2 

y 

−+i 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 2 

complex plane 

 

Figure 3 

imaginary 

real 

Figure 3


How To… 

 

1. 

2. 

3. 

4. 

Example 2 Plotting a Complex Number on the Complex Plane 

−i 

Solution Th−− 

Figure 4 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 4 

Try It #2 

−−i 

Adding and Subtracting Complex Numbers 

 

 

complex numbers: addition and subtraction 

 

a +bi+c+di=a + c+b+di 

 

a+bi−c+di=a − c+b−di 

How To… 

, fiff 

1. 

2. 

3. 

Example 3 Adding and Subtracting Complex Numbers 

 

a. −i++i b. −+i−−+i

114 


Solution 

 

 

 

 

 

 

a. −i++i=−++i 

=++−i+i 

=++−+i 

=+ 

b. −+i−−+i=−++−i 

=−++i −i 

=−++−i 

=+i 

Try It #3 

+i−i 

Multiplying Complex Numbers 

Thff 

 

Multiplying a Complex Number by a Real Number 

 

+i 

+i=·+·i 

=+i 

How To… 

o fi 

1. 

2. 

Example 4 

Multiplying a Complex Number by a Real Number 

+i 

Solution 

 

 

+i=ċ+ċi 

=+i 

Try It #4 

 

−i 

Multiplying Complex Numbers Together 

fi 

 

. Thffi − 

 

a+bic+di=ac+adi+bci+bdi 

=ac+adi+bci−bd i =− 

=ac−bd+ad+bci


How To… 

o fi 

1. 

2. i =− 

3. 

Example 5 

+i−i 

Solution 

 

 

 

Try It #5 

−i+i 

Dividing Complex Numbers 

Multiplying a Complex Number by a Complex Number 

+i−i=−i+i−ii 

=−i +i −i 

=++−+i 

=−i 

 

 

a +bifi 

 

Th 

a +bia −bi 

a +bia −bi 

 

a+bia−bi=a −abi+abi−b i 

 

=a +b 

Th 

Tha +bia −bi 

a −bia +biffi 

 

c +dia +bia b fi 

n fi 

c +di 

a≠ b ≠ 

a +bi 

 

 

 

 

i =− 

 

 

+di 

a+bi ċa−bi 

a−bi =+dia−bi 

a+bia−bi 

= ca−bi+adi−bdi 

 

a −abi+abi−b i 

= ca−cbi+adi−bd− 

 

a −abi+abi−b − 

= ca+bd+ad−cbi 

 

a +b

116 


the complex conjugate 

complex conjugatea +bia −bi 

. Th 

 

 

Example 6 

Finding Complex Conjugates 

 

a. +i√ — b. − 

i 

Solution 

a. Tha +bi. Tha −bi−i√ — 

b. a +bi− 

i. Tha−bi+ 

i 

Th 

i 

Analysis Although we have seen that we can find the complex conjugate of an imaginary number, in practice we 

generally find the complex conjugates of only complex numbers with bothreal and an imaginary component. To obtain a 

real number from an imaginary number, we can simply multiply by i. 

Try It #6 

−+i 

How To… 

 

1. 

2. 

3. 

4. 

Example 7 

+i−i 

Dividing Complex Numbers 

Solution 

+i 

−i 

Th 

+i 

−i ċ+i 

+i 

 

 

+i 

−i ċ+i 

+i =+i +i +i 

+i −i −i 

 

 

 

 

 

 

= +i +i 

+− 

+i −i −− i =− 

= +i 

 

 

= 

+ i


Simplifying Powers of i 

Th i i 

 

i = 

 

i =− 

 

i =i ċi =−ċi =−i 

 

i =i ċi =−iċi =−i =−−= 

 

i =i ċi =ċi = 

fiftifi i 

i 

 

i =i ċi =i ċi =i =− 

 

i =i ċi =i ċi =i =−i 

 

i =i ċi =i ċi =i = 

 

i =i ċi =i ċi =i = 

Thi−−i 

Example 8 

i 

Simplifying Powers of i 

Solution i =i fi 

=ċ+ 

i =i ċ+ =i ċ ċi =i ċi = ċi =i =−i 

Try It #7 

i 

Q & A… 

Can we write i 35 in other helpful ways? 

Example 8i i fi 

fii Table 1 

 

Factorization of i 35 i ċi i ċi i ċi i ċi 

Reduced form i ċi i ċ− i ċ i ċi 

Simplified form − ċi −i i i 

Table 1 

 

 

 

Adding and Subtracting Complex Numbers (http://openstaxcollege.org/l/addsubcomplex) 

Multiply Complex Numbers (http://openstaxcollege.org/l/multiplycomplex) 

Multiplying Complex Conjugates (http://openstaxcollege.org/l/multcompconj) 

Raising i to Powers (http://openstaxcollege.org/l/raisingi)



VERBAL 

1. 2. 

 

3. 

 

ALGEBRAIC 

 

4. 

 

5. y=x +x−yix=i 6. y=x −yix=i 

7. y=x +x+yix=+i 8. y=x +x−yix=−i 

9. y = x + y ix =i 

−x 

10. y =+x 

 

x + y ix =i 

GRAPHICAL 

 

11. −i 12. −+i 13. i 14. −−i 

NUMERIC 

fi 

15. +i+−i 16. −−i++i 17. −+i−−i 18. −i−+i 

19. −+i−−+i 20. +ii 21. −ii 22. −i 

23. −+i 24. +i−i 25. −+i−+i 26. −i+i 

27. +i−i 28. +i 

29. −i 30. −+i 

i 

31. +i 

i 

32. −i 

+i 

33. +i 

−i 

 

34. +i 

−i 

35. √ — −+√ — − 36. −√ — −−√ — − 37. +√— − 38. − +√— 

39. i 40. i 41. i 

TECHNOLOGY 

 

42. +i k k= 

k= 

44. l +i k −l −i k k = 

k = 

43. −i k k= 

k= 

45. x += √— 

+ 

i 

46. x −= √— 

+√— 

i 

EXTENSIONS 

fi 

47. 

i + 

i 

48. 

i − 

i 49. i +i 50. i − +i 

51. +i−i 

+i 52. +i−i 

+i 53. +i 

+i 

55. +i 

+ −i 

i −i 

56. +i 

+i −−i 

+i 

54. +i +i ++i

SECTION 2.5 QUADRATIC EQUATIONS 119 



Solve quadratic equations by factoring. 

Solve quadratic equations by the square root property. 

Solve quadratic equations by completing the square. 

Solve quadratic equations by using the quadratic formula. 

2.5 QUADRATIC EQUATIONS 

Figure 1 

Th Figure 1 

 

 

ff 

Solving Quadratic Equations by Factoring 

 

x +x −=x −=Thfi 

, fi 

ftfi 

 

 

aċb =a =b =a b 

 

 

 

 

x−x+ 

 

x−x+=x +x −x − 

 

=x +x − 

Thx +x −= 

 

 

fifi 

ax +bx+c =ab c a≠. Thx +x −= 

fi 

ff

120 


the zero-product property and quadratic equations 

zero-product property 

aċb =a =b = 

a b 

quadratic equation 

ax +bx+c = 

ab c a≠ 

Solving Quadratics with a Leading Coefficient of 1 

x +x −=ffiffix 

 

How To… 

ffi 

1. c b 

2. x+kx−k, k 

−x+x− 

3. 

Example 1 Factoring and Solving a Quadratic with Leading Coefficient of 1 

x +x −= 

Solution x +x −=− 

− 

ċ− 

−ċ 

ċ− 

ċ− 

Thċ−Th 

 

 

x−x+= 

 

 

x−x+= 

 

x−= 

 

x = 

 

x+= 

 

x =− 

Th−Figure 2Th 

x- y =x +x −= 

y 

x +x− = 

− 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

x 

Figure 2


Try It #1 

x −x−= 

Example 2 

Solve the Quadratic Equation by Factoring 

x +x += 

Solution 

ċ 

ċ 

−ċ− 

−ċ− 

Th. Th 

 

x+x+= 

 

x+= 

 

x =− 

 

x+= 

 

x =− 

Th−− 

Try It #2 

x −x−= 

Example 3 

Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares 

ffx −= 

Solution ff 

 

 

 

x −= 

 

x−x+= 

 

x−= 

 

x = 

 

x+= 

 

x =− 

Th− 

Try It #3 

x −= 

Factoring and Solving a Quadratic Equation of Higher Order 

ffi 

 

1. ax +bx+c =a ċc 

2. acb 

3. bx x 

4. Th 

 

5. 

6.

122 


Example 4 

Solving a Quadratic Equation Using Grouping 

x +x += 

Solution ac=. Th 

 

ċ 

 

ċ 

 

ċ 

 

ċ 

 

ċ 

Th+ b x 

ffixe fi 

 

x +x +x += 

 

xx ++x += 

 

x +x+= 

 

 

x +x+= 

 

x += 

 

 

 

Th− −Figure 

3 

 

x =− 

 

x+= 

x =− 

y 

x 

+x+ = 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 3 

Try It #4 

x +x+= 

Example 5 

Solving a Higher Degree Quadratic Equation by Factoring 

−x −x −x = 

Solution Thfi 

−x 

 

 

−x −x −x = 

 

−xx +x += 

 

 

−xx +x +x += 

 

−xxx++x+= 

 

−xx +x+=


 

 

−x= 

 

x = 

 

x += 

 

x =− 

 

 

x += 

 

x =− 

Th− 

− 

Try It #5 

x +x +x= 

Using the Square Root Property 

square 

root propertyx 

x 

 

the square root property 

x 

x =kx =±√ — k 

k 

How To… 

x x 

1. x 

2. ± 

 

3. ± 

Example 6 

Solving a Simple Quadratic Equation Using the Square Root Property 

x = 

Solution ± 

 

 

x = 

 

x =±√ — 

 

=±√ — 

Th√ — −√ — 

Example 7 Solving a Quadratic Equation Using the Square Root Property 

x += 

Solution x . Th 

 

x += 

 

x = 

 

 

Th √— 

−√— 

 

x = 

 

x =± √—

124 


Try It #6 

x− = 

Completing the Square 

 

completing the square 

 

ffial 

aTh 

 

x +x += 

1. a = 

 

 

x +x =− 

 

 

 

 

 

 

 

 

 

2. b 

 

 

= 

= 

3. ( 

b ) 

x +x +=−+ 

x +x += 

4. Th 

x +x += 

x+ = 

5. 

√ — x+ =±√ — 

6. Th−+√ — −−√ — 

Example 8 

x +=±√ — 

Solving a Quadratic by Completing the Square 

x =−±√ — 

x −x −= 

Solution 

 

x −x = 

Th b 

 

 

 

−=− 

 

 

( − 

) = 

 

 

 

x −x +( − 

) =+( − 

) 

 

x −x + 

=+ 

 

 

 

( x − 

) =


 

 

 

 

Th +√— 

−√— 

Try It #7 

x −x= 

Using the Quadratic Formula 

√ ( x − 

 

 

) =± 

√ 

( x − 

) =± √— 

 

x = 

±√— 

 

Th 

 

 

 

ffi 

−aax +bx+c =a≠ 

 

1. 

 

ax +bx=−c 

2. a 

 

x + a b x =−c 

a 

3. Th 

( b __ 

a ) = b 

a 

 

x + a b x +b 

 

 

 

a =b a −c 

a 

4. eft 

 

 

( x + b a ) = b −ac 

a 

 

5. 

 

x + b 

a =± √ b −ac 

a 

 

 

x + b 

a b −ac =±√— a 

 

6. − b . Th 

a 

x = −b±√— b −ac 

a 

 

the quadratic formula 

ax +bx+c =quadratic formula 

 

x = −b±√— b −ac 

 

a 

ab c a≠

126 


How To… 

 

1. ax +bx+c = 

2. ffiabc 

3. 

 

4. 

Example 9 Solve the Quadratic Equation Using the Quadratic Formula 

x +x += 

Solution ffi: a =b =c =. Th 

 

 

 

x = −±√— − 

 

 

= −±√— − 

 

= −±√— 

 

Example 10 Solving a Quadratic Equation with the Quadratic Formula 

x +x += 

Solution ffi: a =b =c = 

 

 

 

 

 

 

b 

x = −b±√— −ac 

 

a 

 

= 

−±√— −ċċ 

 

ċ 

 

= 

− −±√— 

= 

− −±√— 

= −±i√— 

Th −+i√— 

−−i√— 

Try It #8 

x +x−= 

The Discriminant 

 

b −acTh 

Table 1


Value of Discriminant 

b −ac= 

b −ac> 

b −ac> 

b −ac< 

Results 

 

 

 

 

Table 1 

the discriminant 

ax +bx+c =ab c discriminant 

b −ac 

 

Example 11 

Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation 

o fi 

a. x +x += b. x +x += c. x −x −= d. x −x += 

Solution b −ac 

a. x +x += 

b −ac= −=. Th 

b. x +x += 

b −ac= −= 

c. x −x −= 

b −ac=− −−= 

d. x −x += 

b −ac=− −=−. Th 

Using the Pythagorean Theorem 

Pythagorean Theorem 

a +b =c a b 

c 

 

Th 

 

 

 

Thn Th 

 

a +b =c 

a b c 

Figure 4 

b 

c 

a 

Figure 4

128 


Example 12 

Finding the Length of the Missing Side of a Right Triangle 

Figure 5 

 

 

Figure 5 

Solution b a 

a 

 

 

 

 

 

 

a +b =c 

a + = 

a += 

a = 

a =√ — 

=√ — 

Try It #9 

n Thab 

 

 

Solving Quadratic Equations by Factoring (http://openstaxcollege.org/l/quadreqfactor) 

The Zero-Product Property (http://openstaxcollege.org/l/zeroprodprop) 

Completing the Square (http://openstaxcollege.org/l/complthesqr) 

Quadratic Formula with Two Rational Solutions (http://openstaxcollege.org/l/quadrformrat) 

Length of a Leg of a Right Triangle (http://openstaxcollege.org/l/leglengthtri)



VERBAL 

1. 

 

3. 

 

 

5. 

 

 

2. 

 

 

ax +bx +c = 

y =ax +bx +cx 

4. 

b −ac 

 

 

ALGEBRAIC 

 

6. x +x−= 7. x −x += 8. x +x −= 9. x +x += 

10. x −x+= 11. x −= 12. x +x −= 13. x = 

14. x +x = 15. x =x + 16. x =x 17. x +x = 

18. x 

− 

x = 

 

19. x = 20. x = 21. x − = 22. x − = 

23. x + = 24. x − = 

 

25. x −x −= 26. x −x −= 27. x −x = 28. x + 

x − 

= 

29. +z =z 30. p +p −= 31. x −x −= 

 

 

32. x −x += 33. x +x += 34. x +x −= 35. x −x += 

36. x −x −= 37. x −x −= 

 

No Real Solution 

38. x +x += 39. x +x = 40. x −x −= 41. x −x += 

42. x +x += 43. + x − 

x = 

TECHNOLOGY 

fi 

x-2 nd CALC 2:zerofi 

,, 

 

44. 

=x +x − 45. 

=−x +x − 46. 

=x +x −


47. x +x −= 

 

 

=x +x − 

= 

2 nd CALC 

5:intersection 

 

EXTENSIONS 

49. 

ax +bx +c = x 

 

 

51. 

 

 

119 ft. 

 

53. 

p =−x +x − x 

p 

fi 

fi 

 

 

2 nd CALC maximum 

x y 


54. 

A 

P =A −A + 

 

 

56. 

d =t +t t 

 

 

48. x +x −= 

 

 

=x +x − 

= 

2 nd CALC 

5:intersection 

 

50. 

− a b 

52. 

P =t −t + t 

t = 

 

 

55. Th 

C =x + 

R =x −x fi 

 

x fi 

57. 

den. Th 

f 378 ft 

 

x 

 

x 

x 

 

58. 

infl 

P 

e fl t ft 

P =−t +t +≤ t ≤ 

e fl 

t 

x

SECTION 2.6 OTHER TYPES OF EQUATIONS 131 



Solve equations involving rational exponents. 

Solve equations using factoring. 

Solve radical equations. 

Solve absolute value equations. 

Solve other types of equations. 

2.6 OTHER TYPES OF EQUATIONS 

 

 

 

 

 

Solving Equations Involving Rational Exponents 

 

 

 

√ — √ 

— Th 

 

 

Th 

 

 

( 

) =( 

) = 

rational exponents 

Th 

 

a m 

n =( a 

n ) 

m 

=a m 

n = √ n — a m =( √ n — a ) m 

Example 1 

Evaluating a Number Raised to a Rational Exponent 

 

 

Solution fififi 

 

( 

 

 

Try It #1 

− 

 

 

 

) 

 

( 

 

) 

= 

=

132 


Example 2 

Solve the Equation Including a Variable Raised to a Rational Exponent 

x 

= 

Solution Th x 

 

 

 

 

x 

= 

 

( x 

) 

 

= 

 

x = Th 

 

= 

Try It #2 

x 

= 

Example 3 

Solving an Equation Involving Rational Exponents and Factoring 

x 

=x 

 

Solution Th 

 

 

 

x 

−( x 

) =x 

−( x 

−x 

x 

= 

 

x 

x 

. Thx 

ft 

 

x 

−x 

= 

 

x 

( x 

−) = 

x 

 

Thx 

 

 

Th 

x 

 

 

 

 

 

 

 

x 

( x 

−) = 

Th 

 

Try It #3 

x+ 

 

= 

x 

= 

x = 

) 

x 

−= 

x 

= 

x 

= 

 

( x 

 

) =( 

) 

 

x =


Solving Equations Using Factoring 

 

ffi 

ft 

polynomial equations 

n 

a n 

x n +a n− 

x n − +ċċċ+a 

x +a 

x +a 

n a n 

a 

a n 

≠ 

polynomial equationTh 

n 

Example 4 

Solving a Polynomial by Factoring 

x =x 

Solution . Th 

 

x −x = 

 

x x −= 

ffx − 

Thfix 

x = x −= 

x = x−x+= 

x −= x += 

x = x =− 

Th− 

Analysis We can see the solutions on the graph in Figure 1. The x-coordinates of the points where the graph crosses the 

x-axis are the solutions—the x-intercepts. Notice on the graph that at , the graph touches the x-axis and bounces back. 

It does not cross the x-axis. This is typical of double solutions. 

y 

– – – 

 

 

 

 

 

– – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 1 

Try It #4 

x =x

134 


Example 5 

Solve a Polynomial by Grouping 

x +x −x −= 

Solution Th 

fi 

 

 

x +x −x −= 

 

x x+−x+= 

 

x −x+= 

Thx −ff 

 

x −x+= 

 

x−x+x+= 

x −= x += x += 

x = x =− x =− 

Th−− 

x-Figure 2 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 2 

Analysis We looked at solving quadratic equations by factoring when the leading coefficient is . When the leading 

coefficient is not , we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term 

of quadratic equations. We can also use grouping for some polynomials of degree higher than , as we saw here, since 

there were already four terms. 

Solving Radical Equations 

Radical equations, 

 

 

 

√ — x +=x 

√ — x +=x − 

√ — x +−√ — x −= 

 

fiextraneous solutions 

Th 

 

fi


radical equations 

radical equation 

How To… 

 

1. 

2. 

nn 

 

3. 

4. 

5. fi 

Example 6 

Solving an Equation with One Radical 

√ — −x=x 

Solution Th 

 

√ — −x =x 

 

 

( √ — −x ) =x 

−x =x 

 

 

=x +x − 

 

=x+x− 

=x+ =x − 

−=x =x 

Th−− 

 

 

 

 

√ — −x =x 

√ — −−=− 

√ — =− 

≠− 

Th 

 

 

 

√ — −x=x 

 

√ — −= 

 

√ — = 

 

= 

Th 

Try It #5 

√ — x+=x−

136 


Example 7 

Solving a Radical Equation Containing Two Radicals 

√ — x ++√ — x −= 

Solution 

 

 

√ — x ++√ — x −= 

√ — x +=−√ — x − √ — x − 

( √ — x +) =( −√ — x −) 

a − b =a −ab+b 

 

x += −√ — x −+( √ — x −) 

 

x +=−√ — x −+x− 

x +=+x −√ — x − 

x −=−√ — x − 

x− =( −√ — x −) 

x −x +=x− 

 

 

x −x +=x − 

 

x −x += 

x−x−= 

x −= x −= 

x = x = 

Th 

 

 

 

 

 

 

 

 

 

 

 

 

√ — x ++√ — x −= 

√ — x +=−√ — x − 

√ — +=−√ — − 

√ — =−√ — 

= 

√ — x ++√ — x −= 

√ — x +=−√ — x − 

√ — +=−√ — − 

√ — =−√ — 

≠− 

Th 

Try It #6 

√ — x++√ — x+=


Solving an Absolute Value Equation 

|x −|= 

−Thff 

 

x −= x −=− 

x = x =− 

x = x =− 

 

fi 

absolute value equations 

Th x |x| 

 

x ≥|x|=x 

 

x

138 


c. |x −|−= 

 

 

|x −|−= 

 

|x −|= 

x −= x −=− 

x = x =− 

x = x =− 

 

Th− 

 

d. |−x +|= 

Th 

 

−x += 

 

−x =− 

 

x = 

Th 

Try It #7 

|−x|+= 

Solving Other Types of Equations 

Th 

 

 

Solving Equations in Quadratic Form 

Equations in quadratic formThfiTh 

Th 

x −x +=x +x −= 

x 

+x 

+= 

 

quadratic form 

equation in 

quadratic form 

 

How To… 

 

1. 

2. u 

3. 

4. 

5. 

6.


Example 9 

Solving a Fourth-degree Equation in Quadratic Form 

x −x −= 

Solution Thfi 

u =x u 

 

 

 

 

u −u −= 

u −u −= 

u +u−= 

u 

u += u −= 

 

u =− 

u = 

 

u =− x 

= 

 

x =± 

 

x =− 

 

 

x =±i 

√ 

 

Th±i 

√ 

± 

Try It #8 

x −x −= 

Example 10 

Solving an Equation in Quadratic Form Containing a Binomial 

x+ +x+−= 

Solution ThTh 

fiff 

u =x +. Thu 

 

u +u −= 

 

u+−= 

u 

 

u += 

 

u =− 

 

x +=− 

 

x =− 

Th 

 

u −= 

 

u = 

 

x += 

 

x =− 

−− 

Try It #9 

x− −x−−=

140 


Solving Rational Equations Resulting in a Quadratic 

 

 

 

Example 11 Solving a Rational Equation Leading to a Quadratic 

−x 

x − + 

x + = − x − 

Solution o fi 

x −=x+x−Thx+x−x 

 

 

x+x−( −x 

x − + 

x + ) = ( 

− 

x+x− ) x+x− 

 

−xx++x−=− 

 

−x −x +x −=− 

 

−x += 

 

−x −= 

 

−x+x−= 

 

x =−x = 

. Th 

Try It #10 

x+ 

x− + 

x = − 

x −x 

ff 

Rational Equation with No Solution (http://openstaxcollege.org/l/rateqnosoln) 

Solving Equations with Rational Exponents Using Reciprocal Powers (http://openstaxcollege.org/l/ratexprecpexp) 

Solving Radical Equations part 1 of 2 (http://openstaxcollege.org/l/radeqsolvepart1) 

Solving Radical Equations part 2 of 2 (http://openstaxcollege.org/l/radeqsolvepart2)



VERBAL 

1. 

 

3. e − 

 

ERROR 

 

2. must 

 

4. |x +|=− 

5. 

 

ALGEBRAIC 

 

 

6. x = 

 

7. x = 

 

8. x 

 

−x 

= 

9. x − 

= 

10. x + 

= 

 

11. x 

−x 

+= 

 

12. x 

−x 

 

−x 

= 

 

13. x +x −x −= 14. x −x −x += 15. y −y = 

16. x +x −x −= 17. m +m −m −= 18. x −x = 

19. x +x =x + 

 

20. √ — x −−= 21. √ — x −= 22. √ — x −=x − 

23. √ — t += 24. √ — t ++= 25. √ — −x =x 

26. √ — x +−√ — x += 27. √ — x ++√ — x += 28. √ — x +−√ — x += 

 

29. |x −|= 30. |x −|=− 31. |−x|−= 32. |x +|−= 

33. |x −|−=− 34. |x +|−=− 35. |x +|= 36. −|x +|=− 

fi 

 

37. x −x += 38. t − −t −=− 39. x − +x −−= 

40. x + −x +−= 41. x − −= 

EXTENSIONS 

 

42. x − −x − −= 43. √ — |x| =x 44. t −t += 45. |x +x −|= 


TT =π 

√ L 

g 

L g 

46. 

 

= 

47. s 1 s, 

= 

 

BSA= 

√ wh w = 

 

h = 

48. 

BSA = 

49. 

BSA =

142 




Use interval notation. 

Use properties of inequalities. 

Solve inequalities in one variable algebraically. 

Solve absolute value inequalities. 

2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 

Figure 1 

 

 

ff 

 

Using Interval Notation 

x ≥ 

Figure 2Thx = 

fi 

 

Figure 2 

x|x≥ x x 

 

Thinterval notationTh 

x ≥∞Th 

 

Th 

 

fififi 

interval 

−−−− 

−−∞Table 1 

 

Set Indicated Set-Builder Notation Interval Notation 

abab x|a< x a a,∞ 

bb x| x

SECTION 2.7 LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 143 

Set Indicated Set-Builder Notation Interval Notation 

bb x| x ≤b −∞b 

aba x|a≤ x ba≥b 

a

144 


Solution Th 

 

a. x−< 

x −+ 

x +−>− 

 

x > 

Try It #3 

x−< 

Example 4 

Demonstrating the Multiplication Property 

 

a. x < b. −x −≥ c. −x > 

Solution 

 

 

a. x< 

 

x< 

 

x < 

b. −x−≥ 

 

−x ≥ 

( − 

) −x≥( − 

) 

− 

 

x ≤− 

c. −x> 

 

−x> 

−−x>− − 

x


Example 5 

Solving an Inequality Algebraically 

−x ≥x − 

Solution 

 

−x ≥x − 

−x ≥− 

−x ≥− 

x ≤ − 

Th−∞ 

Try It #5 

−x+< 

x+ 

Example 6 

Solving an Inequality with Fractions 

− 

x ≥− 

+ 

x 

Solution 

 

 

 

 

 

− 

x ≥− 

+ 

x 

− 

x − 

x ≥− 

 

− 

x − 

 

x ≥− 

 

− 

x ≥− 

 

x ≤− 

( − 

) 

 

x ≤ 

 

Th( −∞ 

] 

Try It #6 

 

− 

x≤ 

+ 

x 

Understanding Compound Inequalities 

 

 

compound inequality< x ≤< x 

x ≤Th 

 

 

Example 7 

Solving a Compound Inequality 

≤x +< 

Solution ≤x +x +< 

 

≤x + x +< 

≤x x < 

 

 

≤x 

x

146 


Th 

 

≤ x < 

 

[ 

) 

Th 

 

 

≤x +< 

≤x < 

 

≤ x < 

 

[ 

) 

Try It #7 

x −>x − 

Solution e fi 

+x >x − x −>x − 

>x − x −>− 

>x x >− 

 

 

>x 

x >− 

 

x < 

 

−< x 

Th−< x < 

( − 

) 

fi 

Figure 3 

 

 

 

Figure 3 

Try It #8 

y


ThTh 

Th 

ffi 

 

x- x 

x |x−||x−|≤ 

 

−≤x −≤ 

 

−+≤x −+≤+ 

 

≤ x ≤ 

Th 

 

 

absolute value inequalities 

X k >absolute value inequality 

 

|X|< k −k < X < k 

 

|X|> k X k 

Th|X|≤ k |X|≥k 

Example 9 

Determining a Number within a Prescribed Distance 

x 

Solution x 

Figure 4fi 

 

 

Figure 4 

Th x |x−| x 

 

|x−|≤ 

 

x −≤ x −≥− 

x ≤ x ≥ 

x ≤ x ≥ 

 

|x−|≤ 

 

Try It #9 

x 

Example 10 

|x−|≤ 

Solution 

 

 

 

Solving an Absolute Value Inequality 

|x−|≤ 

−≤x −≤ 

−≤ x ≤ 

−

148 


Example 11 

Using a Graphical Approach to Solve Absolute Value Inequalities 

 

y =− |x −|+ x- y- 

 

Solution y



VERBAL 

1. 

 

−x > 

x 

 

5. y =|x −| 

ALGEBRAIC 

r fi 

6. x −≤ 7. x +≥x − 8. −x +>x − 

9. x +≥x − 10. − 

x≤− 

+ 

x 

12. −x +>−x + 13. x+ 

−x+ 

≥ 

 

 

11. −x −+>x −−x 

14. x− 

+x+ 

≤ 

 

fi 

15. |x+|≥− 16. |x+|< 17. |x−|> 

18. |x +|+≤ 19. |x−|+≥ 20. |−x+|≤ 

21. |x−|− 23. | x− | < 

x- 

24. 25. 

26. 27. 

 

 

28. −x−>x− 

30. y


GRAPHICAL 

x- 

r fi 

33. |x−|> 34. |x+|≥ 35. |x+|≤ 36. |x −|< 37. |x−|< 

fty 

y 

 

y- 

 

38. x +x + 40. x +>x + 41. x +> x − 

 

42. x +< x + 

 

NUMERIC 

 

43. x|−< x < 44. x|x≥ 45. x|x< 46. x| x 

 

47. −∞ 48. ∞ 49. − 50. −∪∞ 

 

51. 

− 

− 

52. 

− 

− 

53. 

 

TECHNOLOGY 

fts 1 

Y2=MATHNum1:abs( 

2 nd CALC 5:intersection1st curve, enter, 2 nd curve, enter, guess, enter 

x-fi 

54. |x +|−< 55. − |x +|< 56. |x +|−> 57. |x −|< 

 

58. |x +|≥ 

EXTENSIONS 

59. |x +|=|x +| 60. x −x> 

61. x − ≤ x ≠− 

x + 62. 


63. 

V =T V T 

 

 

p=−x +x −fi 

x- 

fi 

64. 

 

. Th 

C =+x −



Key Terms 

absolute value equation 

 

area fi 

A =LW 

Cartesian coordinate system 

completing the square 

 

complex conjugate 

 

complex number a +bia 

b 

complex plane 

i 

compound inequality 

conditional equation 

discriminant 

 

distance formula o fi 

equation in two variables xy 

equations in quadratic form 

 

extraneous solutions 

graph in two variables 

 

identity equation 

imaginary number −i =√ — − 

inconsistent equation 

intercepts xy 

interval 

interval notation 

 

linear equation fi 

 

linear inequality 

midpoint formula o fi 

ordered pair 

xy 

origin 

perimeter fi 

P =L +W 

polynomial equation ffi


Pythagorean Theorem 

 

quadrant 

quadratic equation 

quadratic formula 

radical equation 

rational equation 

slope yx 

solution set 

square root property x 

x 

volume V =LWH 

x-axis 

x-coordinate 

x-intercept xy 

y-axis 

y-coordinate 

y-intercept yx 

zero-product property 

 

Key Equations 

 

x = −b±√— b −ac a 

 

Key Concepts 

2.1 The Rectangular Coordinate Systems and Graphs 

fi 

xyExample 1 

Example 2 

 

y = Example 3 

xyfiTh 

Example 4 

ThThfi 


Thfi 

xyExample 7Example 8 

2.2 Linear Equations in One Variable 

ax+b = 

e Example 1 Example 2 

 


fifi 

Example 5, Example 6, Example 7 

n fiExample 8


yExample 9 

n fiExample 10 

fi 

Example 11 

ThExample 12 

fiy =c 

fifix =c 

Example 13 

ffyExample 14Example 15 

 

Example 16 

2.3 Models and Applications 

Example 1 

 

Example 2 

Th 

d =rtExample 3 

P =L +WA =LW 

V =LWHExample 4Example 5Example 6 

2.4 Complex Numbers 

ThiExample 1 

Th 

Example 2 

 

Example 3 

 

◦Example 4Example 5 

◦ 


ThExample 8 

2.5 Quadratic Equations 

ffi 

ffThfiExample 1 


ffi 


Th 

Th 


 

Example 8 

ffi 


Th 

Example 11


ThTh 

fi 

Example 12 

2.6 Other Types of Equations 

 

 

Example 1Example 2Example 3 

 


 


 

Example 8 

fi 

 


Example 11 

2.7 Linear Inequalities and Absolute Value Inequalities 

 

 

Table 1 Example 1 Example 2 

Th 

Example 3Example 4Example 5Example 6 

ft 

 


 

Example 9 

Example 10 

 

Example 11



THE RECTANGULAR COORDINATE SYSTEMS AND GRAPHS 

, fi x- y- 

1. x −y = 2. y −=x 

y x 

3. x =y − 4. x −y = 

, fi 

5. −− 6. −−− 

7. − 

 

, fi 

8. − 9. − 

 

10. y = x + 

 

11. x −y = 

LINEAR EQUATIONS IN ONE VARIABLE 

x 

12. x +=x − 13. x +−=x + 14. x −= 

15. −x +=x − 16. x 

− 

=x 

+ 

 

x x- 

17. 

x 

x − + 

x + = x ≠− 

x − 18. 

+ 

x = 

 

, fi 

19. − 20. −− 

 

 

21. − 22. 

y = x + 

 

MODELS AND APPLICATIONS 

 

 

23. Ths fi 

 

 

 

24.


25. 

 

COMPLEX NUMBERS 

 

26. x −x += 27. x +x += 

 

28. −i 29. −−i 

 

30. −i−−i 31. +i−−−i 32. √ — −+√ — 

33. √ — −+√ — − 34. −ii − 35. −i 

36. √ — −√ — − 37. √ — −( √ — −−√ — ) 

 

38. 

−i 

39. +i 

i 

 

QUADRATIC EQUATIONS 

 

40. x −x −= 41. x +x += 42. x −= 

43. x −x = 

 

44. x = 45. x − = 

 

46. x +x −= 47. x +x −= 

 

No real solution 

48. x −x += 49. x −x −= 

 

50. x − = 51. x =x +


OTHER TYPES OF EQUATIONS 

 

 

52. x = 

53. x 

 

−x 

= 

54. x +x −x −= 

55. x −x = 56. √ — x +=x − 57. √ — x ++√ — x += 

58. |x −|= 59. |x +|−= 

LINEAR INEQUALITIES AND ABSOLUTE VALUE INEQUALITIES 

r fi 

60. x −≤ 61. −x +>x − 62. x − +x + ≤ 

63. |x +|+≤ 64. |x −|> 65. |x −|



1. yx 2. x- y- 

x −y = 

3. x- y- 

 

x −y = 

4. −− 

 

 

5. 

x|x≤ 

6. xx +=x − 

7. xx −−x −=x − 8. x x 

+= 

x 

 

9. x 

x + =+ 

 

x − 

10. Ths 30 in. Th 

 

 

 

11. x 12. x −≤ 

x 

−x =− 

 

13. |x +|< 14. |x −|≥ 

, fi 

15. −− 16. 

 

17. 

18. 

y =− x + 

 

−+−i 

19. √ — −+√ — − 20. i− 

21. −i 

+i 

22. 

a +bix −x += 

23. x − −= 24. x −x = 

25. x −x −= 26. √ — x −=x − 

27. +√ — −x =x 28. x − 

= 

, fi 

29. x −x −x += 30. x + −x +−=

3 

Functions 

P 

 

 

 

y 

 

Figure 1 Standard and Poor’s Index with dividends reinvested 

(credit "bull": modification of work by Prayitno Hadinata; credit "graph": modification of work by MeasuringWorth) 


3.1 Functions and Function Notation 

3.2 Domain and Range 

3.3 Rates of Change and Behavior of Graphs 

3.4 Composition of Functions 

3.5 Transformation of Functions 

3.6 Absolute Value Functions 

3.7 Inverse Functions 

Introduction 

 

Figure 1 

 

Thfi 

 

Th 

 

fi 

 

 

159

160 

CHAPTER 3 FUNCTIONS 



Determine whether a relation represents a function. 

Find the value of a function. 

Determine whether a function is one-to-one. 

Use the vertical line test to identify functions. 

Graph the functions listed in the library of functions. 

3.1 FUNCTIONS AND FUNCTION NOTATION 

flTh 

Th 

 

Determining Whether a Relation Represents a Function 

relationThfidomain 

range 

ThfififiTh 

e fi 

 

ThTh 

inputindependent variableft 

xoutputdependent variable 

fty 

f 

xfifi 

 

 

fifi 

 

 

not 

 

ThfiFigure 

1 

 

 

p 

p 

x 

x 

m 

p 

q 

q 

y 

y 

n 

q 

r 

r 

z 

z 

 

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. 

( b ) This relationship is also a function. In this case, each input is associated with a single output. 

( c ) This relationship is not a function because input q is associated with two different outputs.

SECTION 3.1 FUNCTIONS AND FUNCTION NOTATION 161 

function 

function 

 

inputdomainoutputrange 

How To… 

 

1. 

2. 

3. 

 

Example 1 Determining If Menu Price Lists Are Functions 

ThffFigure 2 

a. b. 

Menu 

Item 

Price 

Plain Donut ..............................1.49 

Jelly Donut ..............................1.99 

Chocolate Donut ..........................1.99 

Figure 2 

Solution 

a. ThFigure 2 

 

b. 

Figure 3 

Menu 

Item 

Price 

Plain Donut ..............................1.49 

Jelly Donut ..............................1.99 

Chocolate Donut .......................... 

Th 

Figure 3 

Example 2 

Determining If Class Grade Rules Are Functions 

 

Table 1

162 


Percent grade 

Grade-point average 

Table 1 

Solution 

fi 

 

 

. Th 

Try It #1 

Table 2 e fi 

Player 

 

 

 

 

 

Rank 

 

 

 

 

 

a. 

b. 

Table 2 

Using Function Notation 

fi 

 

Th 

 

ha 

ThfghftxyzAB 

C 

hfa f 

h =f a 

f a 

f fa 

hahaTha 

hTh 

 

f a +bfiab 

fTh 

function notation 

Thy =f xfifThyxThx 

. Thyf x


Example 3 

Using Function Notation for Days in a Month 

 

 

Solution Thf 

=f d =f mThfi 

 

output 

31f (January) 

rule 

Figure 4 

f =Thd =f m 

d m 

Analysis Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels 

of geometric objects, or any other element that determines some kind of output. However, most of the functions we will 

work with in this book will have numbers as inputs and outputs. 

Example 4 

Interpreting Function Notation 

N =f yffiNyf = 

Solution f =Th 

ffiNN =f yThf = 

ffi 

input 

Try It #2 

d 

Q & A… 

Instead of a notation such as y = f (x), could we use the same symbol for the output as for the function, such as 

y = y (x), meaning “y is a function of x?” 

 

f 

yfxTh 

y =f xP =Wd 

Representing Functions Using Tables 

 

 

 

Table 3== 

Th 

fifD =f mfi 

 

Month number, m (input) 

Days in month, D (output) 

Table 3

164 


Table 4fiQ =g ng 

nQ 

n 

Q 

Table 4 

Table 5Th 

 

ff 

Age in years, a (input) 

Height in inches, h (output) 

Table 5 

How To… 

 

1. 

2. 

Example 5 

Identifying Tables that Represent Functions 

Table 6Table 7Table 8 

Input 

 

 

 

Output 

 

 

 

Input 

− 

 

 

Output 

 

 

 

Input 

 

 

 

Table 6 Table 7 Table 8 

Output 

 

 

 

Solution Table 6Table 7fi 

Table 8fiff 

fi 

 

ThTable 6 

f =f =f = 

 

g −=g =g = 

Table 7. 

Table 8 

Try It #3 

Table 9 

Input 

 

 

 

Table 9 

Output


Finding Input and Output Values of a Function 

 

 

 

 

 

ff 

Evaluation of Functions in Algebraic Forms 

 

f x=−x 

 

How To… 

 

1. 

2. 

Example 6 

Evaluating Functions at Specific Values 

fx=x +x − 

a. b. a c. a +h d. f a +h−f a __ 

 

 

h 

Solution xfi 

a. 

 

f = +− 

 

=+− 

 

= 

b. 

 

f a=a +a − 

c. a +h 

 

f a +h=a +h +a +h− 

 

=a +ah +h +a +h − 

d. 

 

 

f a +h=a +ah+h +a +h − 

 

 

fa=a +a − 

 

 

 

f a +h−fa __ 

 

=____ 

a +ah+h +a +h −−a +a − 

 

h 

h 

=___________ 

ah+h +h 

 

h 

 

=___________ 

ha +h 

+ h 

h 

=a +h +

166 


Example 7 Evaluating Functions 

hp=p +ph 

Solution hp 

 

hp=p +p 

 

h= + 

 

=+ 

 

= 

Th 

Try It #4 

gm=√ — m−g 

Example 8 

Solving Functions 

hp=p +php= 

Solution 

hp= 

p +p = hp=p +p 

p +p −= 

p+p−= 0 

p+p−=p+=p−= 

p 

p+= p =− 

p−= p = 

ThThhp=p =p =− 

Figure 5. Thfih=h−=h= 

h(p) 

 

 

 

 

 

 

 

 

p 

Try It #5 

gm=√ — m−gm= 

p 

h(p) 

Figure 5 

Evaluating Functions Expressed in Formulas 

fi 

fi 

n +p =np 

pn


How To… 

 

1. 

only 

2. 

 

Example 9 

Finding an Equation of a Function 

n +p =p =f n 

Solution p 

np =n 

 

n +p = 

p =−n n 

 

 

 

Thpn 

 

Example 10 

p = −n _______ 

 

p =__ 

 

−n ___ 

 

p =− 

n 

p =f n=− 

__ n 

Expressing the Equation of a Circle as a Function 

x +y =xy 

y =f x 

Solution x 

y 

y =−x 

y =±√ — −x 

=+√ — −x −√ — −x 

 

y =f x 

Try It #6 

x−y =yx 

Q & A… 

Are there relationships expressed by an equation that do represent a function but that still cannot be represented 

by an algebraic formula? 

x =y + y yx 

xyxy 

y 

yx

168 


Evaluating a Function Given in Tabular Form 

 

 

Thfifi 

fi 

Th 

 

Th 

Table 10 Pet Memory span in hours 

 

 

 

fi 

a fi 

 

 

 

 

 

Table 10 

P 

 

PfiPfi= 

 

ThP 

 

How To… 

fi 

1. 

2. 

3. 

4. 

Example 11 

Evaluating and Solving a Tabular Function 

Table 11 

a. g b. gn= 

n 

g (n) 

Solution 

Table 11 

a. g gn =Th 

n =g = 

b. g n=nTable 11 

gg


Try It #7 

Table 11g 

Finding Function Values from a Graph 

fi 

fifi 

 

Example 12 

Reading Function Values from a Graph 

Figure 6 

a. f 

b. f x= 

f x 

 

 

 

 

 

 

 

–– – – – – 

 

 

 

 

 

– 

– 

Solution 

Figure 6 

a. f x =yTh 

f =Figure 7 

f x 

 

 

 

 

 

 

 

–– – – – – 

– 

– 

 

Figure 7 

 

 

 

f = 

b. f x= y = 

−Th 

f x=− f −=f =−Figure 8 

f x 

 

 

− 

 

 

 

 

 

 

 

–– – – – – 

– 

– 

 

Figure 8

170 


Try It #8 

Figure 7f x= 

Determining Whether a Function is One-to-One 

 

Figure 1fiff 

e fiff 

 

 

Table 12 

Letter grade 

 

 

 

 

Grade-point average 

 

 

 

 

Table 12 

Th 

 

Figure 1(a)Figure 1(b)Th 

qr 

nTh 

 

one-to-one function 

one-to-one functionTh 

xy 

Example 13 

Determining Whether a Relationship Is a One-to-One Function 

 

Solution rA =πr r 

A. Thr 

 

AA =πr 

r = 

√ A 

π 

Try It #9 

a. 

b. 

c. 

Try It #10 

a. 

 

b.


Using the Vertical Line Test 

 

Thft 

 

 

Thxyyxy =f x 

fThxyfi 

y =f xfi 

xy 

Figure 9f =f = 

xyy =f xTh 

 

y 

x 

Figure 9 

vertical line test 

fi 

Figure 10 

 

Figure 10 

How To… 

 

1. 

2.

172 


Example 14 

Applying the Vertical Line Test 

Figure 11y =f x 

f x 

y 

f x 

 

 

 

 

x 

 

 

x 

x 

 

Figure 11 

Solution 

Figure 

11Th 

xFigure 12 

y 

x 

Figure 12 

Try It #11 

Figure 13 

y 

x 

Figure 13


Using the Horizontal Line Test 

fi 

horizontal line test 

 

How To… 

 

1. 

2. 

Example 15 

Applying the Horizontal Line Test 

Figure 11(a)Figure 11(b) 

Solution ThFigure 11(a)ThFigure 14 

n fi 

f x 

x 

Figure 14 

ThFigure 11(b) 

Try It #12 

 

y 

x 

Identifying Basic Toolkit Functions

174 


 

fixy =f x 

 

 

Th 

Table 13 

Toolkit Functions 

Name Function Graph 

f x 

f x=cc x 

x 

 

 

 

f(x) 

 

 

 

f x 

f x=x x 

x 

 

 

 

f(x) 

 

 

 

f x 

f x=x x 

x 

 

 

 

f(x) 

 

 

 

f x=x x 

f x 

x 

 

 

 

 

 

f(x) 

 

 

 

 

 

f x=x x 

f x 

x 

 

 

 

 

 

f(x)


f x 

x 

f(x) 

 

 

 

 

 

f x=__ 

x 

x 

 

 

 

 

 

 

 

 

f x 

x 

f(x) 

 

 

 

 

 

f x=_ 

 

x 

x 

 

 

 

 

 

 

 

 

f x 

f x=√ — x x 

x 

 

 

 

f(x) 

 

 

 

f x 

x 

f(x) 

 

 

 

f x= √ 

— x 

x 

 

 

 

 

 

 

 

 

Table 13 

 

Determine if a Relation is a Function (http://openstaxcollege.org/l/relationfunction) 

Vertical Line Test (http://openstaxcollege.org/l/vertlinetest) 

Introduction to Functions (http://openstaxcollege.org/l/introtofunction) 

Vertical Line Test of Graph (http://openstaxcollege.org/l/vertlinegraph) 

One-to-one Functions (http://openstaxcollege.org/l/onetoone) 

Graphs as One-to-one Functions (http://openstaxcollege.org/l/graphonetoone)

176 CHAPTER 3 FUNCTIONS 


VERBAL 

1. e diff 

 

3. 

 

5. 

 

2. e diff 

 

4. 

 

ALGEBRAIC 

 

6. abcdac 7. abbcc 

yx 

8. x + y = 9. y=x 10. x=y 

11. x +y= 12. x+y = 13. y=−x +x 

14. y= 

__ 

x 

15. x=y+ 

y− 

16. x=√— −y 

17. y = 

______ x+ x− 18. x +y = 19. xy= 

20. x=y 21. y=x 22. y=√ — −x 

23. x=± √ — −y 24. y=± √ — −x 25. y =x 

26. y =x 

ff −f f −a−f af a +h 

27. f x=x− 28. fx=−x +x− 29. fx=√ — −x + 

30. fx= 

______ x− x+ 

31. fx=∣x−∣−∣x+∣ 

32. gx=−x gx + h−gx __ 

 

h h≠ 

33. gx=x +x gx−ga _ 

x−a 

x≠a 

34. kt=t− 

a. k 

b. kt= 

36. pc=c +c 

a. p− 

b. pc= 

38. f x= √ — x+ 

a. f 

b. f x= 

35. f x=−x 

a. f− 

b. fx=− 

37. f x=x −x 

a. f 

b. f x= 

39. r+t= 

a. r=f t 

b. f − 

c. ft=


GRAPHICAL 

 

40. 

y 41. 

y 42. 

y 

x 

x 

x 

43. 

y 44. y 

45. 

y 

x 

x 

x 

46. 

y 47. 

y 48. 

y 

x 

x 

x 

49. 

y 50. 

y 51. 

y 

x 

x 

x


52. 

a. f − 

b. fx= 

y 

53. 

a. f 

b. fx=− 

y 

54. 

a. f 

b. fx= 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

55. 

y 56. 

y 57. 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

58. 

y 59. 

y 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

 

 

 

 

 

x 

π 

 

 

 

 

 

 

 

 

 

 

π 

x 

NUMERIC 

 

60. −−−−−− 61. 62. 

yx 

63. x 

y 

64. x 

y 

65. x 

y 

fTable 14 

x 

f (x) 

66. f 67. fx= 

Table 14


ff −f −f f f 

68. f x=−x 69. f x=−x 70. f x=x −x+ 

71. f x=+√ — x+ 72. f x= x− 

x+ 

73. f x= x 

fgh 

f x=x − gx=−x hx=−x +x − 

74. f −g− 75. f ( 

) −h− 

TECHNOLOGY 

y =x 

 

76. − 77. − 78. − 

y =x 

 

79. − 80. − 81. − 

y =√ — x 

 

82. 83. 84. 

y = √ 

— x 

 

85. − 86. − 87. − 


88. ThG 

pG =f pG 

p 

 

a. 

 

f 

b. f = 

89. ThD 

a 

D =ga 

a. 

g 

b. g= 

90. f ttft 

 

a. f = 

b. f = 

91. ht 

tft 

 

a. h= 

b. h= 

92. f x=x − +

180 




Find the domain of a function defined by an equation. 

Graph piecewise-defined functions. 

3.2 DOMAIN AND RANGE 

fi 

I am LegendHannibal The Ring The GrudgeThe ConjuringFigure 1 

 

 

ff 

 

 

Inflation-adjusted Gross, 

in Millions of Dollars 

 

 

 

 

 

 

 

 

 

 

 

Top-Five Grossing Horror Movies 

for years 2000–2013 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1 Based on data compiled by www.the-numbers.com. 

Finding the Domain of a Function Defined by an Equation 

Market Share of Horror Moveis, by Year 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Functions and Function Notation 

fi 

 

 

 

 

 

 

Figure 2 

 

a 

b 

c 

 

 

Figure 2 

interval notation 

 

 

 

 

Thffi 

 

x 

y 

z

SECTION 3.2 DOMAIN AND RANGE 181 

fiftfi 

ff 

 

Th 

 

 

Th 

Th 

 

 

Figure 3 

Inequality Interval Notation Graph on Number Line Description 

x>a 

a∞ 

a 

xa 

x

182 


Example 2 

Finding the Domain of a Function 

f x=x − 

Solution Thx 

 

. Th 

f−∞∞ 

Try It #2 

fx=−x+x 

How To… 

n, fi 

1. 

2. 

x 

 

3. 

Example 3 

Finding the Domain of a Function Involving a Denominator 

f x= x + −x 

Solution 

x 

 

−x = 

 

−x=− 

x = 

Thx Figure 

4∪ 

−∞∪∞ 

Try It #3 

f x= +x ______ 

x− 

 

x 

↓ ↓ 

−∞∪∞ 

Figure 4 

How To… 

, fi 

1. 

2. 

x 

3. Th


Example 4 

Finding the Domain of a Function with an Even Root 

f x=√ — −x 

Solution 

 

x 

 

−x ≥ 

 

−x≥− 

x ≤ 

Th 

−∞ 

Try It #4 

f x=√ — +x 

Q & A… 

Can there be functions in which the domain and range do not intersect at all? 

 

f x=− _ 

 

√ — x 

 

ff 

 

Using Notations to Specify Domain and Range 

 

fi 

set-builder notationx|≤x

184 


 

∪ 

or 

fi 

 

 

x||x |≥=−∞−∪∞ 

set-builder notation interval notation 

Set-builder notation 

x|xxx 

 

x| 

x |≤x ≤x > 

∪∞


Try It #5 

s fi 

a. 

b. 

c. 

 

Figure 7 

Finding Domain and Range from Graphs 

 

xTh 

y 

Figure 8 

y 

 

 

 

 

 

 

 

– –– 

– – – – 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

Figure 8 

−−∞ 

Th−∞ 

 

 

Example 6 Finding Domain and Range from a Graph 

fFigure 9 

y 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

f 

– 

– 

– 

– 

Figure 9

186 


Solution −f− 

Th−−Figure 10 

y 

– 

– 

– 

 

 

– – 

– 

 

 

 

 

 

x 

f 

– 

– 

– 

– 

Figure 10 

 

Example 7 

Finding Domain and Range from a Graph of Oil Production 

fFigure 11. 

Thousand barrels per day 

 

 

 

 

 

 

 

 

 

 

 

 

Alaska Crude Oil Production 

 

Figure 11 (credit: modification of work by the U.S. Energy Information Administration) 

Solution ThtTh 

b. Th 

 

≤t ≤≤b ≤ 

 

 

Try It #6 

Figure 12 

Millions of people 

 

 

 

 

 

 

 

 

 

 

 

World Population Increase 

 

Year 

Figure 12 

 

Q & A… 

Can a function’s domain and range be the same? 

 

===


Finding Domains and Ranges of the Toolkit Functions 

 

f x 

f x=c 

−∞∞ 

cc 

x 

constant functionfx=c 

Th 

cc 

 

c, cc. 

Figure 13 

f x 

−∞∞ 

−∞∞ 

x 

identity functionfx=x 

x 

Figure 14 

f x 

−∞∞ 

∞ 

x 

absolute value functionfx=∣ x ∣ 

xfi 

 

 

Figure 15 

f x 

−∞∞ 

∞ 

x 

quadratic functionfx=x 

 

 

 

 

Figure 16 

f x 

−∞∞ 

−∞∞ 

x 

cubic functionfx=x 

 

Th 

 

 

Figure 17

188 


f x 

−∞∪∞ 

−∞∪∞ 

x 

reciprocal functionfx= x 

 

 

x|x ≠ 

 

Figure 18 

f x 

Figure 19 

f x 

Figure 20 

f x 

−∞∪∞ 

∞ 

x 

x 

∞ 

∞ 

−∞∞ 

−∞∞ 

x 

reciprocal squared functionfx= 

x 

Th 

x 

 

 

 

square root functionfx=√ — x 

 

Th 

xfi 

 

−√ — x x. 

cube root functionfx= √ 

— x 

 

 

 

 

Figure 21 

How To… 

 

1. 

2. fi 

3. 

4. fi 

Example 8 Finding the Domain and Range Using Toolkit Functions 

f x=x −x 

Th 

. Th−∞∞−∞∞


Example 9 Finding the Domain and Range 

 

f x=____ 

 

x + 

Solution −fiTh−∞−∪−∞ 

. Th−∞∪∞ 

Example 10 

Finding the Domain and Range 

f x=√ — x + 

Solution 

x+≥x ≥− 

Thf x−∞ 

fif −=x 

f∞ 

Analysis Figure 22 represents the function f. 

f x 

 

 

 

 

 

– – – – – – 

f 

 

 

 

 

 

x 

Figure 22 

Try It #7 

f x=−√ — −x 

Graphing Piecewise-Defined Functions 

 

f x=|x| 

 

 

 

 

f x=xx ≥ 

 

f x=−xx < 

ff 

piecewise functionfiff 

 

 

ft 

 

 

ThSSS ≤+S− 

S >

190 


piecewise function 

piecewise functionfi 

 

{ 

x 

f x= x 

x 

 

x x ≥ 

|x|= −x x < 

How To… 

 

1. h diff 

2. 

3. 

Example 11 

Writing a Piecewise Function 

fi 

nC 

Solution 

C = 

ffnC =nn 

n


Analysis The function is represented in Figure 24. We can see where the function changes from a constant to a shifted 

and stretched identity at g=. We plot the graphs for the different formulas on a common set of axes, making sure each 

formula is applied on its proper domain. 

Cg 

 

 

 

Cg 

 

 

– – – – 

– 

– 

 

 

 

 

 

g 

Figure 24 

How To… 

 

1. x 

2. 

 

Example 13 

 

Graphing a Piecewise Function 

 

x x≤ 

f x= 

Solution 

 

 

 

 

Figure 25 

f x 

f x 

f x 

 

 

 

 

 

 

 

 

 

 

 

 

– 

– 

– 

 

– 

– 

 

 

 

 

x 

– 

– 

– 

 

– 

– 

 

 

 

 

x 

– 

– 

– 

 

– 

– 

 

 

 

 

x 

 

 

Figure 25 ( a) f (x)=x 2 if x≤1; ( b) f (x)=3 if 12

192 


Figure 26 

f x 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

x 

Figure 26 

Analysis Note that the graph does pass the vertical line test even at x= and x= because the points and 

are not part of the graph of the function, though and are. 

Try It #8 

 

 

x x



VERBAL 

1. in diffr diff 2. 

 

3. fx= √ 

— x diff 

fx=√ — x 

4. 

 

 

5. 

ALGEBRAIC 

, fi 

6. f x=−xx − x − 7. f x=−x 8. f x=√ — x− 

9. f x=−√ — −x 10. f x=√ — −x 11. f x=√ — x + 

12. f x= √ 

— −x 

15. f x= ______ 

x + x + 

 

18. f x= ________ 

 

x −x − 

21. f x= _ 

x+ √ — − x 

24. f x= x __ 

x 

13. f x= √ 

— x− 

16. f x= 

_______ x + 

√— x − 

x 

19. f x= __________ 

− x −x− 

22. f x= _ 

√— x − √ — x − 

25. f x= x −x _ 

x − 

 

14. f x= _____ 

 

x − 

x − 

17. f x= __________ 

 

x +x − 

 

20. f x= _ 

 

√ — x − 

23. f x= _ x − √— √ — x − 

26. f x=√ — x −x 

a. 

b. x 

 

GRAPHICAL 

 

27. 

y 

28. 

y 

29. 

y 

 

 

 

 

 

–– 

– – – – 

 

 

 

 

 

x 

 

 

 

 

 

–– 

– – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

–


30. 

y 

31. 

y 

32. 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

33. 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

34. 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

35. 

y 

 

 

 

 

 

 

– – – – – – – 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

36. 

y 

37. 

y 

 

 

 

 

 

–– 

– – – – 

 

 

 

 

 

x 

 

 

 

 

 

–– 

– – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

x + x 

41. f x= 

 

x < 

√ — x x ≥ 

42. f x= 

 

x x < 

− x x > 

43. f x= 

 

x x < 

x + x ≥ 

44. f x= 

 

x + x < 

x x ≥ 

| x | x < 

45. f x= x ≥


NUMERIC 

ff −f −f −f 

x + x − 

f x= 

 

 

 

−x + x ≤− 

x − x >− 

ff −f f f 

x + x < 

x 

49. f x= 

− x < 

50. 

x + x ≥ 

f x= 

 

x x < 

51. f x= ≤x ≤ 

+ | x − |x ≥ 

x x > 

 

x + x 

f x= −x x ≥ 

TECHNOLOGY 

55. y= 

__ 

x −− 

 

56. y=_ 

x 

−− 

 

EXTENSION 

57. f−fx 

58. 

59. x> 


60. h 

tt 

ht=−t +t 

 

 

61. x 

Cx=x+ 

a. 

 

b. 

c. 

Cx

196 




Find the average rate of change of a function. 

Use a graph to determine where a function is increasing, decreasing, or constant. 

Use a graph to locate local maxima and local minima. 

Use a graph to locate the absolute maximum and absolute minimum. 

3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 

flTable 1 

Th 

y 

C(y) 

Table 1 

 

 

per year 

Finding the Average Rate of Change of a Function 

Thrate of change 

Table 1 

fiaverage 

rate of changefifi 

 

 

= __ 

 

 

= Δy _ 

Δx 

= y _ −y x 

−x 

 

 

= f x −f x __ 

 

x 

−x 

 

 

ThΔfiyx 

yxΔf Δy 

 

 

 

 

Δy _ 

Δx = _ 

≈ 

 

 

 

 

 

Th 

Th 

=

SECTION 3.3 RATES OF CHANGE AND BEHAVIOR OF GRAPHS 197 

rate of change 

Th 

 

Th 

 

Δy _ 

Δx =f x −f x __ 

 

x 

−x 

 

 

How To… 

ff 

x 

x 

 

1. e diffy 

−y 

=Δy 

2. ffx 

−x 

=Δx 

3. Δy _ 

Δx 

Example 1 

Computing an Average Rate of Change 

Table 1, fi 

Solution . Th 

 

 

___ Δy 

Δx =y _ −y x 

−x 

 

 

 

=__ 

− 

− 

 

=______ 

− 

 

 

=− 

Analysis Note that a decrease is expressed by a negative change or “negative increase.” A rate of change is negative when 

the output decreases as the input increases or when the output increases as the input decreases. 

Try It #1 

Table 1 

Example 2 Computing Average Rate of Change from a Graph 

g tFigure 1, fi− 

gt 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

t 

– 

– 

– 

– 

Figure 1

198 


Solution t =−Figure 2g −=t=g = 

gt 

 

 

 

Δgt=– 

 

 

– – – – – – 

– 

– 

– 

– 

 

 

Δt= 

t 

Figure 2 

ThΔ t =Δ gt=− 

ThTh– 

 

− 

_______ 

 

−− =− ___ 

 

=− 

Analysis Note that the order we choose is very important. If, for example, we use y − _ y 

x 

− x 

, we will not get the correct 

 

answer. Decide which point will be and which point will be , and keep the coordinates fixed as ( x 

, y 

) and ( x 

, y 

). 

Example 3 

Computing Average Rate of Change from a Table 

ft 

. ThTable 2e fi 

t (hours) 

D(t) (miles) 

Table 2 

Solution 

 

 

Th 

_______ 

− 

− = ___ 

 

 

= 

Analysis Because the speed is not constant, the average speed depends on the interval chosen. For the interval , 

the average speed is miles per hour. 

Example 4 

Computing Average Rate of Change for a Function Expressed as a Formula 

f x=x − __ 

 

x 

Solution 

 

f = − 

__ 

=−__ 

 

 

= 

__ 

 

f= − 

__ 

=− __ 

 

= __


 

 

= f −f _ 

− 

__ 

 

− __ 

=_ 

− 

 

 

Try It #2 

__ 

 

=_ 

 

 

=__ 

 

 

fxx−√ — x 

Example 5 

Finding the Average Rate of Change of a Force 

F 

dFd= _ 

 

d 

 

 

Solution 

F d= 

__ 

d l [ 

 

 

=__ 

F−F − 

_ 

 

 

= 

− _ 

 

_ 

− 

 

 

 

 

__ 

− __ 

= 

_ 

 

 

 

=− __ 

 

Th− __ 

 

Example 6 

Finding an Average Rate of Change as an Expression 

− __ 

=_ 

 

 

gt=t +t +aTh 

a 

Solution 

 

 

= ga−g _ 

a − 

=___ 

a +a +− ++ 

 

a − 

 

=_____________ 

a +a +− 

 

a 

 

 

 

= 

_______ aa+ 

a 

=a + 

a 

That=t=a 

+=

200 


Try It #3 

f x=x +x−aa 

Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant 

fi 

 

 

Th 

Figure 3 

 

fx 

 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

 

 

 

x 

 

fb>fa 

b>a 

fba 

fb>fa 

b>a 

Figure 3 The function f(x) = x 3 − 12x is increasing on (−∞, −2) ∪ (2, ∞) and is decreasing on (−2, 2). 

 

 

local maximum 

local 

minimumThlocal extrema 

Thftlocal 

relativelocal 

 

local 

fi 

Figure 4x =−Th 

−x = 

= 

x=– 

fx 

 

– 

 

 

 

 

 

– – – – – – 

 

 

 

x 

– 

– 

– 

– 

– 

 

=– 

x= 

Figure 4


 

 

Th 

Figure 5 

fx 

fb 

 

 

 

 

 

a 

b 

c 

Figure 5 Definition of a local maximum 

x 

Thfi 

local minima and local maxima 

fincreasing functionf b>f aab 

b>a 

fdecreasing functionf ba 

fx=baca

202 


Analysis Notice in this example that we used open intervals (intervals that do not include the endpoints), because the 

function is neither increasing nor decreasing at t=, t=, and t=. These points are the local extrema (two minima 

and a maximum). 

Example 8 

Finding Local Extrema from a Graph 

f x= 

__ 

x +x __ 

Th 

 

Solution fiFigure 7 

x =x = 

x=−x=− 

fx 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

Figure 7 

Analysis Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 8 

provides screen images from two different technologies, showing the estimate for the local maximum and minimum. 

 

y 

 

 

 

– 

 

 

 

 

 

x 

 

=– 

Figure 8 

 

=– 

Based on these estimates, the function is increasing on the interval (−∞, −) and (, ∞). Notice that, while we 

expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing 

approximation algorithms used by each. (The exact location of the extrema is at ± √ — , but determining this requires 

calculus.) 

Try It #4 

f x=x −x −x+


Example 9 Finding Local Maxima and Minima from a Graph 

fFigure 9, fi 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

f 

Figure 9 

Solution fThx= 

x=. Thyx= 

Thx=−x=− 

Thyx=−− 

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals 

Figure 10Figure 11Figure 12 

Function Increasing/Decreasing Example 

y 

 

fx=c 

 

 

x 

y 

 

fx=x 

 

x 

y 

 

fx=x 

∞ 

−∞ 

x= 

x 

Figure 10

204 



y 

 

fx=x 

 

x 

y 

 

fx= 

__ 

x 

 

−∞∪∞ 

x 

y 

 

fx= 

_ 

x 

−∞ 

∞ 

x 

Figure 11 


y 

 

fx= √ 

— x 

 

x 

y 

 

fx=√ — x 

∞ 

x 

y 

 

fx=∣ x ∣ 

∞ 

−∞ 

x 

Figure 12


Use A Graph to Locate the Absolute Maximum and Absolute Minimum 

Thff 

Thy 

absolute maximumabsolute minimum 

 

Figure 13 

y 

 

 

 

 

 

– – – – – – 

f 

 

 

 

 

 

f = 

x 

– 

– 

– 

– 

f = − 

Figure 13 

Thf x=x 

absolute maxima and minima 

absolute maximumfx=cf cf c≥f xxf 

absolute minimumfx=df df d≤f xxf 

Example 10 Finding Absolute Maxima and Minima from a Graph 

fFigure 14, fi 

y 

 

 

 

 

 

– – – – – – 

 

 

 

f 

 

 

x 

– 

– 

– 

– 

Figure 14 

Solution fThx=−x= 

Thy 

x=−x= 

Thx= 

Thyx=− 

 

Average Rate of Change (http://openstaxcollege.org/l/aroc)

206 



VERBAL 

1. 

 

3. 

d diff 

2. fab 

bc 

fac 

4. 

 

y=x 

 

ALGEBRAIC 

fifi 

bh 

5. f x=x −b 6. g x=x − 

7. px=x++h 8. kx=x−+h 

9. f x=x +xx+h 10. gx=x −xx+h 

 

11. at= 

 

t+ +h 

 

12. bx= 

 

x+ +h 

13. jx=x +h 14. rt=t +h 

f x+h−fx 15. __ 

f(x=x −xx, x+h 

h 

GRAPHICAL 

fFigure 15 

16. x=x= 

 

 

 

 

 

 

 

y 

 

17. x=x= 

 

 

 

– 

 

 

 

Figure 15 

 

 

 

x 

 

 

18. 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

y 

 

 

 

 

 

x 

19. 

 

 

 

 

 

y 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

20. 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

y 

 

 

 

 

 

x


21. 

y 

 

 

 

 

 

 

– – – – – – – 

 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

Figure 16 

y 

22. 

23. f 

 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 17. 

Figure 16 

y 

24. 

 

25. 

 

 

 

 

 

 

–– 

– – – – 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 17 

NUMERIC 

26. Table 3 

ab 

Year 

Sales (millions of dollars) 

Table 3 

27. Table 4 

ab 

Year 

Population (thousands) 

Table 4


, fifi 

28. f x=x 29. hx=−x − 

30. qx=x − 31. g x=x −− 

32. y= x 

34. k t=t + 

__ 

t − 

33. pt= t −t+ 

 

t + − 

TECHNOLOGY 

 

 

35. f x=x −x + 36. hx=x +x +x +x − 

37. gt=t √ — 

t+ 38. kt=t −t 

39. mx=x +x −x −x+ 40. nx=x −x +x −x+ 

EXTENSION 

41. ThfFigure 18 

 

= 

Figure 18 

= 

42. f x= c 

x 

f c 

− __ 

 


44. 

 

 

 

 

46. 

 

dt=t tdt 

 

 

t=t= 

 

 

a. 

b. 

c. 

d. 

43. f(x)= __ 

 

x .b 

fb 

− 

__ 

 

45. o fi 

 

 

 

 

 

 

47. ThFigure 19 

t 

Amount (milligrams) 

 

 

 

 

 

 

A 

 

 

 

Time (days) 

Figure 19 

 

t=t= 

 

 

t

SECTION 3.4 COMPOSITION OF FUNCTIONS 209 



Combine functions using algebraic operations. 

Create a new function by composition of functions. 

Evaluate composite functions. 

Find the domain of a composite function. 

Decompose a composite function into its component functions. 

3.4 COMPOSITION OF FUNCTIONS 

Th 

 

fiTh 

 

ThCTC 

TThTd 

d=CTd 

ThTd 

Te Th 

CT 

 

CT 

 

 

 

Combining Functions Using Algebraic Operations 

 

 

fi 

 

 

wyhy 

yTfi 

Ty=hy+wy 

 

 

T=h+w 

fiff

210 


f xgxfif + gf − gfg f _ 

g 

 

 

 

f + gx = fx + gx 

 

 

f − gx = fx − gx 

fgx = fxgx 

 

Example 1 

( f _ 

g ) x=fx 

gx 

Performing Algebraic Operations on Functions 

x≠ 

g − f x 

( g _ 

f ) xf x=x −gx=x − 

 

Solution 

 

 

 

 

g−f x=gx−f x 

g−f x=x −−x− 

g−f x=x −x 

g−f x=xx− 

 

 

( g _ 

f ) x=gx 

f x 

 

( g _ − 

f ) x=x 

x − 

x ≠ 

 

 

 

( g _ 

f ) x=x+x− x ≠ 

x − 

( g _ 

x=x + 

f ) 

x ≠ 

 

 

( g _ 

xx ≠x = 

f ) 

fi 

Try It #1 

fgxf−gx 

fx=x −gx=x − 

 

Create a Function by Composition of Functions 

 

 

Th 

. Th 

composite function 

f∘gx=f gx


ft fgxfgxTh 

Th∘ 

 

 

 

f gx≠f xgx 

 

fi 

gxfigxThfgx 

f gx 

gxg 

f 

f ° gx = fgx 

xg 

f∘gg∘ffff gx≠gfxx 

fi 

f x=x gx=x + 

f gx=f x+ 

=x+ 

 

=x +x + 

gfx=gx 

=x + 

Thx 

x =− 

 

fi 

 

composition of functions 

 

xfgficomposite functionf∘g 

 

f∘gx=f gx 

Thf∘gxxggx 

ffgf gx 

f xgx≠f gx 

Example 2 Determining whether Composition of Functions is Commutative 

fif gxgfx 

 

fx=x + gx=−x

212 


Solution gxf x 

 

 

f gx=−x+ 

=−x + 

=−x 

f xgx 

gfx=−x + 

 

=−x − 

 

=−x + 

e figfx≠f gx 

Example 3 

Interpreting Composite Functions 

Thcssst 

tcs 

Solution Thsst = 

s 

scs 

 

Example 4 

Investigating the Order of Function Composition 

f xxgyy 

f gygfx 

Solution Thy =f x 

 

=f 

Thgy 

. Th 

=g 

ThgyThf x 

Thf gy 

Thf xThgy 

f xgy 

Thgfx 

gf xx 

Q & A… 

Are there any situations where f (g(y)) and g(f (x)) would both be meaningful or useful expressions ? 

 

ff 

 

Try It #2 

ThrGrTh 

FaF


Evaluating Composite Functions 

 

fi 

 

 

Evaluating Composite Functions Using Tables 

 

fi 

 

Example 5 Using a Table to Evaluate a Composite Function 

Table 1f ggf 

x f (x) g(x) 

 

 

 

 

Table 1 

Solution f g 

gfigg=f 

gf . Thfife fif = 

g= 

fg=f = 

gffif fif =Th 

g 

gf=g= 

Table 2f∘gg∘f 

x g(x) f (g(x)) f (x) g(f (x)) 

 

Table 2 

Try It #3 

Table 1fggf 

Evaluating Composite Functions Using Graphs 

 

xy

214 


How To… 

 

 

1. x 

2. y 

3. x 

4. y. Th 

Example 6 

Using a Graph to Evaluate a Composite Function 

Figure 1f g 

gx 

fx 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

x 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

x 

Figure 1 

Solution f gFigure 2 

gx 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

g= 

x 

f x 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

f = 

x 

Figure 2 

ggx x 

g=f 

fg=f 

f x x 

f =f g=


Analysis 

Figure 3 shows how we can mark the graphs with arrows to trace the path from the input value to the output value. 

gx 

fx 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

Figure 3 

Try It #4 

Figure 1gf 

Evaluating Composite Functions Using Formulas 

 

Th 

 

fi 

f gx 

f t=t −t 

 

How To… 

 

1. 

2. 

Example 7 

Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input 

f t=t −thx=x +f h 

Solution hhx 

f h=f f t 

h=+ 

h= 

f h=f 

f h= − 

f h= 

Analysis It makes no difference what the input variables t and x were called in this problem because we evaluated for 

specific numerical values.

216 


Try It #5 

f t=t −thx=x + 

a. hf b. hf− 

Finding the Domain of a Composite Function 

f∘gg 

f 

f ∘gfg 

xf gxx 

g 

gxff gx 

fiThf ∘ g 

ggffg 

xxggxf 

domain of a composite function 

Thfgxxggx 

f 

How To… 

f gx 

1. g. 

2. f. 

3. xggxf.Thx 

ggxf. Thf∘g. 

Example 8 

 

Finding the Domain of a Composite Function 

 

f∘gxf x =____ 

gx =_____ 

 

x − x − 

Solution Thgxx = 

__ 

fgx 

xgx = 

 

 

_____ 

 

x − = 

 

=x − 

 

=x 

x = 

f∘g__ 

. Th 

 

x≠ 

__ x ≠ 

 

 

( −∞ __ 

 

) ∪ ( __ 

 

) ∪∞


Example 9 

 

Finding the Domain of a Composite Function Involving Radicals 

f∘gxf x=√ — x +gx =√ — −x 

Solution g−∞ 

 

 

 

f∘gx=√√ — −x + 

f∘gx=√ √ — −x +√ 

— −x +≥ 

√ — −x≥−x ≥−∞ 

Analysis This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful 

in finding the domain of a composite function. It also shows that the domain of f∘g can contain values that are not in 

the domain of f, though they must be in the domain of g. 

Try It #6 

 

f∘gxfx=____ 

 

x− gx=√— x+ 

Decomposing a Composite Function into its Component Functions 

 

Th 

 

Example 10 

Decomposing a Function 

f x=√ — −x 

Solution ghf x=ghx 

f x−x 

 

 

Try It #7 

hx=−x gx=√ — x 

ghx=g−x =√ — −x 

 

fx=__ 

 

−√ — +x 

 

Composite Functions (http://openstaxcollege.org/l/compfunction) 

Composite Function Notation Application (http://openstaxcollege.org/l/compfuncnot) 

Composite Functions Using Graphs (http://openstaxcollege.org/l/compfuncgraph) 

Decompose Functions (http://openstaxcollege.org/l/decompfunction) 

Composite Function Values (http://openstaxcollege.org/l/compfuncvalue)



VERBAL 

1. 

f 

g 

3. 

 

 

 

2. f∘g 

4. 

f∘g 

ALGEBRAIC 

 

5. fx=x +xgx=−x f+g 

f−gfg, f 

g 

6. fx=−x +xgx= f+g 

f−gfg, f 

g 

7. fx=x +xgx= f+g 

x 

f−gfg, f 

g 

 

8. fx= 

x− gx= 

−x ,fi 

f+gf−gfg, f 

g 

9. fx=x gx=√ — x− f+g 

f−gfg, f 

g 

10. fx=√ — x gx=x− g 

f 

11. fx=x +gx=x− 

a. fg b. fgx c gfx d. g∘gx e. f∘f − 

o fif gxgfx 

12. fx=x +gx=√ — x+ 13. fx=√ — x +gx=x + 

14. fx=xgx=x+ 15. fx= √ 

— 

x gx= x+ x 

 

16. fx= 

x − gx= 

x + 

 

17. fx=___ 

 

x− gx= x + 

o fif ghx 

18. fx=x +gx=x−hx=√ — x 19. fx=x +gx= x hx=x+ 

20. fx= x ,gx=x− 

 

a. f∘gx 

b. f∘gx 

c. g∘f x 

d. g∘f x 

e. ( f 

g ) x 

21. fx=√ — −x gx=− x 

 

a. g∘f x 

b. g∘f x


22. fx= −x 

 

x gx= 

+x 

 

a. g∘f x 

b. g∘f 

 

 

23. px = 

√ — x mx=x − 

 

 

px 

a. 

mx 

b. pmx 

c. mpx 

24. qx= 

√ — x hx=x − 

 

 

25. fx= x gx=√— x− 

f∘gx 

a. qx 

hx 

b. qhx 

c. hqx 

, fif xgxhx=f x 

 

26. hx=x+ 27. hx=x− 28. hx= 

x − 

30. hx=+ √ 

— x 

_______ 

31. hx= √ 

x− 

 

 

 

32. hx= 

x − − 

 

 

29. hx= 

x+ 

 

_______ 

33. hx= 

√ x− x+ 

34. hx = ( +x 

 

−x ) 

35. hx=√ — x+ 36. hx=x− 37. hx= √ 

— x− 

 

38. hx=x + 39. hx= 

x− 

 

 

40. hx=( 

x− ) 

41. hx= 

√ x− 

x+ 

GRAPHICAL 

fFigure 4g,Figure 5 

f(x) 

f(x) 

 

 

 

 

 

 

f 

 

 

g 

 

 

 

– 

– 

– 

 

 

 

 

 

 

 

x 

 

– 

– 

– 

 

 

 

 

 

 

 

x 

Figure 4 

Figure 5 

42. fg 43. fg 44. gf 45. gf 

46. ff 47. ff 48. gg 49. gg


f x,Figure 6gx,Figure 7hx,Figure 8 

 

f x 

f x 

f x 

 

 

 

fx 

 

 

 

 

 

 

gx 

hx 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

x 

– 

– 

– 

– 

– 

 

 

 

 

x 

– 

– 

– 

– 

– 

 

 

 

 

x 

Figure 6 

Figure 7 

Figure 8 

50. gf 51. gf 52. fg 53. fg 

54. fh 55. hf 56. fgh 57. fgf− 

NUMERIC 

fgTable 3 

x 

fx 

gx 

Table 3 

58. fg 59. fg 60. gf 61. gf 

62. ff 63. ff 64. gg 65. gg 

fgTable 4 

x − − − 

f (x) − 

g(x) − − − − 

Table 4 

66. f∘g 67. f∘g 68. g∘f 

69. g∘f 70. g∘g 71. f∘f 

o fif ggf 

72. fx=x+gx=−x 73. fx=x+gx=−x 

74. fx=√ — x+gx=−x 

75. fx=_ 

 

x + gx=x+ 

f x=x +gx=x +fi 

 

76. fg 77. fgx 78. g(f− 79. g∘g x


EXTENSIONS 

f x=x +x= √ 

— x − 

80. f∘gxg∘fx 81. f∘gg∘f 

82. g∘fx 83. f∘gx 

84. fx= 

__ 

x 

a. f∘fx 

b. f∘f xf 

F x=x+ f x=x gx=x + 

85. g∘fx=Fx 86. f∘gx=Fx 

, fif x=x +x ≥gx=√ — x − 

87. f∘gg∘f 88. g∘faf∘ga 89. f∘gg∘f 


90. ThDp 

p. Th 

Cxx 

 

 

a. DC 

b. CD 

c. DCx= 

d. CDp= 

92. ff 

xs. Th 

 

Px 

x 

 

94. 

 

 

rt=t+ 

t 

96. Thr 

VrV= ___ 

√ 

V 

π . 

ftt 

Vt=+t 

a. rVt 

b. exact 

10 in 

91. ThAd 

d mi 

em. Th 

ftt 

m(t) 

 

 

a. Am 

b. mA 

c. Amt= 

d. mAd= 

93. 

 

rt= √ 

— t+ 

 

t= 

95. 

ft 

97. Th 

 

NT=T −T+

222 




Graph functions using vertical and horizontal shifts. 

Graph functions using reflections about the x-axis and the y-axis. 

Determine whether a function is even, odd, or neither from its graph. 

Graph functions using compressions and stretches. 

Combine transformations. 

3.5 TRANSFORMATION OF FUNCTIONS 

Figure 1 (credit: "Misko"/Flickr) 

fl 

fl 

 

 

 

Graphing Functions Using Vertical and Horizontal Shifts 

ft 

 

Th 

 

Identifying Vertical Shifts 

ftftTh 

vertical shift 

 

g x=f x+kf xftkFigure 2

SECTION 3.5 TRANSFORMATION OF FUNCTIONS 223 

fx 

 

 

 

 

 

– – – – – – 

fx+ 

fx 

 

 

x 

– 

– 

– 

– 

Figure 2 Vertical shift by k = 1 of the cube root function f ( x) = 3 √ — x . 

fty =f xThf x+ky +k 

yy +kykTh 

 

vertical shift 

f xgx=f x+kkvertical shift 

f xkk k 

 

 

Example 1 

Adding a Constant to a Function 

flFigure 

3V ftt 

 

 

 

 

 

 

 

V 

 

– 

– 

– 

 

 

 

 

 

 

 

t 

Figure 3 

Solution 

ThffftFigure 4 

 

 

 

 

 

V 

 

 

 

– 

– 

– 

 

 

 

 

 

 

t 

Figure 4

224 


Figure 4St 

 

St=Vt+ 

ThtStV 

ThfiSV 

ftTable 1 

t 

V(t) 

S(t) 

Table 1 

How To… 

ft 

1. 

2. ft 

3. 

Example 2 

Shifting a Tabular Function Vertically 

f xTable 2gx=f x− 

x 

f (x) 

Table 2 

Solution Thg x=f x−fig 

f 

f = 

 

gx=f x− 

g=f − 

 

=− 

 

=− 

f xgxTable 3 

x 

f (x) 

g(x) − 

Table 3 

Analysis As with the earlier vertical shift, notice the input values stay the same and only the output values change. 

Try It #1 

Thht=–t +thftt 

btht 

bt


Identifying Horizontal Shifts 

. 

 

horizontal shiftFigure 5 

fx 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

fx+ 

fx 

 

 

 

 

 

x 

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. 

f x=x gx=x− 

Thft 

f 

horizontal shift 

fgx=f x−hhhorizontal shiftf 

h h ft 

Example 3 

Adding a Constant to an Input 

flFigure 3 

 

 

Solution VtF t 

Vt= 

 

Ft= 

flV2 

Vfl8 Ffl 

ThV=FFigure 6fi 

ft 0 ft t 8 aV=F 

F th =−Th 

F t=Vt−−=Vt + 

 

 

 

 

 

 

V 

 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

t 

Figure 6

226 


Analysis Note that Vt + has the effect of shifting the graph to the ft. 

Horizontal changes or “inside changes” affect the domain of a function (the input) instead of the range and often seem 

counterintuitive. The new function F(t) uses the same outputs as V(t), but matches those outputs to inputs 2 hours 

earlier than those of V(t). Said another way, we must add 2 hours to the input of V to find the corresponding output 

for FFt=Vt+ 

How To… 

ft 

1. 

2. ft 

3. 

Example 4 

Shifting a Tabular Function Horizontally 

f xTable 4g x=f x− 

x 

f (x) 

Table 4 

Solution Thgx=f x−gf 

f = 

glargerg x 

f 

g =f − 

 

=f 

 

= 

Table 5 

x 

x − 3 

f (x) 

g(x) 

Table 5 

Thg xftg x 

f xxftfift 

ftftft 

Analysis Figure 7 represents both of the functions. We can see the horizontal shift in each point. 

y 

 

 

 

 

 

 

––– 

– – – – 

– 

– 

– 

– 

– 

f x 

 

 

 

g x=fx – 

Figure 7 

 

x


Example 5 

Identifying a Horizontal Shift of a Toolkit Function 

Figure 8f x=x g xf x 

fig x 

 

 

 

 

 

f (x) 

 

– 

– 

– 

 

 

 

 

 

 

 

x 

Figure 8 

Solution f x=x xft 

ThTh 

ft 

g x=f x− 

x =y =x 

f x 

g xf x− 

f x=x 

g x=f x− 

g x=f x−=x− 

Analysis To determine whether the ft is + or −, consider a single reference point on the graph. For a quadratic, 

looking at the vertex point is convenient. In the original function, f = . In our shifted function, g = . To 

obtain the output value of from the function f, we need to decide whether a plus or a minus sign will work to satisfy 

g () = f x − = f = . For this to work, we will need to subtract units from our input values. 

Example 6 

Interpreting Horizontal versus Vertical Shifts 

ThG mmG m+G m+ 

Solution G m+Thm 

. Thft 

Gm+ 

m. Thft 

Try It #2 

fx=√ — x fxgx = fx+ 

hift 

Combining Vertical and Horizontal Shifts 

ftff 

y ftffx 

ftftftft 

andft

228 


How To… 

ft 

1. 

2. Th 

 

3. Thftft 

 

4. ft 

Example 7 

Graphing Combined Vertical and Horizontal Shifts 

f x= x h x=f x+− 

Solution Thf 

Thhff x+ 

f x+− 

. ThFigure 9 

f x= x 

→− 

−ft−→−− 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

y 

y = x+ 

y = x| 

y = x+– 

 

x 

Figure 10h 

Figure 9 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

y=x+– 

 

x 

Figure 10 

Try It #3 

fx= x hx=fx − +


Example 8 

Identifying Combined Vertical and Horizontal Shifts 

Figure 11 

 

 

 

 

 

 

– 

– 

– 

h(x) 

 

 

 

Figure 11 

Solution Thft 

 

hx=f x−+ 

 

hx=√ — x −+ 

 

 

 

 

x 

Analysis Note that this transformation has changed the domain and range of the function. This new graph has domain 

∞ and range ∞. 

Try It #4 

fx= 

__ 

x 

 

Graphing Functions Using Reflections about the Axes 

flxyvertical reflection 

flxhorizontal reflectionfly 

ThflFigure 12 

y 

 

eflectio 

fx 

 

 

 

eflectio 

f–x 

x 

–fx 

Figure 12 Vertical and horizontal reflections of a function. 

fl 

x-Thfl 

y

230 


reflections 

f xgx=−fxvertical reflectionf x 

flx 

f xgx=f −xhorizontal reflectionf tf x 

fly 

How To… 

fl 

1. – Th x 

2. –flThfl 

y 

Example 9 

flst=√ — t 

Solution 

Reflecting a Graph Horizontally and Vertically 

a.b. 

a. t- 

Figure 13 

st 

vt 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

Figure 13 Vertical reflection of the square root function 

 

Vt=−stVt=−√ — t 

ffst 

 

b. Figure 14 

st 

Ht 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

Figure 14 Horizontal reflection of the square root function


 

Ht=s−tHt=√ — −t 

ff 

 

ff 

∞∞ Vt−∞ 

Ht−∞ 

Try It #5 

flf x=|x−|a.b. 

Example 10 

Reflecting a Tabular Function Horizontally and Vertically 

f xTable 6 

a. gx=−f x b. hx=f −x 

Solution 

x 

f (x) 

Table 6 

a. gx x 

Table 7 

x 

g (x) – – – – 

Table 7 

b. hx 

hxf xTable 8 

x – – – – 

h(x) 

Table 8 

Try It #6 

f xTable 9 

x − 

f(x) 

a. gx=−f x 

b. hx=f −x 

Table 9

232 


Example 11 

Applying a Learning Model Equation 

kt=− −t +k 

fttThf t= t Figure 15 

kt 

f t 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

Figure 15 

Solution Th 

f −t= −t 

−f−t=− −t 

−f−t+=− −t + 

 

 

1. − 

2. Th −− 

3. 

Thft 

Figure 16fiflThflTh 

 

f t 

f t 

kt 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

t 

 

Figure 16 

Analysis As a model for learning, this function would be limited to a domain of t ≥ , with corresponding range . 

Try It #7 

fx=x gx=−fxhx=f−x


Determining Even and Odd Functions 

flfl 

f x=x f x=⎪x⎪ 

yyeven function 

f x=x f x=_ 

x 

flboth 

Figure 17 

y 

f x=x 

f −x 

y 

y 

−f −x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function 

(c) Horizontal and vertical reflections reproduce the original cubic function. 

 

odd function 

f x= x 

f x= 

even and odd functions 

even functionx: f x=f −x 

Thy 

odd functionx: f x=−f−x 

Th 

How To… 

 

1. f x=f −x 

2. fif x=−f−x 

3. 

Example 12 

Determining whether a Function Is Even, Odd, or Neither 

f x=x +x 

Solution fi 

fl 

f −x=−x +−x=−x −x 

Th 

−f−x=−−x −x=x +x 

−f−x=f x

234 


Analysis Consider the graph of fin Figure 18. Notice that the graph 

is symmetric about the origin. For every point (x, y) on the graph, the 

corresponding point −x, −y is also on the graph. For example, is on 

the graph of f, and the corresponding point −− is also on the graph. 

– – – – 

−− 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

f x f 

 

 

 

 

 

 

x 

Figure 18 

Try It #8 

f s=s +s + 

Graphing Functions Using Stretches and Compressions 

 

ffff 

 

fiff 

Vertical Stretches and Compressions 

 

vertical stretch 

vertical compressionFigure 19 

 

y 

 

 

fx 

fx 

fx 

 

 

x 

Figure 19 Vertical stretch and compression 

vertical stretches and compressions 

fxgx=afxavertical stretchvertical 

compressionfx 

a > 


How To… 

 

1. a 

2. a 

3. a >a 

236 


How To… 

 

 

1. a 

2. a 

Example 14 

Finding a Vertical Compression of a Tabular Function 

fTable 10gx= __ 

f x 

 

x 

f (x) 

Table 10 

Solution Thgx= __ 

f xgf 

 

f =Th 

g= __ 

f = __ 

= __ 

 

Table 11 

x 

g(x) 

 

__ 

__ 

 

__ 

 

__ 

 

 

Table 11 

Analysis The result is that the function gx has been compressed vertically by __ 

so the graph is half the original height. 

Try It #9 

fTable 12gx= __ 

f x 

 

x 

f(x) 

Table 12 

. Each output value is divided in half, 

 

Example 15 

Recognizing a Vertical Stretch 

ThFigure 22f x=x gxf x 

n figx 

gx 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 22


Solution ft 

g=f = = 

g 

__ fg= __ 

f 

 

gx= __ 

f x 

 

gfif 

gx= __ 

f x= __ 

x 

 

Try It #10 

 

 

Horizontal Stretches and Compressions 

 

 

horizontal stretchhorizontal 

compression 

– 

y = x 

– 

– 

 

 

 

 

 

 

 

 

 

 

y 

– – 

– 

 

 

y = x 

Figure 23 

 

y = x 

 

 

y =f xy =f bx 

y =x Figure 23Thy =x y =x 

. Thy =x y =x 

x 

horizontal stretches and compressions 

f xgx=f bxbhorizontal stretch horizontal 

compressionf x 

b >__ 

 

b 

238 


Example 16 

Graphing a Horizontal Compression 

fl 

R 

 

4 h 

Solution 

R=P 

R=P 

Rt=Pt 

Figure 24 

f (x) 

f (x) 

 

 

 

 

Pt 

 

 

 

 

Rt 

 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

 

 

x 

 

– 

– 

– 

 

 

 

 

 

 

 

 

x 

Figure 24 (a) Original population graph (b) Compressed population graph 

Analysis Note that the effect on the graph is a horizontal compression where all input values are half of their original 

distance from the vertical axis. 

Example 17 

Finding a Horizontal Stretch for a Tabular Function 

f xTable 13gx=f ( __ 

x ) 

x 

f (x) 

Table 13 

Solution Thgx=f ( __ 

 

x ) g 

fg 

g=f ( __ 

=f f g 

fg 

ċ ) 

Table 14 

g=f ( __ 

ċ ) =f = 

x 

g(x) 

Table 14


Figure 25 

y 

y 

 

 

 

 

 

 

––––– 

– 

– 

– 

– 

– 

– 

 

 

 

x 

 

 

 

 

 

 

––––– 

– 

– 

– 

– 

– 

– 

 

 

 

x 

 

 

Figure 25 

Analysis Because each input value has been doubled, the result is that the function gx has been stretched horizontally 

by a factor of . 

Example 18 Recognizing a Horizontal Compression on a Graph 

gxf xFigure 26 

f (x) 

 

 

 

 

 

g 

f 

 

– 

– 

– 

 

 

 

 

 

 

 

x 

Figure 26 

Solution Thgxf xf xgx 

x__ 

 

( __ 

 

) =g=f 

g=f gx=f x. Th__ 

 

Analysis 

Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or 

compression. So to stretch the graph horizontally by a scale factor of , we need a coefficient of 

__ in our function: f ( __ x ) . 

This means that the input values must be four times larger to produce the same result, requiring the input to be larger, 

causing the horizontal stretching. 

Try It #11

240 


Performing a Sequence of Transformations 

 

ft 

ft fi 

h fi 

f x+Th 

f xfi 

ft 

gx=f x + 

gff = 

gxgx=f x += 

x +=xfift 

 

Thffi 

ft 

 

 

fbx+p=f ( b ( x + p _ 

b) ) 

fx=x + 

fx=x+ 

 

h fi 

combining transformations 

afx+kfia 

k 

f bx−h, fi h 

__ 

 

b 

f bx−h, fi_ 

 

b 

h 

 

d fi 

Example 19 

Finding a Triple Transformation of a Tabular Function 

Table 15f xgx=f x+ 

x 

f (x) 

Table 15 

Solution Th 

f x__ 

x __ 

Table 16 

 

x 

f (3x) 

Table 16


 

Table 17 

x 

2f (3x) 

Table 17 

ftTable 18 

x 

g(x) = 2f (3x) + 1 

Table 18 

Example 20 

Finding a Triple Transformation of a Graph 

f xFigure 27kx=f ( __ 

x + ) − 

f x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 27 

Solution 

f ( __ 

x + ) −=f ( __ 

 

x+ ) − 

fi__ 

 

 

 

Figure 28 

f x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 28

242 


x +Figure 29 

f x 

– – – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

Figure 29 

− 

Figure 30 

f x 

– – – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

Figure 30 

 

Function Transformations (http://openstaxcollege.org/l/functrans)



VERBAL 

1. 

 

 

3. 

 

 

 

5. 

 

2. 

 

 

4. 

 

x 

y 

ALGEBRAIC 

ft 

6. fx) =√ — x hift 7. fx=xhift 

 

8. fx= __ 

 

x hift 

 

9. fx= 

__ 

xhift 

 

f 

10. y = f x − 11. y = f x + 12. y = f x + 

13. y = f x − 14. y = f x + 15. y = f x + 

16. y = f x − 17. y = f x − 18. y = f x − + 

19. y = f x + − 

s 

20. f x = x + − 21. gx=x+ − 22. ax=√ — −x+ 

23. kx = −√ — x − 

y 

GRAPHICAL 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

Figure 31 

f 

 

 

 

 

x 

f x= x Figure 31 

f x 

24. gx = x + 25. hx = x − 

26. wx = x − 

 

 

27. f t = t + − 28. hx = |x − | + 

29. kx = x − − 30. mt = + √ — t +


NUMERIC 

31. fghgxhx 

fx 

x – – 

f (x) – – – 

x – 

g(x) – – – 

x – – 

h(x) – – 

32. fghgxhx 

f x 

x – – 

f (x) – – 

x – – – 

g(x) – – 

x – – 

h(x) – – 

 

 

33. 

y 

34. 

y 

35. 

y 

f 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

f 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

f 

 

 

 

 

x 

36. 

y 

37. 

y 

38. 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

f 

 

 

 

x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

f 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

f 

 

 

 

 

x


39. 

 

x 

f 

y 

– 

– 

– 

– 

–– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

 

 

40. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

fi 

 

41. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

42. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

 

 

43. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

44. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

45. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

46. 

 

x 

f 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

–


 

47. fx=x 48. gx=√ — x 49. hx= 

__ 

x +x 

50. fx=x− 51. gx=x 52. hx=x−x 

 

f 

53. gx=−fx 54. gx=f−x 55. gx=fx 56. gx=fx 

57. gx=fx 58. gx=fx 59. gx=f ( __ 

x ) 

61. gx=f−x 62. gx=−fx 

60. gx=f ( __ 

x ) 

g 

 

63. Thfx=∣x ∣ y 

__ 

 

65. Thfx= 

__ 

x 

__ 

hift 

 

3 uni 

67. Thfx=x 

__ 

hift 

 

64. Thfx=√ — x x 

 

66. Thfx= 

__ 

x 

hift 

 

68. Thfx=x 

hift 

 

Th 

 

69. gx=x+ − 70. gx=x+ − 71. hx=−∣x−∣+ 

72. kx=−√ — x − 73. mx= __ 

 

x 

74. nx= __ 

x− 

 

75. px= ( __ 

x ) 

− 

76. qx= ( __ 

x ) 

+ 

77. ax=√ — −x+ 

Figure 32 

y 

 

 

 

 

 

– – – – – – 

 

 

f 

 

 

 

x 

– 

– 

– 

– 

Figure 32 

78. gx=fx− 79. gx=−fx 80. gx=fx+ 81. gx=fx−

SECTION 3.6 ABSOLUTE VALUE FUNCTIONS 247 



Graph an absolute value function. 

Solve an absolute value equation. 

3.6 ABSOLUTE VALUE FUNCTIONS 

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) 

 

Th 

 

 

 

Understanding Absolute Value 

f x=| x |Th 

 

 

 

absolute value function 

fi 

x x ≥ 

f x=∣ x ∣ = −x x < 

Example 1 

Using Absolute Value to Determine Resistance 

fi 

 

Th 

fift±±± 

± 

 

Solution fiThff 

R 

| R − |≤ 

Try It #1 

20

248 


Graphing an Absolute Value Function 

ThfiTh 

Figure 2 

y 

y=x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

Figure 2 

Figure 3y =| x −|+Thy =| x| ft 

ftTh 

y = x − 

 

y = x 

– – – 

 

 

 

 

 

 

 

 

 

 

– – – – 

y 

 

 

 

 

y = x −+ 

 

 

 

y = x − 

 

x 

Figure 3 

Example 2 Writing an Equation for an Absolute Value Function Given a Graph 

Figure 4 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

Figure 4 

Solution Thft 

Figure 5


y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

, – 

Figure 5 

fi 

 

Figure 6 

y 

 

 

 

 

 

 

– – – – – – 

 

 

 

x 

 

– 

– 

– 

– 

, − 

Figure 6 

 

f x=| x −|− 

f x=| x−|− 

Analysis Note that these equations are algebraically equivalent—the stretch for an absolute value function can be 

written interchangeably as a vertical or horizontal stretch or compression. 

Q & A… 

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it? 

 

x f x 

f x=a| x −|− 

 

 

=a| −|− 

 

=a 

a = 

Try It #2 

hift 2 

hift

250 


Q & A… 

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis? 

s. Th 

 

Th 

ftfl 

Figure 7 

y 

y 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

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) 

The absolute value function intersects the horizontal axis at two points. 

Solving an Absolute Value Equation 

Other Types of Equations 

 

=| x −| 

−Th 

ff 

x −= x −=− 

x = x =− 

x = x =− 

 

fi 

 

| x|= 

| x −|= 

| x +|−= 

solutions to absolute value equations 

AB| A|=BB≥A=BA=−B 

B 

3. x


Example 3 

Finding the Zeros of an Absolute Value Function 

f x=| x +|−7 , fixf x= 

Solution 

=| x +|− 

=| x +| 

=x +−=x + 

=x −=x 

x= 

__ = x = ___ − 

=− 

Thx =x =−Figure 8 

y 

f x 

 

 

x 

fx = 

 

 

 

 

− 

–– 

––– 

– 

– 

– 

– 

– 

 

x 

 

x 

fx = 

f 

Try It #3 

Figure 8 

f x=| x−|− xf x= 

Q & A… 

Should we always expect two answers when solving A= B? 

y fi+| x −|= 

 

Graphing Absolute Value Functions (http://openstaxcollege.org/l/graphabsvalue) 

Graphing Absolute Value Functions 2 (http://openstaxcollege.org/l/graphabsvalue2)



VERBAL 

1. 2. 

x 

3. 

 

 

 

4. 

x 

 

ALGEBRAIC 

5. x 

 

 

7. 

x 

 

6. x__ 

 

− 

 

8. f x 

f x 

 

, fixy 

9. fx=∣ x−∣+ 10. fx=−∣ x−∣− 11. fx=−∣ x+∣+ 

12. fx=−∣ x+∣+ 13. fx=∣ x−∣− 14. fx=∣ −x+∣− 

15. fx=−∣ x−∣+ 

GRAPHICAL 

t fi 

16. y=| x− | 17. y=| x+| 18. y=| x |+ 

 

19. y=| x |− 20. y=−| x| 21. y=−| x |− 

22. y=−| x− |− 23. f x=−| x− |− 24. f x=−| x+ |+ 

25. f x=| x+ |+ 26. f x=| x− |+ 27. f x=| x− |− 

28. f x=| x+ |+ 29. f x=−| x− |− 30. f x=−| x+ | − 

31. f x= 

__ | x+ |−


TECHNOLOGY 

32. f x=| x−| 

 

 

33. f x=−| x|+ 

− 

 

 

34. f x=−| −x|+ 

35. f x=× ∣ x−× ∣ +× 

EXTENSIONS 

 

36. a 

xf x=| x+|+a 

37. a 

yf x=| x+|+a 


38. 

 

 

x 

 

40. 

 

x 

 

39. Thp 

 

 

 

41. 

 

x 

 

42. Th 

 

x

254 




Verify inverse functions. 

Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one. 

Find or evaluate the inverse of a function. 

Use the graph of a one-to-one function to graph its inverse function on the same axes. 

3.7 INVERSE FUNCTIONS 

 

 

ffi 

 

 

Figure 1 

 

x 

y 

f 

 

y 

x 

Figure 1 Can a function “machine” operate in reverse? 

Verifying That Two Functions Are Inverse Functions 

 

 

e fi 

F 

C= 

__ F− 

 

__ −≈ 

 

Figure 2 

 

 

Figure 2

SECTION 3.7 INVERSE FUNCTIONS 255 

fift 

FftC 

 

 

= __ 

F− 

 

 

ċ 

__ =F − 

 

 

F =ċ 

__ +≈ 

ft 

 

ff 

Thinverse function 

 

 

f xf − xfxTh− 

−f − xnot ___ 

 

f x ___ 

f x 

f 

Tha − 

a =af − ∘f 

 

f − ∘f x=f − f x=f − y=x 

Thxf 

 

f xg xf x 

g f x=xf g x=x 

 

y =xy = __ 

x 

 

f − ∘f x=f − x= __ 

x=x 

 

 

f∘f − x=f ( __ 

x ) = ( __ 

 

x ) =x 

y =x−− 

y = 

__ x −− 

 

 

inverse function 

f x=yf − xinverse functionff − y=xTh 

f − fx=xxfff − x=xxf − 

f − f 

Thf − ff − 

ftf − xfx 

 

f − 

x≠___ 

 

f x

256 


Example 1 

Identifying an Inverse Function for a Given Input-Output Pair 

f =f = 

 

Solution Th 

f=f − = 

f=f − = 

gg =g = 

Analysis Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. See 

Table 1. 

Try It #1 

(x, f (x)) 

 

 

Table 1 

(x, g (x)) 

 

 

h − =h 

How To… 

f xg x 

1. f gx=xgfx=x 

2. g =f − f =g − 

g ≠f − f ≠g − 

Example 2 

Testing Inverse Relationships Algebraically 

 

f x=____ 

 

x + g x= __ 

 

x −g =f − 

 

Solution 

g fx=_ 

− 

( _ 

x + ) 

=x +− 

=x 

 

g =f − f =g − 

Th 

 

f g x=_ 

 

 

__ 

 

x −+ 

=_ 

 

 

__ 

 

x 

=x 

Analysis Notice the inverse operations are in reverse order of the operations from the original function. 

Try It #2 

fx=x −g x= √ 

— x− g=f −


Example 3 

Determining Inverse Relationships for Power Functions 

f x=x g x= __ 

 

Solution f g x= 

__ x 

≠x 

 

xg =f − 

Analysis The correct inverse to the cube is, of course, the cube root √ — x = x that is, the one-third is an exponent, 

not a multiplier. 

Try It #3 

f x=x− g x= √ 

— x +g=f − 

Finding Domain and Range of Inverse Functions 

Thff − ff − 

ff − ff − Figure 3 

f 

f x 

f 

a 

b 

f – x 

f – f – 

Figure 3 Domain and range of a function and its inverse 

 

f x=√ — x f − x=x 

∞ 

f x=√ — x 

f x=x 

 

 

− 

 

ff 

 

 

f x=x 

∞ 

 

f x=x− ∞f − x=√ — x + 

Thf =f − =∞ 

Thf − =f =∞ 

Q & A… 

Is it possible for a function to have more than one inverse? 

ffghfi 

fg =h 

ff

258 


domain and range of inverse functions 

Thf xf − xThf xf − x 

How To… 

n, fi 

1. 

 

2. 

 

Example 4 

Finding the Inverses of Toolkit Functions 

fi 

ThTable 2 

y 

Constant Identity Quadratic Cubic Reciprocal 

fx=c fx=x fx=x fx=x fx=_ 

x 

Reciprocal squared Cube root Square root Absolute value 

fx=_ 

 

x 

fx= √ 

— x fx=√ — x fx=x 

Table 2 

Solution Th 

 

Th∞ 

Th∞ 

Analysis We can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown in Figure 

4. They both would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes 

the horizontal line test, then in that restricted domain, it can have an inverse. 

f x 

f x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

Try It #4 

 

 

Figure 4 (a ) Absolute value (b ) Reciprocal squared 

Thf∞f−∞−


Finding and Evaluating Inverse Functions 

fi 

 

Inverting Tabular Functions 

fi 

 

 

 

 

Example 5 

Interpreting the Inverse of a Tabular Function 

f tTable 3tf − 

t (minutes) 

f (t) (miles) 

Table 3 

Solution Thfff − 

Th 

ff − =Th 

fif a=bf − b=afi 

f − =aaf a=tf t= 

t = 

Try It #5 

Table 4, fia.f b.f − 

tminutes 

ftmiles 

Table 4 

Evaluating the Inverse of a Function, Given a Graph of the Original Function 


fivertical 

fi 

horizontal 

fi 

 

How To… 

fi 

1. y 

2. x

260 


Example 6 Evaluating a Function and Its Inverse from a Graph at Specific Points 

gxFigure 5gg − 

g(x) 

 

 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

 

x 

Figure 5 

Solution gfixfiyTh 

g= 

g − fig − xgx= 

fig=fig − =Figure 6 

 

 

 

g(x) 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

 

 

x 

Try It #6 

Figure 6 

Figure 6a.fig − b.g − 

Finding Inverses of Functions Represented by Formulas 

 

yxftfi 

xy 

How To… 

, fi 

1. f 

2. x 

3. xy 

Example 7 

Inverting the Fahrenheit-to-Celsius Function 

 

C= 

__ 

F−


Solution 

C = 

__ F− 

C ċ 

__ =F − 

 

F = 

__ C + 

 

 

C =hF= 

__ F− 

 

F =h − C= 

__ C + 

 

h 

C − 

Try It #7 

xyy= 

__ x− 

Example 8 

Solving to Find an Inverse Function 

 

f x=____ 

 

x − + 

 

Solution 

y =____ 

+ 

x − 

 

y −= 

_____ 

x − 

 

x −=____ 

 

y − 

 

x =____ 

 

y − + 

 

x −y − 

 

f − 

y= 

_____ 

y − +f − 

x=____ 

 

x − + 

Analysis The domain and range of f exclude the values and , respectively. f and f −1 are equal at two points but are 

not the same function, as we can see by creating Table 5. 

Example 9 

Solving to Find an Inverse with Radicals 

f x=+√ — x − 

Solution 

 

f − x=x− + 

x f − y 

f (x) y 

Table 5 

y =+√ — x − 

y− =x − 

x =y− + 

Thf∞f∞ 

f − ∞ 

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 

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 

domain of f −1 is exactly the range of f.

262 


Try It #8 

fx=−√ — x 

Finding Inverse Functions and Their Graphs 

fi 

f x=x ∞ 

Figure 7 

f x 

– – – 

 

 

 

 

 

– – – 

 

 

 

 

 

x 

– 

Figure 7 Quadratic function with domain restricted to [0, ∞). 

∞ 

 

f − x=√ — x 

ff − xff − 

Thf − xf xfly =x 

Figure 8 

y 

 

 

 

 

 

– – – – – – 

f x 

 

 

y = x 

f – x 

x 

 

– 

– 

– 

– 

Figure 8 Square and square-root functions on the non-negative domain 

Th 

. Th 

Example 10 

Finding the Inverse of a Function Using Reflection about the Identity Line 

f xFigure 9f − x


y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

Figure 9 

Solution Th 

∞−∞∞−∞∞∞ 

fly =xflfl 

Figure 10 

y 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

f – x y = x 

f x 

 

x 

Figure 10 The function and its inverse, showing reflection about the identity line 

Try It #9 

ff − Example 8 

Q & A… 

Is there any function that is equal to its own inverse? 

f =f − f f x=xTh 

 

_ 

 

 

_ =x 

x 

 

f x = c − x, c 

 

Inverse Functions (http://openstaxcollege.org/l/inversefunction) 

One-to-one Functions (http://openstaxcollege.org/l/onetoone) 

Inverse Function Values Using Graph (http://openstaxcollege.org/l/inversfuncgraph) 

Restricting the Domain and Finding the Inverse (http://openstaxcollege.org/l/restrictdomain)

264 



VERBAL 

1. n eff 

 

2. 

fx=x 

3. 4. 

 

5. 

 

ALGEBRAIC 

6. fx=a−xa 

, fif − x 

7. fx=x+ 8. fx=x+ 9. fx=−x 

10. fx=−x x 

11. fx= 

x+ 

 

12. fx= x+ 

x+ 

fif 

. Thn fif 

13. fx=x+ 14. fx=x− 15. fx=x − 

x 

16. fx= 

 

+x gx= −x 

a. fgxg fx 

x 

b. fxgx 

f xgx 

17. fx= √ 

— x−g x=x + 

18. fx=−x+g x= 

x− 

− 

GRAPHICAL 

 

19. fx=√ — x 20. fx= √ 

x+ 

21. fx=−x+ 22. fx=x − 

 

23. 

y 

24. 

y 

 

 

 

 

 

––––– – 

– 

– 

– 

– 

f 

 

x 

 

 

 

 

 

– – – – – 

– 

f – 

– 

– 

– 

 

 

 

 

x


fFigure 11 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

y 

 

f 

 

 

 

 

x 

25. f 

26. fx= 

27. f − 

28. f − x= 

Figure 11 

Figure 12 

y 

29. f − 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

Figure 12 

 

f 

 

 

x 

30. ff − 

31. f 

f 

32. f 

f 

NUMERIC 

f 

33. f= f − 34. f= f − 

35. f − −=− f− 36. f − −=− f− 

Table 6 

x 

f (x) 

Table 6 

37. f 38. fx= 

39. f − 40. f − x= 

41. fTable 7f − x 

x 

f (x) 

Table 7


TECHNOLOGY 

, fi. Th 

 

42. fx= 

_____ 

x− 

43. fx=x − 

 

44. fx=_ 

 

x− 

 


45. xy 

fx= 

__ x+ 

 

 

46. ThC 

Cr=πr 

 

rCrπ 

47. 

Th 

tdt=t 

 

td 

t



Key Terms 

absolute maximum 

absolute minimum 

average rate of change e diff 

diff 

composite function 

 

decreasing function f ba 

dependent variable 

domain 

even function f x=f −xy 

function 

horizontal compression 

b > 

horizontal line test 

 

horizontal reflection y− 

horizontal shift 

 

horizontal stretch 

f aab 

b >a 

independent variable 

input 

interval notation 

 

 

inverse function f xf − xf − fx=xx 

ff f − x=xxf − 

local extrema 

local maximum 

local minimum 

odd function f x=−f −x 

 

one-to-one function 

output 

piecewise function 

range 

rate of change 

relation


set-builder notation 

x x 

vertical compression 

 

Key Equations 

 

 

 

 

 

f x=cc 

f x=x 

f x=x 

f x=x 

f x=x 

f x=_ 

x 

 

f x=_ 

 

x 

f x=√ — x 

 

 

 

ft 

ft 

fl 

fl 

 

 

 

 

f x= √ 

— x 

Δy _ 

Δx =f x −f x _ 

 

x 

−x 

 

 

f∘gx=f gx 

gx=f x+k k > 

gx=f x−hh > 

gx=−fx 

gx=f −x 

gx=afxa> 

gx=afx


Key Concepts 

3.1 Functions and Function Notation 

 


y =f xExample 3 

Example 4 

 

Example 5 

 


 

Example 8 


Example 11 

Example 12 

Example 13 

 

Example 14 

Th Example 15 

3.2 Domain and Range 

Th 

 

ThExample 1. 

Th 


 

Example 5 


Example 8 



Example 13 

3.3 Rates of Change and Behavior of Graphs 

. Th 

Example 1 

Example 2 

 

Example 3 

 


ThExample 6 

 

Example 7


 

 

 


ThExample 10 

3.4 Composition of Functions 

Example 1 

 

 

ThExample 2Example 3 

Th 

Example 4 

 

 

Example 5 

Example 6 

Example 7 

Th 


 

 

ftExample 10 

3.5 Transformation of Functions 

hiftExample 1Example 2 

hiftExample 3Example 4Example 5 

 

Example 6 

ftExample 7Example 8 

x 

– 

y 

– 

. Th 

ff Example 9 

 

Example 10 

Example 11 

y 

f x=f −x 

f x=−f−x 

Example 12 

Example 13 


Example 16 

Example 17Example 18


Thh diffff 

 


3.6 Absolute Value Functions 

 

Example 1 

Th 

Example 2 

 

 

Example 3 

3.7 Inverse Functions 

gxf xgfx=f gx=xExample 1Example 2Example 3 

Example 4 

 

 

Example 5 

Th Example 6 

y =f xxy. Thxy 


Th y =x 

Example 10



FUNCTIONS AND FUNCTION NOTATION 

 

1. abcded 

2. 

3. y +=xxy 

4. Figure 1 

 

f −f f −a−faf a+h 

5. fx=−x +x 

Figure 

 

– 

7. fx=−x+ 8. fx=x− 

 

9. 

y 

10. 

y 

11. 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

12. fx=x+ 13. fx=x −


Figure 2 

y 

14. f 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

15. f− 

16. fx=−x 

17. fx=x 

Figure 2 

ht=−t +to fi 

18. 

h−h 19. 

ha−h 

 

− a− 

DOMAIN AND RANGE 

, fi 

 

20. fx= 

 

x+ 

x− 

21. fx= 

 

x −x− 

22. fx= 

√— x− √ — x− 

23. fx= { 

x+ x


32. Figure 3 

−3, 3]. Th− 

 

33. 

Figure 3 

– – – – 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 3 

COMPOSITION OF FUNCTIONS 

, fif∘gxg ∘fx 

34. fx=−xgx=−x 35. fx=x+gx=−x 36. fx=x +xgx=x+ 

37. fx=√ — x+gx=_ 

x 

 

38. fx= 

x+ 

 

−x 

gx=√— 

, fif∘gf∘gx 

39. fx= x+ 

x+ gx= __ 

 

x 

 

40. fx= 

x+ gx= 

x− 

41. fx= 

x gx=√— x 

 

42. fx= x+ 

x − gx=√— 

HfgHx=f∘gx 

43. Hx= 

√ x− 

x+ 

 

44. Hx= 

x − − 

 

TRANSFORMATION OF FUNCTIONS 

 

45. fx=x− 46. fx=x+ 47. fx=√ — x + 

48. fx=−x 49. fx= √ 

— −x 

50. fx=√ — −x − 

51. fx=x−− 52. fx=−x+ − 

gfFigure 4 

– – – – 

 

 

 

 

 

y 

– 

– 

– 

 

 

 

 

 

x 

53. gx=fx− 

54. gx=fx 

 

Figure 4


 

55. 

y 

56. 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

57. fx=x 58. gx=√ — x 59. hx=_ 

x 

+x 

 

60. 

y 

61. 

y 

62. 

y 

 

 

 

 

 

––––– 

– 

– 

– 

– 

– 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

ABSOLUTE VALUE FUNCTIONS 

f x=| x | 

63. 

y 

64. 

y 

65. 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

66. fx=| x−| 67. fx=−| x−| 68. fx=| x−|


INVERSE FUNCTIONS 

, fif − x 

x 

69. fx=+x 70. fx= 

x+ 

fif 

. Thn fif 

71. fx=x + 

72. fx=x −gx= √ 

— x+ 

a. fgxgfx 

b. fxgx 

 

73. fx= x 74. fx=−x +x 75. f= f − 

76. f= f −


277 


 

1. y=x+ 2. −− 

f x=−x +x 

3. f− 4. fa 

5. fx=−x− + 

 

6. fx=√ — −x 

 

7. fx=x −x fa+−f 8. fx= { 

x+−


Figure 2 

y 

25. f 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

f 

 

 

 

 

 

x 

26. f− 

27. 

Figure 2 

Table 1 

x 

F (x) 

Table 1 

28. F 29. Fx= 

30. 31. 

32. F − 33. fx=−x+ f − x

4 

Linear Functions 


4.1 Linear Functions 

4.2 Modeling with Linear Functions 

4.3 Fitting Linear Models to Data 

Figure 1 A bamboo forest in China (credit: “JFXie”/Flickr) 

Introduction 

 

Th 

1.5 

36 A 

 


fi 

 

 

 

279

280 

CHAPTER 4 LINEAR FUNCTIONS 



Represent a linear function. 

Determine whether a linear function is increasing, decreasing, or constant. 

Interpret slope as a rate of change. 

Write and interpret an equation for a linear function. 

Graph linear functions. 

Determine whether lines are parallel or perpendicular. 

Write the equation of a line parallel or perpendicular to a given line. 

4.1 LINEAR FUNCTIONS 

Figure 1 Shanghai MagLev Train (credit: “kanegen”/Flickr) 

 

fiFigure 1 

 

 

 

 

 

Representing Linear Functions 

Thfi 

Th 

 

 

Representing a Linear Function in Word Form 

 

 

The train’s distance from the station is a function of the time during which the train moves at a constant speed plus 

its original distance from the station when it began moving at constant speed. 

Th 

Th 

 

. Th

SECTION 4.1 LINEAR FUNCTIONS 281 

Representing a Linear Function in Function Notation 

 

xmb 

 

y =mx+b 

f x=mx+b 

DtDt 

mThb 

 

Dt=t + 

Representing a Linear Function in Tabular Form 

The 

Figure 2 

y 1 s 

 

t 

D(t) 

 

 

 

 

 

 

 

 

 

Figure 2 Tabular representation of the function D showing selected input and output values 

Q & A… 

Can the input in the previous example be any real number? 

Th 

. Th 

Representing a Linear Function in Graphical Form 

 

Dt=t +Figure 3 

 

ThTh 

yy 

 

 

 

 

 

 

 

 

 

 

Figure 3 The graph of D(t) = 83t + 250. Graphs of linear functions are lines because the rate of change is constant. 

 

f x=x + 

Th

282 


linear function 

linear functionslope-intercept form 

 

fx=mx+b 

bx =m 

. Thyb 

Example 1 

Using a Linear Function to Find the Pressure on a Diver 

ThPFigure 4 

dThPd=d + 

 

Figure 4 (credit: Ilse Reijs and Jan-Noud Hutten) 

Solution Th 

 

Analysis The initial value, , is the pressure in PSI on the diver at a depth of feet, which is the surface of the 

water. The rate of change, or slope, is PSI per foot. This tells us that the pressure on the diver increases PSI 

for each foot her depth increases. 

Determining Whether a Linear Function Is Increasing, Decreasing, or Constant 

Th 

 

Th 

Figure5(a) 

Th 

Figure 5(b) 

Figure 5(c) 

f x 

Increasing function 

Decreasing function 

f x 

f x 

Constant function 

f 

f 

f 

 

x x x 

 

 

Figure 5


increasing and decreasing functions 

Thincreasing linear functiondecreasing linear function 

 

fx=mx+bm > 

fx=mx+bm < 

fx=mx+bm = 

Example 2 

Deciding Whether a Function Is Increasing, Decreasing, or Constant 

 

fiTh 

 

a. ThTh 

 

b. Th 

 

c. Th 

 

Solution 

a. Thf x=xxTh 

 

b. Thf x=−xx 

 

ftx 

c. Thf x=ffTh 

 

Interpreting Slope as a Rate of Change 

ft 

x 

x 

 

y 

y 

x 

y 

x 

y 

m 

m= __ 

 

=Δy _ 

Δx =y _ −y x 

−x 

 

 

y 

y 

fy 

=f x 

y 

=f x 

 

 

m= f x −f x _ 

x 

−x 

 

 

Figure 6x 

y 

x 

y 

 

. Th

284 


1. 

x – x 

– 

y 

 

 

 

 

 

 

 

 

 

 

y – y 

 

x 

, y 

x , y 

 

Δy 

m = = 

Δx 

x 

y 

– y 

x 

– x 

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 

is used as the (x 2 

, y 2 

) and which is the (x 1 

, y 1 

), as long as each calculation is started with the elements from the same coordinate pair. 

Q & A… 

Are the units for slope always 

units for the output 

__ 

units for the input ? 

Th 

 

calculate slope 

m 

m= __ 

 

=Δy _ 

Δx =y _ −y x 

−x 

 

 

x 

x 

y 

y 

 

How To… 

 

1. 

2. 

3. 

Example 3 

Finding the Slope of a Linear Function 

f x−fi 

 

Solution Th−fi 

 

m= __ 

 

=−− ________ 

− = __ 

 

m =. Thm > 

Analysis As noted earlier, the order in which we write the points does not matter when we compute the slope of the line 

as long as the first output value, or y-coordinate, used corresponds with the first input value, or x-coordinate, used. Note 

that if we had reversed them, we would have obtained the same slope. 

m= −− 

− =− 

− =


Try It #1 

fx 

 

Example 4 Finding the Population Change from a Linear Function 

Th 

 

Solution ThTh 

−=fi 

 

__ 

 

= __ 

 

 

Analysis Because we are told that the population increased, we would expect the slope to be positive. This positive slope 

we calculated is therefore reasonable. 

Try It #2 

Th 

 

Writing and Interpreting an Equation for a Linear Function 

Equations and Inequalities 

 

Th 

fFigure 7. 

f 

y 

–– – – 

 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

x 

Figure 7 

fi 

 

m = y −y x 

 

−x 

 

 

= − 

− 

=− 

 

 

y −y 

=mx−x 

 

y −=− 

 

x−

286 


d fi 

y −=− 

x− 

y −=− x + 

 

y =− x + 

 

fifi 

yThb =b 

mmb 

fx=mx+b 

↑ ↑ 

− 

 

fx=− x + 

 

f x=− 

x +y =− x + 

 

How To… 

 

1. 

2. 

3. yy 

4. y 

Example 5 

Writing an Equation for a Linear Function 

fFigure 8 

y 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

f 

 

 

 

x 

Figure 8 

Solution −− 

m = y −y x 

 

−x 

 

 

= −− 

−− 

= − 

− 

= 

 

y −y 

=mx−x 

 

y −−=x−− 

y +=x+


 

y +=x+ 

y +=x + 

y =x + 

Analysis This makes sense because we can see from Figure 9 that the line crosses the y-axis at the point , which is 

the y-intercept, so b = . 

y 

 

 

 

 

 

–– – – – 

– 

−− – 

– 

– 

– 

 

 

 

x 

Figure 9 

Example 6 

Writing an Equation for a Linear Cost Function 

fi 

ffiCCxx 

 

Solution ThfiTh 

ThTh 

Cx=+x 

Analysis If Ben produces items in a month, his monthly cost is found by substitution for x. 

C=+ 

= 

So his monthly cost would be 

Example 7 

Writing an Equation for a Linear Function Given Two Points 

ff =−f =1 , fi 

Solution 

f =−→− 

f =→ 

 

m = y −y x 

 

−x 

 

 

= −− − 

= 

 

 

y −y 

=mx−x 

 

y −−= 

 

x−

288 


 

Try It #3 

y += 

x− 

y += 

x − 

 

y = 

x − 

 

fxf=−f=− 

 

Modeling Real-World Problems with Linear Functions 

 

fi 

fi 

ff 

How To… 

ff c 

1. 

2. f x=mx+b 

3. x =c 

Example 8 Using a Linear Function to Determine the Number of Songs in a Music Collection 

 

Nt 

 

Solution ThN= 

b = 

ThTh 

m = 

f x=mx+b 

↑ ↑ 

 

 

Nt=t + 

Nt=t + 

Figure 10 

 

t = 

N=+ 

=+ 

= 

 

Analysis Notice that N is an increasing linear function. As the input (the number of months) increases, the output 

(number of songs) increases as well.


Example 9 

Using a Linear Function to Calculate Salary Based on Commission 

Th 

In 

ThIn 

 

Solution Thfi 

 

m = ________ − − 

 

 

= 

_______ 

 

= 

 

Th 

 

 

In=n +b 

=+b n =I= 

 

−=b 

 

=b 

Thbn = 

 

e fi 

In=n + 

fi 

 

Example 10 

Using Tabular Form to Write an Equation for a Linear Function 

Table 1 

Number of weeks, w 

Number of rats, P(w) 

Table 1 

Solution b = 

m 

. Thfi 

Pw=w + 

 

 

m = __________ − − 

= ___ 

 

= 

Q & A… 

Is the initial value always provided in a table of values like Table 1? 

 

 

fif x=mx+bb

290 


Try It #4 

ffTable 2 

xHxx 

 

x 

H(x) 

Table 2 

Graphing Linear Functions 

Th 

Thfi 

Thy 

f x=x 

Graphing a Function by Plotting Points 

fi 

Th 

fi 

f x=x 

 

 

ft 

How To… 

 

1. 

2. 

3. 

4. 

5. 

Example 11 

Graphing by Plotting Points 

f x=− __ x + 

 

Solution Th 

 

 

x = 

x = 

f =− __ 

+=⇒ 

f =− __ 

+=⇒ 

x = f =− __ 

+=⇒ 

Figure11 

f x=− __ x +


f(x) 

 

f 

 

 

 

 

 

– 

– 

 

 

 

x 

Analysis 

Figure 11 The graph of the linear function f (x) = − 2 __ 

3 x + 5. 

The graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant, 

which indicates a negative slope. This is also expected from the negative, constant rate of change in the equation for the function. 

Try It #5 

fx=− 

x+ 

Graphing a Function Using y-intercept and Slope 

fi 

Thfiyfiy 

x = 

Th 

 

fx= x + 

 

Th 

Thy 

 

x =Thyy 

m = 

m = 

y 

 

Figure 12 

– 

 

 

– 

 

 

 

y 

y 

 

←= 

↑= 

 

 

 

Figure 12 

 

f 

 

 

x 

graphical interpretation of a linear function 

fx=mx+b 

byby 

m 

 

m= 

 

=Δy 

Δx =y −y x 

 

−x

292 


Q & A… 

Do all linear functions have y-intercepts? 

y-y-.Note y- 

y- 

How To… 

y 

1. o fiy 

2. 

3. y 

4. 

5. 

Example 12 

Graphing by Using the y-intercept and Slope 

f x=− x +y 

 

Solution x =fiyThx = 

y 

− 

Th 

− 

fiyFigure 13 

 

 

 

 

 

 

f(x) 

f 

 

– 

– 

 

 

 

 

 

 

 

x 

Figure 13 Graph of f (x ) = − 2 __ 

3 

x + 5 and shows how to calculate the rise over run for the slope. 

Analysis The graph slants downward from left to right, which means it has a negative slope as expected. 

Try It #6 

Example 12x 

Graphing a Function Using Transformations 

f x=x 

ftfl 

Vertical Stretch or Compression 

f (x=mxmm 

flFigure14f x=x 

mfmm >fm 


y 

 

 

 

 

 

 

– – – – – – 

– 

 

– 

– 

– 

– 

– 

Figure 14 Vertical stretches and compressions and reflections on the function f (x) = x. 

x 

Vertical Shift 

f x=mx+bbftff 

Figure 15bf x=xftfbb 

bb 

– – – – 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

fx = x + 

fx = x + 

fx = x 

fx = x – 

fx = x – 

x 

Figure 15 This graph illustrates vertical shifts of the function f (x) = x. 

ftff 

 

 

How To… 

f x=mx+b 

1. f x=x 

2. m 

3. b 

Example 13 

Graphing by Using Transformations 

f x= __ x − 

 

Solution Thm = __ 

__ 

Th 

b =−hift 

Figure16 

Th Figure 17

294 


1. 

y 

y 

Try It #7 

 

 

 

y = x 

 

y = x 

 

 

 

y = x 

 

 

 

 

 

y = x − 

 

x 

x 

– – – – – – – 

– 

– – – – – – – 

– 

Figure 16 The function, y = x, compressed by a factor of 

__ 1 2 . Figure 17 The function y = __ 

1 x, shifted down 3 units. 

2 

– 

– 

– 

– 

– 

– 

– 

– 

fx=+x 

Q & A… 

In Example 15, could we have sketched the graph by reversing the order of the transformations? 

Th 

Th 

fi 

f = __ 

− 

=− 

=− 

Writing the Equation for a Function from the Graph of a Line 

 

Figure18 

yy 

y 

f 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 18 

Thfi 

−y 

 

m= = 

= 

y 

y=x +


How To… 

n, fi 

1. y 

2. 

3. y 

Example 14 Matching Linear Functions to Their Graphs 

Figure19 

a. f x=x + b. gx=x − c. hx=−x + d. jx= x + 

 

y 

– – – – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

x 

Figure 19 

Solution 

a. y 

 

gdiffy 

f 

b. y−− 

 

c. −y 

 

d. 

y 

(0, jfj 

e fl 

Figure20 

y 

gx = x − 

 

jx = 

 

x + 

– – – – – – 

f x = x + 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

Figure 20 

 

 

 

 

 

x 

hx = −x +

296 


Finding the x-intercept of a Line 

fiy 

yxx 

x 

fixf xx 

 

f x=x − 

x 

 

=x − 

 

=x 

 

=x 

x = 

Thx 

Q & A… 

Do all linear functions have x-intercepts? 

y =cc 

xy =xTh 

xFigure 21 

y 

 

y= 

 

 

 

 

– – – – – – 

 

– 

– 

– 

– 

 

 

x 

Figure 21 

x-intercept 

x-interceptxf x==mx+b 

Example 15 Finding an x-intercept 

xf x= x − 

 

Solution x 

 

 

 

Thx 

= x − 

 

= 

x 

=x 

x = 

Analysis A graph of the function is shown in Figure 22. We can see that the x-intercept is as we expected.


y 

– – – – 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

Figure 22 

Try It #8 

xfx= 

x− 

Describing Horizontal and Vertical Lines 

Th 

yFigure23Th 

m = 

f x=mx+bfif x=b 

Thf x= 

y 

– – – – – 

 

 

 

 

 

 

 

 

f 

 

 

x 

x − − 

y 

Figure 23 A horizontal line representing the function f (x) = 2. 

x 

 

 

fi 

m = __ 

 

← 

← 

 

Figure 24 Example of how a line has a vertical slope. 0 in the denominator of the slope. 

Figure25xyx = 

Thx = 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

x 

y − − 

– 

– 

– 

– 

Figure 25 The vertical line, x = 2, which does not represent a function.

298 


horizontal and vertical lines 

 

horizontal linefif x=b 

vertical linefix =a 

Example 16 

Writing the Equation of a Horizontal Line 

Figure26 

y 

– – – – 

– 

– 

– 

– 

– 

– 

 

 

 

f 

 

x 

Figure 26 

Solution xy−y =− 

Example 17 

Writing the Equation of a Vertical Line 

Figure27 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

f 

 

x 

Figure 27 

Solution Thxx = 

Determining Whether Lines are Parallel or Perpendicular 

ThFigure28Th 

Thffyft 

y 

– – – – – 

y 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

Figure 28 Parallel lines 

 

yffff 

 

 

 

 

 

 

x


} } 

f x=−x + 

f x=x + 

f x=−x − f x=x + 

ThTh 

Figure29 

y 

– – – – – 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

Figure 29 Perpendicular lines 

Thff 

fiThTh 

m 

m 

− 

m 

m 

=− 

fi 

 

 

fi, fit fi 

 

Th 

 

Th− 

f x= 

x + 

− 

f x=−x +− 

 

−( 

) =− 

parallel and perpendicular lines 

parallel lines. Th 

f x=m 

x +b 

gx=m 

x +b 

m 

=m 

 

b 

=b 

m 

=m 

 

perpendicular lines 

f x=m 

x +b 

gx=m 

x +b 

m 

m 

=−m 

=− 

m 

Example 18 Identifying Parallel and Perpendicular Lines 

 

 

f x=x + 

hx=−x + 

gx= x − 

 

jx=x −

300 


Solution f x=x +jx=x − 

− 

 

gx= x −hx=−x + 

 

Analysis A graph of the lines is shown in Figure 30. 

hx = −x + 

– – – – 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

f x = x + 

 

 

jx = x − 

 

 

gx = 

x − 

x 

Figure 30 

f x=x +jx=x −gx= x − 

 

hx=−x + 

Writing the Equation of a Line Parallel or Perpendicular to a Given Line 

 

 

Writing Equations of Parallel Lines 

 

f x=x + 

y 

f xf x 

gx=x + hx=x + px=x + __ 

 

fTh 

ft 

b 

 

y −y 

=mx−x 

 

y −=x− 

y −=x − 

y =x + 

gx=x +f x=x + 

How To… 

 

 

1. 

2. 

3. 

Example 19 

Finding a Line Parallel to a Given Line 

f x=x +


Solution Thm =x = 

f x=o fiy 

gx=x +b 

 

=+b 

b =− 

Thf xgx=x − 

y 

Analysis We can confirm that the two lines are parallel by graphing them. 

Figure 31 shows that the two lines will never intersect. 

 

 

 

 

 

– – – – – – 

– 

y = x + 

 

– 

– 

– y = x − 

– 

x 

Figure 31 

Writing Equations of Perpendicular Lines 

 

 

f x=x + 

Th− __ 

− __ 

 

f xf x 

gx=− __ 

x + hx=− __ 

x + px=− __ 

x − __ 

 

 

f x 

− __ 

fiy 

b 

g x=mx+b 

 

=− __ 

+b 

 

=−+b 

 

=b 

b = 

Th− __ 

y 

gx=− __ x + 

 

gx=− __ x +f x=x + 

 

 

Q & A… 

A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular, 

but the product of their slopes is not −1. Doesn’t this fact contradict the definition of perpendicular lines? 

− 

fi

302 


How To… 

 

 

1. 

2. 

3. xygx=mx+b 

4. b 

5. 

Example 20 

Finding the Equation of a Perpendicular Line 

f x=x + 

Solution Thm = 

− __ n 

fi 

 

gx=− __ x +b 

 

 

=− __ 

+b 

 

=b 

b = 

Thf xgx=− __ x + 

 

y 

f x = x + 

Analysis A graph of the two lines is shown in Figure 32. 

– – – – – 

Figure 32 

Note that that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear 

to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely. 

 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

 

gx = − 

 

x + 

x 

 

Try It #9 

hx=x − 

a. hx b. hx 

How To… 

 

1. 

2. 

3. 

4.


Example 21 

Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point 

− 

 

Solution 

− 

m 

= 

−− 

_______ 

= ___ − 

 

 

=− __ 

 

m 

= _ − − 

__ 

 

 

=−( − __ 

) 

= 

y 

g x=x +b 

 

=+b 

 

=+b 

−=b 

b =− 

Th 

 

y =x − 

Try It #10 

−−− 

 

 

Linear Functions (http://openstaxcollege.org/l/linearfunctions) 

Finding Input of Function from the Output and Graph (http://openstaxcollege.org/l/findinginput) 

Graphing Functions Using Tables (http://openstaxcollege.org/l/graphwithtable)

304 CHAPTER 4 LINEAR FUNCTIONS 


VERBAL 

1. Et 

fttEt=−t 

 

 

2. 

ft 

ft 

 

3. 

 

 

ftt 

4. 

 

y- 

5. f x=a 

x=a 

 

 

ALGEBRAIC 

 

6. y= __ 

x+ 7. y=x− 8. y=x − 

9. x+y= 10. x +y= 11. x+y = 

12. −x +y = 13. − ______ 

x− 

=y 

 

14. fx=x+ 15. gx=x+ 16. ax=−x 

17. bx=−x 18. hx=−x+ 19. kx=−x+ 

20. jx= __ 

x− 

23. mx=− __ 

x+ 

21. px= __ 

x− 

22. nx=− __ 

x− 

, fi 

24. 25. 26. − 

27. − 28. − 

fi 

29. f−=−f= 30. f−=f= 

31. 32. 

33. − 34. − 

35. x−y− 36. x−y 

 

 

37. x−y= 

x +y = 

38. y+x= 

−y=x + 

39. y+x= 

−y =x + 

40. x−y= 

x +y =


, fixy 

41. fx=−x+ 42. gx=x+ 43. hx=x− 

44. kx=−x+ 45. −x+y= 46. x+y= 

fi 

 

47. − 

−− 

48. −− 

−− 

49. − 

 

50. 

−− 

51. − 

−− 

 

52. f x=−x− 

− 

54. 

h t=−t+−− 

53. g x=x− 

 

55. 

p t=t+ 

GRAPHICAL 

, fi 

56. 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

x 

57. 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

x 

 

58. 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

x 

59. 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

x 

60. 

– – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

 

x


61. 

– – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

 

x 

62. 

– – 

– – – 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

x 

63. 

– – 

– – – 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

x 

 

Figure33 

A B 

 

 

 

C 

 

– – – – – – 

– 

 

– 

– 

D 

– 

F 

E 

64. fx=−x− 

65. fx=−x− 

66. fx=− __ 

x− 

67. fx= 

68. fx=+x 

69. fx=x+ 

Figure 33 

 

70. x−y− 71. x−y 

72. y− __ 

 

73. y__ 

 

74. −−− 75. −− 

 

76. fx=−x− 77. gx=−x+ 78. hx= __ 

x+ 

79. kx= __ 

x− 80. ft=+t 81. pt=−+t 

 

82. x= 83. x=− 84. rx= 

 

85. 

y 

 

 

 

 

 

 

– – – – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

86. 

y 

 

 

 

 

 

 

– – – – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

87. 

y 

 

 

 

 

 

 

– – – – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

88. 

y 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

 

 

 

 

 

x


NUMERIC 

fi 

 

89. x 

g(x) − − − 

90. x 

h(x) 

91. x 

f (x) − 

92. x 

k(x) 

93. x 

g(x) − − − 

94. x 

h (x) 

95. x 

f (x) − 

96. x 

k(x) 

TECHNOLOGY 

 

97. ff=f=− 

98. f−fx=x− 

x−x 

99. f −fx=x+ 

100. Table 3wk 

ka. 

b.k 

 

w − b 

k − a − 

Table 3 

101. Table 4pq 

qa. 

b.k 

p b 

q a 

Table 4 

102. f−__ 

y __ 

 

− 

103. f−y 

−− 

104. ffx=ax+b− 

ab 

a. a=b= b. a=b= c. a=b=− d. a=b=− 

EXTENSIONS 

105. x 

 

x−m= 

107. 

abab+ 

106. y 

 

ym=− 

108. 

abab+ 

109. 

acd 

110. 

gx=−x+


111. gxx+ 

f x=−x+gx=x + 

112. fg 113. f xg xg x 

f x 


114. 

 

 

n 

116. 

n 

p 

 

 

 

pn=mn+bp 

n 

118. 

n 

y 

 

28 o 

y=mn+b 

n 

120. 

 

 

Pt 

tft 

122. 

 

 

 

C 

FC 

a. 

 

b. F 

c. F− 

115. 

 

fi 

 

117. 

Cn=+nn 

Cn 

 

 

119. 

 

 

 

 

121. 

 

Ix=x+x 

ft 

 

a. 

b. 

 

c. 

 

d.

SECTION 4.2 MODELING WITH LINEAR FUNCTIONS 309 



Build linear models from verbal descriptions. 

Model a set of data with a linear function. 

4.2 MODELING WITH LINEAR FUNCTIONS 

Figure 1 (credit: EEK Photography/Flickr) 

 

 

x 

 

 

Building Linear Models from Verbal Descriptions 

 

fl 

1. 

 

2. 

 

3. 

4. . Oft 

 

 

5. 

6. 

7. 

8.

310 


Th 

 

fi 

M 

t 

Mt 

 

M 

− 

 

Th 

 

ThMt=mt+bTh 

 

Mt=mt +b 

↑ ↑ 

− 

Mt = − t + 

o fix 

 

=−t + 

t =____ 

 

 

= 

Thx 

ft 

 

fi 

 

 

ftx 

Th 

≤t ≤ 

 

 

 

 

Using a Given Intercept to Build a Model 

yy 

x 

Thy 

Th− 

 

f x=mx+b 

=−x + 

xo fix


 

 

 

=−x + 

=x 

=x 

x = 

xThx 

 

Using a Given Input and Output to Build a Model 

 

 

fi 

How To… 

 

1. 

2. 

3. 

4. 

5. x 

6. xy 

7. 

Example 1 

Using a Linear Model to Investigate a Town’s Population 

 

 

a. 

b. 

Solution Th 

y 

 

fi 

t 

Pt 

t=fifi 

 

 

 

m= __ 

 

 

Thfi 

t =fi 

y. Tht = 

Th

312 


m =__________ 

− − 

=____ 

 

= 

y 

Pt=t + 

t = 

P=+ 

= 

 

o fiPt=t 

 

=t + 

 

=t 

 

t ≈ 

ft 

 

Try It #1 

. Thr a fi 

 

a. t tCx 

b. y 

Try It #2 

 

 

a. 

b. 

Using a Diagram to Model a Problem 

 

fi 

ThThft 

fiftTh 

 

Example 2 

Using a Diagram to Model Distance Walked 

 

Thft 

 

Solution


 

fi 

t 

AtEt 

fiFigure 2 

 

 

 

 

Figure 2 

Tht = 

. Th 

 

. ThAE 

 

At=t 

Et=t 

fi 

ThA 

fi 

E 

fifi 

 

fiD 

ftFigure 3 

D 

tAtEtDt 

A 

E 

D 

Figure 3 

Figure 3n Th

314 


n Th 

Dt =At +Et 

=t +t 

=t +t 

=t 

Dt=±√ — t Dt 

=±|t | 

tDt 

Dt=tTh 

D 

Dt=t 

Dt= 

 

t = 

t = __ 

= 

Th 

Q & A… 

Should I draw diagrams when given information based on a geometric shape? 

e fi 

Example 3 

Using a Diagram to Model Distance Between Cities 

Th 

fi 

 

 

Solution Figure 4 

Th 

 

 

 

 

 

 

Figure 4 

 

 

n fi 

 

m=______ 

− − = __ 

 

 

 

Wx= __ 

x 

m =− 

n fi 

Ex=−x +b 

=−+b 

b = 

Ex=−x +


fifi 

 

 

 

__ 

x =−x + 

 

__ 

x = 

 

x = 

x = 

y =W 

Wx 

= __ 

 

= 

Thfi 

 

= √ —— 

x 

−x 

+y 

−y 

 

=√ — − +− 

≈ 

Analysis One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This 

means real-world applications discussing maps need linear functions to model the distances between reference points. 

Try It #3 

Th 

 

 

Modeling a Set of Data with Linear Functions 

 

 

Figure 5 

y y y 

 

 

 

f 

g 

x 

 

 

 

x 

g 

f 

x 

 

 

 

Figure 5 

How To… 

 

1. 

2. y 

3. xfi

316 


Example 4 

Building a System of Linear Models to Choose a Truck Rental Company 

Thfi 

Th 

 

Solution Th 

fiTable 1 

Input 

Outputs 

Initial Value 

Rate of Change 

d 

Kd 

Md 

K=M= 

Kd=Pd= 

Table 1 

f x=mx+b 

Kd=d + 

Md=d + 

 

Kd 

 

Interpreting a Linear Function (http://openstaxcollege.org/l/interpretlinear)



VERBAL 

1. 

 

2. 

 

3. 

 

4. 

 

ALGEBRAIC 

5. 

yx=f x=+x 

f(x 

6. x 

f(x=–__ 

x 

f(x 

7. y 

f(x=–__ 

x 

f(x 

8. 

xg x=f(x=x 

f x 

 

 

9. 10. 

 

 

11. 12. 

 

 

 

13. 

Ptt 

 

14. 

P 

15. P 

xy 

16. P 

 

17. 18.


: Th. Th 

s fi 

19. 

W, 

t 

20. 

W 

21. W 

xy 

22. W 

 

23. 24. 

 

: Thffl 

ffl 

25. 

C, 

t 

27. C 

xy 

26. 

C 

28. C 

 

29. 30. 

GRAPHICAL 

Figure 7fiy 

tt 

y 

 

 

 

 

 

 

 

 

 

Figure 7 

t 

31. yyt 

32. y 

33. x 

34.


Figure 8fiy 

tt 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

t 

Figure 8 

35. yyt 

36. y 

37. x 

38. 

NUMERIC 

fl 

Table 2 

Year Mississippi Hawaii 

 

 

Table 2 

39. 

40. 

41. ft 

r? (Th 

fl 

Table 3 

Year Indiana Alabama 

 

 

Table 3 

42. 

43. 

44. ft 

r? (Th



45. 

 

 

a. 

 

b. 

 

c. 

d. 

e. P 

t 

f. 

 

46. 

 

 

a. 

 

b. 

 

c. 

d. 

e. Pt 

 

f. 

 

47. 

ys a fl 

 

 

 

 

a. 

x 

 

b. y 

c. 

 

48. 

ys a fl 

 

 

 

 

a. 

x 

 

b. y 

c. 

 

49. 

 

 

 

a. 

P 

b. 

 

50. 

 

e 285. Th 

 

a. P 

 

b. 

 

51. Th 

 

 

a. 

Rt 

 

b. 

c. 

 

52. 

 

 

a. 

Rt 

b. 

 

c.


53. o diff 

s. Th 

. Th 

plus 

 

 

54. o diff 

nies. Th 

Th 

 

 

55. 

 

 

 

 

 

 

 

56. 

 

 

 

 

 

 

 

57. 

 

 

 

 

 

 

 

58.

322 




Draw and interpret scatter plots. 

Use a graphing utility to find the line of best fit. 

Distinguish between linear and nonlinear relations. 

Fit a regression line to a set of data and use the linear model to make predictions. 

4.3 FITTING LINEAR MODELS TO DATA 

fi 

fi 

 

 

Drawing and Interpreting Scatter Plots 

 

 

Figure 1 

Final Exam Score 

 

 

 

 

 

 

 

 

 

 

Final Exam Score vs. Age 

 

 

Age 

Figure 1 A scatter plot of age and final exam score variables. 

notTh 

e fi 

Example 1 

Using a Scatter Plot to Investigate Cricket Chirps 

Table 1ff 

 

Chirps 

Temperature 

Table 1 Cricket Chirps vs Air Temperature 


Th

SECTION 4.3 FITTING LINEAR MODELS TO DATA 323 

 

Cricket Chirps vs. Temperature 

Temperature (°F) 

 

 

 

 

 

 

 

 

Cricket Chirps in 15 Seconds 

Figure 2 

Finding the Line of Best Fit 

 

fiTh 

y 

 

Example 2 

Finding a Line of Best Fit 

t fiTable 1o fi 

Solution 

 

y. Th 

m=__ 

= 

Tc=c + 

cTcTh 

Figure 3 

 


 

T(c), Temperature (°F) 

 

 

 

 

 

 

 

 

 

c, Number of Chirps 

Analysis 

Figure 3 

This linear equation can then be used to approximate answers to various questions we might ask about the trend.

324 


Recognizing Interpolation or Extrapolation 

 

interpolation 

Thextrapolation 

 

Figure 4 Example 2 

 

 

 

Thff 

 

ftmodel breakdown 

 

x = 

s fift 

 


 


 

 

 

 

Extrapolation 

Interpolation 

 

 

 

 

 


Figure 4 Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside. 

interpolation and extrapolation 

ff 

Thinterpolation 

 

Thextrapolation 

 

Example 3 

Understanding Interpolation and Extrapolation 

Table 1 

a. 15 

 

b.


Solution 

a. Th 

 

T=+ 

= 

 

b. Th 

 

 

 

=+c 

=c 

c ≈ 

Figure 5 

 



 

 

 

 

 

 

Extrapolation 

Interpolation 

 

 

 

 


Figure 5 

Analysis Our model predicts the crickets would chirp times in seconds. While this might be possible, we have 

no reason to believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether 

below around degrees. 

Try It #1 

Table 1 

Finding the Line of Best Fit Using a Graphing Utility 

fi 

ff least squares regression 

ftft 

fi 

 

ff

326 


How To… 

n, fit fi 

1. List 1 (L1) 

2. List 2 (L2) 

3. Linear Regression (LinReg) 

Example 4 

Finding a Least Squares Regression Line 

Table 2 

Solution 

1. List 1 (L1) 

2. List 2 (L2)Table 2 

L1 

L2 

Table 2 

3. Linear Regression (LinReg) 

 

Tc=+c 

Analysis Notice that this line is quite similar to the equation we “eyeballed” but should fit the data better. Notice also 

that using this equation would change our prediction for the temperature when hearing chirps in seconds from 66 

degrees to: 

T=+ 

= 

≈ 

The graph of the scatter plot with the least squares regression line is shown in Figure 6. 

 

Number of Cricket Chirps 

vs. Temperature 


 

 

 

 

 

 

 

 

 


Figure 6 

Q & A… 

Will there ever be a case where two different lines will serve as the best fit for the data? 

. Tht fi


Distinguishing Between Linear and Non-Linear Models 

fi 

ft 

ffifi 

fiffir 

ffi 

ffi 

ffi 

rFigure 7 

ffi 

 

 

− 

− 

− 

 

− − − 

0 0 0 

Figure 7 Plotted data and related correlation coefficients. (credit: “DenisBoigelot,” Wikimedia Commons) 

correlation coefficient 

correlation coefficientr− 

r> 

r< 

Th 

Th− 

Example 5 

Finding a Correlation Coefficient 

ffiTable 1 

Solution r 

Th 

effir =. Th 

ffi 

2nd>0>alphax − 1DIAGNOSTICSON 

Fitting a Regression Line to a Set of Data 

ffi 

 

t fi

328 


Example 6 

Using a Regression Line to Make Predictions 

 

Table 3 fi 

 

Year 

Consumption 

(billions of gallons) 

 

Table 3 

ThFigure 8 

 

Gas Consumption vs. Year 

Gas Consumption 

(billions of gallons) 

 

 

 

 

 

 

 

Years After 1994 

Figure 8 

Solution t 

Th 

Ct=+t 

ffi 

 

t= 

C=+ 

= 

Th 

Try It #2 

Example 6 

 

h fi 

Introduction to Regression Analysis (http://openstaxcollege.org/l/introregress) 

Linear Regression (http://openstaxcollege.org/l/linearregress)



VERBAL 

1. 

 

2. 

3. 4. e diff 

 

5. 

 

ALGEBRAIC 

6. 

 

xy). Th 

 

 

 

y=ax+b 

a=− 

b= 

r=− 

7. s 

 

xys). Th 

 

 

y=ax+b 

a= 

b=− 

r=− 

 

8. 

 

− − − − − − 

9. 

 

 

10. 

 

11. 

 

 

 

12. 

 

 

 

 

Year 

Population 

13. 

 

° 

 

 

Temperature, °F 

Time, seconds


GRAPHICAL 

fiFigure 9Figure 10 

 

Figure 9 

 

 

Figure 10 

 

14. r= 15. r=− 

16. r=− 17. r=− 

fi 

18. 

 

19.


20. 21. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NUMERIC 

22. Thes. Th 

Table 4 

 

Year 

Percent Graduates 

Table 4 

23. ThTable 5 

 

Year 

Imports 

Table 5 

24. Table 6 

 

 

Year 

Number Unemployed 

TECHNOLOGY 

Table 6 

 

ffi 

25. x 

y 

26. x 

y


27. x 

y 

x 

y 

28. x 

y 

29. 

x 

y − − − 

30. x 

y 

31. x 

y 

EXTENSIONS 

32. f x=x+ 

x=− 

d fi 

 

33. f x=−x− 

x=− 

 

Thfi 

fi 

fi 

 

34. P 

fi 

 

35. 

x 

36. 

y



Th 

Thfi 

 

 

37. y 

 

 

38. 

ThfiTh 

fi 

fifi 

 

39. y 

fi 

 

40. fi 

 

Thfi 

Thfi 

fifi 

 

41. y 

fi 

 

42. fi



Key Terms 

correlation coefficient r− 

e fi 

decreasing linear function f x=mx+bm < 

extrapolation 

horizontal line fif x=bb. Th 

increasing linear function f x=mx+bm > 

interpolation 

least squares regression fiff 

 

linear function 

model breakdown ft 

parallel lines 

perpendicular lines 

point-slope form y −y 

=mx−x 

 

slope 

slope-intercept form f x=mx+b 

vertical line fix =aa. Thfi 

Key Concepts 

4.1 Linear Functions 

Example 1 

 

 

Example 2 

Thdiff 

e diffxExample 3Example 4 

Example 5 

Thm bExample 6 

Example 7 

diffExample 8 


Example 11 

Example 12 

 

Example 13 

ThExample 14 

xxExample 15 

f x=bExample 16


x =bExample 17 

 

Example 18 

 

xyf x=mx+b 

Example 19 

 


4.2 Modeling with Linear Functions 

 

 

yExample 1 


 

y 

◦ 

xy =mx+b 

◦ 

Thxy 

Example 4 

◦ 

 

4.3 Fitting Linear Models to Data 

Example 1 

 

Tht fiftExample 2 

 

Example 3 

Th rExample 4 

t fiExample 5 

Th 

Example 6


CHAPTER 4 

REVIEW EXERCISES 

LINEAR FUNCTIONS 

1. 

x+y= 

3. 

fx=x− 

5. 

 

 

7. 

– – – – – 

– 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

9. 

 

– – – – – 

– 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

y 

11. 

 

x 

g(x) – – – – 

 

 

 

 

 

 

 

 

 

 

 

 

x 

x 

2. 

x −y= 

4. 

gx=−x+ 

6. 

 

xy 

8. 

– – – – – 

– 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

10. 

 

x – 

g(x) – – – 

12. fi 

 

ft 

n ft 

 

 

 

 

 

 

x 

 

 

13. x−y= 

−x+y = 

14. y=__ 

x− 

x +y =− 

, fixy 

15. x+y=− 16. fx=x−


fi 

 

17. 

−− 

18. −− 

 

19. 

fx=x− 

20. y 

− __ 

 

21. ft=t− 22. 

x=y+ 

x −y = 

23. ff 

 

 

 

MODELING WITH LINEAR FUNCTIONS 

24. yfx=−xf 

 

25. 

 

26. Th 

 

fflC 

tffl 

Figure 1fiy 

xx 

 

 

 

 

 

 

 

y 

 

Figure 1 

27. yyx 

28. y 

 

 

29. 

a. 

b. 

c. Pt 

x


 

 

30. P 

31. 

Th 

flTable 1 

Year Pima Central East Valley 

 

 

Table 1 

32. 

33. 

FITTING LINEAR MODELS TO DATA 

34. Table 2. Th 

0 2 4 6 8 10 

– – 

Table 2 

35. Table 3 

 

Year 

Population 

Table 3 

36. Th 

Table 4t fi 

Predicted 

Actual 

Table 4 

37. t-fi 

y 

 

 

 

 

 

 

 

 

x


Table 5 

5 y 

Year 


Table 5 

38. 

 

 

 

39. 

40. Table 6 

 

 

 

x 

y 

Table 6 

41. Table 7 

 

 

 

x 

y 

Table 7 

Th 

The fi 

 

 

42. y 

 

 

43. 

44. 

 

45. 

in 2014?


CHAPTER 4 

PRACTICE TEST 

1. 

 

x+y= 

3. 

fx=x+ 

2. 

fx=−x+ 

4. 

 

− 

5. 

x−y− 

6. Figure 1 

y 

– – – – – 

– 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

Figure 1 

 

 

 

 

 

x 

7. Figure 2 

y 

– – – – – 

– 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

Figure 2 

 

 

 

 

 

x 

8. Table 1 

 

x − 

g(x) 

Table 1 

9. Table 2 

 

x 

g(x) 

Table 2 

10. 

nft 

 

 

11. y=__ 

x− 

−x −y = 

13. xy 

x+y=− 

12. −x+y= 

x +__ 

y = 

 

14. 

 

−−


341 

15. 

fx=x+ 

16. y− __ 

 

17. fx=−x+ 18. 

x=y+ 

 

x −y =− 

19. ff 

 

 

 

 

 

20. y 

fx=−x 

f 

21. 

 

 

 

22. Th 

 

 

 

 

Ct 

 

Figure 3fiy 

xx 

 

 

 

 

 

 

 

 

y 

 

Figure 3 

x 

23. yyx 

 

24. y 

25. 

 

a. 

b. 

c. Pt


26. Table 3 

Th 

 

0 2 4 6 8 10 

− − 

Table 3 

27. t-fi 

y 

 

 

 

 

 

 

 

 

 

x 

Table 4 

 

Year 


Table 4 

28. 

 

29. 

30. Table 5 

 

x 

y 

Table 5 

ThTh 

 

fi 

 

31. y 

 

32. 

33.

Polynomial and Rational Functions 

5 

Figure 1 35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. 

(credit “film”: modification of work by Horia Varlan; credit “memory cards”: modification of work by Paul Hudson) 


5.1 Quadratic Functions 

5.2 Power Functions and Polynomial Functions 

5.3 Graphs of Polynomial Functions 

5.4 Dividing Polynomials 

5.5 Zeros of Polynomial Functions 

5.6 Rational Functions 

5.7 Inverses and Radical Functions 

5.8 Modeling Using Variation 

Introduction 

 

fi 

 

fift 

ft 

eff 

fi 

 

343

344 

CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS 



Recognize characteristics of parabolas. 

Understand how the graph of a parabola is related to its quadratic function. 

Determine a quadratic function’s minimum or maximum value. 

Solve problems involving a quadratic function’s minimum or maximum value. 

5.1 QUADRATIC FUNCTIONS 

Figure 1 An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr) 

Figure 1 

Th 

 

 

 

 

Recognizing Characteristics of Parabolas 

Th 

vertex 

 

Th 

axis of symmetry. ThFigure 2 

y 

 

 

 

– 

– 

– 

 

 

x 

x 

 

– 

y 

 

– 

 

Figure 2

SECTION 5.1 QUADRATIC FUNCTIONS 345 

yyThx 

xxzerosroots 

xy = 

Example 1 

Identifying the Characteristics of a Parabola 

yFigure 3 

y 

 

 

 

 

 

– – 

x 

Figure 3 

Solution Th 

x = 

Thxyy 

Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions 

general form of a quadratic function 

fx=ax +bx+c 

abca ≠a >a < 

o fi 

Thfix =− b 

a x b −ac 

=−b±√— 

a 

ax +bx+c =xfixx =− b 

a 

 

Figure 4y = x +x + 

 

a =b =c =a >Thx =− 

 

=−Th 

x =−Th 

 

−−Thxx 

−− 

x 

x 

– – 

– 

– 

 

 

 

y y = x + x + 

Figure 4

346 


standard form of a quadratic function 

fx=ax−h +k 

hk 

vertex form of a quadratic function 

a >a < 

Figure 5 

y =−x+ +x −h =x +h =−a =−h =−k = 

a ftk 

fth >fth


Thfi 

ah +k =c 

k =c −ah 

=c −a ( __ 

a) b 

=c − b 

a 

kh 

fh=k 

forms of quadratic functions 

f de. Th 

general form of a quadratic functionfx=ax +bx+cabca≠ 

standard form of a quadratic functionfx=ax−h +ka≠ 

Thhk 

h=− 

b 

a k=fh=f ( 

−b 

a ) 

How To… 

 

1. h k 

2. hkfx=ax−h +k 

3. xf x 

4. a 

5. a >a

348 


− 

−=a+ − 

 

=a 

a = __ 

 

g x=__ 

 

x+ − 

 

gx=__ 

 

x+ − 

=__ 

x +x +− 

 

=__ 

x +x +− 

=__ 

x +x +− 

=__ 

x +x − 

ft 

ff 

Analysis We can check our work using the table feature on a graphing utility. First enter Y1 = 1 __ 

2 (x + 2) 2 − 3. Next, 

select TBLSET, then use TblStart = − 6 and ΔTbl = 2, and select TABLE. See Table 1. 

x − − − 

y − − − 

Table 1 

The ordered pairs in the table correspond to points on the graph. 

Try It #1 

Figure 8 

 

Figure 8 (credit: modification of work by Dan Meyer) 

How To… 

, fi 

1. abc 

2. hxabh =− b __ 

a 

3. kyk =f h=f ( − b __ 

a)


Example 3 

Finding the Vertex of a Quadratic Function 

f x=x −x + 

Solution Th 

h =− b 

a 

=− − 

 

Th 

= 

 

= 

 

k =fh 

=f( 

) 

=( 

) −( 

) + 

= 

 

afi 

Thfi 

a 

f x=ax +bx+c 

f x=x −x + 

Th 

f x=( x − 

) + 

 

Analysis One reason we may want to identify the vertex of the parabola is that this point will inform us where the 

maximum or minimum value of the output occurs, k, and where it occurs, x. 

Try It #2 

g x=+x −x 

Finding the Domain and Range of a Quadratic Function 

Th 

 

y 

yy 

 

domain and range of a quadratic function 

Th 

. Th fx = ax + bx + c 

f x ≥ f ( − 

__ 

 

a) b [ f ( − 

__ 

 

a) b ∞ ) 

Thaf x≤f ( − 

 

 

a) b ( −∞f ( − 

 

 

a) b 

] 

Thf x=ax−h +ka 

f x≥kaf x≤k

350 


How To… 

n, fi 

1. 

2. aaa 

 

3. k 

4. fx≥kk∞ 

f x≤k−∞k 

Example 4 

Finding the Domain and Range of a Quadratic Function 

f x=−x +x − 

Solution 

a 

y fix 

Thf h 

Thf x≤ 

( −∞ 

] 

f( 

h =− b 

a 

 

=− 

− 

= 

 

 

) +( ) − 

= 

 

) =− ( 

Try It #3 

f x=( x− 

) + 

Determining the Maximum and Minimum Values of Quadratic Functions 

Th 

Figure 9 

 

 

y 

f x = x − + 

– 

 

 

y 

gx = −x + + 

– 

– 

– 

 

– 

– 

– 

 

 

 

 

x 

 

x = 

 

Figure 9 

– 

– 

– 

 

– 

– 

– 

 

 

 

x 

 

x = − 

Thfi


Example 5 

Finding the Maximum Value of a Quadratic Function 

 

 

a. 

L 

b. d a 


W 

 

W 

L 

 

Figure 10 

a. L+W+L=L+W= 

WL 

W =−L 

 

 

A =LW=L−L 

AL=L −L 

LTh 

 

AL=−L +L 

b. 

 

 

aa=−b=c= 

o fi 

h =− b k =A 

a 

 

h =− =− 

− 

= = 

ThL = 

 

 

Analysis This problem also could be solved by graphing the quadratic function. We can see where the maximum area 

occurs on a graph of the quadratic function in Figure 11. 

Area (A) 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 

 

Length (L) 

Figure 11 

 

x

352 


How To… 

o fi 

1. 

2. 

3. y 

Example 6 

Finding Maximum Revenue 

ThffTh 

 

 

 

 

Solution 

p 

Q=pQ 

fi 

p =Q = 

p =Q =fi 

. Th 

m = − _____________ 

 

− 

=_______ 

− 

=− 

Th 

y 

Q =−p +b Q =p = 

=−+b b 

b = 

ThQ =−p + 

 

 

=pQ 

 

=p−p + 

 

=−p +p 

fi 

n fi 

_________ 

h =− − 

= 

Thfi 

 

 

=− + 

= 

Analysis This could also be solved by graphing the quadratic as in Figure 12. We can see the maximum revenue on a 

graph of the quadratic function.


Revenue ($1,000) 

y 

 

 

 

 

 

 

 

 

 

Price (p) 

Figure 12 

x 

Finding the x- and y-Intercepts of a Quadratic Function 

fi 

fiyfi 

xFigure 13x 

 

y 

y 

y 

 

 

 

 

 

– – – 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – 

– 

– 

– 

 

 

 

 

 

x 

x 

x 

Figure 13 Number of x-intercepts of a parabola 

x 

How To… 

f x, fiyx 

1. f o fiy 

2. f x=o fix 

Example 7 

Finding the y- and x-Intercepts of a Parabola 

yxfx=x +x − 

Solution e fiyf 

f= +− 

=− 

y− 

xe fifx= 

 

=x +x − 

 

 

=x −x + 

=x − =x + 

x =__ 

x=− 

 

x( __ 

) −

354 


y 

f x = x 

Analysis By graphing the function, we can confirm that the graph crosses 

+ x − 

– 

the y-axis at −. We can also confirm that the graph crosses the xaxis 

 

) 

at ( __ 

) and −. See Figure 14. x 

– – – 

– 

) 

– 

– 

– 

– 

– 

Figure 14 

Rewriting Quadratics in Standard Form 

Example 7 

y fi 

How To… 

n, fix 

1. abh =− b __ 

a 

2. x =ho fik 

3. hk 

4. o fix 

Example 8 

Finding the x-Intercepts of a Parabola 

xf x=x +x − 

Solution 

=x +x − 

fi 

 

f x=ax−h +k 

a =. Thhk 

h =− b __ 

a 

k =f − 

 

=− ___ 

 

=− +−− 

=− 

=− 

 

fx=x + − 

 

 

=x + − 

 

=x + 

 

=x + 

x +=±√ — 

x =−±√ — 

Thx−−√ — −+√ —


 

x Figure 15 

 

 

y 

− 

– – – 

– 

– 

– 

 

x 

Figure 15 

Analysis We could have achieved the same results using the quadratic formula. Identify a = , b = , and c = −. 

 

 

 

 

 

b 

x = −b±√— −ac 

 

a 

 

x = −±√— −− 

 

 

x = −±√— 

 

x = 

−±√— 

x =−±√ — 

So the x-intercepts occur at−−√ — and−+√ — 

Try It #4 

Try Itgx=+x −x y 

x 

Example 9 

Applying the Vertex and x-Intercepts of a Parabola 

Th 

Ht=−t +t + 

a. 

b. 

c. 

Solution 

a. Th 

 

h =− 

− 

= 

 

= 

 

= 

Thft

356 


b. y 

k =H ( − b 

a) 

=H 

=− ++ 

= 

Th 

Ht= 

 

t = −±√— −− 

 

− 

 

= −±√— − 

 

 

 

t = −−√— 

 

− ≈t = 

 

−+√— − ≈− 

Th 

ftFigure 16. 

H 

 

 

 

Ht = −t + t + 

 

 

 

 

 

Figure 16. 

t 

Try It #5 

 

 

 

Th 

Ht=−t +t + 

a. 

b. 

c. 

 

Graphing Quadratic Functions in General Form (http://openstaxcollege.org/l/graphquadgen) 

Graphing Quadratic Functions in Standard Form (http://openstaxcollege.org/l/graphquadstan) 

Quadratic Function Review (http://openstaxcollege.org/l/quadfuncrev) 

Characteristics of a Quadratic Function (http://openstaxcollege.org/l/characterquad)



VERBAL 

1. 

 

3. a≠ 

 

2. 

 

4. 

 

5. 

 

ALGEBRAIC 

 

6. f x=x −x+ 7. gx=x +x− 8. f x=x −x 

9. f x=x +x− 10. hx=x +x− 11. kx=x −x− 

12. f x=x −x 13. f x=x −x− 

 

 

14. yx=x +x+ 15. fx=x −x+ 16. fx=−x +x+ 

17. fx=x +x− 18. ht=−t +t− 19. fx=__ 

x +x+ 

20. fx=− __ 

x −x+ 

 

21. fx=x − + 22. fx=−x + − 23. fx=x +x+ 

24. fx=x −x+ 25. kx=x −x− 

hkxyo fi 

 

26. hk=xy= 27. hk=−−xy=− 28. hk=xy= 

29. hk=xy= 30. hk=−xy= 31. hk=xy= 

32. hk=xy= 33. hk=xy= 

GRAPHICAL 

 

 

34. fx=x −x 35. fx=x −x− 36. fx=x −x− 

37. fx=x −x+ 38. fx=−x +x− 39. fx=x −x−

358 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS 

 

40. 

y 41. 

y 42. 

y 

 

 

 

 

 

– – – – 

– 

 

– 

– 

– 

– 

x 

 

 

 

 

 

 

 

 

– – – – – – 

– 

 

– 

x 

 

 

 

 

 

 

 

– – – 

– 

 

– 

– 

x 

43. 

y 44. y 

45. 

y 

 

 

 

– – – – – – 

– 

 

– 

– 

– 

– 

– 

– 

x 

 

 

 

 

 

 

 

 

– – 

– 

 

– 

x 

 

 

 

 

 

– – – – – – – 

– 

 

– 

– 

– 

– 

x 

NUMERIC 

 

, fi 

46. x − − 47. x − − 48. 

y 

y 

x − − 

y − − 

49. x − − 50. 

y − − 

x − − 

y 

TECHNOLOGY 

o fi 

51. 

f x=x fx=x fx= 

x 

ffffi 

53. fx=x 

fx=x − fx − fx=x + 

e eff 

 

52. fx=x fx=x + 

fx=x fx=x +fx=x − 

e eff 

54. Th 

 

hx= − 

x +xx 

hx 

TRACE] 

 

 

55. hx=x −≤x≤ 

∣ x ∣ hxTRACE]


EXTENSIONS 

 

o fi 

56. − 57. − 

58. − 59. − 

 

 

60. fx=x 

y 

62. fx=x 

y 

64. fx=x 

y 


61. −fx=x 

y 

63. −fx=−x 

y 

65. −fx=x 

x− 

66. 

 

 

68. 

 

 

70. e diff 

 

 

72. 

 

ht=−t +t+ 

 

74. 

 

 

 

 

 

67. 

 

 

69. 

 

 

71. 

 

p=−xx 

 

R=xċp 

 

73. 

 

 

ht=−t +t+ 

 

75.

360 




Identify power functions. 

Identify end behavior of power functions. 

Identify polynomial functions. 

Identify the degree and leading coefficient of polynomial functions. 

5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 

Figure 1 (credit: Jason Bay, Flickr) 

Table1 

Year 

Bird Population 

Table 1 

ThPt=−t +t +Pt 

tft 

 

 

Identifying Power Functions 

ffpower 

functioncoefficient 

fiffi 

. Thr 

Ar=πr 

r 

Vr= __ 

πr 

ffiπ _ 

π 

r 

power function 

power function 

fx=kx p 

kpkcoefficient

SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 361 

Q & A… 

Is f (x) = 2 x a power function? 

fiTh 

. Th 

Example 1 

Identifying Power Functions 

 

f x= 

 

f x=x 

 

f x=x 

 

f x=x 

 

f x= __ 

 

x 

 

f x=__ 

 

x 

 

f x=√ — x 

f x= √ 

x 

Solution 

Thf x=x f x=x 

 

Thf x=x f x=x 

 

f x=x − f x=x − 

Th 

f x=x f x=x 

Try It #1 

 

fx=x ċx gx=−x +x hx= _ x − 

x + 

Identifying End Behavior of Power Functions 

Figure 2f x=x gx=x hx=x 

 

fl 

f x = x 

 

y 

gx = x 

 

hx = x 

 

 

– – 

x 

Figure 2 Even-power functions

362 


fi∞ 

fi−∞fixfi 

x →∞x 

 

x 

fif x 

x →±∞ f x→∞ 

Figure 3f x=x gx=x hx=x 

 

s fl 

 

y 

gx = x 

 

f x = x x 

hx = x 

– – 

– 

– 

Figure 3 Odd-power functions 

Thf x=x n Figure 

2f x=x n nyFigure 3 

f x=x n n 

xfif xx 

fif x 

x →−∞ f x→−∞ x →∞ f x→∞ 

Thx→−∞x→∞ 

end behavior 

Figure 4 f x=kx n n 

 

Even power 

y 

Odd power 

y 

 

k > 

x 

x 

x→−∞ f x→∞ 

x →∞ f x→∞ 

y 

x→−∞ f x→∞ 

x →∞ f x→−∞ 

y 

 

k < 

x 

x 

x→−∞ f x→∞ 

x →∞ f x→−∞ 

Figure 4 

x→−∞ f x→−∞ 

x →∞ f x→−∞


How To… 

f x=kx n n 

1. 

2. 

3. Figure 4 

Example 2 

Identifying the End Behavior of a Power Function 

f x=x 

Solution Thffix 

fif xx →∞f x→∞x 

fix →−∞f x→∞ 

Figure 5 

Example 3 

y 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

 

Figure 5 

Identifying the End Behavior of a Power Function 

f x=−x 

Solution Thffi− 

flxf x=x Figure 6xfi 

xfi 

 

x →−∞ f x→∞ 

x →∞ f x→−∞ 

y 

 

 

 

 

x 

 

 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 6

364 


Analysis We can check our work by using the table feature on a graphing utility. 

x 

− 

− 

 

 

 

f x 

 

 

 

− 

− 

Table 2 

We can see from Table 2 that, when we substitute very small values for x, the output is very large, and when we substitute 

very large values for x, the output is very small (meaning that it is a very large negative value). 

Try It #2 

fx=−x 

Identifying Polynomial Functions 

Th 

8 

Thrw 

. Th 

rw=+w 

A 

Ar=πr 

 

Aw=Arw 

=A+w 

=π+w 

 

Aw=π +πw+πw 

Thpolynomial function 

fiffi 

 

polynomial functions 

npolynomial function 

f x=a n 

x n ++ a 

x +a 

x +a 

Tha i 

ffia n 

 

=a i 

x i term of a polynomial function 

Example 4 

Identifying Polynomial Functions 

 

fx=x ċx + gx=−xx − hx=√ — x + 

Solution Thfi 

f x=a n 

x n ++a 

x +a 

x +a 

ffi


fxf x=x + 

gxgx=−x +x 

hx 

Identifying the Degree and Leading Coefficient of a Polynomial Function 

fi 

 

Thdegree 

e fi 

Thleading term 

. Thleading coefficientffi 

terminology of polynomial functions 

ft 

ffi 

 

fx=a n 

x n ++a 

x +a 

x +a 

 

 

How To… 

ffi 

1. x 

2. xo fi 

3. ffi 

Example 5 

Identifying the Degree and Leading Coefficient of a Polynomial Function 

ffi 

fx=+x −x 

gt=t −t +t 

hp=p −p − 

Solution f xxTh 

−x . Thffiffi− 

gttTh 

t . Thffiffi 

hppTh 

−p ffiffi− 

Try It #3 

 

fx=x −x +x−

366 


Identifying End Behavior of Polynomial Functions 

 

 

fix 

 

Table3 

Polynomial Function Leading Term Graph of Polynomial Function 

fx=x +x −x − x 

 

y 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

y 

 

 

 

 

 

x 

fx=−x −x +x +x −x 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

y 

fx=x −x +x + x 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

y 

fx=−x +x +x + −x 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Table 3


Example 6 Identifying End Behavior and Degree of a Polynomial Function 

Figure 7 

y 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 7 

Solution xf xx 

f x 

x →−∞ f x→−∞ 

x →∞ f x→∞ 

xfifix 

fifi 

fl 

ffi 

Try It #4 

 

Figure 8 

y 

 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 8 

Example 7 

Identifying End Behavior and Degree of a Polynomial Function 

f x=−x x−x+ 

 

Solution f x 

f x=−x x−x+ 

=−x x +x − 

=−x −x +x 

Thf x=−x −x +x Th−x 

Thffi− 

x →−∞ f x→−∞ 

x →∞ f x→−∞

368 


Try It #5 

fx=x−x+x− 

 

Identifying Local Behavior of Polynomial Functions 

 

turning point 

 

y 

Th 

y 

a 

Thx 

xFigure 9 

y 

 

x 

x 

 

 

←y 

Figure 9 

intercepts and turning points of polynomial functions 

turning point 

ThyTh 

x 

How To… 

 

1. yx =d fi 

2. x 

Example 8 

Determining the Intercepts of a Polynomial Function 

f x=x−x+x− 

yx 

Solution yx 

f =−+− 

=−− 

= 

y


x 

 

=x−x+x− 

x −= x += x −= 

x = x =− x = 

x 

Figure 10 

y 

y 

 

 

 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

 

 

x 

 

x 

− 

Example 9 

Figure 10 

Determining the Intercepts of a Polynomial Function with Factoring 

f x=x −x −yx 

Solution y 

f = − − 

=− 

y− 

x 

 

f x=x −x − 

=x −x + 

=x−x+x + 

 

=x−x+x + 

x −= x += x += 

x = x =− 

x− 

Figure 11 

f x=f −x 

y 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

 

Figure 11 

 

 

x 

− 

x 

 

 

y 

− 

Try It #6 

fx=x −x −xyx

370 


Comparing Smooth and Continuous Graphs 

Thx 

n th nn 

xThnn −Th 

 

continuous functionft 

smooth curveTh 

. Th 

intercepts and turning points of polynomials 

nnxn − 

Example 10 

Determining the Number of Intercepts and Turning Points of a Polynomial 

fi 

xf x=−x +x −x +x 

Solution 

Try It #7 

Thx 

x 

fx=−x −x +x +x 

Example 11 

Drawing Conclusions about a Polynomial Function from the Graph 

Figure 12 

 

y 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 12 

Solution ThFigure 13 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

x 

x 

 

 

Figure 13 

Thx


Try It #8 

Figure 14 

 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 14 

Example 12 Drawing Conclusions about a Polynomial Function from the Factors 

f x=−xx+x− 

Solution yf 

f =−+− 

= 

y 

x 

 

=−xx+x− 

x = x += x −= 

x = x =− x = 

x− 

Th 

Try It #9 

fx=x−x+x− 

 

Find Key Information About a Given Polynomial Function (http://openstaxcollege.org/l/keyinfopoly) 

End Behavior of a Polynomial Function (http://openstaxcollege.org/l/endbehavior) 

Turning Points and x-intercepts of Polynomial Functions (http://openstaxcollege.org/l/turningpoints) 

Least Possible Degree of a Polynomial Function (http://openstaxcollege.org/l/leastposdegree)



VERBAL 

1. e diff 

 

3. 

 

 

5. 

 

x→−∞fx→−∞x→∞ 

fx→−∞ 

2. 

 

 

4. 

 

 

ALGEBRAIC 

 

6. fx=x 7. fx=x 8. fx=x−x 

9. fx= 

_____ x 

x − 10. fx=xx+x− 11. fx= x+ 

, fiffi 

12. −x 13. −x 14. −x −x +x− 

15. x−x x+ 16. x x− 

 

17. fx=x 18. fx=x 19. fx=−x 

20. fx=−x 21. fx=−x −x +x− 22. fx=x +x− 

23. fx=x x −x+ 24. fx=−x 

, fi 

25. ft=t−t+t− 26. gn=−n−n+ 27. fx=x − 

28. fx=x + 29. fx=xx −x− 30. fx=x+x − 

GRAPHICAL 

 

31. 

y 

32. 

y 

33. 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x


34. 

 

x 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

35. 

 

x 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

36. 

 

x 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

37. 

 

x 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

 

38. 

 

x 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

 

 

 

39. 

x 

y 

40. 

x 

y 

41. 

x 

y 

42. 

x 

y 

43. 

x 

y 

44. 

x 

y 

45. 

x 

y


NUMERIC 

fi 

46. fx=−x 47. fx=x −x 48. fx=x −x 

49. fx=x−x−−x 50. fx=__ 

x 

−x 

TECHNOLOGY 

 

 

51. fx=x x− 52. fx=xx−x+ 53. fx=x−x−x 

54. fx=x−x−x 55. fx=x −x 56. fx=x − 

57. fx=x − 58. fx=−x +x +x 59. fx=x −x −x 

60. fx=x −x 

EXTENSIONS 

 

ffi−. Th 

61. y−4). Thx− 

x→−∞ 

fx→∞x→∞fx→∞ 

63. ys (0, 0). Thx 

x→−∞ 

fx→−∞x→∞fx→∞ 

65. ys (0, 1). Thx 

x→−∞fx→∞ 

x→∞fx→∞ 

62. ys (0, 9). Thx− 

x→−∞ 

fx→−∞x→∞fx→−∞ 

64. ys (0, 1). Thx 

x→−∞fx→∞ 

x→∞fx→−∞ 


 

 

66. . Th 

 

d 

 

68. 

x 

 

x 

70. 

. Th 

 

x 

67. et. Th 

 

m 

 

69. 

x 

 

 

x

SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 375 



Recognize characteristics of graphs of polynomial functions. 

Use factoring to find zeros of polynomial functions. 

Identify zeros and their multiplicities. 

Determine end behavior. 

Understand the relationship between degree and turning points. 

Graph polynomial functions. 

Use the Intermediate Value Theorem. 

5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 

Thr a fiTable 1 

Year 

Revenues 

Th 

Table 1 

Rt=−t +t −t +t − 

Rtt = 

 

Th 

 

 

Recognizing Characteristics of Graphs of Polynomial Functions 

 

 

Figure 1 

 

y 

y 

f 

x 

x 

f 

Figure 1

376 


Example 1 

Recognizing Polynomial Functions 

Figure 2 

y 

y 

g 

x 

x 

f 

y 

y 

x 

x 

h 

k 

Figure 2 

Solution Thfh. Th 

ThgkThg 

Thk 

Q & A… 

Do all polynomial functions have as their domain all real numbers? 

 

Using Factoring to Find Zeros of Polynomial Functions 

fxf x=f 

 

fixxfi 

 

 

 

 

1. Th 

2. Th 

3. 

How To… 

f, fix 

1. f x= 

2. 

a. 

b. 

3. o fix


Example 2 

Finding the x-Intercepts of a Polynomial Function by Factoring 

xf x=x −x +x 

Solution o fif x= 

x −x +x = 

x x −x += 

x x −x −= 

x −= x −= 

x = x = x = 

x = x =±x =±√ — 

Thfix−( √ — ) ( −√ — ) Figure 3 

y 

 

y 

f x = x – x + x 

 

– √ 

√ 

x 

– 

 

– 

– 

– 

Example 3 

Figure 3 

Finding the x-Intercepts of a Polynomial Function by Factoring 

xf x=x −x −x + 

Solution f x= 

x −x −x += 

x x−−x−= 

x −x−= e diff 

x+x−x−= 

x+= x −= x −= 

x=− x = x = 

Thx−Figure 4 

y 

 

 

– 

– 

– 

 

 

 

 

f x = x − x − x + 

x 

– 

– 

– 

Figure 4

378 


Example 4 

Finding the y- and x-Intercepts of a Polynomial in Factored Form 

yxgx=x− x + 

Solution yg 

g=− + 

= 

y 

xgx= 

x− x += 

x− = x += 

x( − 

) 

x −= x =− __ 

 

x = 

Analysis We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial 

as shown in Figure 5. 

y 

gx = (x − ) (x + ) 

 

 

 

 

 

 

– 

 

– – – – 

– 

x 

Figure 5 

Example 5 

Finding the x-Intercepts of a Polynomial Function Using a Graph 

xhx=x +x +x − 

Solution Th 

fi 

ffi 

Figure 6xx =−− 

 

y hx = x + x + x − 

– – 

x 

– 

– 

– 

– 

Figure 6


xg t 

h−=h−=h= 

hx=x +x +x − 

h−=− +− +−−=−+−−= 

h−=− +− +−−=−+−−= 

h= + +−=++−= 

x 

 

hx=x +x +x − 

=x+x+x− 

Try It #1 

yxfx=x −x +x 

Identifying Zeros and Their Multiplicities 

ffx 

ff 

 

fx=x+x− x+ 

Figure 7x 

 

 

 

 

– – 

– 

– 

– 

– 

y 

f x = (x + )(x − ) (x + ) 

x 

Figure 7 Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero. 

xx =−x+=Thx 

x =−Th 

 

xx =x− =Th 

The 

 

x− =x−x− 

Thx−Th 

multiplicityThx = 

x−

380 


xx =−x+ =Th 

flfiThe 

f x=x 

 

touchx 

crossxFigure 8 

 

y 

y 

y 

p = p = p = 

x 

x 

x 

 

 

Figure 8 

 

 

r flx 

 

r flx 

graphical behavior of polynomials at x-intercepts 

x−h p x h 

px =h multiplicityp 

Thx. Th 

x 

Th 

How To… 

n 

1. x 

2. x 

3. x 

4. Thn 

Example 6 

Identifying Zeros and Their Multiplicities 

Figure 9


y 

 

 

 

– 

– 

– 

 

 

 

x 

– 

– 

– 

Figure 9 

Solution Th. Th 

ftfix =−Thx 

. Th− 

Thx =−. Th. Th 

Thx =Thx 

 

Try It #2 

Figure 10 

y 

 

 

 

– 

– 

– 

 

 

 

x 

– 

– 

– 

Figure 10 

Determining End Behavior 

 

fx=a n 

x n +a n− 

x n − ++ a 

x +a 

xx 

ThTh 

−− 

end behavior 

a n 

x n x 

f xx 

f xxf x 

Figure 11

382 


Even Degree 

ffia n 

> 

y 

Odd Degree 

ffia n 

> 

y 

x 

x 

 

x→ ∞f x→ ∞ 

x→ −∞f x→ ∞ 

ffia n 

< 

y 

 

x→ ∞f x→ ∞ 

x→ −∞f x→ ∞ 

ffia n 

< 

y 

x 

x 

 

x→ ∞f x→ −∞ 

x→ −∞f x→ −∞ 

 

x→ ∞f x→ −∞ 

x→ −∞f x→ ∞ 

Figure 11 

Understanding the Relationship Between Degree and Turning Points 

 

 

f x=x −x −x +xFigure 12Th 

 

y 

 

 

 

 

x 

 

Figure 12 

Thf Th


interpreting turning points 

 

 

nn − 

Example 7 

Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function 

 

a. f x=−x +x −x + b. f x=−x− +x 

Solution 

a. f x=−x +x −x + 

f x=x −x −x + 

 

Th−= 

b. f x=−x− +x 

 

f x=−x− +x 

a n 

=−x x −x 

Th. Th 

Th−= 

Graphing Polynomial Functions 

 

 

How To… 

 

1. 

2. y 

f −x=f xf −x=−fx 

3. x 

4. 

5. 

6. 

7. 

Example 8 

Sketching the Graph of a Polynomial Function 

f x=−x+ x− 

Solution Thxx =−Th 

xx = 

 

yf 

y 

f =−+ − 

=−ċċ− 

=

384 


−x 

flfi 

fiFigure 13 

y 

x 

Figure 13 

 

x →−∞f x→∞ 

x 

f −x=−−x+ −x−f x 

− x 

yyFigure 14 

y 

 

− 

x 

Figure 14 

ft 

Figure 15 

y 

 

− 

 

x 

Figure 15 

x →∞f x→−∞


Figure 16 

Figure 15 

 

y 

f x = −(x + ) (x − ) 

 

 

– 

– 

– 

 

x 

– 

– 

– 

Figure 16 The complete graph of the polynomial function f (x ) = −2(x + 3) 2 (x − 5) 

Try It #3 

fx= 

xx− x+ 

Using the Intermediate Value Theorem 

 

xfif 

ThIntermediate Value Theoremabfa

386 


Example 9 

Using the Intermediate Value Theorem 

f x=x −x +x +x =x = 

Solution f xx =Table 2 

x 

f x − 

Table 2 

x =f f Th 

 

x =x = 

Analysis We can also see on the graph of the function in Figure 18 that there are two real zeros between x = and 

x = . 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

f = 

 

 

 

 

 

f =− 

f = 

Figure 18 

Try It #4 

fx=x −x −x x=x= 

Writing Formulas for Polynomial Functions 

fi 

x 

x 

factored form of polynomials 

e px =x 

x 

x n 

n 

f x=ax−x p x−x p x−x n p n pi 

a 

x 

How To… 

 

1. xo fi 

2. x 

3. s s 

4. y


Example 10 Writing a Formula for a Polynomial Function from the Graph 

Figure 19 

y 

 

 

 

– – – 

– 

– 

– 

 

x 

Figure 19 

Solution Thxx =−Thy, −x =−x = 

 

x =e 

 

f x=ax+x− x− 

y−a 

f =a+− − 

−=a+− − 

−=−a 

a =__ 

 

 

Thf x= 

x+x− x− 

Try It #5 

Figure 20 

 

 

 

y 

– 

– 

– 

 

 

 

x 

– 

– 

– 

Figure 20 

Using Local and Global Extrema 

fifi 

fi 

fi 

 

 

global maximumglobal minimum 

Th

388 


local and global extrema 

x =a 

x =a 

af a≥f xxx =aa 

f a≤f xxx =a 

global maximumglobal minimum 

s a gaf a≥f xxaf a≤f x 

x 

ffFigure 21 

– –– 

– – – – 

– 

– 

– 

– 

– 

y 

 

 

 

 

 

 

 

 

Figure 21 

 

 

 

 

 

 

 

x 

Q & A… 

Do all polynomial functions have a global minimum or maximum? 

f x=x 

 

Example 11 

Using Local Extrema to Solve Applications 

4 

 


w 

w 

w 

w 

w 

w 

w 

w 

w 

Figure 22 

ft−w−w 

w. Th 

Vw=−w−ww 

=w −w +w


w−w−w 

Th 

wTh 



VERBAL 

1. e diffx 

f 

3. ue Th 

 

5. x 

 

 

2. nn 

 

 

4. 

 

ALGEBRAIC 

, fixt 

6. Ct=t−t+t− 7. Ct=t+t−t+ 8. Ct=tt− t+ 

9. Ct=tt−t+ 10. Ct=t −t +t 11. Ct=t +t −t 

12. fx=x −x 13. fx=x +x −x 14. fx=x +x −x 

15. fx=x +x −x− 16. fx=x +x −x− 17. fx=x −x −x+ 

18. fx=x −x − 19. fx=x +x − 20. fx=x −x −x+ 

21. fx=x −x −x 22. fx=x −x −x 23. fx=x −x +x 

Thfi 

 

24. fx=x −xx=−x=− 25. fx=x −xx=x= 

26. fx=x −xx=x= 27. fx=−x +x=x= 

28. fx=−x −xx=−x= 29. fx=x −x+x=x= 

, fi 

30. fx=x+ x− 31. fx=x x+ x− 

32. fx=x x− x+ 33. fx=x x +x+ 

34. fx=x+ x −x+ 35. fx=x+ x −x+ 

36. fx=xx −x+x +x+ 37. fx=x −x −x 

38. fx=x +x +x 39. fx=x −x +x 

40. fx=x x −x +x 41. fx=x x −x +x 

GRAPHICAL 

xy 

42. fx=x+ x− 43. gx=x+x− 44. hx=x− x+ 

45. kx=x− x− 46. mx=−xx−x+ 47. nx=−xx+x−


 

48. 

fx 

49. 

fx 

50. 

fx 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

51. 

fx 

52. 

fx 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

 

53. 

y 

54. 

y 

55. 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

56. 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

x 

– 

– 

– 

– 

 

57. x=−x=x= 

y− 

59. x=x= 

x=−y 

 

58. x=−x=−x= 

y 

60. x= 

x=x=−y 

−


61. x= 

x = 

63. x=−x=−x= 

y 

65. x=_ 

 

x=x=−y 

 

62. x=x=x= 

y− 

64. x=−x= 

x=−y 

66. x=−x= 

 

TECHNOLOGY 

 

 

67. fx=x −x− 68. fx=x −x− 69. fx=x +x 

70. fx=−x +x− 71. fx=x −x + 

EXTENSIONS 

 

72. 

74. 

– 

 

– 

– 

– 

f x 

 

 

– 

– 

– 

 

 

 

 

 

 

 

fx 

 

 

– – – – 

– – 

x 

x 

73. 

 

 

 

 

 

 

fx 

– – – 

– 

– 

– 

– 

– 

– 

– 

 

 

x 

– 

– 

– 


 

75. 

xx 

 

 

x 

77. x+ 

x+ 

 

x 

79. x+ 

 

n. Th 

V=_ 

 

πr hrh 

76. 

xx 

 

x 

78. x+

SECTION 5.4 DIVIDING POLYNOMIALS 393 



Use long division to divide polynomials. 

Use synthetic division to divide polynomials. 

5.4 DIVIDING POLYNOMIALS 

Figure 1 Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr) 

Th 

[] y fi 

V =l ċw ċh 

=ċċ 

= 

fi 

 

h = V ____ 

l ċw 

_______ 

= ċ 

= 

fifi 

 

x −x −x +xTh 

xx −fi 

 

Using Long Division to Divide Polynomials 

 

 

 

Long Division 

×=−= 

 

− ×=−= 

 

R 

−

394 


Th 

 

 

=ċ+ 

 

=ċ+ 

=+ 

= 

Division Algorithmft 

 

Th 

Th 

x −x +x +x + 

 

 

 

 

 

 

 

 

 

 

 

 

 

x+x −x +x + 

x 

x+x −x +x + 

x 

x+x −x +x + 

−x +x 

−x +x 

x −x 

x+x −x +x + 

−x +x 

−x +x 

−−x +x 

x + 

x −x + 

x+x −x +x + 

−x +x 

−x +x 

−−x +x 

x + 

−x + 

− 

 

x xx 

x +x 

 

 

−x x−x 

x +−x 

 

xx 

x + 

 

x −x +x + 

 

x + =x −x +− 

x + 

 

 

 

x −x +x + 

 

x + =x+x −x +− 

 

x – x + x + = x+ x – x + + –


the Division Algorithm 

Division Algorithmf xdx 

dxf xqxrx 

fx=dxqx+rx 

qxrx. Th 

dx 

rx=dxf x. Thdxqxf x 

How To… 

 

1. 

2. fi 

 

3. 

4. 

5. 

6. 

7. 

Example 1 

x +x −x + 

Solution 

 

 

 

 

 

 

 

Using Long Division to Divide a Second-Degree Polynomial 

x+x +x − 

x 

x+x +x − 

x 

x+x +x − 

−x +x 

−x − 

x − 

x+x +x − 

−x +x 

−x − 

−−x − 

 

Thx −. Th 

 

x xx 

x +x 

 

−xx− 

x +− 

 

 

 

 

__________ 

x +x − 

x + =x − 

x +x −=x+x − 

Analysis This division problem had a remainder of . This tells us that the dividend is divided evenly by the divisor, 

and that the divisor is a factor of the dividend.

396 


Example 2 

Using Long Division to Divide a Third-Degree Polynomial 

x +x −x +x − 

Solution 

 

 

 

 

 

 

 

x +x − 

x −)x +x −x + 

−x −x 

x −x 

−x −x 

−x + 

−−x + 

 

Th 

x xx 

x − x 

x xx 

x −x 

−xx− 

x −− 

. Th 

 

__________________ 

x +x −x + 

x − =x +x 

_____ 

−+ 

x − 

Analysis We can check our work by using the Division Algorithm to rewrite the solution. Then multiply. 

 

x −x +x −+=x +x −x + 

Notice, as we write our result, 

the dividend isx +x −x + 

the divisor isx − 

the quotient isx +x − 

the remainder is 

Try It #1 

x −x +x−x+ 

Using Synthetic Division to Divide Polynomials 

Synthetic division 

ffi 

 

 

x −x +x +x + 

The fi 

x +x + 

x+x −x +x + 

−x +x 

−x +x 

−−x −x 

x + 

−x + 

− 

Thffi


− 

− − 

− 

− 

− 

fi 

fi 

−Th 

ffi 

− − 

− − 

− − 

ftTh 

ffi 

x −x +−. ThExample 3 

synthetic division 

x −kk 

synthetic divisione cffi 

How To… 

 

1. k 

2. ffi 

3. ffi 

4. ffik 

5. 

6. k 

7. 

8. Th 

 

Example 3 

Using Synthetic Division to Divide a Second-Degree Polynomial 

x −x −x − 

Solution kffi 

− − 

ffiffik 

− − 

 

 

k 

. Th 

− − 

 

 

Thx +. Thx −

398 


Analysis Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the 

remainder. 

x−x ++=x −x − 

Example 4 

Using Synthetic Division to Divide a Third-Degree Polynomial 

x +x −x −x + 

Solution Thx +k =−− 

 

− − − 

− − 

− 

Thx +x −. Th. Thx +x +x −x − 

Analysis 

The graph of the polynomial function f x = x + x − x − in Figure 2 shows a zero at x = k = −. 

This confirms that x + is a factor of 4x + x − x − . 

y 

− − 

– – – – 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 2 

Example 5 

Using Synthetic Division to Divide a Fourth-Degree Polynomial 

−x +x +x −x − 

Solution xffi 

Th−x +x +x 

_ 

++ 

x − 

Try It #2 

− − 

− 

− 

x +x −x+x+ 

Using Polynomial Division to Solve Application Problems


Example 6 

Using Polynomial Division in an Application Problem 

Thx −x −x +xTh 

xx −t 

Solution Th 

Figure 3 

 

x 

x− 

Figure 3 

 

V =l ċw ċh 

 

x −x −x +x =x ċx−ċh 

h, fix 

 

 

x ċx−ċh 

x 

= x −x −x +x 

x 

 

x−h=x −x −x + 

h 

h = x −x −x + 

x − 

 

− − 

− 

− 

Thx +x −. Thx +x − 

Try It #3 

Thx +x −x+Thx+ 

 

 

Dividing a Trinomial by a Binomial Using Long Division (http://openstaxcollege.org/l/dividetribild) 

Dividing a Polynomial by a Binomial Using Long Division (http://openstaxcollege.org/l/dividepolybild) 

Ex 2: Dividing a Polynomial by a Binomial Using Synthetic Division (http://openstaxcollege.org/l/dividepolybisd2) 

Ex 4: Dividing a Polynomial by a Binomial Using Synthetic Division (http://openstaxcollege.org/l/dividepolybisd4)



VERBAL 

1. 

 

2. n 

 

ALGEBRAIC 

 

3. x +x−÷x− 4. x −x−÷x− 5. x +x+÷x+ 

6. x −x+÷x+ 7. x −x−÷x+ 8. −x −÷x+ 

9. x −x+÷x+ 10. x −÷x− 11. x −x+÷x+ 

12. x −x +x−÷x− 13. x +x −x+÷x+ 

o fi 

14. x −x +x−÷x+ 15. x −x −x+÷x− 

16. x −x −x−÷x+ 17. x −x −x−÷x+ 

18. x −x +x+÷x− 19. x −x +x−÷x+ 

20. −x +x −÷x− 21. x +x −x−÷x− 

22. x −x +x+÷x+ 23. x −x +÷x+ 

24. x −x+÷x+ 25. x −x +x−÷x− 

26. x −x +x−÷x− 27. x −x+÷x− 

28. x −x +x+÷x+ 29. x +x −x −x+÷x+ 

30. x −x +÷x− 31. x +x −x +x+÷x+ 

32. x −x +x −x+÷x− 33. x −x +x −x+÷x− 

34. x +x −x −x+÷x+ 35. x −x +x −x+÷x− 

36. x −x −x+÷x− 37. x +x −x +x+÷x+ 

e fi 

 

38. x−x −x −x+ 39. x−x −x −x+ 40. x+−x +x + 

41. x−x −x − 42. x− 

x −x +x− 

43. x+ 

x +x −x+ 

GRAPHICAL 

 

. Thffi 

44. x −x+ 

y 

45. x +x+ 

y 

46. x +x+ 

y 

 

 

 

 

 

 

 

 

 

– 

– 

– 

 

 

 

x 

– 

– 

– 

 

 

 

x 

– 

– 

– 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

–


47. x +x+ 

y 

 

 

 

48. x +x+ 

y 

 

 

 

– 

– 

– 

 

 

 

x 

– 

– 

– 

 

 

 

x 

– 

– 

– 

– 

– 

– 

o fi 

49. _______ 

x − x− 

50. + x _______ 

x+ 

52. ____________ 

−x −x 

− x+ 53. ______ x − 

x+ 

51. +x− x __________ 

x− 

TECHNOLOGY 

 

54. xk − 

x− k= 

k= 

56. x −k 

x−k k= 

k= 

x k 

58. 

x− k= 

k= 

EXTENSIONS 

55. xk + 

x+ k= 

k= 

x k 

57. 

x+ k= 

k= 

 

59. _____ x+ 

x−i 

62. _____ x + 

x+i 


60. _____ x + 

x−i 

63. _____ x + 

x−i 

61. _____ x+ 

x+i 

 

64. x+x +x− 65. x+x +x +x+ 

66. x−x −x +x −x− 

 

 

67. x +x −x− 

x+x− 

69. x +x +x−x− 

x+ 

68. x −x −x+ 

x−x− 

70. x +x −x− 

x+ 

 

 

71. πx −x −x−x+ 72. πx +x −x−x+ 

73. πx +x +x −x− 

x+

402 




Evaluate a polynomial using the Remainder Theorem. 

Use the Factor Theorem to solve a polynomial equation. 

Use the Rational Zero Theorem to find rational zeros. 

Find zeros of a polynomial function. 

Use the Linear Factorization Theorem to find polynomials with given zeros. 

Use Decartes' Rule of Signs. 

Solve real-world applications of polynomial equations. 

5.5 ZEROS OF POLYNOMIAL FUNCTIONS 

ffTh 

ThTh 

 

 

Th 

 

Evaluating a Polynomial Using the Remainder Theorem 

 

Remainder Theoremx −k 

kf k 

f xdx 

dxf xqxrx 

f x=dxqx+rx 

dxx −k 

fx=x−kqx+r 

x −krx =k 

f k=k−kqk+r 

=ċqk+r 

=r 

f kf xx −k 

the Remainder Theorem 

f xx −kf k 

How To… 

ff xx =kTh 

1. x −k 

2. Thf k

SECTION 5.5 ZEROS OF POLYNOMIAL FUNCTIONS 403 

Example 1 

Using the Remainder Theorem to Evaluate a Polynomial 

r Thf x=x −x −x +x −x = 

Solution fiTh 

x − 

Th. Thf = 

Analysis We can check our answer by evaluating f (). 

− − − 

 

 

 

f x = x − x −x + x − 

f = − − + − 

= 

Try It #1 

der Thfx=x −x −x +x +x=− 

Using the Factor Theorem to Solve a Polynomial Equation 

Factor Theorem 

 

fx=x−kqx+r 

krf k=f x=x−kqx+f x=x−kqx 

x −kf xkf xx −kf x 

x −kf xf x) =x−kqx+rTh 

k 

ThThn 

nThn 

 

the Factor Theorem 

Factor Theoremkf xx−kf x 

How To… 

r Th 

1. x−k 

2. fi 

3. x−k 

4. 

5. 

Example 2 Using the Factor Theorem to Solve a Polynomial Equation 

x+x −x −x +

404 


Solution x+ 

− − − 

− − 

− 

Thx+ 

 

x+x −x + 

 

x+x−x− 

r Thx −x −x +− 

Try It #2 

r Th fx=x +x −x−x− 

Using the Rational Zero Theorem to Find Rational Zeros 

Th 

fiThRational ZeroTheorem 

ffi 

 

x = 

x = 

Th 

 

 

x −__ 

=x − __ 

= 

x −=x −= 

f x=x −x − 

f x=x −x + 

 

 

f x=ċx −x +ċ 

 

ffi 

 

 

ffiThTh 

 

the Rational Zero Theorem 

Rational Zero Theoremf x=a n 

x n +a n− 

x n− ++a 

x +a 

 

ffif x p _ 

q 

pa 

q 

ffia n 

 

ffi


How To… 

f xo Tho fi 

1. ffi 

2. p _ 

q pqffi 

 

3. f ( p _ 

q) 

Example 3 

Listing All Possible Rational Zeros 

f x=x −x +x − 

Solution Thf x, − 

ffi 

Th−−p =±±± 

Thffiq =±± 

− 

 

p 

_ 

q =± __ 

± __ 

 

p _ 

q =± __ 

± __ 

 

p _ 

q =± __ 

± __ 

 

 

= 

= 

 

p _ 

q = __ 

 

=±±±± __ 

 

Example 4 

Using the Rational Zero Theorem to Find Rational Zeros 

o Tho fif x=x +x −x + 

Solution 

Tho Th p _ 

q 

f xpq 

p _ 

q = ___ 

 

ffi 

_ 

= 

Th±±±Th p _ 

q ±± _ 

Th 

 

xf x 

f −=− +− −−+= 

f = + −+= 

f ( − __ 

 

) = ( − __ 

 

f ( __ 

 

) = ( __ 

 

) +( − __ 

) −( − __ 

) += 

) +( __ 

) −( __ 

) +=− _ 

 

−− 

f 

xf x 

 

Try It #3 

o Th fx=x −x +x+

406 


Finding the Zeros of Polynomial Functions 

ThTh 

zeros 

How To… 

fo fi 

1. o Th 

2. 

 

3. 

 

4. 

 

Example 5 

Finding the Zeros of a Polynomial Function with Repeated Real Zeros 

f x=x −x − 

Solution Tho Th p _ 

q 

f xp−q 

 

p _ 

q = ___ 

 

ffi 

__ 

= − 

 

Th−±±±±Th p _ 

q ±± 

± 

 

 

fi 

− − 

 

 

x−. Th 

x−x +x + 

Thf x 

x−x + 

Thfi 

 

 

x += 

x =− __ 

 

Th− 

 

Analysis Look at the graph of the function f in Figure 1. Notice, at x = −, the graph bounces off the x-axis, indicating 

the even multiplicity for the zero − At x = , the graph crosses the x-axis, indicating the odd multiplicity 

for the zero x =. 

y 

 

 

 

 

–2.–––– 

– 

– 

– 

– 

– 

 

Figure 1 

 

 

x


Using the Fundamental Theorem of Algebra 

fi 

ThFundamental Theorem of Algebra 

. Th 

ff x=ThTh 

c 

Thf xx −c 

 

x −c 

 

Th 

c 

x −c 

 

ThTh 

f x 

the Fundamental Theorem of Algebra 

ThThf xn >f x 

 

f xn >a 

f xn 

fx=ax−c 

x−c 

x−c n 

 

c 

c 

c n 

Thf xn 

Q & A… 

Does every polynomial have at least one imaginary zero? 

 

 

Example 6 

Finding the Zeros of a Polynomial Function with Complex Zeros 

f x=x +x +x + 

Solution Tho Th p _ 

q 

f xpq 

 

p _ 

q = ___ 

 

ffi 

_ 

= 

Th±±Th p _ 

q , 

±±± 

fi 

− 

− 

− − 

 

x+−. Th 

x+x + 

o fi 

 

x += 

x =− __ 

 

Thf x−± _ 

i √— 

x =± 

√ − =±i _ √—

408 


Analysis Look at the graph of the function f in Figure 2. Notice that, at x = −, the graph crosses the x-axis, indicating an 

odd multiplicity () for the zero x = −. Also note the presence of the two turning points. This means that, since there is a 

3 rd degree polynomial, we are looking at the maximum number of turning points. So, the end behavior of increasing without 

bound to the right and decreasing without bound to the left will continue. Thus, all the x-intercepts for the function are shown. 

So either the multiplicity of x = − is and there are two complex solutions, which is what we found, or the multiplicity at 

x = − is three. Either way, our result is correct. 

 

 

y 

 

 

– 

– 

– 

 

 

 

x 

– 

– 

– 

Figure 2 

Try It #4 

fx=x +x −x+ 

Using the Linear Factorization Theorem to Find Polynomials with Given Zeros 

Th 

nnTh 

nThLinear Factorization Theorem 

x−cc 

fffia +bib ≠f xTh 

Thx −a+bif xfffix −a−bif xTh 

x −a−bix −a+bi 

ffi 

fffia +bi 

a −bif x. ThTh 

complex conjugate theorem 

Linear Factorization Theorem, 

x−cc 

fffia +bi 

a −bi 

How To… 

fcf cfTh 

o fi 

1. 

2. 

3. cf cffi 

4.

Example 7 


Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros 

ffi−if −= 

Solution x =iThx =−iTh 

x+x−x−ix+i 

 

f x=ax+x−x−ix+i 

f x=ax +x −x + 

f x=ax +x −x +x − 

o fiaf −=x =−f =f x 

 

=a− +− −− +−− 

 

=a− 

−=a 

 

f x=−x +x −x +x − 

 

f x=−x −x +x −x + 

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 

polynomial with real coefficients, then −i must also be a zero of the polynomial because −i is the complex conjugate of i. 

Q & A… 

If 2 + 3i were given as a zero of a polynomial with real coefficients, would 2 − 3i also need to be a zero? 

ffi 

 

Try It #5 

 

−if= 

Using Descartes’ Rule of Signs 

Th 

Descartes’ Rule of Signs 

f x 

 

f x=x +x +x +x− 

Th 

Thf −x 

f −x=−x +−x +−x +−x− 

f −x=+x −x + x −x − 

f −x. Thf x

410 


Descartes’ Rule of Signs 

Descartes’ Rule of Signs,f x=a n 

x n +a n− 

x n− ++a 

x +a 

 

ffi 

Thes f x 

 

Thf −x 

 

Example 8 Using Descartes’ Rule of Signs 

 

f x=−x −x +x −x − 

Solution 

f x=−x −x +x − x − 

Figure 3 

Thf −x 

 

f −x=−−x −−x +−x −−x− 

f −x=−x +x +x +x − 

f −x=−x +x +x +x− 

Figure 4 

 

ThTable 1 

Positive Real Zeros Negative Real Zeros Complex Zeros Total Zeros 

 

 

 

 

Table 1 

Analysis We can confirm the numbers of positive and 

negative real roots by examining a graph of the function. See 

Figure 5. We can see from the graph that the function has 

positive real roots and negative real roots. 

Try It #6 

x = − 

 

 

 

 

 

 

y 

– – – – – 

– 

– 

– 

fx = − x − x + x − x − 

x = − 

 

 

 

Figure 5 

 

fx=x −x +x −x+ 

 

Solving Real-World Applications 

 

 

 

 

x


Example 9 

Solving Polynomial Equations 

ffTh 

ThTh 

 

 

Solution ThV =lwh 

l =w + 

 

h = 

w 

V =w+w( __ 

w ) 

V =__ 

w +__ 

w 

 

 

=__ 

w +__ 

w V 

=w +w 

=w +w − 

ThTh 

±±±±±±±±±± 

 

x = 

x = 

− 

 

− 

− 

 

− 

x = 

− 

 

 

 

 

l=w +=+=h = __ 

w = __ 

= 

Th 

Try It #7 

s. Th 

 

 

 

Real Zeros, Factors, and Graphs of Polynomial Functions (http://openstaxcollege.org/l/realzeros) 

Complex Factorization Theorem (http://openstaxcollege.org/l/factortheorem) 

Find the Zeros of a Polynomial Function (http://openstaxcollege.org/l/findthezeros) 

Find the Zeros of a Polynomial Function 2 (http://openstaxcollege.org/l/findthezeros2) 

Find the Zeros of a Polynomial Function 3 (http://openstaxcollege.org/l/findthezeros3)



VERBAL 

1. der Th 2. o Th 

 

3. e diff 4. 

 

 

 

5. 

 

ALGEBRAIC 

r Tho fi 

6. x −x +÷x− 7. x −x +x−÷x+ 

8. x +x −x−÷x+ 9. −x +x+÷x− 

10. x −x +x −x +x−÷x+ 11. x −÷x− 

12. x +x −x+÷x− 13. x +x −x+÷x+ 

Thfi 

 

14. fx=x −x +x−x− 15. fx=x +x −x+x+ 

16. fx=x +x −x+x+ 17. fx=x +x +x+x+ 

18. fx=−x +x −x− 19. x +x +x+x+ 

20. x −x+x− 21. x +x −x−x+ 

o Tho fi 

22. x −x −x+= 23. x +x −x−= 24. x +x −x−= 

25. x +x −x−= 26. x −x −x+= 27. x −x −x−= 

28. x +x −x−= 29. x −x −x+= 30. x −x −x−= 

31. x −x +x−= 32. x −x +x+= 33. x −x −x +x+= 

34. x +x −x −x+= 35. x +x −x −x+= 36. x −x −x +x−= 

37. x +x −x −x−= 38. x −x+= 39. x +x +x +x+ 

, fi 

40. x +x +x+= 41. x −x +x−= 42. x +x +x+= 

43. x −x +x+= 44. x +x +x +x−= 45. x −x +x+= 

GRAPHICAL 

Thfi 

 

46. fx=x − 47. fx=x −x − 48. fx=x −x −x+ 

49. fx=x −x +x− 50. fx=x +x −x +x− 51. fx=x +x +x+


52. fx=x −x −x+ 53. fx=x −x −x +x+ 54. fx=x −x −x +x+ 

55. fx=x −x + 

NUMERIC 

 

56. fx=x +x −x+ 57. fx=x +x −x+ 58. fx=x +x −x+ 

59. fx=x −x +x+ 60. fx=x −x +x +x − 

TECHNOLOGY 

fi 

 

61. fx=x −x + 62. fx=x −x −x− 

63. fx=x −x −x+ 64. fx=x +x +x −x+ 

65. fx=x −x +x −x+ 

EXTENSIONS 

 

66. −f= 67. − 

f= 

68. − 

 

−f−=− 

70. −−−f−=− 

69. − 

 

−f−=− 


, fi 

71. Thh. Th 

h. Th 

 

73. Th 

t. Th 

 

75. Thh. Th 

t. Th 

 

72. Th 

s. Th 

74. Th 

h. Th 

 

, fi 

76. Tht. Th 

π 

78. Tht diff. Th 

π 

80. Th t. 

Th 

 

 

π π 

77. Th 

Thπ 

79. Tht diff 

Thπ

414 




Use arrow notation. 

Solve applied problems involving rational functions. 

Find the domains of rational functions. 

Identify vertical asymptotes. 

Identify horizontal asymptotes. 

Graph rational functions. 

5.6 RATIONAL FUNCTIONS 

xTh 

Cx=x −x +x 

x 

Thx 

f x=__________________ 

x −x + 

x 

 

 

 

 

Using Arrow Notation 

 

Figure 1 

Graphs of Toolkit Functions 

y 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

f x = 

 

x 

 

x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

f x = 

 

x 

 

 

 

 

 

x 

Figure 1 

f x=_ x 

 

1. xy=x →−∞ 

2. x= h zer 

 

3. xy=x→∞ 

arrow notationxf xTable 1

SECTION 5.6 RATIONAL FUNCTIONS 415 

Symbol 

x→a − 

x→a + 

x→∞ 

x→−∞ 

f x→∞ 

f x→−∞ 

fx→a 

Meaning 

xa xaa 

x x 

x x 

Th 

Th 

Tha 

Table 1 Arrow Notation 

Local Behavior of f (x) = 

__ 1 x 

f x= _ x 

 

fix = 

fi 

Table 2 

x − − − − 

f (x) = 1 __ 

x 

− − − − 

 

Table 2 

x → − f x→−∞ 

 

fiTable 3 

x 

f (x) = 1 __ 

x 

 

 

Figure 2 

Table 3 

x → + f x→∞ 

x → + 

f x → ∞ 

y 

x → −∞ 

f x → 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

x → ∞ 

f x → 

x → – 

f x → −∞ 

Figure 2

416 


Thvertical asymptote 

x =Figure 3 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

x = 

 

 

 

x 

Figure 3 

vertical asymptote 

vertical asymptotex =afi 

a 

x →af x→∞x →af x→−∞ 

End Behavior of f(x ) = 1 _ 

x 

xfixfi 

Figure 4 

x →∞f x→x →−∞f x→ 

 

 

 

 

 

– – – – – 

– 

x → −∞ 

– 

f x → – 

– 

– 

y 

x → + 

f x → ∞ 

x → ∞ 

f x → 

x 

 

x → – 

f x → −∞ 

Figure 4 

 

ffThhorizontal asymptote 

 

y =Figure 5 

y 

y = 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

x = 

 

 

 

x 

Figure 5


horizontal asymptote 

horizontal asymptotey =b 

 

x →∞x →−∞f x→b 

Example 1 

Using Arrow Notation 

Figure 6 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

Figure 6 

Solution x =fi 

x = 

x → − f x→−∞x → + f x→∞ 

 

y = 

x →∞f x→x →−∞f x→ 

Try It #1 

 

Example 2 

 

 

 

Using Transformations to Graph a Rational Function 

ft 

 

Solution ft 

 

fx=_____ 

 

x + + 

 

f x=______ 

x + 

x + 

ThftFigure 7 

x = − 

– – – – – – 

– 

y 

 

 

 

 

 

 

 

– 

– 

– 

 

Figure 7 

 

 

 

x 

y = 

x

418 


fix =−x =− 

x →− − f x→−∞x →− + f x→∞ 

 

y = 

x →±∞f x→ 

Analysis Notice that horizontal and vertical asymptotes are shifted left and up along with the function. 

Try It #2 

 

hift 

Solving Applied Problems Involving Rational Functions 

Example2ftf x) = x + 

x + Th 

rational function 

fi 

ft 

rational function 

rational functionPxQx 

fx= Px 

Qx =a x p +a 

p p− 

x p − ++a 

x +a 

___ 

Qx≠ 

b q 

x q +b q− 

x q − ++b 

x +b 

Example 3 Solving an Applied Problem Involving a Rational Function 

 

 

r ft 

 

Solution t 

Th 

 

 

Wt=+t 

 

St=+t 

ThC 

+t 

Ct=________ 

+t 

ThftCtt = 

+ 

C=__________ 

 

+ 

=___ 

 

 

Th 

 

+ 

C=_________ 

 

+ 

=__ 

 

 

___ 

 

≈> __ 

 

=ft


Try It #3 

Thft.m., 20 f 

y fio s 

Finding the Domains of Rational Functions 

fi 

 

o fi 

domain of a rational function 

Th 

How To… 

n, fi 

1. 

2. o fix 

3. Th 

Example 4 

Finding the Domain of a Rational Function 

f x=_ 

x + x − 

Solution 

x −= 

x = 

x =± 

Thx =±. Thx =± 

Analysis A graph of this function, as shown in Figure 8, confirms that the function is not defined when x = ±. 

y 

y = 

– – – – – 

 

 

 

 

– 

– 

– 

– 

– 

 

 

x = 

x 

Figure 8 

There is a vertical asymptote atx = and a hole in the graph atx =−We will discuss these types of holes in greater 

detail later in this section 

Try It #4 

x 

fx= 

x−x−

420 


Identifying Vertical Asymptotes of Rational Functions 

 

 

 

Vertical Asymptotes 

Th 

 

How To… 

 

1. 

2. 

3. 

4. fiTh 

 

5. Th 

Example 5 

Identifying Vertical Asymptotes 

+x 

kx= 

 

−x −x 

Solution 

 

+x 

kx=________ 

 

−x −x 

+x 

=___________ 

 

+x−x 

fifi 

 

 

+x−x= 

x =− 

x =−x =Th 

Figure 9fi 

y 

x = − 

 

x = 

 

 

 

 

– – – – – – – 

 

y = − 

– 

– 

– 

– 

– 

– 

 

 

 

x 

Figure 9


Removable Discontinuities 

fi 

removable discontinuity 

x 

f x= 

− 

x −x − 

fx= x+x− 

 

x+x− 

x +Thx =− 

x − 

Thx =Figure 10 

 

y 

 

 

x = − 

 

 

 

– – 

– 

x 

– 

– 

x = 

Figure 10 

removable discontinuities of rational functions 

A removable discontinuityx =aa 

e 

fiTh 

Th 

 

t v 

Example 6 

Identifying Vertical Asymptotes and Removable Discontinuities for a Graph 

kx= x − x − 

Solution 

x − 

kx=___________ 

 

x−x+ 

x −Thx = 

Th 

x +Thx =− 

Thx =−Figure 11

422 


y 

x = − 

 

 

 

– – – – 

– 

x 

– 

– 

Figure 11 

Thx =−x = 

Try It #5 

x 

fx= 

− 

x −x +x 

Identifying Horizontal Asymptotes of Rational Functions 

 

output 

input 

 

 

Th 

Case 1:>y = 

x + 

f x=_________ 

 

x +x − 

f x≈_ 

x 

x = _ 

x 

Th 

gx=_ x 

 

y =Figure 12 

 

 

 

y 

y = 

– – – – 

– 

– 

x 

x = − 

– 

x = 

Figure 12 Horizontal asymptote y = 0 when f(x ) = p(x ____ ) , q(x ) ≠ 0 where degree of p < degree of q. 

q(x ) 

Case 2:< 

f x=__________ 

x −x + x − 

f x≈_ x 

x 

=xTh 

gx=xff 

gx=xf 

gy =x. Th


fi x −x + 

Thx + 

x − 

gx=x +e Figure 13 

 

 

 

 

 

 

 

– – 

– 

– 

– 

y 

y = x + 

x = 

Figure 13 Slant asymptote when f( x ) = p( ____ x ) , q( x ) ≠ 0 where degree of p > degree of q by 1 

q( x ) 

Case 3:=y = a _ n 

a 

b n 

b n 

 

ffipxqxf x= px 

n 

qx qx≠ 

x 

f x=_________ 

+ 

x +x − 

f x≈_ 

x 

x =Th 

gx=x →±∞f x→y = 

Figure 14 

– – – – – 

– 

– 

x = − x = 

Figure 14 Horizontal asymptote when f ( x ) = p( _ 

x ) , q( x ) ≠ 0 where degree of p = degree of q. 

q( x ) 

 

 

 

 

 

 

fx=_______ 

x −x 

x + 

 

f x≈___ 

x 

x =x 

ffi 

x→±∞f x→∞ 

 

 

 

 

– 

y 

x 

x 

y = 

horizontal asymptotes of rational functions 

The 

 

is less thany = 

is greater than degree of denominator by one 

is equal to

424 


Example 7 

Identifying Horizontal and Slant Asymptotes 

 

a. gx= x −x ________ 

x +x 

b. −x + hx=x _________ 

x + c. +x 

kx=x ______ 

x − 

Solution f x= px 

qx qx≠ 

a. gx=_ 

x −x 

x +x: Thp =q = 

s. Thy = _ y = 

b. hx=_ x −x + 

Thp =q =p >q 

x + 

_ x −x + 

 

 

− 

x + 

− 

− 

Thx −s 13. Thy =x − 

c. kx=_ 

x +x 

: Thp =


Try It #6 

 

fx= _____________ 

x−x+ 

x−x+ 

intercepts of rational functions 

yfi 

yfi 

x 

n ix 

o z 

Example 10 

Finding the Intercepts of a Rational Function 

x−x+ 

f x= 

 

x−x+x− 

Solution n fiy 

−+ 

f = _________________ 

 

−+− 

=___ 

− 

 

=− 

 

=− 

x 

x−x+ 

 

=_________________ 

 

x−x+x− 

 

=x−x+ 

x =− 

y−x−Figure 16 

(− 

y 

 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

– 

– 

x = − 

 

( 

 

(– 

x = 

 

Figure 16 

 

x = 

x y = 

Try It #7 

hiftt 3 uni 

Th xy

426 


Graphing Rational Functions 

Example 9x 

 

ff 

 

Th 

 

fifi 

Figure 17 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

y = x 

 

 

x = 

 

 

 

x 

Figure 17 

 

fifi 

Figure 18 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

y = x 

 

 

x = 

 

 

 

x 

Figure 18 

f x= x+ x− Figure 

19 

x+ x− 

 

y 

f x = x + x − 

x + x − 

 

 

– – – – 

(− – ( 

– 

y = 

x 

x = − 

– 

x = 

Figure 19


xx =−x+ 

 

xx =x− 

 

x =−x+ e 

f x=_ 

 

x 

x =x− 

e 

f x= 

x 

How To… 

 

1. o fiy 

2. 

3. 

o fix 

4. x 

5. 

, fi 

6. fi 

 

7. 

8. 

Example 11 

Graphing a Rational Function 

f x= x+x− 

 

x+ x− 

Solution 

l fiy 

f=__ 

+− 

+ − 

= 

fixfi 

xx =−x = 

 

yx− 

fiThx += 

x −=x =−x = 

ThTh 

 

y = 

x 

y 

s fiFigure 20

428 


y 

– – – – – 

 

 

 

 

 

 

– 

– 

– 

 

 

 

 

x 

Figure 20 

Thx =− 

Thfi 

fi 

x = 

Figure 21ftx 

 

y = 

y 

 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

x = – 

Figure 21 

 

 

 

x = 

 

x 

Try It #8 

fx= x+ x− 

x− x− 

 

Writing Rational Functions 

 

 

x 

x 

 

 

 

writing rational functions from intercepts and asymptotes 

xx =x 

x 

x n 

x =v 

v 

v m 

x i 

=v j 

 

 

fx=a x−x p x−x 

p x−x n 

p n 

___ 

x−v 

q x−v 

q x−v m 

q n 

p i 

q i 

 

a 

x


How To… 

 

1. x 

Thfi 

ffi 

2. 

 

3. o fi 

Example 12 Writing a Rational Function from Intercepts and Asymptotes 

Figure 22 

y 

– – – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

Figure 22 

Solution Thxx =−x = 

ThThx =− 

 

x fifi 

Thx =_ 

 

x 

Figure 23 

fi 

y 

 

 

 

 

 

– – – 

– 

x 

– 

x 

– 

Figure 23 

 

x+x− 

fx=a 

__ 

x+x−

430 


o fiy− 

−=a__ 

+− 

+− 

− 

−=a 

_ 

 

a 

_ 

= − − = _ 

 

Ths a fif x= x+x− 

 

x+x− 

 

Graphing Rational Functions (http://openstaxcollege.org/l/graphrational) 

Find the Equation of a Rational Function (http://openstaxcollege.org/l/equatrational) 

Determining Vertical and Horizontal Asymptotes (http://openstaxcollege.org/l/asymptote) 

Find the Intercepts, Asymptotes, and Hole of a Rational Function (http://openstaxcollege.org/l/interasymptote)



VERBAL 

1. l diff 

 

 

3. 

 

 

5. 

x 

2. l diff 

 

4. 

 

ALGEBRAIC 

, fi 

6. fx=_____ 

x− 

x+ 

9. fx= 

_________ 

x +x− 

x −x + 

7. fx=x+ _____ 

x − 

8. fx=_________ 

x + 

x −x− 

, fi 

10. fx=____ 

 

x− 

13. fx=__________ 

x 

x +x− 

11. fx=_____ 

 

x+ 

14. fx=______ 

+x 

x − 

12. fx=_____ 

x 

x − 

15. fx= _______ x− 

x −x 

16. fx=___________ 

x − 

x +x +x 

19. fx= −x ______ 

x− 

17. fx=______ 

x+ x − 

18. fx=x− _____ 

 

x− 

, fixy 

20. fx=_____ 

x+ x + 

21. fx=_____ 

x 

x −x 

22. fx= 

___________ 

x +x+ x +x+ 

23. fx= 

___________ 

x +x+ x −x+ 

24. fx=−x _______ 

 

x − 

 

25. fx=_____ 

x 

x+ 

26. fx=_____ 

x 

x− 

27. fx=−x 

_____ 

x− 

28. fx= 

_________ 

x −x+ 

x −x− 

29. fx=___________ 

x − 

x +x− 

, fi 

30. fx= x +x ________ 

x+ 

31. − fx=x _______ 

x− 

32. − fx=x ________ 

x− 

33. fx= x −x _______ 

x + 

34. +x+ fx=x _________ 

x−


GRAPHICAL 

 

 

35. Thhift 36. Thhift 

 

37. Thhift 

 

38. Thhift 

 

, fi 

 

39. px=______ 

x− 

x+ 

40. qx=x− _____ 

x− 

41. sx=______ 

 

x− 

42. rx=______ 

 

x+ 

43. fx=x __ 

−x− 

x +x− 

44. gx=x __ 

+x− 

x −+ 

45. ax=_ 

x +x− x − 46. bx=x _ 

−x− x − 47. hx=x _ +x− 

 

x− 

48. kx=__ 

x −x− 49. wx=__ 

x−x+x− 50. zx=__ 

x+ x− 

x− x+ x− x−x+x+ 

 

51. x= 

x=−x 

−y 

52. x=− 

x=−x 

y 

53. x=− 

x=−x 

− 

y= 

54. x=− 

x=x− 

 

y=− 

55. x=− 

x=y 

 

56. x= 

x=y 

 

 

57. y 

58. y 

59. y 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

x


60. y 

61. y 

62. y 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

63. y 

64. y 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

–– – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

NUMERIC 

fl 

 

65. fx=_____ 

 

x− 

66. fx=_____ 

x 

x− 

67. fx=_____ 

x 

x+ 

68. fx=______ 

x 

x− 

69. fx=_________ 

x 

x +x+ 

TECHNOLOGY 

f xf x> 

70. fx=_____ 

 

x+ 

71. fx=_____ 

 

x− 

72. fx=___________ 

 

x−x+ 

73. fx=___________ 

x+ 

x−x− 

74. fx=____________ 

x+ 

x− x+ 

EXTENSIONS 

 

75. fx=_____ 

x − 

x− 

76. + fx=x _____ 

x+ 

77. +x− fx=x ________ 

x− 

78. fx= 

__________ 

x +x− x+ 79. +x 

fx=x ______ 

 

x+



 

80. 

 

 

 

 

 

ftt 

 

81. 

 

 

 

 

 

ftt 

82. ThC 

tft 

t 

Ct=_ 

+t 

t 

83. ThC 

tft 

t 

Ct=_ 

t + 

s hig 

Th 

 

84. 

 

x= 

 

86. 

 

x= 

85. 

et. Th 

t. Th 

t. Th 

 

x= 

 

87. 

 

x= 

88. 

 

o co 

 

x=

SECTION 5.7 INVERSES AND RADICAL FUNCTIONS 435 



Find the inverse of an invertible polynomial function. 

Restrict the domain to find the inverse of a polynomial function. 

5.7 INVERSES AND RADICAL FUNCTIONS 

 

Figure 1 

Th 

V =__ 

πr h 

=__ 

πr r 

=__ 

πr 

 

Vr 

fi 

 

√ 

r = 

V 

π 

ThVr 

 

 

Finding the Inverse of a Polynomial Function 

fgfab 

gba 

 

 

Figure 2 

fifi 

 

 

 

 

Figure 2

436 


 

xy 

Figure 3 

y 

 

 

 

 

 

 

 

 

 

–– 

– – – – 

 

– 

– 

 

x 

Figure 3 

fi 

yx=ax 

a 

 

=a 

a =__ 

 

=__ 

 

 

 

yx=__ 

x 

 

 

yxx 

fi 

 

fiis 

xfi 

 

y =__ 

 

x 

 

y =x 

x =±√ — y 

Thx 

 

y =__ 

x x > 

 

xx 

Th 

 

=l ċw 

=ċx 

=x 

=√ — y 

Th 

1. 

2.Thdiff


ftfi 

invertible functions 

f − x 

f − xf xTh− 

f x f x − =_ 

 

f x 

f − f 

ff − fxf − 

 

f − f x=xxf 

 

f f − x=xxf − 

 

verifying two functions are inverses of one another 

fgxfg 

g f x=f gx=x 

How To… 

fi 

 

1. f xy 

2. xy 

3. yf − x 

Example 1 Verifying Inverse Functions 

 

f x=_ 

 

x + f − x= −x ≠− 

x 

Solution f − f x=xf f − x=x 

f − fx=f ( − _____ 

x + ) 

 

=_ 

− 

 

 

_____ 

x + 

=x+− 

=x 

f f x=f( − __ 

 

x − ) 

 

=__ 

 

( __ 

 

x − ) + 

=_ 

 

 

__ 

x 

=x 

 

Thf x= 

 

x + f − x= 

x − 

Try It #1 

fx= x+ 

f − x=x−

438 


Example 2 

Finding the Inverse of a Cubic Function 

f x=x + 

Solution Th 

g fx 

y =x + 

x =y + 

x −=y 

_____ x − 

=y 

√ 

— 

f − x= _____ x − 

 

 

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 

always the case when graphing a function and its inverse function. 

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 

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 

, is on the graph of f − . See Figure 4 

 

y 

f x= x + 1 

 

y = x 

 

 

x 

– – – 

– 

 

– 

– 

Figure 4 

Try It #2 

fx= √ — x+ 

Restricting the Domain to Find the Inverse of a Polynomial Function 

fi 

 

Th 

 

o fi 

restricting the domain


How To… 

n fi 

1. 

2. f xy 

3. xy 

4. yf − x 

5. f − x 

 

Example 3 

Restricting the Domain to Find the Inverse of a Polynomial Function 

f 

a. f x=x− x ≥ b. f x=x− x ≤ 

Solution Thf x=x− x ≥ 

x ≤Figure 5 

y 

y 

 

 

 

 

 

–– 

– – – 

– 

f x = x − x ≥ 

 

x 

 

 

 

 

 

–– 

– – – 

– 

f x = x − x ≤ 

 

x 

Figure 5 

o fif xy 

y =x− xy 

x =y− 

±√ — x =y − 

 

±√ — x =y 

Th 

xyf x 

xxyy 

x ≥y ≥ 

a. Thx ≥ 

f x≥e + 

f − x=+√ — x 

b. Thx ≤ 

f x≤− 

f − x=−√ — x 

Analysis On the graphs in Figure 6, we see the original function graphed on the same set of axes as its inverse function. 

Notice that together the graphs show symmetry about the line y = x. The coordinate pair is on the graph of f and the 

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 

f − . Finally, observe that the graph of f intersects the graph of f − on the line y = x. Points of intersection for the graphs 

of f and f − will always lie on the line y = x.

440 


Example 4 

 

 

 

 

 

–– 

– – – 

– 

y 

 

f x 

 

 

 

y = x 

f – x 

x 

Figure 6 

 

 

 

 

 

–– 

– – – 

– 

y 

f x 

 

 

 

 

y = x 

f – x 

x 

Finding the Inverse of a Quadratic Function When the Restriction Is Not Specified 

n fi 

f x=x− − 

Solution − 

 

x ≥ 

fif xy 

x 

y =x− − xy 

x =y− − 

x +=y− 

± √ — x +=y − 

± √ — x +=y 

 

f − x=± √ — x + 

x ≥ 

+ 

f − x=+√ — x + 

fiTh 

 

Analysis Notice that we arbitrarily decided to restrict the domain on x ≥ . We could just have easily opted to restrict the 

domain on x ≤ , in which casef − (x) = − √ — x + . Observe the original function graphed on the same set of axes as its 

inverse function in Figure 7. Notice that both graphs show symmetry about the line y = x. The coordinate pair (, −) is on 

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 

of axes that 

 

 

 

f =f − =∞ 

f − =f =−∞ 

ff − y =x 

y 

y = x 

− 

 

 

 

 

 

–– 

– – – 

– 

– 

– 

– 

– 

f x 

f – x 

 

− 

x 

Figure 7


Try It #3 

fx=x +x ≥ 

Solving Applications of Radical Functions 

 

fi 

 

How To… 

n, fi 

1. 

2. f xyx 

3. 

Example 5 

Finding the Inverse of a Radical Function 

f x=√ — x −n fi 

Solution f x≥f xyx 

y =√ — x − 

x =√ — y − 

x =√ — y − 

x =y − 

x +=y 

f − x=x + 

f xy 

xy 

 

 

f − x 

 

f − x=x +x ≥ 

Analysis Notice in Figure 8 that the inverse is a reflection of the original function over the line y = x. Because the 

original function has only positive outputs, the inverse function has only positive inputs. 

 

 

 

 

y 

f – x 

y = x 

 

 

 

f x 

x 

Figure 8 

Try It #4 

fx=√ — x+

442 


Solving Applications of Radical Functions 

 

 

Example 6 Solving an Application with a Cubic Function 

Th 

 

V =__ 

πr 

 

V = 

πr Vr 

Thπ = 

Solution Vr > 

 

V ≥rV 

fi 

 

V =__ 

πr 

 

r =___ 

V 

π r 

r =√ 

 

___ V r 

π 

ThV =π = 

Th3 ft 

r = 

 

√ V 

= 

π 

√ 

______ ċ 

ċ 

≈ √ 

— 

≈ 

Determining the Domain of a Radical Function Composed with Other Functions 

 

Example 7 

Finding the Domain of a Radical Function Composed with a Rational Function 

f x= √ ___________ x+x− 

 

x− 

Solution 

 

fi 

 

x+x− 

 

x− ≥Th 

xx =− 

 

Figure 9 

 

 

 

 

− 

– – – – – – – 

– 

– 

– 

– 

– 

y 

x= 

 

 

 

 

 

 

x 

Figure 9


ThxxTh 

 

Th 

y√ — 

yxx =− 

 

 

f xfif x−≤x



VERBAL 

1. 

 

3. 

 

2. 

 

4. Th 

 

ALGEBRAIC 

, fi 

5. fx=x− ∞ 6. fx=x+ −∞ 7. fx=x+ −−∞ 

8. fx=x +−∞ 9. fx=−x ∞ 10. fx=−x ∞ 

11. fx=x +∞ 

, fi 

12. fx=x + 13. fx=x + 14. fx=−x 

15. fx=−x 

, fi 

16. fx=√ — x+ 17. fx=√ — −x 18. fx=+√ — x− 

19. fx=√ — x−+ 20. fx=+ √ 

— x 

21. fx=− √ 

— x 

22. fx=_____ 

 

x+ 

25. fx=_____ 

x− 

x+ 

23. fx=_____ 

 

x− 

26. fx=x+ ______ 

 

−x 

24. fx=x+ _____ 

 

x+ 

27. fx=______ 

x+ 

−x 

28. fx=x +x−∞ 29. fx=x +x+−∞ 30. fx=x −x+∞ 

GRAPHICAL 

, fi 

31. fx=x +x≥ 32. fx=−x x≥ 33. fx=x+ x≥− 

34. fx=x− x≥ 35. fx=x + 36. fx=−x 

37. fx=x +xx≥− 38. fx=x −x+x≥ 39. fx= __ 

x 

40. fx=__ 

 

x x≥ 

 

41. fx= √ __ 

x+x− x 42. fx= √ __ 

x+x− 

43. fx= x− √ _ xx+ x− 

44. fx= √ _ x −x− 

45. fx= x− √ _ −x 

x+


TECHNOLOGY 

Th 

y 

46. fx=x −x−y= 47. fx=x +x−y= 48. fx=x +x−y= 

49. fx=x +x−y=− 50. fx=x +x+y=− 

EXTENSIONS 

, fiabc 

51. fx=ax +b 52. fx=x +bx 53. fx=√ — ax +b 

54. fx= √ — ax+b 

55. fx= ax+b ______ 

x+c 


 

56. 

htftt 

ht=−t t 

h 

50 m 

58. ThVr 

Vr= 

πr r 

V 

 

60. 

n 

Cn= +n 

+n 

C 

nnC 

 

 

62. ThVr 

hV=πr h 

 

V 

 

64. ThV 

rhV=_ πr h 

rh 

12 f 

 

57. 

htftt 

ht=−t t 

h 

 

59. ThA 

rAr=πr r 

V 

 

61. ThT 

l 

Tl=π 

√ 

l 

lT 

 

 

63. ThA 

rhA=πr +πrh 

 

V 

 

65. 

rV

446 




Solve direct variation problems. 

Solve inverse variation problems. 

Solve problems involving joint variation. 

5.8 MODELING USING VARIATION 

ffThff 

 

ff 

 

Solving Direct Variation Problems 

Th 

e =se 

 

Table 1 

s, sales price e = 0.16s Interpretation 

e== a 

e== a 

e== a 

Table 1 

 

 

 

direct variationvaries directly 

Figure 1 

Thy =kx n Thk 

constant of variationk =n = 

 

 

 

e, Earnings, $ 

 

 

 

 

 

 

 

s, Sales Prices in Dollars 

Figure 1

SECTION 5.8 MODELING USING VARIATION 447 

direct variation 

xy 

y=kx n 

direct variationyvaries directlyn 

xk = y _ 

x 

n k 

constant of variationfi 

How To… 

 

1. xy 

2. yfix 

 

3. 

4. o fi 

Example 1 

Solving a Direct Variation Problem 

Thyxy =x =, fiyx 

Solution Thy =kx Thy 

x 

k = y _ 

x 

=__ 

 

 

=__ 

 

 

 

x =y 

y =__ 

 

x 

y =__ 

 

 

= 

Analysis The graph of this equation is a simple cubic, as shown in Figure 2. 

y 

 

 

 

 

 

 

 

 

Figure 2 

x

448 


Q & A… 

Do the graphs of all direct variation equations look like Example 1? 

 

 

Try It #1 

Thyxy=x= yx 

Solving Inverse Variation Problems 

 

T = 

 

d 

° 

Table 2 

d, depth T = 14,000 _ 

d 

Interpretation 

0 ft 

_ = 

 

0 ft° 

ft 

_ 

 

= 

0 ft° 

ft 

_ 

 

= 

0 ft° 

Table 2 

Th 

inversely proportionalvaries inversely 

inverse variations 

Figure 3 

e Thy = k _ 

x 

 

k = 

 

Temperature, T 

(°Fahrenheit) 

 

 

 

 

 

 

 

 

 

 

Depth, d (ft) 

Figure 3 

inverse variation 

xy 

y=__ 

k x n 

kyvaries inverselynxinversely 

proportionalinverse variationsk =x n y

SECTION 5.8 MODELING USING VARIATION 449 

Example 2 

Writing a Formula for an Inversely Proportional Relationship 

 

 

Solution tv 

vt=fi 

vt=t = v 

tv 

tv=___ 

 

v 

=v − 

 

 

How To… 

 

1. xy 

2. yfix 

 

3. 

4. o fi 

Example 3 

Solving an Inverse Variation Problem 

yxy =x =, fiyx 

Solution Thy = k 

x 

yx 

Th 

k =x y 

= ċ 

= 

 

y=__ 

k xk = 

y =___ 

 

x 

x =y 

y =___ 

 

 

=__ 

 

Analysis The graph of this equation is a rational function, as shown in Figure 4. 

 

 

 

 

 

 

 

y 

 

 

 

 

 

 

Figure 4 

 

Try It #2 

yxy=x= yx 

x

450 


Solving Problems Involving Joint Variation 

ft 

 

joint variation 

Thc 

nd 

joint variation 

 

xyzx =kyzxy 

zx = ky 

z 

Example 4 

Solving Problems Involving Joint Variation 

xyzx =y =z = 

fixy =z = 

Solution 

x =_ 

ky 

√ 

— z 

x =y =z =o fik 

 

=_ 

k 

√ 

— 

 

=__ 

k 

 

 

=k 

 

x =_ 

y 

√ 

— z 

o fixy =z =yz 

Try It #3 

x =_ 

√ 

— 

= 

xyzx=y=z= x 

y=z= 

 

Direct Variation (http://openstaxcollege.org/l/directvariation) 

Inverse Variation (http://openstaxcollege.org/l/inversevariatio) 

Direct and Inverse Variation (http://openstaxcollege.org/l/directinverse)



VERBAL 

1. 

 

3. 

 

2. 

 

ALGEBRAIC 

 

4. yxx=y= 5. yxx=y= 

6. yx 7. yxx=y= 

x=y= 

8. yx 9. yxx= 

x=y= 

y= 

10. yxx=y= 11. yxx = 3, y= 

12. yx 13. yx 

x=y= 

x=y= 

14. yx 15. yx 

x=y= 

x=y= 

16. yxzx= 17. yxzwx=z= 

z=y= 

w=y= 

18. yx 

zx=z=y= 

20. yxz 

wx=z= 

w=y= 

22. yx 

zw 

x=z=w=y= 

19. yxz 

x=z=y= 

21. yxzwx= 

z=w=y= 

23. yxz 

wtx=z=w= 

t=y= 

NUMERIC 

o fi 

24. yxx=y= 25. yxx= 

yx= 

y=yx= 

26. yxx= 27. yx 

y=yx= 

x=y=yx= 

28. yx 

29. yxx=y= 

x=y=yx= 

yx= 

30. yxx= 

y=yx= 

32. yx 

x=y=yx= 

34. yxzx=z= 

y=yx=z= 

36. yxz 

x=z=y=yx= 

z= 

38. yxzw 

x=z=w=y=y 

x=z=w= 

31. yxx= 

y=yx= 

33. yx 

x=y=yx= 

35. yxzwx=z= 

w=y=yx=z=w= 

37. yx 

zx=z=y=y 

x=z= 

39. yxz 

wx= 

z=w=y=yx= 

z=w=


40. yxzwtx=z= 

w=t=y=yx=z=w=t= 

TECHNOLOGY 

 

41. yxx= 42. yxx=y= 

y= 

43. yx 44. yxx=y= 

x=y= 

45. yx 

x=y= 

EXTENSIONS 

T 

a 

46. 

Ta 

48. 

Ta 

50. 

 

 


 

51. Ths 

t 

 

53. Th 

 

 

 

 

55. Th 

 

 

 

 

57. Th 

 

 

 

59. Th 

 

 

 

 

 

47. 

 

 

49. 

 

 

52. Thv 

tft 

ft 

54. Th 

 

 

 

 

 

56. Th 

 

 

f 3 m 

 

58. Th 

 

 

 

 

 

 

60. ThK 

mv


CHAPTER 5 

REVIEW 

Key Terms 

arrow notation 

 

axis of symmetry 

x =− b 

a 

coefficient 

constant of variation kfi 

 

continuous function lift 

 

degree 

Descartes’ Rule of Signs 

f xf −x 

direct variation 

 

Division Algorithm f xdx 

dxf xqxrxfx=dx 

qx+rxqxrxTh 

dx 

end behavior 

Factor Theorem kf xx−kf x 

Fundamental Theorem of Algebra 

general form of a quadratic function f x=ax +bx+c 

abca ≠ 

global maximum f af a≥f xx 

global minimum f af a≤f xx 

horizontal asymptote y =b 

 

Intermediate Value Theorem abfa


polynomial function 

 

power function f x=kx p k 

p 

rational function 

Rational Zero Theorem p _ 

q 

p 

q 

Remainder Theorem f xx −kf k 

removable discontinuity efifi 

 

roots xy = 

smooth curve 

standard form of a quadratic function f x=ax−h +k 

hk 

synthetic division x −k 

term of a polynomial function a i 

x i f x=a n 

x n ++a 

x +a 

x +a 

turning point 

varies directly 

varies inversely 

vertex 

 

vertex form of a quadratic function 

vertical asymptote x =a a 

zeros xy = 

Key Equations 

 

 

 

 

f x=ax +bx+c 

f x=ax−h +k 

f x=a n 

x n ++a 

x +a 

x +a 

f x=dxqx+rxqx≠ 

 

 

 

f x= Px 

Qx = a x p p− 

+a 

p p− 

x ++a 

x +a 

___ 

Qx≠ 

q q− 

b q 

x 

+b q − 

x 

++b 

x +b 

y =kx n 

k 

y = k _ 

x n k


Key Concepts 

5.1 Quadratic Functions 

 

Th 

ThThx 

xThyyExample 1 


ft 

Example 2 

ThExample 3 

Th. ThExample 4 

y 

Th 

Example 5 

Example 6 

Th Example 9. 

5.2 Power Functions and Polynomial Functions 

Example 1 

Th 

ThExample 2Example 3 

 

Example 4 

ThTh 

Thffi 

ffiExample 5 

Th 


nn xn −Example 8 


5.3 Graphs of Polynomial Functions 

Example 1 

fi 


fix 

xExample 5 

ThxExample 6 

Th 

Th 

Th 

Th 

nn −Example 7 

, fi 

e fin −Example 8Example 10 

Example 11


ThThf af b 

cabf c=Example 9 

5.4 Dividing Polynomials 

 


Th 

 

x −kExample 3 


Example 6 

5.5 Zeros of Polynomial Functions 

fif kf xx −k 

er ThExample 1 

r Thkf xx−kf xExample 2 

Thffi 

ffi Example 3Example 4 

ffi 

o fiExample 5 

l ThExample 6 

 

 

x−ccExample 7 

Th 

 

Thf −x 

Example 8 

 

Example 9 

5.6 Rational Functions 

f x= _ 

x f x= _ 

 

x 

Example 1 

 

Example 2 

ftExample 3 

Th 

Example 4 

Th 

Example 5 

 

Example 6 

 


fi 

Example 11 

xx =x 

x 

x n 

x =v 

v 

v m 

 

x i 

=v j


Example 12 

5.7 Inverses and Radical Functions 

fx=a x−x p x−x p x−x n p ___ n 

 

x−v 

q x−v q x−v m q n 

Th 

f − fff − Example 1 

fi 

Example 2 

fi 


fiExample 5 

Example 7 


5.8 Modeling Using Variation 

Example 1 

 

Example 2 

Example 3 

 

Example 4



QUADRATIC FUNCTIONS 

Th 

 

1. fx=x −x− 2. fx=−x −x 

, fi 

3. Th− 4. Th− 

 

 

5. 

 

 

 

 

6. 

 

h 

xhx= − +x 

x 

 

POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 

 

effi 

7. fx=x −x +x− 8. fx= x+ −x 9. fx=x −x+x 

 

10. fx=x +x −x + 11. fx=x −x + 12. fx=x +x−x 

GRAPHS OF POLYNOMIAL FUNCTIONS 

, fi 

13. fx=x+ x−x+ 14. fx=x +x +x 15. fx=x −x +x− 

 

16. 

y 

17. 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

–– 

– – – 

– 

– 

– 

– 

– 

 

 

 

x


18. ue Th 

fx=x −x+ 

DIVIDING POLYNOMIALS 

o fi 

19. ______________ 

x −x +x+ 

x− 20. _______________ 

−x +x+ x 

x+ 

fi 

 

21. ______________ 

x −x +x− 

x+ 22. __________ 

x +x+ x− 

23. _________________ 

x +x −x− x+ 24. _______________ 

+x +x+ x x+ 

ZEROS OF POLYNOMIAL FUNCTIONS 

o Th 

25. x −x −x−= 26. x +x +x−= 

27. x −x +x −x+= 28. x +x +x +x+= 

fi 

29. x −x −x+= 30. x −x +x −x+= 

RATIONAL FUNCTIONS 

fi 

 

31. fx=_____ 

x+ 

x− 

33. fx=________ 

x − x +x− 

32. + fx=x _____ 

x − 

34. fx=x+ _____ 

x − 

, fi 

35. fx=_____ 

x − 

x+ 

36. −x + fx=x __________ 

x + 

INVERSES AND RADICAL FUNCTIONS 

 

37. fx=x− x≥ 38. fx=x+ −x≥− 39. fx=x +x−x≥− 

40. fx=x − 41. fx=√ — x+− 42. fx=_____ 

x− x+


MODELING USING VARIATION 

, fi 

43. yxx= 

y= yx= 

45. yxz 

x=z=y= yx=z= 

44. yx 

x =y= yx= 

46. yxz 

wx=z=w= 

y = yx=z=w= 

 

47. Th 

 

e e 

n t 

 

 

48. ThV 

TP


461 


ffi 

1. fx=x −x −x 

 

2. fx=x −x +x− 3. fx=−x −x−x 

 

4. fx=x +x− 

, fi 

5. 

 

6. 

 

 

 

7. fx=x− x−x − 8. fx=x −x +x 

 

9. 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

x 

o fi 

10. __________ 

x +x− x+ 

o fi 

11. __________ 

x +x − x− 12. ________________ 

+x −x− x x+ 

o Thu fi 

13. fx=x +x −x− 14. fx=x +x +x +x+


15. fx=x +x +x −x− 16. fx=x +x +x +x +x+ 

n, fi 

17. x=x= 

x=−y 

18. x= 

 

x=− 

 

19. x −x += 

, fi 

20. fx=_________ 

x+ 

x −x− 

21. +x− fx=x _________ 

x − 

 

22. fx=_________ 

x +x− x− 

 

23. fx=√ — x−+ 24. fx=x − 

25. fx=______ 

x+ 

x− 

 

26. yxx= 

y=yx= 

27. yxz 

x=z=y= yx=z= 

 

28. Th

Exponential and Logarithmic 

Functions 

6 

Figure 1 Electron micrograph of E. Coli bacteria (credit: “Mattosaurus,” Wikimedia Commons) 


6.1 Exponential Functions 6.5 Logarithmic Properties 

6.2 Graphs of Exponential Functions 6.6 Exponential and Logarithmic Equations 

6.3 Logarithmic Functions 6.7 Exponential and Logarithmic Models 

6.4 Graphs of Logarithmic Functions 6.8 Fitting Exponential Models to Data 

Introduction 

 

Th 

 

 

fi 

 

ft 

Table 1 

 

y 

 

Hour 

Bacteria 

Table 1 

 

 

 

 

fb 

463

464 

CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 



Evaluate exponential functions. 

Find the equation of an exponential function. 

Use compound interest formulas. 

Evaluate exponential functions with base e. 

6.1 EXPONENTIAL FUNCTIONS 

Th 

 

ft 

exponential growth 

exponential functions 

 

Identifying Exponential Functions 

 

f x=x + 

Thff 

percent 

Defining an Exponential Function 

 

 

fi 

 

grow exponentiallydoublepercent increase 

Th 

 

Percent change changepercent 

Exponential growthincrease 

percent 

Exponential decaydecrease 

percent 

 

Thfi 

Th 

Table 1 

x f (x)=2 x gx=2x 

 

 

 

 

 

 

 

Table 1

SECTION 6.1 EXPONENTIAL FUNCTIONS 465 

Table 2 

Exponential growthsame percentage 

 

Linear growthsame amount 

 

fffi 

 

 

g 2 t 

Thf x=ab x ab 

 

b > 

466 


exponential function 

x 

f(x=ab x 

 

a 

bb ≠ 

Thf 

Thfa > 

Thfa < 

yay = 

Example 1 

Identifying Exponential Functions 

not 

fx= x− g x=x h x=( 

) x j x=− x 

Solution fi 

Thgx=x gx=x 

 

bb ≠Thjx=− x 

− 

Try It #1 

 

f(x=x −x + 

gx= x 

hx=x + 

jx= −x 

Evaluating Exponential Functions 

 

b 

b =−x = 

Thf x=f ( 

 

) =− =√ — − 

 

 

b =1. Thf x= x =x 

f x=b x x 

 

f x= x f 

f x= x 

f = x = 

=


 

f x= x f 

f x= x 

f = x = 

= 

= 

 

f = ≠ = 

Example 2 Evaluating Exponential Functions 

f x= x + f 

Solution 

f x= x + 

f = + x = 

= 

= 

= 

Try It #2 

f x= x− f 

Defining Exponential Growth 

ft 

fi 

 

 



xabb ≠ 

f x=ab 

 

x 

a 

bx 

exponential function 

ff 

Ax=+x 

 

Bx=+ x 

Table 3 

Year, x Stores, Company A Stores, Company B 

+= + = 

+= + = 

+= + = 

+= + = 

x Ax=+x Bx=+ x 

Table 3

468 


ThfiFigure 2 

 

Number of Stores 

 

 

 

 

 

 

 

 

y 

Bx= + x 

 

 

Years 

Ax= + x 

Figure 2 The graph shows the numbers of stores Companies A and B opened over a five-year period. 

∞∞ft 

 

Bx=+ x 

 

+=Bx= x 

basexexponent 

Example 3 

Evaluating a Real-World Exponential Model 

 

ThPt= t t 

 

Solution t =ft 

 

P= ≈ 

Th 

Try It #3 

Th 

Pt= t t 

 

Example 3 

Finding Equations of Exponential Functions 

 

 

o fiab 

How To… 

 

1. aaa 

f x=ab x b 

2. af x=ab x 

o fiab 

3. abf x=ab x 

 

 

x


Example 4 

Writing an Exponential Model When the Initial Value Is Known 

Th 

NtNt 

Solution tftTh 

 

fta = 

Nt=b t o fib 

Nt=b t 

=b 

 

__ 

=b 

b =( __ 

) b 

b ≈ 

NOTE:Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places 

for the remainder of this section. 

ThNt= t 

 

 

 

Figure 3 

∞∞ 

Try It #4 

Deer Population 

 

 

 

 

 

 

 

 

 

 

 

 

Nt 

 

 

 

 

Years 

Figure 3 Graph showing the population of deer over time, N(t) = 80(1.1447) t , t years after 2006. 

 

236 

Nt 

Example 5 

Writing an Exponential Model When the Initial Value is Not Known 

− 

Solution f x=ab x 

ab 

−=ab − 

=ab 

t

470 


e fiab 

 

=ab − 

 

b −=a 

 

a =b 

ab 

 

=ab 

=b b =b a 

b =( 

) b 

b ≈ 

be fia 

a =b ≈ ≈ 

Thf x= x 

Figure 4 

−. Th 

fx 

− 

– – – – 

 

 

 

 

 

 

 

 

 

 

– 

– 

– 

 

 

 

 

x 

Try It #5 

Figure 4 The graph of f (x) = 2.4492(0.6389) x models exponential decay. 

 

 

Q & A… 

Do two points always determine a unique exponential function? 

x xff x 

 

 

x 

How To… 

 

1. y 

 

2. yaaa 

f x=ab x b 

3. af x=ab x 

o fiab 

4. f x=ab x


Example 6 

Writing an Exponential Function Given Its Graph 

Figure 5 

fx 

 

 

 

 

 

 

 

– – –– 

– –– 

 

– 

x 

Figure 5 

Solution yfiTha = 

 

y =ab x 

y =b x a 

=b y x 

=b 

b =± 

bb =ab 

f (x= x 

Try It #6 

Figure 6 

fx 

 

 

 

 

 

– – – – – – 

√ 

 

 

x 

Figure 6 

How To… 

o fi 

1. [STAT] 

2. L1L2 

3. L1x 

4. L2y 

5. [STAT]CALCExpReg (Exponential Regression)[ENTER] 

6. Thab y =a ⋅b x 

Example 7 

Using a Graphing Calculator to Find an Exponential Function 

o fi 

Solution [STAT][EDIT][1: Edit…]L1L2 

L1xL2y 

[STAT][CALC][0: ExpReg][ENTER]Tha =b =Th 

y =⋅ x 

Try It #7

472 


Applying the Compound-Interest Formula 

 

compound interestThcompounding 

 

annual percentage rate (APR)nominal rate 

Thnominal 

ffgreater 

! Th 

 

tPrn 

At=P ( +r 

n) nt 

Table 4 

 

Frequency 

 

 

 

 

 

Table 4 

the compound interest formula 

Compound interest 

At=P ( +r _ 

n) nt 

Value after 1 year 

 

 

 

 

 

 

At 

t 

Pft 

r 

n 

Example 8 

Calculating Compound Interest 

 

 

Solution P =r = 

n = 

At = 

At=P ( +r 

n) nt 

A=( + ____ 

) ⋅ 

≈ 

Th 

Try It #8


Example 9 

Using the Compound Interest Formula to Solve for the Principal 

 

 

 

 

Solution Thr =n = 

fiP 

P 

 

 

At=P ( +r _ 

n) nt 

=P ( + ____ 

) 

=P 

Arnt 

 

______ 

 

=P P 

P ≈ 

 

Try It #9 

Example 9. 

Evaluating Functions with Base e 

Table 5 

 

g. Th 

Table 5 

Frequency 

A(t) = ( 1 + 

_ 

n) 1 n Value 

 

 

 

 

 

 

 

 

( + 

) 

( + 

) 

( + 

) 

( + 

) 

( + 

) 

( + 

 

) 

( + 

 

) 

( + 

 

) 

Table 5

474 


Thnn 

( + n) 

n e 

Th 

 

the number e 

The 

( + __ 

 

n) n n 

Thee 

e ≈ 

 

Example 10 

Using a Calculator to Find Powers of e 

e o fi 

Solution [e x ]Th[e^(] 

[)][ENTER]e ≈fiExp 

fio fie 

Try It #10 

e − o fi 

Investigating Continuous Growth 

e 

econtinuous growth or 

decay modelsfi 

fl 

the continuous growth/decay formula 

tar 

At=ae rt 

 

a 

r 

t 

r>r < 

 

 

At=Pe rt 

 

P 

r 

t


How To… 

t 

1. a 

2. r 

a. r > 

b. r < 

3. t 

4. At 

Example 11 

Calculating Continuous Growth 

 

 

Solution r = 

ThP =fift 

t = 

At=Pe rt 

=e 

≈ 

Thft 

Prt 

 

Try It #11 

 

 

Example 12 Calculating Continuous Decay 

 

Solution r =−Th 

a =o fiftt = 

At=ae rt 

=e − 

≈ 

 

art 

 

Try It #12 

Example 12ft 

 

Exponential Growth Function (http://openstaxcollege.org/l/expgrowth) 

Compound Interest (http://openstaxcollege.org/l/compoundint)

476 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 


VERBAL 

1. 

 

 

3. Th nominal 

 

 

 

nominal rate 

 

 

2. 

 

 

 

ALGEBRAIC 

 

4. Th 

f w 

6. Th 

 

8. Tht 

ht=−t +t+ 

5. 

e 

 

7. 

 

t 

At= t 

Bt= t 

9. 10. 

 

11. 

 

ft 

 

13. 

 

 

 

 

t infl 

 

12. 

 

ft 

 

 

 

14. y=−t 15. y= x 

16. y= x 

 

17. y= t 

, fi 

18. 19. 20. ( − 

) 

21. − 22. 

fi


 

, fi 

23. 

x 

24. 

x 

 

f(x) − 

h(x) 

25. 

x 

26. 

m (x) 

27. x 

g(x) − 

x 

f(x) 

At=P ( + r 

n) nt 

28. ft 

 

10, 250 ( + 

) 

 

30. 

 

32. 

 

 

34. 

 

ft 

 

36. 

r 

38. 

 

 

ft 

29. 

 

31. 

 

 

33. 

P 

35. 

 

5 m 

37. 

 

 

f $9,000 aft 

 

 

39. y=e t 40. y=e 

t 41. y=e −t 

42. 

 

 

ft 

43. 

ft 

 

NUMERIC 

 

44. fx= x f − 45. fx=− x+ f − 46. fx=e x f 

47. fx=−e x− f − 48. fx= −x+ +f − 49. fx=e x −f 

50. fx=− 

−x + f


TECHNOLOGY 

fi 

 

51. 52. 53. 

54. 55. 

EXTENSIONS 

56. annual percentage yield 

 

 

 

 

=( + r 

) − 

58. 

f(x=ab x ab 

b≠ 

bb=e n n 

 

es a b 

60. ThA 

t wr 

tAt=ae rt a 

 

 

t 

It=e rt − 


61. Th 

 

 

 

63. 

 

 

 

 

 

65. 

 

, in o 

 

67. 

 

 

 

 

57. 

 

 

In 

t con 

59. 

d 1. Th 

b> 

fx=a( 

b) x 

b=e n 

fx=ae −nx 

n 

62. 

ft 

 

ft 

64. 

 

 

 

66. 

 

 

 

 

 

Hint : 

 

68. 

f 7% wa 

ft

SECTION 6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 479 



Graph exponential functions. 

Graph exponential functions using transformations. 

6.2 GRAPHS OF EXPONENTIAL FUNCTIONS 

 

fi 

 

 

 

Graphing Exponential Functions 

 

f x=b x f x= x 

Table 1 

x − − − 

f (x) = 2x 

 

 

 

 

 

 

Table 1 

constantratio 

f x=ab x bTh 

a 

 

x 

x 

x 

Figure 1f x= x 

fx 

− 

− 

− 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

f x= x 

x 

Figure 1 Notice that the graph gets close to the x-axis, but never touches it. 

Thf x= x ∞y = 

f x=b x 

gx=( 

) x Table 2 

 

 

 

 

 

x

480 


x − − − 

g(x)=( 1 2 ) x 

 

Table 2 

 

 

 

 

x 

x 

x 

Figure 2gx=( 

) x 

gx= 

− 

– 

x 

– 

 

– – – – 

– 

gx 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 

Figure 2 

Thgx=( 

) x ∞y = 

 

 

 

x 

 

 

 

 

characteristics of the graph of the parent function f (x) = b x 

f x=b x b >b ≠ 

 

fx 

y = 

f x= b x 

–∞∞ 

b > 

∞ 

x 

y 

b 

b > 

 

b < 

x 

Figure 3 

 

Figure 3 

 

fx 

f x= b x 



b 

x 

How To… 

f x=b x 

1. 

2. y 

3. 

4. −∞∞∞y =


Example 1 

Sketching the Graph of an Exponential Function of the Form f (x) = b 

f x= x 

Solution 

b =Th 

y = 

Table 3 

x − − − 

f (x) = 0.25 x 

Table 3 

y− 

Figure 4 

fx= x 

− 

fx 

 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

– 

Figure 4 

 

 

 

Th−∞∞∞y = 

Try It #1 

f x= x 

Graphing Transformations of Exponential Functions 

 

ftfl 

f x=b x 

ftfl 

 

Graphing a Vertical Shift 

Thfidf x=b x d 

f x= x 

ftd =ftgx= x +fthx= x − 

ftFigure 5 

g(x) = 2 x + 3 

f (x) = 2 x 

h(x) = 2 x – – – 

− 3 

– – – – 

– 

– 

y 

 

 

 

 

 

 

 

Figure 5 

 

 

 

 

 

x 

x 

y = 3 

y = 0 

y = −3

482 


ftf x= x 

Th−∞∞ 

hiftgx= x + 

◦ y 

◦ Th y = 

◦ Th∞ 

hifthx= x − 

◦ y − 

◦ Th y =− 

◦ Th−∞ 

Graphing a Horizontal Shift 

Thcf x=b x 

ftcopposite 

f x= x ftc = ftgx= x + ft 

h x= x − ftFigure 6 

y 

 

 

 

 

 

g(x) = 2 x + 3 f (x) = 2 x h(x) = 2 x − 3 

– – – – – – 

 

 

 

 

y = 0 

x 

ftf x= x 

Th−∞∞ 

Thy = 

y 

Figure 6 

◦ hift gx= x + y x + = x 

 

◦ hifthx= y( x − 

) =( x − 

) x 

 

 

shifts of the parent function f (x) = b x 

cdf x=b x +c +dftf x=b x 

dsamed 

coppositec 

y(b c +d) 

Thy =d 

Thd∞ 

Th−∞∞


How To… 

f x=b x +c +d 

1. y =d 

2. −cd f x=b x ftcccc 

3. f x=b x dddd 

4. −∞∞d∞y =d 

Example 2 

Graphing a Shift of an Exponential Function 

f x= x + − 

Solution f x=b x + c +db =c =d =− 

y =dy =− 

−cd −− 

f x=b x 

 

 

 

− 

 

– – – – – – – 

– 

−− – 

– 

– 

f x 

f (x) = 2 x + − 3 

 

 

x 

 

y = −3 

Figure 7 

Th−∞∞−∞y =− 

Try It #2 

f x= x − + 

How To… 

f x=b x +c +dx 

1. [Y=]Y 1 

= 

2. f xY 2 

= 

3. [WINDOW]yY 2 

= 

4. [GRAPH]fif x 

5. fix[2ND][CALC] 

[ENTER]. Thx 

Example 3 

Approximating the Solution of an Exponential Equation 

= x + 

Solution [Y=] x +Y 1 

=ThY 2 

= 

−x−y[GRAPH]. Thx = 

[2ND][CALC][5: intersect][ENTER]Th 

xffff 

ffGuess?x ≈

484 


Try It #3 

= x − 

Graphing a Stretch or Compression 

ft 

f x=b x ∣ a ∣ > 

f x= x a =gx= x 

Figure 8a = 

hx= 

x Figure 8 

– – 

 

y g(x) = 2) x 

f (x) = 2 x 

 

 

 

 

 

– – – – 

– 

 

 

 

 

 

y f (x) = 2 x 

hx= 2) x 

 

 

 

 

 

x 

x 

 

– – – – – 

y = 0 – 

y = 0 

– 

 

 

 

Figure 8 (a) g(x) = 3(2) x stretches the graph of f (x) = 2 x vertically by a factor of 3. 

(b) h(x) = 

__ 1 3 (2)x compresses the graph of f (x) = 2 x vertically by a factor of 

__ 1 3 . 

stretches and compressions of the parent function f ( x ) = b x 

a >f x=ab x 

a∣ a ∣ > 

a∣ a ∣ < 

ya 

y =∞−∞∞ 

 

Example 4 

Graphing the Stretch of an Exponential Function 

f x=( 

) x 

Solution 

b = 

x 

 

xx 

a =f x= ( 

) x 

Table 4 

x − − − 

f (x)=4( 1 

2 ) x 

Table 4 

y−


Figure 9 

fx 

y = 0 

 

– 

 

 

 

 

– – – – – – – 

 

 

 

 

 

f x= x 

 

x 

 

– 

Figure 9 

Th−∞∞∞y = 

Try It #4 

f x= 

x 

Graphing Reflections 

ftflxy 

f x=b x −flx 

−flyf x= x 

flThflxgx=− x Figure 

10flyhx= −x Figure 10 

x 

y 

y 

y 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

 

f (x) = 2 x 

f (x) = 2 x 

h(x) = 2 −x 

 

 

y = 0 

y = 0 

x 

x 

 

– – – – – 

– 

– 

– 

– 

g(x) = −2 x 

– 

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. 

reflections of the parent function f (x) = b x 

Thf x=−b x 

f x=b x x 

y− 

−∞ 

y =−∞∞ 

Thf x=b −x 

f x=b x y 

yy =∞−∞∞

486 


Example 5 

Writing and Graphing the Reflection of an Exponential Function 

g xflf x=( 

) x x 

 

Solution flf x=( 

g x=− ( 

) x Table 5 

) x xf x− 

x − − − 

g(x)= − 

__ 

( 4) 1 x − − − − − − − 

Table 5 

y−−−− 

 

gx 

 

 

 

 

y = 0 

– – – – – – 

−− – 

– 

– 

– 

gx= − x 

 

− 

 

− 

x 

Figure 11 

Th−∞∞−∞y = 

Try It #5 

gx f x= x y 

 

Summarizing Translations of the Exponential Function 

Table 

6 

Translations of the Parent Function f (x) = b x 

Translation 

Form 

ft 

ce left 

fx=b x +c +d 

d 

 

| a |> 


translations of exponential functions 

 

f x=ab x +c +d 

y =b x b > 

hiftc 

∣ a ∣ ∣ a ∣ > 

∣ a ∣


6 .2 SECTION EXERCISES 

VERBAL 

1. 

 

 

ALGEBRAIC 

3. Thf x= x y 

f 4. W 

gx 

y 

5. Thf x= x x 

hift 

e ngxy 

 

7. Thf x=( 

) − x hift 

hift 

x 

gx 

y 

2. 

 

 

4. Thf x= ( 

) −x 

y 

 

gx 

y 

6. Thf x= x hift 

 

xhift 

gx 

y 

 

GRAPHICAL 

fly 

y 

8. fx=( 

) x 9. gx=− x 10. hx= −x 

 

11. fx=( 

) x gx= x hx= x 

12. fx= 

x gx= x hx= x 

Figure 12 

B C D E 

A 

F 

Figure 12 

13. fx= x 14. fx= x 15. fx= x 

16. fx= x 17. fx= x 18. fx= x


Figure 13fx=ab x 

y 

B C D 

A 

E 

F 

x 

Figure 13 

19. b 20. b 

21. a 22. a 

flx 

23. fx= 

x 24. fx=x − 25. fx=− x + 

f x= x 

 

26. fx= −x 27. hx= x + 28. fx= x− 

 

29. fx=− x − 30. fx=( 

) x − 

31. fx= −x + 

f x= x Th 

 

32. fx 33. fx 34. fxts left 

35. fx 36. fxx 37. fxy 

y = x 

38. 

y 

39. 

y 

40. 

y 

 

 

 

 

 

– – – – – 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – 

– 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

–


, fi 

41. 

y 

42. 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

NUMERIC 

x 

43. gx= 

x− g 44. f x= x− −f 45. hx=− 

( 

) x +h− 

TECHNOLOGY 

 

f x=ab x +d 

46. −=− ( 

) −x 

49. =( 

) x− − 

47. = 

( 

) x 

50. −=− x+ + 

48. = x + 

EXTENSIONS 

51. fx=b x 

gx= ( 

b) x . Th 

 

b x ( 

b) x b> 

53. fx= x 

gx= hx=( x− ) x . Th 

 

b ( x 

b ) b x n 

n 

b> 

52. 

54.

SECTION 6.3 LOGARITHMIC FUNCTIONS 491 



Convert from logarithmic to exponential form. 

Convert from exponential to logarithmic form. 

Evaluate logarithms. 

Use common logarithms. 

Use natural logarithms. 

6.3 LOGARITHMIC FUNCTIONS 

Figure 1 Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce) 

 

 

Figure 1 

Th 

Th 

f 

− = = 

 

Converting from Logarithmic to Exponential Form 

ff 

 

 

ffTh x =xff 

x 

 

ffi x = = =x 

2 y = x Figure 2 

y 

– 

– 

 

 

 

 

 

– 

– 

 

 

 

y= x 

y= 

x 

Figure 2

492 


fi 

Figure 2Thy =b x 

x =b y xy 

yfiyxy = b 

x 

Thblogarithmb 

Thbxyfibx 

s ybyx 

s = 

= 

 

 

b 

x=y ⇔b y =xb >b ≠ 

b 

 

= 

b 

x=y 

 

Th 

by =x 

b 

x 

f x 

b 

x 

x 

 

= 

b 

c=a b a =c 

 

 

 

Th y = b 

x y =b x 

definition of the logarithmic function 

logarithm b x fifi 

x >b>b≠ 

y= b 

xb y = x 

 

b 

x b x b x 

y b x 

x y 

n. Th 

b ∞ 

b −∞∞ 

Q & A… 

Can we take the logarithm of a negative number? 

 

 

 

How To… 

b 

x=y 

1. y = b 

xbyx 

2. b 

x= y b y =x


Example 1 

Converting from Logarithmic Form to Exponential Form 

 

a. 

√ — = 

b. 

= 

Solution byx. Thb y =x 

a. 

√ — = 

 

b. 

= 

b = y = 

x =√— Th 

√ — = 

 

=√ — 

b = y = x =9. Th 

= = 

Try It #1 

 

a. 

= b. 

= 

Converting from Exponential to Logarithmic Form 

bx 

y. Th x = b 

y 

Example 2 

Converting from Exponential Form to Logarithmic Form 

 

 

a. = b. = c. − = 

 

 

Solution byx. Th x = b 

y 

a. = b = x = y =8. Th = 

= 

b. = 

b = x = y =Th = 

= 

 

c. − = 

 

Try It #20 

 

 

b = x =− y = 

 

Th− = 

 

 

 

( 

 

) =− 

 

a. = b. = c. − = 

 

Evaluating Logarithms 

 

 

= 

 

= 

 

 

 

=49. Th 

= 

=27. Th 

= 

 

( 

) 

 

 

= =( 

) = 

 

Th ( 

) =

494 


How To… 

y = b 

x 

1. x b b y =x 

2. b y b x 

Example 3 

Solving Logarithms Mentally 

y = 

 

Solution y = 

 

= 

= 

Try It #3 

y = 

 

Example 4 

Evaluating the Logarithm of a Reciprocal 

y = 

( 

) 

Solution y = 

 

 

 

= 

b−a = 

b a 

 

 

− = 

__ 

 

Th 

( 

) =− 

Try It #4 

y = 

( ) 

=__ 

 

 

Using Common Logarithms 

 

x 

xcommon logarithm 

 

 

definition of the common logarithm 

common logarithm 

xxTh 

x fifi 

x > 

y=x y = x 

x xx 

Th y x


How To… 

y =x 

1. x y =x 

2. y 

x 

Example 5 

Finding the Value of a Common Logarithm Mentally 

y = 

Solution y = 

== 

Try It #5 

y = 

How To… 

y =x 

1. [LOG] 

2. x[ ) ] 

3. [ENTER] 

Example 6 

Finding the Value of a Common Logarithm Using a Calculator 

y = 

Solution 

[LOG] 

[ ) ] 

[ENTER] 

≈ 

Analysis Note that = and that = . Since is between and , we know that log must be 

between log and log. This gives us the following: 

< < 

 

< < 

Try It #6 

y = 

Example 7 

Rewriting and Solving a Real-World Exponential Model 

Th 

Th x = x ff 

ff 

Solution 

 

x = 

= x

496 


 

[LOG] 

[ ) ] 

[ENTER] 

≈ 

Thff 

Try It #7 

Th 

Th x = x diff 

e diff 

Using Natural Logarithms 

Theefi 

natural logarithms. The e 

xx 

xTh 

=fi 

o fie 

definition of the natural logarithm 

natural logarithme e 

xx. Th 

x fifi 

x > 

y=xe y = x 

xe xx. 

Th y ex 

y =e y =xe x = x x e = x x > 

How To… 

y =x 

1. [LN] 

2. x[ ) ] 

3. [ENTER] 

Example 8 

Evaluating a Natural Logarithm Using a Calculator 

y = 

Solution 

[LN] 

[ ) ] 

[ENTER] 

≈ 

Try It #8 

− 

 

Introduction to Logarithms (http://openstaxcollege.org/l/intrologarithms)



VERBAL 

1. b 

 

b y = x b 

x = y b > b ≠ 

3. b 

x = y 

x 

5. 

b 

n diff 

2. f x= b 

x 

gx=b x 

 

4. 

b 

n diff 

ALGEBRAIC 

 

6. 

q=m 7. 

b=c 8. 

y=x 9. x 

=y 

10. y 

x=− 11. 

a=b 12. y 

=x 13. 

=a 

14. v=t 15. w=n 

 

16. x =y 17. c d =k 18. m − =n 19. x =y 

20. x − 

=y 21. n = 22. ( 

24. a =b 25. e k =h 

) m =n 

23. y x = 

x 

26. 

x= 27. 

x=− 28. 

x= 29. 

x= 

30. 

x= 31. 

x= 

32. 

x= 33. 

x=− 

34. x= 35. x= 

fi 

36. 37. 38. 39. e 

40. e − 41. e + 

NUMERIC 

b 

42. 

( ) 43. √ — 44. 

( 

) + 

45. 

 

 

46. 47. 48. + 49. −


 

50. ( e ) 51. 52. e − − 53. ( e ) 

TECHNOLOGY 

 

54. 55. 56. ( 

) 57. √— 58. √ — 

EXTENSIONS 

59. x =f(x=x 

x = 

 

61. x x= 

 

63. e 

= 

 

60. f(x=fx=x 

x 

62. 

 

( =− 

 

) 


64. ThEI 

 

EI= 

( f ) 

t 

ft 

 

 

 

66. ThI 

 

I =M 

I 

−M 

 

 

M 

 

 

 

 

 

 

65. 

EI−

SECTION 6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 499 



Identify the domain of a logarithmic function. 

Graph logarithmic functions. 

6.4 GRAPHS OF LOGARITHMIC FUNCTIONS 

Graphs of Exponential Functions 

 

 

 

causeeffect 

ff 

tA =e t 

 

 

 

Figure 1 

Years 

 

 

 

 

 

 

 

 

 

 

 

Logarithmic Model Showing Years as a Function of the Balance in the Account 

 

 

 

 

 

Account balance 

Figure 1 

fi 

 

Finding the Domain of a Logarithmic Function 

 

fi 

fiy =b x xb >b ≠ 

Thy−∞∞ 

Thy∞ 

y = b 

xy =b x 

 

Thy = b 

xy =b x ∞ 

Thy = b 

xy =b x −∞∞

500 


y = b 

x 

ftfl 

 

Graphs of Exponential Functionsrangey =b x 

y = b 

xdomainfi 

only of positive real numbers 

Th 

f x= 

x −Thfix 

x −o fix 

x −> 

x > 

x > 

f x= 

x −∞ 

How To… 

 

1. 

2. x 

3. 

Example 1 Identifying the Domain of a Logarithmic Shift 

f x= 

x+ 

Solution Thfifix +> 

 

x +> Th 

x >− 

Thf x= 

x+−∞ 

Try It #1 

f x= 

x−+ 

Example 2 

Identifying the Domain of a Logarithmic Shift and Reflection 

f x=−x 

Solution Thfifi 

−x > 

 

−x > Th 

−x >− 

x


Try It #2 

f x=x−+ 

Graphing Logarithmic Functions 

fi 

Thy = b 

x 

ftfl 

y = b 

x 

y =b x fly =x 

y = x 

x = 

yTable 1 

x − − − 

2 x = y 

 

 

 

 

 

log 2 

(y) = x − − − 

Table 1 

Table 1 

f x= x gx= 

xTable 2 

f (x) = 2 x 

( − 

) 

( − 

) 

( − 

) 

g(x) = log 2 

(x) 

( 

− ) 

( 

− ) 

( 

− ) 

Table 2 

xyFigure 2fg 

f x= x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

y 

y = x 

gx= x 

 

x 

Figure 2 Notice that the graphs of f (x) = 2 x and g(x) = log 2 

(x) are reflections about the line y = x. 

 

fx= x ygx= 

xx 

Thf x= x −∞∞gx= 

x 

Thf x= x ∞gx= 

x

502 


characteristics of the graph of the parent function, f (x) = log b 

(x) 

xb >b ≠e t 

f x= b 

x 

 

f(x) 

f(x) 

x = 

∞ 

−∞∞ 

x 

b 

ye 

b > 

 

Figure 4bf x= b 

xff 

 

Notex 

e ≈ 

x 

Figure 3 

x = 

–––– – 

– 

b 

 

 

 

 

 

 

– 

– 

– 

– 

– 

y 

x = 

f x= b x 


f x 

–– – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

x= 

 

 

 

f x= x 

x 

Figure 5 

Th∞−∞∞x = 

Try It #3 

f x= x 

 

Graphing Transformations of Logarithmic Functions 

 

ftfly = b 

x 

Graphing a Horizontal Shift of f (x ) = log b 

(x ) 

cf x= b 

x c 

oppositecft 

f x= b 

xc >ftftgx = b 

x+cfthx= b 

x−c 

Figure 6 

Shift left 

g (x) = log b 

(x + c) 

y 

x= –c 

x= gx= b x + c 

x= 

y 

Shift right 

h(x) = log b 

(x − c) 

x= c 

f x= b x 

b − c 

− c 

 

b 

f x= b x 

x 

b 

 

b + c 

+ c 

hx= b x − c 

x 

Thx =−c. 

−c, ∞ 

−∞∞ 

Figure 6 

Thx =c. 

c, ∞ 

−∞∞ 

horizontal shifts of the parent function y = log b 

x 

cf x= b 

x+c 

y = b 

x cc > 

y = b 

xcc < 

x =−c 

−c∞ 

−∞∞

504 


How To… 

f x= b 

x+c 

1. ft 

a. c > f x= b 

x c 

b. c < f x= b 

xc 

2. x =−c 

3. ft 

cx 

4. 

5. Th−c∞−∞∞x =−c 

Example 4 Graphing a Horizontal Shift of the Parent Function y = log b 

( x) 

f x= 

x− 

 

Solution f x= 

x−x +−=x − 

c =−c


Shift up 

g (x) = log b 

(x) + d 

Shift down 

h(x) = log b 

(x) − d 

y 

y 

x= 

gx= b x+ d 

x= 

f x= b x 

b − d 

b −d 

b 

 

f x= b x 

x 

b 

 

b +d 

b d 

hx= b x− d 

x 

Thx = 

∞ 

−∞∞ 

Thx = 

∞ 

−∞∞ 

Figure 8 

vertical shifts of the parent function y = log b 

(x) 

df x= b 

x+d 

y = b 

xdd > 

y = b 

xdd < 

x = 

∞ 

−∞∞ 

How To… 

f x= b 

x+d 

1. ft 

a. d > f x= b 

xd 

b. d < f x= b 

xd 

2. x = 

3. ftd 

y 

4. 

5. Th∞−∞∞x = 

Example 5 

Graphing a Vertical Shift of the Parent Function y = log b 

(x) 

f x= 

x− 

 

Solution f x= 

x−d =−. Thd < 

Th f x= 

x 

Thx =

506 


( 

− ) 

Thy 

( 

− ) −− 

Th∞−∞∞x = 

y 

 

 

 

 

 

 

– – 

– 

– 

– 

– 

 

y= 

x 

 

x 

 

− f x= 

x − 

− 

x= 

Figure 9 

Th∞−∞∞x = 

Try It #5 

f x= 

x+ 

 

Graphing Stretches and Compressions of y = log b 

(x ) 

f x= b 

xa > 

a > 

f x= b 

xg x=a b 

xhx= a x b 

Figure 10 

Vertical Stretch 

g (x) = alog b 

(x), a > 1 

y 

gx= a b x 

x= 

y 

Vertical Compression 

h(x) = 

_ 

a 1 log (x), a > 1 

b 

x= 

a 

b 

b 

 

fx= b x 

x 

b 

 

fx= b x 

 

hx= a b x 

b a 

x 

Thx =. 

x 

∞ 

−∞∞. 

Thx =. 

x 

∞ 

−∞∞ 

Figure 10


vertical stretches and compressions of the parent function y = log b 

(x) 

a > 

f x = alog b 

x 

y = b 

xaa > 

y = b 

xa 

1. 

a. ∣ a ∣ >f x= b 

xa 

b. ∣ a ∣

508 


Try It #6 

f x= 

x 

 

 

Example 7 Combining a Shift and a Stretch 

f x=x+ 

Solution fi 

Figure 12Thftx =−Thx 

−Th−∞−, 

x =xx =x += 

y 

–– – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

x= − 

Figure 12 

 

y= x + 

y= x + 

x 

y= x 

Th−∞−∞∞x =− 

Try It #7 

f x=x−+ 

Graphing Reflections of f (x ) = logb(x ) 

f x= b 

x−flxinput 

−flt yflb > 

e f x= b 

xflxgx=− b 

x 

flyhx= b 

−x 

Reflection about the x-axis 

g (x) = log b 

(x), b > 1 

Reflection about the y-axis 

h(x) = log b 

(−x), b > 1 

y 

y 

x= 

x= 

b − 

 

b 

f x= b x 

gx= − b x 

x 

hx= b −x 

−b 

− 

f x= b x 

 

b 

x 

x 

 

Thx =. 

x 

Thb − , 

∞ 

−∞∞ 

Figure 13 

x 

 

Thx =. 

x− 

Th−b 

−∞ 

−∞∞


reflections of the parent function y = log b 

x 

Thf x=− b 

x 

y = b 

xx 

∞−∞∞x = 

 

Thf x= b 

−x 

y = b 

xy 

−∞ 

−∞∞x = 

How To… 

f x= b 

x 

Iff (x)=−log b 

(x) 

x = 

x 

flf x= b 

x 

x 

 

∞−∞∞ 

x = 

Table 3 

If f (x)=log b 

(−x) 

x = 

x 

flf x= b 

x 

y 

 

−∞−∞∞ 

x = 

Example 8 

Graphing a Reflection of a Logarithmic Function 

f x=−x 

 

Solution f x=−x 

b =input 

−f yThf x=−xx 

x = 

x− 

 

y 

x= 

– 

f x= −x 

y= x 

 

– 

 

x 

Figure 14 

Th−∞−∞∞x = 

Try It #8 

f x=−−x

510 


How To… 

 

1. [Y=]Y 

=Y 

= 

2. [GRAPH][WINDOW]fi 

s 

3. fix[2ND][CALC] 

[ENTER]. Thx 

Example 9 

Approximating the Solution of a Logarithmic Equation 

x+=−x− 

Solution [Y=]+Y 1 

=Th−x−Y 2 

= 

x–yGRAPHThx = 

[2ND][CALC]5: intersect[ENTER]Th 

xffff 

ffGuess?x ≈ 

Try It #9 

x+=−x 

Summarizing Translations of the Logarithmic Function 

Table 

4 

Translations of the Parent Functiony =log b 

(x) 

Translation 

Form 

ft 

ce left y= b 

x+c+d 

d 

 

∣ a ∣ > 

∣ a ∣ < 

flx 

fly 

 

Table 4 

y=a b 

x 

y=− b 

x 

y= b 

−x 

y=a b 

x+c+d 

translations of logarithmic functions 

y = b 

x 

fx=a b 

x+c+d 

y = b 

xb > 

hiftd 

hift c 

∣ a ∣ ∣ a ∣ > 

∣ a ∣


Example 10 

Finding the Vertical Asymptote of a Logarithm Graph 

f x=− 

x++ 

Solution Thx =− 

Analysis The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve units 

to the left shifts the vertical asymptote to x = −. 

Try It #10 

f x=+x− 

Example 11 

Finding the Equation from a Graph 

Figure 15 

– – – – 

– 

– 

– 

– 

fx 

 

 

 

 

 

 

x 

Figure 15 

Solution Thx =−fl 

 

fx=−ax++k 

−−− 

 

=−a−++k − 

=−a+k 

=k = 

 

−=−a++ − 

−=−a 

 

a =_____ 

 

 

a 

 

Thf x=− 

 

x++ 

Analysis You can verify this answer by comparing the function values in Table 5 with the points on the graph in 

Figure 15. 

x − 

f (x) − − − 

x 

f (x) − − − − − 

Table 5

512 


Try It #11 

Figure 16 

fx 

– – – – 

– 

 

 

 

 

– 

– 

– 

– 

– 

 

x 

Figure 16 

Q & A… 

Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph? 

Figure 16Th 

x =−x =− 

x|x >−Th 

s 

. Thx → − + f x→−∞x →∞f x→∞ 

 

 

Graph an Exponential Function and Logarithmic Function (http://openstaxcollege.org/l/graphexplog) 

Match Graphs with Exponential and Logarithmic Functions (http://openstaxcollege.org/l/matchexplog) 

Find the Domain of Logarithmic Functions (http://openstaxcollege.org/l/domainlog)



VERBAL 

1. Th 

 

 

 

3. ff 

 

2. ff 

 

4. 

f (x) = b 

x. x 

5. 

 

ALGEBRAIC 

 

6. f x= 

x+ 7. hx=( 

−x ) 

9. hx=x+− 10. fx= 

−x− 

8. gx= 

x+− 

 

11. fx= b 

x− 12. gx=−x 13. fx=x+ 

14. fx=−x+ 15. gx=−x+− 

 

16. fx=−x 17. fx=( x− 

) 

19. gx=x+− 20. fx= 

−x+ 

18. hx= −x−+ 

xy 

 

21. hx= 

x−+ 22. fx=x++ 23. gx=−x− 

24. fx= 

x+− 25. hx=x− 

GRAPHICAL 

Figure 17 

 

y 

A 

dx=x 

 

 

 

 

 

B 

C 

D 

E 

x 

fx=x 

gx= 

x 

– 

hx= 

x 

– 

jx= 

x 

Figure 17


Figure 18 

–5 –4 –3 –2 

–1 

6 

5 

4 

3 

2 

1 

–1 

–2 

–3 

–4 

–5 

–6 

y 

A 

1 2 3 4 5 

B 

C 

x 

fx= x 

 

gx= 

x 

hx= x 

 

 

 

 

 

Figure 18 

 

3. fx=xgx= x 

3. fx= 

xgx=x 

3. fx=xgx= 

 

 

 

 

3. fx=e x gx=x 

Figure 19 

y 

 

 

B 

–5 –4 –3 –2 

–1 

5 

4 

3 

2 

1 

–1 

–2 

–3 

–4 

–5 

A 

1 2 3 4 5 

C 

x 

Figure 19 

. f x= 

−x+ . gx=− 

x+ 4. hx= 

x+ 

 

4. fx= 

x+ 4. fx=x 4. f x=−x 

4. gx=x++4. gx=−x+ 4. hx=− 

x+− 

 

4. y= 

x 

y 

. fx= 

x 

y 

–5 –4 –3 –2 

–1 

5 

4 

3 

2 

1 

–1 

–2 

–3 

–4 

–5 

1 2 3 4 5 

x 

–5 –4 –3 –2 

–1 

5 

4 

3 

2 

1 

–1 

–2 

–3 

–4 

–5 

1 2 3 4 5 

x


. fx= 

x 

y 

5. fx= 

x 

y 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

x 

– – – – 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

 

x 

TECHNOLOGY 

o fi 

5. x−+=x−+ 

5. x−+=−x−+ 5. x−=−x+ 

5. x+= 

−x+ 5. 

 

−x=x++ 

EXTENSIONS 

5. bb≠ 

b 

 

. 

5. fx= 

x 

 

gx) = − 

x 

 

. 

fx=( x+ 

x− ) 

6. x 

fx=x +x+

516 




Use the product rule for logarithms. 

Use the quotient rule for logarithms. 

Use the power rule for logarithms. 

Expand logarithmic expressions. 

Condense logarithmic expressions. 

Use the change-of-base formula for logarithms. 

6.5 LOGARITHMIC PROPERTIES 

Figure 1 The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan) 

Th 

 

 

 

 

 

 

 

 

 

 

fi 

Thfia 

 

 

=−H + 

 

=( _____ 

H + ) 

Th−H + 

( 

 

H + ) 

Using the Product Rule for Logarithms 

Th 

 

 

 

b 

= 

 

b 

b= 

 

= = 

= =

SECTION 6.5 LOGARITHMIC PROPERTIES 517 

 

 

b 

b x =x 

b x b =xx > 

 

b 

 

b x =x 

= 

e e e b b x =xe e = 

 

 

b 

M= b 

NM =N 

 

x= 

x +x 

x 

x =x + 

x = x 

 

x+ 

x +=Th 

 

product rule of exponentsx a x b =x a +b 

product rule for logarithms 

 

 

xMNbb ≠ 

 

b 

MN= b 

M+ b 

N 

m = b 

Mn = b 

Nb m =Mb n =N 

b 

MN= b 

b m b n MN 

= b 

b m +n 

=m +n 

= b 

M+ b 

N mn 

 

b 

wxyz 

 

b 

wxyz= b 

w+ b 

x+ b 

y+ b 

z 

the product rule for logarithms 

product rule for logarithms 

 

b 

MN= b 

M+ b 

Nb > 

How To… 

 

1. 

2.

518 


Example 1 

 

xx + 

Using the Product Rule for Logarithms 

Solution 

 

 

xx += 

xx + 

 

 

 

xx += 

+ 

+ 

+ 

x+ 

x + 

Try It #1 

b 

k 

Using the Quotient Rule for Logarithms 

quotient rule of exponents 

 

x a 

b =x a−b Thquotient rule for logarithms 

ff 

 

xMNbb ≠ 

 

b 

( M 

N ) = b M− b N 

m = b 

Mn = b 

Nb m =Mb n =N 

 

b 

( M 

N ) = b ( bm 

b ) 

n 

= b 

b m −n 

=m −n 

MN 

 

 

= b 

M− b 

N mn 

( _ 

x +x 

x + ) fi 

 

 

( _______ 

x +x 

x + )=( xx+ ________ 

x+ ) 

=( x __ 

) 

 

 

 

Th 

 

( x __ 

) =x− 

=+x− 

the quotient rule for logarithms 

quotient rule for logarithms ff 

 

b 

( M 

N ) = b M− b N 

How To… 

ff 

1. 

2. 

 

3.


Example 2 

Using the Quotient Rule for Logarithms 

 

( xx− 

 

x +−x ) 

 

Solution 

xx− 

 

( 

 

 

x +−x ) 

= 

xx−− 

x +−x 

 

 

 

xx−− 

x +−x= 

+ 

+ 

x+ 

x−− 

x ++ 

−x 

= 

+ 

+ 

x+ 

x−− 

x +− 

−x 

Analysis There are exceptions to consider in this and later examples. First, because denominators must never be zero, 

this expression is not defined for x = − 

_ and x = . Also, since the argument of a logarithm must be positive, we note 

 

as we observe the expanded logarithm, that x > , x > , x > − 

_ , and x < . Combining these conditions is beyond the 

 

scope of this section, and we will not consider them here or in subsequent exercises. 

Try It #2 

 

( 

x 

 

+x 

 

xx−x− ) 

 

Using the Power Rule for Logarithms 

x 

 

 

b 

x = b 

x⋅x 

= b 

x+ b 

x 

= 

b x 

fi 

power rule for logarithms 

 

 

= √ — = _ 

e 

=e− 

the power rule for logarithms 


 

b 

M n =n b 

M 

How To… 

 

 

1. 

2.

520 


Example 3 

 

x 

Expanding a Logarithm with Powers 

Solution Thx 

 

 

 

x = 

x 

Try It #3 

x 

Example 4 

Rewriting an Expression as a Power before Using the Power Rule 

 

 

Solution 

= 

 

 

 

 

 

= 

 

Try It #4 

( 

x ) 

Example 5 

Using the Power Rule in Reverse 

xffi 

Solution 

x 

x 

x=x 

Try It #5 

 

 

 

Expanding Logarithmic Expressions 

ft 

 

 

b ( x _ 

y ) = b x− b y 

= b 

+ b 

x− b 

y 

 

 

 

b 

( A 

C ) = b AC − 

= b 

A+ b 

C − 

= b 

A+− b 

C 

= b 

A− b 

C 

ff 

fi


Example 6 Expanding Logarithms Using Product, Quotient, and Power Rules 

( x y 

_ 

)ff 

Solution t r 

 

( x y 

 

)=x y− 

The fi 

 

x y−=x +y− 

e fi 

 

x +y−=x+y− 

Try It #6 

( x y 

z ) 

Example 7 

√ — x 

Solution 

 

Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression 

√ — x =x 

Try It #7 

√ 

— x 

= 

__ x 

Q & A… 

Can we expand ln(x 2 + y 2 )? 

. Thff 

Example 8 

Expanding Complex Logarithmic Expressions 

 

( x x + x − ) 

Solution 

Try It #8 

 

( x x + x − ) = 

+ 

x + 

x +− 

x − 

= 

+ 

x + 

x +− 

x − 

( 

x−x + 

√— 

x − ) 

 

Condensing Logarithmic Expressions 

= 

+ 

x+ 

x +− 

x − 

 

ff

522 


How To… 

ff 

 

1. fi 

 

2. 

3. ff 

Example 9 

Using the Product and Quotient Rules to Combine Logarithms 

 

+ 

− 

 

Solution 

 

+ 

= 

= 

 

Th 

 

− 

 

Th 

Try It #9 

−+− 

 

− 

= 

( __ 

 

) = 

 

Example 10 

Condensing Complex Logarithmic Expressions 

 

x + 

x−− x+ 

Solution e fi 

 

 

x + 

__ x−− x+ = 

x + 

√ — x −− 

x+ 

 

 

x + 

√ — x −− 

x+ = 

x √ — x −− 

x+ 

ff 

 

 

x √ — x −− 

x+ = 

( x √ — ________ x − 

x+ ) 

Try It #10 

+x−x −+x− 

Example 11 

Rewriting as a Single Logarithm 

x−x++ x x + 

Solution e fi 

 

x−x++ x x +=x −x+ +( x + x −1 ) 

 

x −x+ +( x + x −1 ) =x −( x+ x + x −1 ) 

ff 

 

x −( x+ x + x −1 ) = 

( 

x 

x+ x + x −1 )


Try It #11 

x+x+−x + 

Example 12 

Applying of the Laws of Logs 

=−H + 

ff 

Solution CPTh 

P =−CC. Th 

 

=−C 

 

=−C=−+C=−−C 

P =−C 

 

=P −≈P − 

 

Try It #12 

 

Using the Change-of-Base Formula for Logarithms 

 

echange-of-base formula 

 


Mbnn ≠b ≠ 

 

b 

M= M n n 

b 

y = b 

Mnb y =M 

 

n 

b y = n 

M 

 

y n 

b= n 

M 

y = n M 

n 

b 

b 

M= n M 

n 

b 

 

y 

y 

5 

fi 

 

 

5 

= 

≈

524 


the change-of-base formula 

change-of-base formula 

Mbnn ≠b ≠ 

b 

M= M n 

 

n 

b 

 

 

b 

M= M _ 

b 

 

b 

M= n M 

n 

b 

How To… 

b 

M 

nn ≠ 

1. nxx 

e 

2. 

a. ThnM 

b. Thnb 

Example 13 

Changing Logarithmic Expressions to Expressions Involving Only Natural Logs 

 

 

Solution 

n =e 

Th 

. Th 

b 

M= M _ 

b 

Try It #13 

 

 

Q & A… 

Can we change common logarithms to natural logarithms? 

 

= 

_ 

 

Example 14 

Using the Change-of-Base Formula with a Calculator 

 

 

 

= 

 

Solution 

e 

Try It #14 

 

= e 

≈ 

 

 

 

The Properties of Logarithms (http://openstaxcollege.org/l/proplog) 

Expand Logarithmic Expressions (http://openstaxcollege.org/l/expandlog) 

Evaluate a Natural Logarithmic Expression (http://openstaxcollege.org/l/evaluatelog)



VERBAL 

1. 

 

n √ — x 

2. 

 

ALGEBRAIC 

ff 

 

3. b 

xy 4. abc 5. b ( 

) 

6. ( 

x 

 

w z ) 

7. ( 

) 

8. 

y x 

 

9. +x+y 10. 

+ 

a+ 

+ 

b 11. b 

− b 

 

12. a−d−c 13. − b ( 

) 

14. 

 

 

ff 

15. ( x y z ) 

16. ( a− 

y 

 

) −y 

b − c ) 17. √— x y − 18. ( y √ 

19. x y √ 

— x y 

 

20. x +x 21. x −x 22. x+x+ 

23. x− 

y+z 24. 

c+ a 

+ b 

 

 

25. 

e 26. 

 

 

=a 

=b 

ab 

27. 

28. 

29. 

( ) 

NUMERIC 

 

30. 

( 

) − 

 

31. 

+ 

 

 

32. 

− 

+ 

( 

) 

 

o fi 

33. 

34. 

35. 

36. 

( 

) 

EXTENSIONS 

38. x 

 

x++ 

x+= 

 

40. 

 

b x =m 

 

42. 

= 

 

 

37. 

 

39. 

 

x+− 

x−= 

 

 

41. b 

n= 

 

n 

b 

b>n>

526 




Use like bases to solve exponential equations. 

Use logarithms to solve exponential equations. 

Use the definition of a logarithm to solve logarithmic equations. 

Use the one-to-one property of logarithms to solve logarithmic equations. 

Solve applied problems involving exponential and logarithmic equations. 

6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS 

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) 

Th 

 

 

 

 

 

Using Like Bases to Solve Exponential Equations 

Thfi 

bSTb >b ≠b S =b T S =T 

Th 

Th 

Th 

 

x − = x 

 

, Th 

x 

 

x − =___ 

x 

 

 

x − =___ 

x 

 

x − = x − 

x −=x − 

x = x 

x =

SECTION 6.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS 527 

using the one-to-one property of exponential functions to solve exponential equations 

STb ≠ 

b S =b T S = 

How To… 

b S =b T ST 

 

1. b S =b T 

2. 

3. S =T 

Example 1 

x − = x − 

Solving an Exponential Equation with a Common Base 

Solution x − = x − Th 

x −=x − 

x = 

x 

Try It #1 

x = x + 

Rewriting Equations So All Powers Have the Same Base 

 

 

= x − Th 

x 

 

= x − 

= x− 

= x − 

=x − 

=x 

x = 

 

How To… 

 

1. 

2. b S =b T 

3. 

4. S =T 

Example 2 

x + = x + 

Solving Equations by Rewriting Them to Have a Common Base 

Solution 

x + = x + 

x+ = x+ 

x + = x + 

x +=x + 

x = 

x

528 


Try It #2 

x = x + 

Example 3 

x =√ — 

Solution 

Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base 

x = 

 

 

 

Try It #3 

x =√ — 

x =__ 

 

x =__ 

 

 

 

x 

Q & A… 

Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problemsolving 

process? 

 

fi 

Example 4 

x + =− 

Solving an Equation with Positive and Negative Powers 

Solution ThTh x 

 

Analysis Figure 2 shows that the two graphs do not cross so the left side is never equal to the right side. Thus the 

equation has no solution. 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

y 

y = x + 

 

 

 

 

 

 

 

x 

y = − 

Figure 2 

Try It #4 

x =− 

Solving Exponential Equations Using Logarithms 

 

a=ba =b


How To… 

 

1. 

a. 

b. 

2. 

Example 5 

x + = x 

Solving an Equation Containing Powers of Different Bases 

Solution x + = x Th 

x + = x 

x+=x 

x+=x 

x−x=− xx 

x−=− x 

x( __ 

 

__ 

) 

 

Try It #5 

x = x + 

Q & A… 

Is there any way to solve 2 x = 3 x ? 

. Th 

Equations Containing e 

) = ( 

( __ 

) 

x = 

_ 

 

x 

( __ 

) 

eTh 

fi e 

 

How To… 

y =Ae kt t 

1. A 

2. 

3. k 

Example 6 

=e t 

Solution 

Solve an Equation of the Form y = Ae k t 

=e t 

=e t 

=t xe x 

 

t =___ 

t

530 


Analysis Using laws of logs, we can also write this answer in the form t = ln√ — . If we want a decimal approximation 

of the answer, we use a calculator. 

Try It #6 

e t = 

Q & A… 

Does every equation of the form y = Ae kt have a solution? 

Thk ≠yA 

=−e t 

Example 7 

e x += 

Solving an Equation That Can Be Simplified to the Form y = Ae k t 

Solution 

e x += 

e x = 

e x =__ 

 

 

 

 

x =( __ 

) 

x =__ 

( __ 

) 

x 

Try It #7 

+ t =e t 

Extraneous Solutions 

extraneous solution 

 

 

 

Example 8 

x −e x = 

Solution 

Solving Exponential Functions in Quadratic Form 

e x − x = 

e x −e x −= 

e x +e x −= 

e x +=e x −= 

e x =−e x = 

e x = 

x = 

Analysis When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because 

zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation 

e x = − because a positive number never equals a negative number. The solution ln(−) is not a real number, and in the 

real number system this solution is rejected as an extraneous solution.


Try It #8 

e x =e x + 

Q & A… 

Does every logarithmic equation have a solution? 

 

Using the Definition of a Logarithm to Solve Logarithmic Equations 

b 

x=yb y =x 

 

 

 

+ 

x −= 

fix 

 

 

+ 

x −= 

 

x −= 

 

x −= 

=x − 

=x − 

=x 

x = 

using the definition of a logarithm to solve logarithmic equations 

Sb cb >b ≠ 

b 

S=cb c =S 

Example 9 Using Algebra to Solve a Logarithmic Equation 

x+= 

Solution 

x+= 

x= 

x= 

x =e 

Try It #9 

+x= 

Example 10 

x= 

Solution 

 

 

Using Algebra Before and After Using the Definition of the Natural Logarithm 

x= 

x=__ 

 

x =e 

x = 

__ e

532 


Try It #10 

x+= 

Example 11 

x= 

Using a Graph to Understand the Solution to a Logarithmic Equation 

Solution 

x= 

x =e 

Figure 3x 

e ≈e ≈ 

y 

 

 

y = x 

y = 

e ≈ 

 

– 

– 

 

x 

Figure 3 The graphs of y = ln(x ) and y = 3 cross at the point (e 3 , 3), which is approximately (20.0855, 3). 

Try It #11 

x = 

 

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations 

Th 

x >S >T > 

bb ≠ 

b 

S= b 

TS =T 

 

 

x−= 

x −= 

x −=xx =x = 

 

−= 

= 

ThTh 

Th 

 

 

x −−=x+ 

x 

 

x −−=x+ 

 

( x ______ − ) =x+ 

 

______ 

x − =x + 

 

 

x −=x + 

x = x


x =x −−=x+ 

 

 

 

−−=+ 

−= 

( __ 

 

) = 

Th 

using the one-to-one property of logarithms to solve logarithmic equations 

STbb ≠ 

b 

S= b 

TS =T 

 

 

How To… 

 

1. 

b 

S= b 

T 

2. 

3. S =T 

Example 12 

Solving an Equation Using the One-to-One Property of Logarithms 

x =x + 

Solution 

 

x =x + 

x =x + 

x −x −= 

x−x+= 

x −=x += 

x =x =− x 

Analysis There are two solutions: or −. The solution − is negative, but it checks when substituted into the original 

equation because the argument of the logarithm functions is still positive. 

Try It #12 

x = 

Solving Applied Problems Using Exponential and Logarithmic Equations 

 

 

 

 

 

Table 1

534 


Substance Use Half-life 

 

 

 

 

 

 

Table 1 

 

ftfi 

 

At=A 

e T t 

 

At=A 

e t 

 

At=A 

e t T At=A 

( __ 

 

) t T 

 

A 

 

T 

t 

yftt 

Example 13 

Using the Formula for Radioactive Decay to Find the Quantity of a Substance 

 

 

T Solution 

y =e 

 

t 

 

 

=e 

=e 

 

t 

 

t 

ft 

 

 

 

 

=( e 

t ) 

 

= 

t 

 

 

eM =M 

 

t =× 

 

t ≈ 

t 

Analysis Ten percent of grams is grams. If grams decay, the amount of uranium- remaining is 

grams. 

Try It #13 

 

 

Solving Logarithmic Equations (http://openstaxcollege.org/l/solvelogeq) 

Solving Exponential Equations with Logarithms (http://openstaxcollege.org/l/solveexplog)



VERBAL 

1. 2. 

 

3. 

 

ALGEBRAIC 

 

4. −v− = −v 5. ⋅ x = 6. x+ ⋅ x = 

7. −n ⋅ 

=n+ 8. ⋅ x+ = 9. b 

 

= −b 

b 

10. ( ) n ⋅= 

 

11. x− = 12. e x = 13. e r+ −=− 

14. ⋅ a = 15. −⋅ p+ −=− 16. e n− +=− 

17. e −k += 18. −e x− −=− 19. −e x+ +=− 

20. x+ = x− 21. e x −e x −= 22. e x+ −=− 

23. e x+ += 24. e x+ −= 25. e −x− −=− 

26. x+ = x − 27. e x −e x −= 28. e −x +=− 

fi 

29. ( 

) =− 

30. 

= 

 

fi 

31. 

n= 32. − 

x= 33. + 

k= 

34. n++= 35. −−x= 

 

36. −x=−x 37. 

n−= 

−n 38. x+−x= 

39. −x=x −x 40. 

−m= 

m 41. x−−x= 

42. 

n −n= 

−+n 43. x −+= 

x 

44. x+=x+ 45. x+x−=x 46. 

x+= 

47. +−x = 48. 

x+− 

x= 

49. −−x= 

50. 

x− 

= 

 

GRAPHICAL 

xTh 

 

51. 

x−=− 52. 

x+= 53. x= 

54. x−= 55. +−x= 56. −+ 

−x=− 

57. x−−=− 58. −x=−x 59. 

−x −x= 

x− 

60. x+=−x 61. 

−x= 

x− 62. x +=x+ 

 

63. 

 

 

−x−= 64. x−x+=


 

65. 

 

ft 

67. Th 

=e t t 

 

 

66. Th 

D D= 

( I 

I 

 

) 

I 

I 

= − 

 

 

⋅ 

TECHNOLOGY 

 

Th 

68. t = 69. e x = 

70. t = 71. x− = 

72. e −t = 

 

 

 

73. e x − += 74. +x+= 75. −x−=+ 

76. P 

P=e −x x 

 

 

 

Hint 

77. ThM 

M= 

( E E 

) E 

 

E 

= 

 

 

· 

EXTENSIONS 

78. 

b b x =x 

80. A=a ( + r 

k) kt 

 

t 

79. 

y=Ae kt 

 

tt 

 

81. 

T 

T=T s 

+ T 

−T s 

e −kt T s 

 

T 

 

k 

 

 

tt

SECTION 6.7 EXPONENTIAL AND LOGARITHMIC MODELS 537 



Model exponential growth and decay. 

Use Newton’s Law of Cooling. 

Use logistic-growth models. 

Choose an appropriate model for data. 

Express an exponential model in base e. 

6. 7 EXPONENTIAL AND LOGARITHMIC MODELS 

Figure 1 A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology campus. (credit: Georgia Tech Research Institute) 

 

 

Modeling Exponential Growth and Decay 

 

 

 

y=A 

e kt 

A 

ek 

doubling time 

fi 

 

s fi 

 

y =A 

e kt A 

e 

k 

half-life

538 


ft 

 

Figure 2Figure 3 

xx 

y 

y = e x 

 

 

 

e 

 

 

− 

e 

y = 

x 

– – – – – 

– 

– 

y = 

y = e −x 

− 

 

e 

– – – – – – 

– 

– 

– 

– 

y 

 

 

 

 

 

 

 

e 

 

x 

Figure 2 A graph showing exponential 

growth. The equation is y = 2e 3x . 

Figure 3 A graph showing exponential 

decay. The equation is y = 3e −2x . 

ftft 

Thorder of magnitudefi 

 

fi× 

 

characteristics of the exponential function, y = A 0 

e kt 

y =A 

e kt 

 

y 

y 

y = 

 

–∞∞ 

k A 

e 

– 

k A e 

∞ 

x 

y = A e kt 

k > 

y = A e kt 

k < 

yA 

 

A 

A 

k >Figure 4 

A 

– 

k e 

k 0 and exponential decay when k < 0. 

Example 1 

Graphing Exponential Growth 

 

 

Solution fifiA 

 

A 

A 

=fikftt= 

. Th 

 

=e k ⋅ 

=e k 

=k 

k =. Thy =e t =e t = t . ThFigure 5


 

 

 

 

 

 

 

 

y 

y= e t 

 

t 

Figure 5 The graph of y = 10e (ln2)t . 

Analysis The population of bacteria after ten hours is We could describe this amount is being of the order of 

magnitude . The population of bacteria after twenty hours is which is of the order of magnitude , so we 

could say that the population has increased by three orders of magnitude in ten hours. 

Half-Life 

 

half-life 

 

 

o fi 

 

__ A =A e kt 

0 

e fikA 

 

Th 

 

__ 

A =A e kt 

0 

 

 

__ 

kt 

=e A 

 

 

( 

)=kt 

− =kt 

 

 

 

 

− 

____ =t 

k 

k 

tk. Th 

How To… 

, fi 

1. A =A 

e kt 

2. A 

A t 

 

3. o fikk 

t =− ____ 

 

k 

Note: It is also possible to find the decay rate usingk =− 

t

540 


Example 2 

Finding the Function that Describes Radioactive Decay 

Tht 

Solution Th 

A =A 

e kt Th 

A 

=A 

e k ⋅ A 

ft 

=e k A 

 

=k 

k =______ 

 

 

k 

A =A 

e ( 

) t k 

Thft=A 

e ( 

) t . t 

 

≈−×− 

Try It #1 

Th 

 

Radiocarbon Dating 

Th 

 

ff 

 

 

 

 

 

 

 

 

 

 

ftt 

A ≈A 

e ( 

) t 

 

A 

A 

 

Th 

A =A 

e kt Th 

A 

=A 

e k ⋅ tA 

f t 

=e k A 

 

=k 

k =______ 

k 

 

A =A 

e ( 

) t r 

o fit 

( A 

t = A 

 

) 

−


 

Th 

r 

A ≈A 

e −t 

r = A ≈e A −t 

t 

 

r 

t = 

 

− 

How To… 

 

1. k 

r 

2. kt = 

 

− t 

Example 3 

Finding the Age of a Bone 

 

Solution =kt 

r 

t = 

 

− 

 

 

= 

− r 

≈ 

 

Th 

Analysis The instruments that measure the percentage of carbon- are e xtremely sensitive and, as we mention above, 

a scientist will need to do much more work than we did in order to be satisfied. Even so, carbon dating is only accurate 

to about , so this age should be given as years ± or years ± years. 

Try It # 

 

 

Calculating Doubling Time 

 

fi 

doubling time 

A =A 

e kt 

A 

=A 

e kt 

Th 

A 

=A 

e kt 

Th 

=e kt 

=kt 

t = 

 

k 

t = 

k 

A 

 

 

 

t

542 


Example 4 Finding a Function That Describes Exponential Growth 

 

 

Solution Th 

t =___ 

 

k 

Th 

=___ 

 

k 

 

k =___ 

 

k 

 

A =A 

e 

t k 

ThA 

e 

t . 

Try It # 

 

 

 

Using Newton’s Law of Cooling 

 

 

 

 

ThNewton’s Law of Coolingfi 

 

Tt=Ae kt +T s 

Th 

Tt=Ab ct +T s 

Tt=Ae bct 

+T s 

Tt=Ae ctb +T s 

 

Tt=Ae kt +T s 

 

 

 

cb)k 

Newton’s law of cooling 

ThTT s 

 

Tt=Ae kt +T s 

 

 

e diff 

 

How To… 

 

1. T s 

y 

2. Tt=Ae kt +T s 

o fiAk 

3. o fio fi


Example 5 

Using Newton’s Law of Cooling 

°° 

ft° 

° 

Solution 

 

Tt=Ae kt + 

T= 

=Ae k + 

 

A = 

A 

T=k 

=e k + 

=e k 

 

___ 

k 

=e 

 

( ___ 

) =k 

 

( ___ 

) 

k = 

_ 

≈− k 

 

ThTt=e −t + 

 

=e −t + Tt 

=e −t 

 

 

___ 

=e−t 

 

( ___ 

) =−t 

 

( ___ 

) 

t = 

_ 

≈ t 

− 

° 

Try It # 

70 

o 45 dego 60 deg 

Using Logistic Growth Models 

 

 

 

 

 

ft

544 


Thlogistic growth modelfi 

carrying capacityabc 

x 

c 

f(x=_______ 

+ae −b x 

ThFigure 6Th 

 

 

f x 

 

y = c 

 

c 

+a 

a) c 

b 

f x= 

 

 

c 

+ ae –bx 

x 

logistic growth 

Th 

 

 

c 

+a 

ccarrying capacity, or limiting value 

b 

Figure 6 

c_ 

f(x= 

+ae −b x 

Example 6 

Using the Logistic-Growth Model 

flTh 

fl 

Th 

 

t =fl 

flfifl 

b =flft 

s flft 

Solution 

c 

f x=_______ 

+ae −b x 

fl 

c =fiat = 

c =a = 

+a 

 

Thftflf (x= 

≈ 

−x 

+e 

fl 

e flu ic = 

Analysis Remember that, because we are dealing with a virus, we cannot predict with certainty the number of people 

infected. The model only approximates the number of people infected and will not give us exact or actual values. The graph in 

Figure 7 gives a good picture of how this model fits the data.


 

 

 

 

 

Cases 

 

 

 

 

 

 

y = 

 

 

 

 

 

 

Days 

 

Figure 7 The graph of f (x) = 

1000 __ 

1 + 999e −0.6030x 

Try It # 

Example 6f fl 

Choosing an Appropriate Model for Data 

 

fl 

fi 

 

ft 

 

 

 

Thft 

 

ft 

 

Th 

 

 

 

 

 

 

 

 

fl 

ftfi

546 


Example 7 

Choosing a Mathematical Model 

fiTable1 

 

Solution 

 

x 

y 

Table 1 

Figure 8fi 

 

 

 

 

 

 

 

 

 

 

 

y 

 

Figure 8 

 

 

y =ab xe fi=ab 

a =b=Thb =y =ax 

a 

y =ax 

 

=a 

a =_____ 

 

 

a = ≈y 

=x 

 

Figure 9 

x 

 

 

 

 

 

 

 

 

 

 

 

y 

x= 

y = (x) 

 

x 

Figure 9 The graph of y = 2lnx. 

d fi 

Figure 9y =x Figure 10


 

 

 

 

 

 

 

 

 

 

 

y 

x= 

y = (x ) 

 

Figure 10 The graph of y = ln(x 2 ) 

Thx >fiy =x =xx > 

x

548 


Example 8 Changing to base e 

y = x y =A 

e kx 

Solution 

Th 

y = x 

=e x 

 

=e x 

=e x 

Try It # 

y = x e 

 

 

 

 

 

 

Logarithm Application – pH (http://openstaxcollege.org/l/logph) 

Exponential Model – Age Using Half-Life (http://openstaxcollege.org/l/expmodelhalf) 

Newton’s Law of Cooling (http://openstaxcollege.org/l/newtoncooling) 

Exponential Growth Given Doubling Time (http://openstaxcollege.org/l/expgrowthdbl) 

Exponential Growth – Find Initial Amount Given Doubling Time (http://openstaxcollege.org/l/initialdouble)



VERBAL 

1. halflife 

 

 

3. 

doubling time 

 

5. 

 

2. 

 

4. . Th 

 

 

NUMERIC 

6. Thftt 

Tt=e −t +ft 

 

f x= 

+e 

 

−x 

7. f 8. f 

9. 10. 

11. 

 

hmic. Th 

 

12. f(x= x 

eo fi 

x 

f(x) 

TECHNOLOGY 

 

 

13. x 

f(x) 

14. 

15. 

16. 

x 

f(x) 

x 

f(x) 

x 

f(x) 

fit 

 

Pt= 

 

+e −t 

17. 18. f fi 

19. 

e fi 

21. 

 

20. e fi 

ft 

22. e fi 

P


EXTENSIONS 

23. 

 

 

 

 

25. 

M= 

( S S 

) 

 

S 

27. b x =e xb b≠ 

24. Th 

Pt=P 

e rt P 

 

r>t 

M 

26. y 

c 

y= 

 

+ae −rx 

 


 

 

28. 29. 

 

ftts. Th 

 

ft 

 

30. 

f 

 

 

 

31. 

 

33. 

ft 

o fi 

 

 

 

35. Th 

 

 

 

37. 

 

Th 

o fi 

 

 

ft 

32. 

ft 

tys. Th 

ft 

 

34. Th 

 

 

 

36. 

 

 

t? (Th


ft 

ft 

38. 

 

39. 

 

 

 

 

ftr fift 

40. 

 

42. 

ft 

41. 

 

 

ft 

43. 44. 

ft 

45. 

 

fi 

 

46. 

x 

47. 

x 

– – – – – 

 

– – – – – 

 

48. 

− 

___ W 

m 

− 

___ W 

m W 

m 

49. 

M= 

( __ S 

S 

) 

 

 

 

 

 

ThNt= 

 

−t 

+e 

ftt 

50. 51. 

ft 

52. tNt 

 

 

53. ft 

 

 

 

a. f t= t b. f t=e t c. f t=e −t d. f t=__________ 

 

+e −t

552 




Build an exponential model from data. 

Build a logarithmic model from data. 

Build a logistic model from data. 

6.8 FITTING EXPONENTIAL MODELS TO DATA 

 

Thfifi 

regression analysisfi 

Th 

o fit fi. Th 

modelftfunctionequationmodel 

fiThmodel 

 

 

fi 

 

fi 

fl 

 

Building an Exponential Model from Data 

 

 

 

way 

 

 

y =ab x y =A 

e kx 

y =ab x fl 

y =ab x a > 

b 

Th y =a 

b >x 

 

SECTION 6.8 FITTING EXPONENTIAL MODELS TO DATA 553 

exponential regression 

Exponential regression 

 

ExpRegfiTh 

y =ab x 

 

b 

b > 

554 


Solution 

a. STATEDITBAC 

ThSTATPLOTFigure 1 

 

 

 

 

 

 

 

 

 

 

 

y 

x 

Figure 1 

ExpRegSTATCALC 

y= x 

 

y= x 

r ≈fi 

d fiFigure 2 

 

 

 

 

 

 

 

 

 

 

 

y 

x 

Figure 2 

b. xy 

y = x 

 

= x 

≈ 

 

ft


Try It #1 

Table 2ft 

Month 

Debt ($) 

a. o fi 

b. ft 

Table 2 

Q & A… 

Is it reasonable to assume that an exponential regression model will represent a situation indefinitely? 

 

 

 

 

Building a Logarithmic Model from Data 

 

 

 

way 

fi 

fl 

fl 

y =a +bx 

x 

Tha 

b > 

b < 

logarithmic regression 

Logarithmic regressionfi 

fi 

Th 

y=a +bx 

 

x 

b > 

b < 

How To… 

 

1. STATEDIT 

a. 

b. 

c.

556 


2. STATPLOT 

a. ZOOM9o fi 

b. 

3. 

a. LnRegSTATCALC 

b. ab y =a +bx 

4. d fi 

Example 2 

Using Logarithmic Regression to Fit a Model to Data 

 

 

Table 3 

Year 

Life Expectancy (Years) 

Year 

Life Expectancy (Years) 

Table 3 

a. xx =x = y 

o fi 

b. 

Solution 

a. STAT EDIT 

Th STATPLOT 

Figure 3 

 

 

 

 

 

 

 

 

 

 

y 

x 

Figure 3 

LnRegSTATCALC 

y=+x 

d fiFigure 4 

Center for Disease Control and Prevention


y 

 

 

 

 

 

 

 

 

 

 

x 

Figure 4 

b. x= y 

y =+x 

=+ x 

≈ 

 

 

Try It #2 

fiTable 4 

 

Year 

 

 

 

 

 

 

 

 

 

 

 

Number Sold (Thousands) 

 

 

 

 

 

 

 

 

 

 

 

Table 4 

a. x x= y 

 

b.

558 


Building a Logistic Model from Data 

ff 

 

limiting value 

 

Th 

 

 

 

c 

y=_______ 

+ae −b x 

 

c 

 

 

+a 

b > ( a 

b 

c ) 

y =c 

ccarrying capacity 

logistic regression 

Logistic regressiont fi 

o fi 

. Th 

c 

y=_______ 

+ae 

 

−b x 

Th 

c 

+a 

y =c 

How To… 

 

1. STATEDIT 

a. 

b. 

c. 

2. STATPLOT 

a. ZOOM9o fi 

b. 

3. 

a. LogisticSTATCALC 

b. abc y = 

+ae −b x 

c 

4. d fi 

Example 3 

Using Logistic Regression to Fit a Model to Data 

 

Table 5 

The World Bankn


Year Americans with Cellular Service (%) Year Americans with Cellular Service (%) 

 

 

 

 

 

 

 

 

 

Table 5 

a. xx = y 

o fi 

b. 

 

c. c 

 

Solution 

a. STATEDIT 

ThSTATPLOT 

Figure 5 

y 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5 

LogisticSTATCALC 

 

y= 

___________________ 

+e −x 

Figure 6d fi 

 

 

 

 

 

 

 

 

 

 

 

y 

 

Figure 6 

x 

x

560 


b. x = 

y 

 

y =____________________ 

 

+e 

−x 

 

=_____________________ 

 

+e 

− 

x 

≈ 

 

c. Th 

 

c = 

ft 

Try It #3 

Table 6 

Year 

Seal Population 

(Thousands) 

Year 

Seal Population 

(Thousands) 

 

 

 

 

 

 

 

 

Table 6 

a. xx=y 

o fi 

b. 

c. 

 

Exponential Regression on a Calculator (http://openstaxcollege.org/l/pregresscalc)



VERBAL 

1. 

 

d fi 

3. 

 

 

2. 

 

 

4. 

 

 

5. y 

 

 

GRAPHICAL 

fiFigure 7 

Figure 11 

 

 

 

 

 

 

 

 

y 

 

 

 

Figure 7 

x 

 

 

 

 

 

 

 

 

y 

 

 

 

Figure 8 

x 

 

 

 

 

 

 

 

 

y 

x 

 

Figure 9 

y 

y 

 

 

 

 

 

 

 

 

 

 

 

x 

 

 

 

 

 

 

 

 

 

 

 

x 

Figure 10 

Figure 11 

6. y=e −x 7. y=−x 8. y= x 

9. y=+x 

 

10. y=__________ 

 

+e −x


NUMERIC 

11. 

 

 

Pt= 

 

+ −t 

13. 

hp=−p 

php= 

12. At= x 

e 

 

14. 

 

Pt= 

 

−t 

+e 

tPt= 

15. y 

 

TECHNOLOGY 

io: ThPx 

 

Px= 

 

+e −x 

16. 

 

18. ft 

 

20. 

 

 

17. 

19. 

 

io: ThP 

 

Px= 

x 

−x 

+e 

21. 

 

22. 

 

23. ft 24. 

 

25. 

 

 

Table 7 

x 

f (x) 

Table 7 

26. 

 

28. 

e 

27. 

t fi 

29.


30. x 

f x= 

Table 8 

x 

f (x) 

Table 8 

31. 

e d 

33. 

 

35. x 

fx= 

32. 

t fi 

34. 

 

Table 9 

x 

f (x) 

Table 9 

36. 

 

38. 

x= 

40. x 

fx= 

Table 10 

37. LOGREG 

 

y=a+bxt fi 

39. 

 

x 

f (x) 

Table 10 

41. 

 

43. 

x= 

45. x 

fx= 

42. LOGREG 

 

y= a+bxt fi 

44. 

 

Table 11 

x 

f (x) 

Table 11


46. 

 

47. LOGISTIC 

y = 

c 

−b x 

t fi 

+ae 

48. 49. 

 

50. x 

 

Table 12 

x 

f (x) 

Table 12 

51. 

 

52. LOGISTIC 

c 

h m y = 

 

−b x 

+ae 

t fi 

53. 54. 

 

55. x 

 

EXTENSIONS 

56. 

c 

Pt= 

 

 

+ae −bt 

t= 

P=P 

 

c−Pt 

Pt =c−P e −bt 

P 

57. 

fx 

x 

 

y =x 

 

 

58. 

 

 

59. f − x 

c 

fx= 

 

−b x 

+ae 

60. 

 

Pt= 

 

−t 

+e



Key Terms 

annual percentage rate (APR) nominal rate 

carrying capacity 

change-of-base formula er b 

common logarithm x 

xx 

compound interest 

doubling time 

exponential growth 

extraneous solution 

half-life 

logarithm bxy = b 

x 

c 

logistic growth model f x= 

 

c_ 

−b x 

+ae +a c 

b 

natural logarithm ex e 

xx 

Newton’s Law of Cooling 

 

nominal rate annual percentage rate 

order of magnitude 

 

power rule for logarithms 

 

product rule for logarithms 

quotient rule for logarithms o a diff 

Key Equations 

definition of the exponential function 

definition of exponential growth 

compound interest formula 

continuous growth formula 

General Form for the Translation of the 

Parent Function f (x)=b x 

Definition of the logarithmic function 

Definition of the common logarithm 

f x=b x b >b ≠ 

f x=ab x a >b >b ≠ 

A =a ( + r 

k) kt 

Att 

t 

Pft 

r 

n 

At=ae rt 

t 

aa 

P 

ee ≈ 

f x=ab x +c +d 

x >b >b ≠ y = b 

xb y =x 

x >y =x y =x


Definition of the natural logarithm 

General Form for the Translation of the 

Parent Logarithmic Function f (x)=log b 

(x) 

The Product Rule for Logarithms 

x >y =xe y =x 

f x=a b 

x+c+d 

b 

MN= b 

M+ b 

N 

The Quotient Rule for Logarithms 

The Power Rule for Logarithms 

b ( M __ 

N ) = b M− b N 

b 

M n =n b 

M 

The Change-of-Base Formula 

b 

M = M _ n 

n >n ≠b ≠ 

n 

b 

One-to-one property for exponential functions 

 

ST 

bb S =b T S =T 

Definition of a logarithm 

Sbc 

b ≠ 

 

b 

S=cb c =S 

One-to-one property for logarithmic functions 

 

ST 

bb ≠ 

 

b 

S= b 

TS =T 

Half-life formula 

A =A 

e kt k t = k 

Tt=Ae kt +T s 

T s 

A =T−T s 

 

k 

 

Example 1 


Example 4 

Example 5 

Example 6 

Example 7 

Tht 

Example 8 

 

Example 9 

Theft 

e ≈ 

[e x ][exp(x)]eExample 10 

e 

Example 11Example 12


6.2 Graphs of Exponential Functions 

Th f x=b x y−∞∞∞ 

y =Example 1 

b >Th y = 

 

 

 

cc < 

Example 4 

Th f x= b 

x+d y = b 

x 

dd > 

dd < 

Example 5


a > f x=a b 

x 

y = b 

xa∣ a ∣ > 

y = b 

xa∣ a ∣ < 


y = b 

x− x 

− y 

Thf x=− b 

x x 

Th f x= b 

−x y 

Example 8 

Example 9 

f x=a b 

x+c+d 

Table 4 

f x=a b 

x+c+d 

x =−cExample 10 

f x=a b 

x+c+d 

Example 11 

6.5 Logarithmic Properties 

 

Example 1 

ff 

Example 2 

 


 


Thdiff 


 

Example 13 

Thfte 

n logs. ThExample 14 

6.6 Exponential and Logarithmic Equations 

 

Th 

 

 

Example 1 

not 

 


 

Example 5 

e 


ft 

Example 8


b 

S=cS 

b c =S 


b 

S=cy = b 

S 

y =cx 

Example 11 

b 

S= b 

TST 

S =TExample 12 

 

Example 13 

6.7 Exponential and Logarithmic Models 

f x=ab x b >k



EXPONENTIAL FUNCTIONS 

1. y= t 

 

 

2. Th 

At= t t 

 

ft 

3. 

4. Table 1 

 

5. 

 

 

 

7. y=e −t 

 

 

GRAPHS OF EXPONENTIAL FUNCTIONS 

9. fx= x 

y 

x 

f(x) 

Table 1 

6. 

 

 

 

 

8. 

 

 

ft 

10. fx=( 

) x 

y 

y 

11. Th fx= x y 

gxy 

12. Th fx= x 

 

LOGARITHMIC FUNCTIONS 

13. 

=x 

 

15. a − =b 

 

17. x 

x=( 

) 

– – – 

y 

 

 

 

 

 

 

 

 

 

– – – 

– 

– 

– 

 

Figure 1 

14. s=t 

 

16. e − =h 

 

18. 

( 

) 

19. 20. 

 

 

 

 

 

 

x


21. e − 22. ( √ 

— ) 

 

GRAPHS OF LOGARITHMIC FUNCTIONS 

23. gx=x+− 24. hx=−x+ 

25. 

gx=x+− 

LOGARITHMIC PROPERTIES 

26. rst 27. 

x+ 

+ 

y+ 

 

 

28. m ( 

) 

30. ( 

x ) 

32. ( r s t ) 

34. b+c+ −a 

 

29. z–x–y 

31. − y ( ) 

33. 

( b √ b+ 

b− ) 

35. 

v+ 

w− u _ 

 

 

36. 

e 37. x− =hm. Th 

x 

 

EXPONENTIAL AND LOGARITHMIC EQUATIONS 

38. x x = x + 39. 

 

( 

 

 

) −x−= 

40. 

−x −=no 

solution 

42. e x −= 

no solution 

41. 

e n − +=−no 

solution 

43. e x− −=− 

no solution 

44. 2x − = x+ 

no solution 

46. 

− 

n= 

48. 

 

+ 

−x= 

 

f tno solution 

50. Th 

D D= 

( I 

I 

 

) I 

I 

= − 

 

 

− 

52. f − 

fx=e x + − 

45. e x −e x −= 

no solution 

47. 

+a+= 

49. 

+x −= 

no solution 

51. Th 

Pt=e t t 

 

 

 

53. f − 

fx= 

x +


EXPONENTIAL AND LOGARITHMIC MODELS 

 

 

54. 

55. ftt 

Th ft 

 

ff 

ftr fift 

56. 

 

57. 

 

 

ThNt= 

 

−t 

+e 

ftt 

58. 

59. 

60. 

 

 

 

61. x 

f(x) 

62. x 

f(x) 

63. −d (0, 4). Th 

e 

FITTING EXPONENTIAL MODELS TO DATA 

 

64. Pt= 

 

−t 

+e 

 

 

65. ThPt= 

t 

−t 

+e 

 

 

 

Thfi 

o fi 

66. x 

f (x) 

67. x 

f (x) 

68. x 

f (x)


573 


1. Th 

At= t t 

 

ft 

3. 

 

 

, co 

 

5. fx= −x 

y 

y 

2. 

 

4. 

 

 

ft 

6. Th 

fx=( 

) x 

 

y 

 

 

 

 

– – – – – 

– 

 

– 

– 

– 

– 

– 

– 

x 

7. 

=a 

 

9. x 

x= 

 

11. 

 

13. 

fx= 

−x+ 

 

8. e =m 

 

10. 

12. gx=−x+ 

14. ab 

 

15. t 

− t 

16. 

( a 

b ) 

17. 

y z √ 

— x− 

19. x− =hm. Th 

x 

 

21. 

−e a− −=− 

no solution 

23. −e −x− −= 

no solution 

25. e x −e x −= 

no solution 

18. 

c+d+ a 

+b+ 

 

 

20. ( ) x 

= ( 

) 

−x− 

 

22. e x+ += 

no solution 

24. x− = x− 

no solution 

26. 

n−=−


27. 

x −+= 

no solution 

28. Th 

D 

D= 

( I __ 

I 

 

) 

I 

I 

= − 

 

 

− 

29. 

ft 

o fi 

 

 

 

31. 

 

 

ft 

 

 

 

ft 

30. 

e 

o fi 

32. Th 

 

Pt= 

t 

−t 

+e 

 

 

 

33. Table 2 

 

x 

f(x) 

Table 2 

 

34. Thf fiPt= 

t 

−t 

+e 

 

 

 

. Tho fi 

o fi 

35. x 

f(x) 

36. x 

f(x) 

37. x 

f(x)

7 

Systems of Equations and Inequalities 


Introduction 

Figure 1 Enigma machines like this one, once owned by Italian dictator Benito Mussolini, were used by government 

and military officials for enciphering and deciphering top-secret communications during World War II. (credit: Dave Addey, Flickr) 

7.1 Systems of Linear Equations: Two Variables 

7.2 Systems of Linear Equations: Three Variables 

7.3 Systems of Nonlinear Equations and Inequalities: Two Variables 

7.4 Partial Fractions 

7.5 Matrices and Matrix Operations 

7.6 Solving Systems with Gaussian Elimination 

7.7 Solving Systems with Inverses 

7.8 Solving Systems with Cramer's Rule 

 

 

 

 

 

 

Th 

 

Th 

 

fi 

 

 

 

 

 

575

576 

CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES 



Solve systems of equations by graphing. 

Solve systems of equations by substitution. 

Solve systems of equations by addition. 

Identify inconsistent systems of equations containing two variables. 

Express the solution of a system of dependent equations containing two variables. 

7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES 

Figure 1 (credit: Thomas Sørenes) 

Th 

 

fi 

fi 

 

Introduction to Systems of Equations 

 

system of linear equations 

 

fifi 

 

fi 

 

 

ff 

 

x +y = 

 

x −y = 

solution 

 

fi 

f fi 

+= 

−= 

 

consistent system 

independent systemTh

SECTION 7.1 SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES 

577 

ffdependent system 

y 

fiTh 

fi 

inconsistent system 

ThffyTh 

 

types of linear systems 

Th 

independent systemxyTh 

 

inconsistent system 

dependent systemfi. Th. Th 

 

Figure 2 

y 

y 

y 

 

− 

 

– – – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

 

 

 

 

– – – – – – – – – – – – 

– 

– 

– 

–– – 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

x 

 

Figure 2 

How To… 

 

1. 

2. 

 

Example 1 

Determining Whether an Ordered Pair Is a Solution to a System of Equations 

 

 

x +y = 

 

x −=y 

Solution 

 

+= 

= 

 

−= 

= 

Thfi 

Analysis We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that 

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 


y 

 

 

 

 

 

 

x +y= 

– – 

– 

– 

x 

– 

− =y 

– 

– 

x 

Figure 3 

Try It #1 

 

 

x −y = 

 

x+=y 

Solving Systems of Equations by Graphing 

Th 

 

 

Example 2 

Solving a System of Equations in Two Variables by Graphing 

 

 

x +y =− 

 

x −y =− 

Solution e fiy 

 

 

y 

 

 

x +y =− 

y =−x − 

x −y =− 

y =x + 

Figure 4 

y 

y = −x− 

 

 

 

 

 

– – – – – – 

 

 

 

y = x+ 

x 

 

– 

– 

−− – 

– 

Figure 4


579 

Th−− 

 

 

−+−=− 

 

−=− 

 

−−−=− 

 

−=− 

Th−− 

Try It #2 

 

 

x −y =− 

 

−x+y= 

Q & A… 

Can graphing be used if the system is inconsistent or dependent? 

 

fi 

 

Solving Systems of Equations by Substitution 

 

 

 

substitution method 

 

 

How To… 

 

1. 

2. 

3. fifi 

 

4. 

Example 3 Solving a System of Equations in Two Variables by Substitution 

 

 

−x+y =− 

 

x −y = 

Solution e fiy 

 

−x+y =− 

 

y =x −

580 


x −y 

 

x −y = 

 

x −x−= 

 

x −x += 

 

−x =− 

 

x = 

x =e fiy 

 

−+y =− 

 

y = 

 

 

 

−x+y =− 

−+=− 

 

x −y = 

−= 

Try It #3 

 

x =y + 

 

=x −y 

Q & A… 

Can the substitution method be used to solve any linear system in two variables? 

ffi− 

 

Solving Systems of Equations in Two Variables by the Addition Method 

addition method 

ffi 

ffift 

 

How To… 

 

1. xy 

2. 

ffi 

 

ffi 

3. 

4. 

5.


581 

Example 4 Solving a System by the Addition Method 

 

 

x +y =− 

 

−x+y = 

Solution ffix 

−ffixfix 

 

 

x +y =− 

 

−x+y = 

 

y = 

xy 

 

y = 

 

y = 

 

Thyx 

 

−x+y = 

 

 

 

 

Th( − 

 

) 

e fi 

 

−x+ 

= 

−x=− 

 

−x= 

 

x =− 

 

x +y =− 

 

 

( − 

) + ( 

) =− 

− 

+ 

=− 

 

− 

=− 

−=− 

Analysis We gain an important perspective on systems of equations by looking at the graphical representation. See 

Figure 5 to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution 

because observing the graph confirms that the system has exactly one solution. 

y 

x + y = − 

−x + y = 

 

– 

 

– – – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

Figure 5

582 


Example 5 

Using the Addition Method When Multiplication of One Equation Is Required 

 

 

x +y =− 

 

x −y = 

Solution fi 

xx−x 

 

x −y = 

−x−y=− − 

−x +y =− 

 

 

x +y =− 

 

−x +y =− 

 

y =− 

 

y =− 

y =−x 

 

x +y =− 

 

x +−=− 

 

x −=− 

 

x = 

 

x = 

−Figure 6 

 

x −y = 

 

−−=+ 

= 

y 

 

 

 

 

 

 

– – – – – – – 

 

 

 

 

 

 

x 

– 

– 

– 

– 

– 

– 

x − y= 

x + y = − 

Figure 6 

Try It #4 

 

 

 

x −y = 

x+y=− 

Example 6 Using the Addition Method When Multiplication of Both Equations Is Required 

 

 

x +y =− 

 

x −y =


583 

Solution xxThx 

xfi 

− 

 

−x +y=−− 

 

−x −y = 

 

x −y= 

 

x −y = 

Th 

 

−x −y = 

 

x −y = 

 

−y = 

 

y =− 

y =−l fi 

 

x +−=− 

 

x −=− 

 

x =− 

 

x =− 

Th−− 

 

x −y = 

 

−−−= 

 

−+= 

 

= 

Figure 7 

y 

 

 

 

 

 

– – – – – – – 

x + y= − 

– 

– 

−− – 

– 

x − y = 

– 

– 

 

 

 

 

 

 

x 

Example 7 

Figure 7 

Using the Addition Method in Systems of Equations Containing Fractions 

 

 

 

x 

+y 

= 

x 

−y 

= 

Solution 

 

 

( x 

+y 

) = 

 

 

 

x +y = 

( x 

−y 

) = 

x −y =

584 


−x 

 

−x −y=− 

 

−x +y =− 

x 

 

x +y = 

 

−x +y =− 

 

y = 

 

y = 

y =e fi 

 

x += 

 

x = 

 

x = 

 

 

= 

Th( 

) 

 

x 

−y 

= 

 

 

 

Try It #5 

 

 

 

 

 

 

− 

= 

 

− 

= 

 

= 

x +y = 

x+y= 

Identifying Inconsistent Systems of Equations Containing Two Variables 

 

ffy 

Th 

= 

Example 8 

Solving an Inconsistent System of Equations 

 

 

 

x =−y 

x +y = 

Solution x 

 

 

x +y = 

 

−y+y = 

 

+y = 

 

= 

≠. Th


585 

Thfi 

e fi 

 

x =−y 

 

y =−x + 

 

y =− 

x + 

 

 

 

x +y = 

 

y =−x + 

 

y =− 

x + 

 

ffyTh 

 

 

y =− 

x + 

 

 

y =− 

x + 

 

Analysis Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will 

intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in 

common. The graphs of the equations in this example are shown in Figure 8. 

Try It #6 

 

 

y = − x + 

 

 

y 

 

 

 

 

 

 

––– 

– – – – 

– 

– 

– 

– 

– 

Figure 8 

 

 

y −x = 

 

y−x= 

 

y = − 

x + 

 

x 

 

Expressing the Solution of a System of Dependent Equations Containing Two Variables 

 

fi 

ft= 

Example 9 

Finding a Solution to a Dependent System of Linear Equations 

 

 

x +y = 

 

x +y = 

Solution 

xfi− 

x

586 


 

x +y = 

 

−x+y=− 

 

−x −y =− 

 

 

−x −y =− 

 

+x +y = 

 

= 

fi 

Analysis If we rewrote both equations in the slope-intercept form, we might know what the solution would look like 

before adding. Let’s look at what happens when we convert the system to slope-intercept form. 

 

x +y = 

 

y =−x + 

 

 

 

y =− 

x + 

 

x +y = 

y =−x + 

 

y =− 

x + 

 

 

y =− 

x + 

 

See Figure 9. Notice the results are the same. The general solution to the system is( x− 

x + 

) 

y 

 

 

 

 

x + y= 

 

– – – – – – 

– 

– 

– 

– 

 

 

 

x 

 

x + y = 

Figure 9 

Try It #7 

 

y −x = 

−y+x=− 

Using Systems of Equations to Investigate Profits 

 

Threvenue function 

R =xpx = 

p =. ThFigure 10 

Thcost functionfi 

ThFigure 10Thx 

. Thy


587 

Money (in hundreds of dollars) 

 

 

 

 

 

 

 

 

– 

 

 

 

 

 

 

 

Quantity (in hundreds of units) 

Figure 10 

Thbreak-even point 

 

. Th 

Thfi 

Th ffThprofit function 

Px=Rx−Cx 

 

Example 10 

Finding the Break-Even Point and the Profit Function Using Substitution 

Cx=x +Rx=xfi 

rofi 

Solution y 

 

y =x + 

 

y =x 

x +e fix 

 

x +=x 

 

=x 

 

=x 

Thx = 

 

= 

Th 

ThfiPx=Rx−Cx 

 

Px=x −x + 

 

=x − 

ThfiPx=x − 

Analysis The cost to produce units is , and the revenue from the sales of units is also . To 

make a profit, the business must produce and sell more than units. See Figure 11. 

We see from the graph in Figure 12 that the profit function has a negative value until x = , when the graph crosses 

the x-axis. Then, the graph emerges into positive y-values and continues on this path as the profit function is a straight 

line. This illustrates that the break-even point for businesses occurs when the profit function is . The area to the left of 

the break-even point represents operating at a loss.

588 


Dollars 

 

 

 

 

 

 

Example 11 

 

Quantity 

 

 

 

 

Cx=x+ 

 

Rx=x 

Figure 11 

Dollars profit 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

Writing and Solving a System of Equations in Two Variables 

 

 

 

 

 

Profit 

 

Px=x− 

Figure 12 

Quantity 

 

 

 

 

 

 

 

 

 

 

Th 

 

Solution c=a= 

Th 

 

c +a = 

ThcTh 

a. Th 

 

 

c +a = 

 

 

c +a = 

 

c +a = 

fiffifica 

a 

 

c +a = 

 

a =−c 

−cac 

 

c +−= 

 

c +−c = 

 

−c =− 

 

c = 

c =e fia 

 

+a = 

 

a = 

e fi 

Try It #8 

 

 

 

Solving Systems of Equations Using Substitution (http://openstaxcollege.org/l/syssubst) 

Solving Systems of Equations Using Elimination (http://openstaxcollege.org/l/syselim) 

Applications of Systems of Equations (http://openstaxcollege.org/l/sysapp)



VERBAL 

1. 

 

3. 

 

 

5. 

diff 

2. 

 

 

fi 

4. 

 

 

 

ALGEBRAIC 

 

6. x−y= 

x+y= 

9. −x+y= 

x+y=− 

7. −x−y= 

−x+y=− 

10. x+y= 

x −y=− 

 

8. x+y= 

x+y= 

11. x+y= 

x+y= 

12. x −y= 

x+y=− 

13. x+y=− 

x+y= 

14. x+y=− 

x −y= 

15. −x+y= 

−x−y= 

16. x−y= 

−x+y= 

17. x+y= 

x+y=− 

18. −x+y= 

x −y=− 

19. 

x+ 

y= 

20. − 

x+ 

y= 

 

x+ 

y= 

− 

x+ 

y= 

 

21. −x+y=− 

x+y= 

22. x−y=− 

x+y= 

23. x−y=− 

−x−y= 

24. x−y= 

x+y= 

25. −x+y=− 

x−y= 

29. −x+y= 

x−y=− 

26. x+y= 

−x−y=− 

30. −x+y= 

x−y= 

27. 

x+ y = 

 

 

x− 

y=− 

28. 

x+ 

y= 

 

− 

x+ 

y=− 

 

 

31. x+y= 

x+y= 

32. x −y=− 

x+y= 

33. x−y= 

x−y= 

34. x− 

y=− 

 

−x+ 

y=

590 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES 

35. x−y= 

 

x+y= 

 

36. x+y= 

x+y= 

37. 

x− 

y= 

− 

x+ 

y=− 

38. 

x+ 

y= 

 

 

x+ 

y=− 

 

39. x+y=− 

x+y= 

40. x+y= 

x −y= 

GRAPHICAL 

 

fi 

41. x−y= 

x−y= 

44. x−y= 

x−y= 

TECHNOLOGY 

42. −x+y= 

x−y= 

45. x−y= 

−x+y=− 

43. x+y= 

x+y= 

 

 

46. x+y= 

−x+y= 

47. −x+y= 

x+y= 

48. x+y= 

x −y= 

49. x+y= 

−x+y= 

50. −x+y= 

x+y= 

EXTENSIONS 

ABCDEFAF 

A≠BAE≠BD 

51. x+y=A 

x−y=B 

52. x+Ay= 

x+By= 

53. Ax+y= 

Bx+y= 

54. Ax+By=C 

x+y= 

55. Ax+By=C 

Dx+Ey=F 


 

56. uff 

C=x+ 

R=x 

58. 

Cx=x+ 

Rx=x 

60. 

Cx=x+ 

ft 

 

 

57. 

Cx=x+Rx=x 

fi 

59. Cx=x+x 

t. Th 

ft


 

61. d diff 

 

63. Th 

 

ft 

 

65. 

. Th 

 

 

 

67. Th 

 

 

69. 

 

 

 

71. 

 

 

 

 

 

fi 

d de 

73. 

 

 

 

 

75. s. Th 

 

 

 

 

77. 

 

g a diff 

 

t t 

62. 

 

 

64. es a fl 

 

 

 

 

l cos 

66. 

 

 

 

 

68. 

 

Th 

n t 

ft 

 

 

 

70. fi 

 

fi 

h y 

72. 

 

 

 

 

 

74. 

 

 

76. 

ftt. Th 

 

Th

592 




Solve systems of three equations in three variables. 

Identify inconsistent systems of equations containing three variables. 

Express the solution of a system of dependent equations containing three variables. 

7.2 SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES 

Figure 1 (credit: “Elembis,” Wikimedia Commons) 

 

 

fi 

 

fi 

 

 

fi 

Solving Systems of Three Equations in Three Variables 

 

ftfi 

fifi 

Th 

 

fixyz 

 

Ax+By+Cz=D 

Ey+Fz=G 

Hz=K 

Thzfiyx 

 

1. 

2. 

3.

SECTION 7.2 SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES 

593 

solution setxyzfi 

 

fifl 

e fl 

number of possible solutions 

Figure 2Figure 3 

ftsolution set 

xyz 

ft 

= 

 

ft 

= 

 

Figure 2 ( a)Three planes intersect at a single point, representing a three-by-three system with a single solution. 

( b) Three planes intersect in a line, representing a three-by-three system with infinite solutions. 

 

 

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. 

(b) Two of the planes are parallel and intersect with the third plane, but not with each other. 

( c) All three planes are parallel, so there is no point of intersection. 

Example 1 Determining Whether an Ordered Triple Is a Solution to a System 

− 

 

x +y +z = 

 

x −y +z = 

 

x +y +z = 

Solution xyz 

 

x +y +z = 

 

+−+= 

 

 

 

x −y +z = 

 

−−+= 

 

++=

594 


 

x +y +z = 

 

+−+= 

 

−+= 

 

 

Th− 

How To… 

 

1. 

2. 

3. 

4. 

Example 2 

Solving a System of Three Equations in Three Variables by Elimination 

 

x −y +z = 

−x+y −z =− 

x −y +z = 

Solution Thfi 

x 

x −y +z = 

−x+y −z =− 

y +z = 

Th−Th 

x 

−x +y −z =− − 

x −y +z = 

−y −z =− 

z 

y +z = 

−y−z =− 

z = 

 

x −y +z = 

y +z = 

z = 

z =y 

 

y += 

 

y += 

 

y =− 

z =y =−. Thx 

 

x −−+= 

 

x ++= 

 

x =


595 

Th−Figure 4 

− 

x = 

z = 2 

Example 3 

y = −2 

Figure 4 

Solving a Real-World Problem Using a System of Three Equations in Three Variables 

ff 

 

 

Th 

Solution 

 

 

x = 

 

y = 

 

z = 

The fi 

 

x +y +z = 

 

 

 

z =y + 

Th 

 

x +y +z = 

Th 

 

x +y +z = 

 

−y +z = 

 

x +y +z = 

. Th 

x +y +z = 

−y +z = 

x +y +z = 

 

 

x +y +z = 

 

x +y +z = 

 

−y +z = 

− 

 

x +y +z = 

 

y +z = 

 

−y +z =

596 


 

 

x +y +z = 

 

y +z = 

 

z = 

zyTh 

zyx 

 

z = 

 

z = 

 

y += 

 

y = 

 

x ++= 

 

x = 

 

Try It #1 

 

 

x +y −z =− 

 

x −y −z = 

x−y+z= 

Identifying Inconsistent Systems of Equations Containing Three Variables 

 

fiTh 

 

Th= 

 

Example 4 

Solving an Inconsistent System of Three Equations in Three Variables 

 

x −y +z = 

−x+y −z = 

x −y +z = 

Solution ffixx 

x −y +z = 

−x+y −z = 

−y −z = 

− 

−x +y −z =− − 

x −y +z = 

y +z =− 

Th 

−y −z = 

y +z =− 

 

=


597 

Thfi= 

 

Analysis In this system, each plane intersects the other two, but not at the same location. Therefore, the system is 

inconsistent. 

Try It #2 

 

 

x +y +z = 

y −z = 

x+y+z= 

Expressing the Solution of a System of Dependent Equations Containing Three Variables 

fi 

Thfi 

Th 

ff 

fi 

Example 5 

Finding the Solution to a Dependent System of Equations 

 

x +y −z = 

x +y −z = 

x −y +z = 

Solution − 

−x −y +z = − 

x +y −z = 

 

= 

Th= 

fiTh− 

d fi= 

fi 

 

 

 

x +y −z = 

x −y +z = 

x −z = 

z 

 

x −z = 

 

z = 

x 

zy

598 


 

 

 

 

x +y −( 

x ) = 

x +y − x = 

 

y = x −x 

 

y = 

x 

( x 

x 

x ) xThyz 

x 

Analysis As shown in Figure 5, two of the planes are the same and they intersect the third plane on a line. The solution 

set is infinite, as all points along the intersection line will satisfy all three equations. 

x − y + z = 0 

−4x − 2y + 6z = 0 

4x + 2y − 6z = 0 

Figure 5 

Q & A… 

Does the generic solution to a dependent system always have to be written in terms of x? 

x 

xy 

Try It #3 

 

 

x +y +z = 

x −y −z = 

x+y+z= 

 

Ex 1: System of Three Equations with Three Unknowns Using Elimination (http://openstaxcollege.org/l/systhree) 

Ex. 2: System of Three Equations with Three Unknowns Using Elimination (http://openstaxcollege.org/l/systhelim)



VERBAL 

1. 

 

3. 

 

 

2. 

 

 

4. 

 

5. 

 

f y 

f n 

ALGEBRAIC 

 

6. x −y+z=− 

x+y+z=− − 

−x+y+z=− 

7. x−y+z= 

x+y+z= 

x+y= 

−− 

8. x −y+z= 

−x−y+z= 

x+y−z= 

− 

9. x−y= 

x−z= − 

x−y+z=− 

10. −x−y+z= 

x+y −z= − 

−x+y −z=− 

 

11. x−y+z=− 

x+y+z= 

x+y+z= 

14. x−y+z= 

−x+y+z= 

x+y−z=− 

12. x−y+z= 

x−y−z=− 

x+y−z= 

15. x−y+z= 

−x+y−z=− 

x+y−z= 

13. x+y+z= 

−x+y+z= 

x−y+z=− 

16. x+y+z= 

−x+y−z= 

x−y+z=− 

 

17. x−y+z= 

−x+y−z=− 

y+z=− 

20. x+y−z= 

x+y−z= 

−x−y+z=− 

23. x+y+z= 

y+z=− 

−y−z=− 

18. x−y+z= 

−x+y= 

x−z= 

21. x+y−z= 

−x+y+z= 

−x+y+z= 

24. x−y+z=− 

−x+y−z= 

−x+y+z=− 

19. x+y−z= 

−x−y+z=− 

x+y+z= 

22. x+y−z= 

−x−y−z=− 

−x−y+z=− 

25. x+y+z= 

x−y+z= 

x−z=


26. x+y−z= 

x−y+z=− 

x+y−z= 

29. x−y+z= 

 

x− 

y+ 

z= 

−x− 

y−z=− 

27. x+y+z= 

x−y+z= 

x−z= 

30. 

x− 

y+ 

z=− 

 

 

x− 

y− 

z=− 

− 

x− 

y− 

z=− 

 

28. x− 

y−z=− 

 

x+z= 

−x+ 

y= 

 

31. − 

x− 

y− 

z= 

 

− 

x− 

y− 

z= 

− 

x− 

y− 

z=− 

 

32. 

x− 

y+ 

z= 

 

x− 

y+ 

z=− 

 

x+ 

y− 

z= 

33. 

x− 

y+ 

z= 

− 

x− 

y+ 

z=− 

− 

x− 

y+ 

z=− 

34. − 

x− 

y+ 

z=− 

 

− 

x− 

y+ 

z=− 

 

− 

x− 

y+ 

z= 

35. − 

x− 

y+ 

z=− 

− 

x− 

y+ 

z= 

 

− 

x− 

y+ 

z= 

 

38. x+y−z= 

x+y−z= 

x+y−z= 

41. x+y+z= 

x−y−z=− 

x−y−z=− 

44. x+y+z= 

x+y+z= 

x+y+z= 

36. 

x+ 

y+ 

z= 

− 

x− 

y− 

z=− 

 

 

x+ 

y+ 

z= 

39. x+y−z=− 

x+y−z=− 

x+y−z=− 

42. x−y−z= 

x−y−z= 

x−y−z=− 

45. x+y+z= 

x−y+z= 

x+y+z= 

37. x−y+z= 

x−y+z= 

x−y+z= 

40. x−y+z= 

x−y+z= 

x−y+z= 

43. x+y−z= 

x−y+z= 

x−y−z= 

EXTENSIONS 

xyz 

46. x+y+z= 

x− 

+y− 

+z+ 

= 

x− 

+y+ 

+z− 

= 

 

47. x−y− z+ 

= 

 

x+ y− 

+z=− 

x+ 

−y+z= 

48. x+ 

−y− 

+z+ 

= 

x− 

+y+ 

−z+ 

= 

x+ 

−y+ 

+z+ 

= 

49. x− 

+y+ 

−z− 

= 

x+ 

+y− 

+z+ 

= 

x+ 

−y− 

+z+= 

50. x− 

+y+ 

+z+ 

= 

x+y−z= 

x+y−z=



51. o 108. Th 

 

 

 

53. 

 

. Th. Th 

 

 

 

55. ff 

. Th 

 

 

 

 

 

 

57. . Th 

 

. Th 

 

Th 

. Th 

 

 

 

59. t. Th 

f $28,112.50. Th 

 

 

 

 

61. 

 

 

o infl 

$151,830. Th 

 

 

 

 

52. o 147. Th 

 

 

54. 

 

 

 

 

56. ff 

 

 

diffax. Th 

 

 

ax. Th 

 

 

 

 

58. 

. Th 

 

s $15.25. Th 

 

 

 

60. h $19.50. Th 

 

 

 

 

62. 

 

ft


63. 

. Th 

 

 

ft 

 

 

 

65. Th 

 

 

 

. Th 

 

Th 

 

 

 

67. Th 

 

a. Th 

ts. Th 

 

 

 

 

69. 

t. Th 

d fi 

 

n fid fi 

 

 

d fi 

64. 

. Th 

 

 

ft 

t fi 

 

 

66. Th 

 

 

 

 

 

 

 

 

68. Th 

 

ka. Th 

n. Th 

 

 

 

 

70. 

 

fif fi 

 

 

 

ffi 

ffi 

ffi 

: Th 

Th

SECTION 7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES 

603 



Solve a system of nonlinear equations using substitution. 

Solve a system of nonlinear equations using elimination. 

Graph a nonlinear inequality. 

Graph a system of nonlinear inequalities. 

7.3 SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES 

Figure 1 

ThTh 

ff 

Figure 1 Halley’s Comet (credit: "NASA Blueshift"/Flickr) 

 

Th 

Solving a System of Nonlinear Equations Using Substitution 

system of nonlinear equations 

Ax+By+C = 

. Th 

 

. Th 

Intersection of a Parabola and a Line 

Th 

possible types of solutions for points of intersection of a parabola and a line 

Figure 2 

n. Th 

n. Th 

s. Th 

 

y 

y 

y 

 

 

 

– – – – – 

– 

– 

– 

 

 

 

 

 

x 

 

– – – 

– 

– 

– 

– 

– 

 

 

Figure 2 

x 

 

 

 

 

 

 

– – – – 

 

 

 

 

 

x

604 


How To… 

, fi 

1. 

2. 

3. 

4. 

Example 1 

 

 

 

Solving a System of Nonlinear Equations Representing a Parabola and a Line 

x −y =− 

y =x + 

Solution e fix 

 

x −y =− 

x =y − x 

 

y =x + 

y =y− + x 

 

 

y =y− 

 

=y −y ++ 

 

=y −y + 

 

=y −y + 

 

=y−y− 

yy =y =yfix 

 

 

x −y =− 

 

x −=− 

 

x = 

 

x −=− 

 

x = 

Thfixy 

Figure 3 

– – 

y =x 

 

+ 

y 

 

 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

 

 

Figure 3 

x − y = − 

 

 

 

x


605 

Q & A… 

Could we have substituted values for y into the second equation to solve for x in Example 1? 

xx 

y = 

 

y =x + 

 

=x + 

 

x = 

 

x =±√ — = 

Th 

y = 

 

y =x + 

 

=x + 

 

x = 

 

x =±√ — =± 

− 

Try It #1 

 

 

x −y =− 

 

x −y= 

Intersection of a Circle and a Line 

 

 

possible types of solutions for the points of intersection of a circle and a line 

Figure 4 

n. Th 

n. Th 

s. Th 

 

Figure 4 

How To… 

, fi 

1. 

2. 

3. 

4.

606 


Example 2 

Finding the Intersection of a Circle and a Line by Substitution 

 

 

x +y = 

 

y =x − 

Solution yy =x − 

 

 

x +x − = 

 

x +x −x += 

 

x −x += 

x 

 

x −x += 

 

x−x−= 

 

x = 

 

x = 

xy 

 

y =− 

 

= 

y =− 

 

=− 

Th−fixy 

Figure 5 

 

x y 

 

 

 

y x 

 

 

 

 

 

 

 

 

 

 

 

Figure 5 

Try It #2 

 

 

x +y = 

x−y=− 

Solving a System of Nonlinear Equations Using Elimination 

ft 

 

ft


607 

possible types of solutions for the points of intersection of a circle and an ellipse 

Figure 6 

Th 

 

n. Th 

s. Th 

s. Th 

s. Th 

Thr 

Figure 6 

Example 3 

Solving a System of Nonlinear Equations Representing a Circle and an Ellipse 

 

x +y = 

x +y = 

Solution − 

 

−x +y =− 

 

−x −y =− 

 

x +y = 

 

y = 

fty 

 

y = 

 

y =±√ — =± 

y =±x 

x + = x +− = 

 

x += 

x += 

 

x = 

 

x ==± 

 

x =±√ — =± 

Th−−−−Figure 7 

− 

– – 

−− 

y 

 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

Figure 7 

 

 

 

x 

 

−

608 


Try It #3 

 

x +y = 

x +y = 

Graphing a Nonlinear Inequality 

 

 

 

nonlinear inequality 

 

y >ay a; (b) an example of y ≥ a; (c) an example of y < a; (d) an example of y ≤ a 

How To… 

 

1. . Th 

2. ≤≥ 

3. 

4. fi 

 

 

5. 

Example 4 Graphing an Inequality for a Parabola 

y >x + 

Solution y =x +y >x + 

Th 

 

y >x + 

> + 

> 

> + 

>


609 

ThFigure 9 

 

y 

– – – – – 

 

 

 

 

 

 

– – 

 

 

 

x 

Graphing a System of Nonlinear Inequalities 

Figure 9 

 

system of nonlinear inequalities 

 

Thfffi 

Th 

feasible region 

How To… 

 

1. 

2. 

3. 

4. 

 

Example 5 

Graphing a System of Inequalities 

 

 

 

x −y ≤ 

x +y ≤ 

Solution Thfi 

y. Thx 

 

x −y = 

 

x +y = 

 

x = 

 

x = 

 

x =± 

xy 

 

x −y = 

 

−y = 

 

−y = 

 

y = 

 

− −y = 

 

−y = 

 

y =

610 


Th− 

 

x −y ≤ 

 

x ≤y 

 

y ≥x 

 

x +y ≤ 

 

y ≤−x + 

Figure 10Th 

x +y ≤x −y ≤ 

y 

 

 

 

 

 

– 

 

– – 

– – – 

– – 

 

Figure 10 

 

 

 

 

x 

Try It #4 

 

y ≥x − 

x−y≥− 

 

Solve a System of Nonlinear Equations Using Substitution (http://openstaxcollege.org/l/nonlinsub) 

Solve a System of Nonlinear Equations Using Elimination (http://openstaxcollege.org/l/nonlinelim)



VERBAL 

1. 

 

 

 

 

3. 

 

 

 

 

5. 

 

xfie n 

2. 

 

 

4. 

fi 

ALGEBRAIC 

 

6. x+y= 

x +y = 

9. y=−x 

x +y = 

7. y=x − 

x +y = 

10. x= 

x −y = 

8. y=x 

x +y = 

 

11. x −y = 

x +y = 

14. y −x = 

x +y = 

12. x +y = 

x −y = 

15. x +y + 

= 

y=x 

13. x +y = 

x −y =x − 

 

16. −x +y=− 

x−y= 

19. x +y = 

y=−x 

22. x −x =y 

x +y= 

17. −x +y= 

−x+y= 

20. x −x =y 

y= 

−x 

23. x −x =y 

x +y= 

18. x +y = 

y=x − 

21. x +y = 

x − +y = 

 

24. x +y = 

y=−x 

27. x −y = 

x−y= 

30. x +y = 

x −y = 

25. x −y = 

x= 

28. −x +y= 

−x+y=− 

31. x +y = 

y =x 

26. x −y = 

y= 

29. −x +y= 

y=−x 

32. x −y += 

y +x =


33. x −y = 

x − +y = 

36. x −y −x−y−= 

−x +y = 

34. x −y = 

x − +y = 

37. x +y −y= 

x +y= 

35. x −y = 

x +y = 

38. x +y = 

xy= 

GRAPHICAL 

 

39. x +y< 40. x +y < 

 

41. x +y< 

y>x 

44. x −y >− 

x +y < 

42. x +yx+ 

45. x +y > 

x −y < 

43. x +y < 

x −y > 

EXTENSIONS 

 

46. y≥e x 

y≤x+ 

47. y≤−x 

y≤e x 

, fi 

48. x + 

y = 

x − 

y += 

49. x − 

y = 

x − 

y = 

 

50. x −xy+y −= 

x+y= 

51. x −xy −y −= 

x +y = 

52. x +xy −y −= 

x=y+ 

TECHNOLOGY 

fi 

53. xy< 

y>√ — x 

54. x +y< 

y>x 


 

 

55. 

 

 

56. Tho 360. Th 

 

 

57. 

Cx=x −x+ 

Rx=−x +x+ 

fi 

 

fi 

58. 

Cx=x − x+ 

Rx=−x +x 

fi 

fi

SECTION 7.4 PARTIAL FRACTIONS 

613 



Decompose P (x )Q (x ), where Q (x ) has only nonrepeated linear factors. 

Decompose P (x )Q (x ), where Q (x ) has repeated linear factors. 

Decompose P (x )Q (x ), where Q (x ) has a nonrepeated irreducible quadratic factor. 

Decompose P (x )Q (x ), where Q (x ) has a repeated irreducible quadratic factor. 

7.4 PARTIAL FRACTIONS 

 

 

 

Th 

 

Decomposing P(x ) ____ 

Q(x ) 

Where Q(x ) Has Only Nonrepeated Linear Factors 

Thfi 

fffi 

 

fi 

partial fractions 

 

 

x− + − x+ 

d fio fix+x− 

d fi 

 

 

x− ( x+ 

x+ ) + − x+ ( x− 

x− ) =x+−x+ 

 

x+x− = 

x+ 

x −x− 

 

 

x+ 

 

x −x − = 

 

fi 

x − + − x+ 

 

 

 

e fi 

fi 

 

x −x −x−x+ 

 

fiTh

614 


partial fraction decomposition of 

____ P(x) : Q(x) has nonrepeated linear factors 

Q(x) 

Th ____ Px Qx Qx 

PxQx 

Px 

Qx = 

A 

 

( a 

x+b 

) + 

A 

 

( a 

x+b 

) + 

A 

 

( a 

x+b 

) ++ 

A n 

 

 

( a n 

x+b n 

) 

 

How To… 

 

1. ABC 

fiA n 

 

Px 

Qx = 

A 

 

( a 

x+b 

) + 

A 

 

( a 

x+b 

) ++ 

A n 

 

( a n 

x+b n 

) 

2. 

3. 

4. ffi 

 

Example 1 Decomposing a Rational Function with Distinct Linear Factors 

 

 

x 

 

x+x− 

Solution ABC 

 

x 

 

x+x− = 

A 

 

x+ + 

B 

 

x − 

 

 

x+x− 

[ 

x 

 

x+x − ] = x+x − 

[ 

A 

x+ ] +x+ 

− 

[ 

B 

x − ] 

Th 

 

x=Ax−+Bx+ 

 

 

x=Ax−A+Bx+B 

 

x=A+Bx−A+B 

ffi 

 

=A+B 

 

=−A+B 

B 

 

=+B 

 

=−A+B 

 

=+B 

 

=B 

B= 

 

=A+ 

 

=A 

Th 

 

x 

 

x+x− = 

 

x+ + 

 

x−


615 

AB 

xABx=A 

B 

 

x=Ax−+Bx+ 

 

=A−+B+ 

 

=+B 

 

=B 

B=ABx=− 

 

 

x=Ax−+Bx+ 

 

−=A−−+B−+ 

 

−=−A+ 

 

− 

− =A 

 

=A 

AB 

 

x 

 

x+x− = 

 

x+ + 

 

x− 

ft 

ft 

 

Try It #1 

 

x 

 

x−x− 

Decomposing ____ P(x ) 

Q(x ) 

Where Q(x ) Has Repeated Linear Factors 

 

 

partial fraction decomposition of 

____ P(x) : Q(x) has repeated linear factors 

Q(x) 

Th Px 

Qx Qxn 

PxQx 

Px 

Qx = 

A 

 

ax+b + 

A 

 

ax+b + 

A 

 

ax+b ++ 

A n 

 

ax+b n 

 

How To… 

 

1. ABC 

Px 

Qx = 

A 

 

ax+b + 

A 

 

ax+b ++ 

A n 

 

ax+b n 

2. 

3. 

4. ffi

616 


Example 2 

Decomposing with Repeated Linear Factors 

 

 

−x +x+ 

x −x +x 

Solution Thxx− x− 

xx−x− 

 

−x +x+ 

 

x −x +x =A x 

+ 

B 

 

x− + 

C 

 

x− 

 

 

 

xx− [ 

−x +x+ 

 

xx− 

] 

= 

[ A 

x + 

B 

 

x− + 

C 

x− ] xx− 

−x +x+=Ax − +Bxx−+Cx 

 

 

−x +x+=Ax −x++Bx −x+Cx 

 

=Ax −Ax+A+Bx −Bx+Cx 

 

=A+Bx +−A−B+Cx+A 

ffi. Th 

 

−x +x+=A+Bx +−A −B+Cx+A 

A+B=− 

−A −B+C= 

A= 

A 

 

A= 

 

A= 

A= 

 

A+B=− 

 

+B=− 

 

B=− 

ThCAB 

 

 

 

 

 

 

−A−B+C= 

−−−+C= 

−++C= 

C= 

−x +x+ 

x 

−x +x = x − 

 

x− + 

 

x− 

Try It #2 

 

 

x − 

x−


617 


Q(x ) 

Where Q(x ) Has a Nonrepeated Irreducible Quadratic Factor 

 

ABC 

Th 

Ax+BBx+C 

decomposition of 

_____ P(x) Q(x) has a nonrepeated irreducible quadratic factor 

Q(x): 

Th Px 

Qx Qx 

degree of PxQxwritten as 

Px 

Qx = 

A 

x + B 

 

 

 

a 

x +b 

x +c 

+ 

A 

x + B 

 

 

 

a 

x +b 

x +c 

++ 

A n 

x + B n 

 

 

a n 

x +b n 

x +c n 

 

Th 

ABC 

How To… 

 

1. ABCA 

 

x+B 

A 

x+B 

 

Px 

Qx = 

A 

 

ax+b + 

A 

x + B 

 

 

 

a 

x +b 

x +c 

+ 

A 

x + B 

 

 

 

a 

x +b 

x +c 

++ 

A n 

x + B n 

 

 

a n 

x +b n 

x +c n 

 

2. 

3. 

4. ffi 

 

Example 3 

Decomposing ____ P(x) When Q(x) Contains a Nonrepeated Irreducible Quadratic Factor 

Q(x) 

 

 

x 

+x− 

x+x +x+ 

Solution 

. Th 

 

x 

+x− 

x+x +x+ = 

A 

 

x+ + 

Bx+C 

 

x +x+ 

 

 

 

 

x+x +x+ [ 

x 

+x − 

x+x +x+ ] 

= 

[ 

A 

 

x+ + 

Bx+C 

x +x+ ] x+x +x+ 

x +x−=Ax +x++Bx+Cx+

618 


AxBx+Cx=− 

 

 

x +x−=Ax +x++Bx+Cx+ 

 

− +−−=A− +−++B−+C−+ 

 

=A 

A= 

ATh 

 

 

 

x +x −=x +x++Bx+Cx+ 

x +x −=x +x++Bx +B+Cx+C 

x +x −=+Bx ++B+Cx++C 

ffiffi 

 

+B= 

+B+C= 

+C=− 

BC 

+B= 

 

B= 

+C=− 

 

C=− 

 

C=− 

Th 

 

x 

+x− 

x+x +x+ = 

 

x+ + 

x− 

x +x+ 

Q & A… 

Could we have just set up a system of equations to solve Example 3? 

AfiTh 

 

 

x +x −=Ax +Ax+A+Bx +B+Cx+C 

 

x +x −=A+Bx +A+B+Cx+A+C 

 

A+B= 

A+B+C= 

 

A+C=− 

Try It #3 

 

x −x+ 

x−x +


619 


Q(x ) 

Where Q(x ) Has a Repeated Irreducible Quadratic Factor 

fi 

fi 

Th 

 

decomposition of 

____ P(x) : when Q(x) has a repeated irreducible quadratic factor 

Q(x) 

Th Px 

Qx Qx 

PxQx 

Px 

 

ax 

+bx +c n = 

A 

x + B 

 

 

 

ax +bx +c + 

A 

x + B 

 

 

 

ax +bx +c + 

A 

x + B 

 

 

 

ax +bx +c ++ 

A n 

x + B n 

 

 

 

ax +bx +c n 

 

How To… 

 

1. ABCA 

x+B 

 

A 

x+B 

 

Px 

Qx = 

A 

 

ax+b + 

A 

x + B 

 

 

 

ax +bx +c + 

A 

x + B 

 

 

 

ax +bx +c ++ 

A n 

x + B n 

 

 

 

ax +bx +c n 

2. 

3. 

4. ffi 

 

Example 4 

Decomposing a Rational Function with a Repeated Irreducible Quadratic Factor in the Denominator 

 

x +x +x −x+ 

xx + 

Solution Thxx +x + 

Ax+B 

 

 

x +x +x −x+ 

 

xx + =A x 

+Bx+C + 

Dx+E 

 

x + x + 

xx + 

 

x +x +x −x+=Ax + +Bx+Cxx ++Dx+Ex 

 

x +x +x −x+=Ax +x ++Bx +Bx +Cx +Cx+Dx +Ex 

 

=Ax +Ax +A+Bx +Bx +Cx +Cx+Dx +Ex 

 

x +x +x −x+=A+Bx +Cx +A+B+Dx +C+Ex+A

620 


ffi 

 

A+B= 

 

C= 

 

A+B+D= 

 

C+E=− 

 

A= 

A=e fi 

 

+B= 

 

B= 

A=B= 

 

++D= 

 

D=− 

C= 

 

+E=− 

 

E=− 

A=B=C=D=− 

E=− 

x +x +x −x+ 

 

xx + = x + 

 

x + − 

x+ 

x + 

Try It #4 

 

 

x −x +x− 

x −x+ 

 

Partial Fraction Decomposition (http://openstaxcollege.org/l/partdecomp) 

Partial Fraction Decomposition With Repeated Linear Factors (http://openstaxcollege.org/l/partdecomprlf) 

Partial Fraction Decomposition With Linear and Quadratic Factors (http://openstaxcollege.org/l/partdecomlqu)



VERBAL 

1. 

 

 

3. 

 

2. 

 

 

4. 

 

 

5. 

 

 

x+ 

 

x +x+ = 

A 

 

x+ + 

B 

 

x+ 

x+=Ax++Bx+ 

 

xAB 

AB 

ALGEBRAIC 

, fi 

x+ 

6. 

x +x+ 

7. 

x − 

x −x− 

8. 

−x− 

x −x− 

x+ 

9. 

x +x+ 

x+ 

12. 

x +x+ 

15. x 

 

x − 

x+ 

18. 

x +x+ 

10. 

x 

 

x +x+ 

13. x 

 

x − 

16. 

x− 

x −x+ 

19. 

x− 

x −x+ 

11. 

x− 

 

x −x+ 

14. 

x 

 

x − 

17. x− x −x− 

, fi 

20. −x− 

x+ 

21. 

x 

 

x− 

22. x+ 

x+ 

23. −x− 

x+ 

24. −x− 

x− 

25. 

−x 

 

x− 

x+ 

26. 

x +x+ 

27. x +x+ 

xx+ 28. x +x+ 

xx+ 

29. x +x +x+ 

x x+ 30. x −x +x+ 

x x +x+


fi 

 

x 

31. +x+ 

x+x +x+ 

x 

34. +x+ 

x+x +x− 

32. 

x +x+ 

x−x +x+ 

35. 

x +x− 

x+x +x+ 

33. 

−x +x+ 

x−x +x+ 

36. 

x 

 

 

x+x +x− 

37. x +x+ x − 38. −x +x− 

x + 39. x −x+ 

x + 

40. x +x+ x − 41. x +x+ 

x − 42. −x +x− 

x − 

43. −x −x +x+ 

 

x +x 

fi 

44. x +x +x+ 

x + 45. x +x +x+ 

x + 46. x −x +x− 

x − 

47. x +x+ x+ 48. x +x +x 

 

x +x+ 

49. 

x + 

 

x +x+ 

50. x +x+x+ 

x +x+ 51. 

x+ 

xx + 

52. x +x +x +x+ 

xx + 

53. x − 

x −x 

54. x −x+ x +x 

EXTENSIONS 

, fi 

55. x + 

x+ 

56. x −x +x+ 

x − 

n fi 

57. 

x+ + 

 

x − − 

x − 

x −x − 

58. 

 

x − − 

 

x+ − 

x+ 

x +x − 

59. x 

 

x − − 

−x 

 

x +x+ − x− x −x

SECTION 7.5 MATRICES AND MATRIX OPERATIONS 

623 



Find the sum and difference of two matrices. 

Find scalar multiples of a matrix. 

Find the product of two matrices. 

7.5 MATRICES AND MATRIX OPERATIONS 

Figure 1 (credit: “SD Dirk,” Flickr) 

 

Table 1 

Wildcats Mud Cats 

 

 

 

Table 1 

fi 

 

. Th 

Finding the Sum and Difference of Two Matrices 

 

 

 

ABC 

 

A= [ 

] B =[ 

− 

] C = 

− 

[ 

]

624 


Describing Matrices 

ftm ×nmn 

fifiAfia ij 

 

ijAa 

 

 

a 

a 

a 

A= a 

 

[ 

a 

a 

a 

a 

a 

] 

n ×nTh 

× 

×n 

a 

a 

a 

 

m × 

 

a 

 

a [ 

a 

] 

ffi 

ft 

fi 

 

matrices 

matrixABC 

a ij 

ijft 

m ×nmn 

Example 1 

Finding the Dimensions of the Given Matrix and Locating Entries 

A 

a. A 

b. a 

a 

 

 

A 

 

=[ 

] − 

Solution 

a. Th× 

b. a 

Tha 

 

 

Adding and Subtracting Matrices 

 

 

Thaddition and subtraction of matrices is only possible when the 

matrices have the same dimensions×× 

××


625 

adding and subtracting matrices 

ABABC 

D 

+B =a ij 

+b ij 

=c ij 

 

A−B =Da ij 

−b ij 

=d ij 

 

 

A +B =B +A 

 

 

A+B+C =A +B+C 

Example 2 

AB 

Finding the Sum of Matrices 

A = [ a b 

Solution 

A +B = [ a b 

h ] 

 

c d ] B = [ 

e f 

g 

h ] 

 

c d ] + [ 

e f 

g 

= [ 

a + e b + f 

c + g d + h ] 

 

Example 3 

AB 

Adding Matrix A and Matrix B 

A = [ 

 

] B = [ ] 

Solution a 

A 

n 1b 

B 

A +B = [ 

 

] + [ ] 

= [ 

+ + + + ] 

 

= [ ] 

Example 4 

ffAB 

Finding the Difference of Two Matrices 

A = [ − 

Solution 

A −B = [ − 

 

] B = [ ] 

 

] − [ ] 

= [ −− − 

− − ] 

=[ − − − ]

626 


Example 5 Finding the Sum and Difference of Two 3 x 3 Matrices 

AB 

a. 

b. e diff 

 

− − 

A=[ 

 

 

− 

] [ B = 

 

 

− 

− − ] 

 

− − 

a. 

− 

A +B =[ 

− 

 

 

 

] 

[ + 

 

 

− − 

− ] 

 

− − − 

+ −+ 

 

−− 

=[ 

+ − 

− 

 

− − + − 

− 

 

=[ 

 

− 

] 

− − 

b. 

− 

A −B =[ 

− 

 

 

 

] 

[ − 

 

 

− − 

− ] 

 

− − − 

− −− 

 

−+ 

=[ 

− + 

+ 

 

+ −− + 

− − 

 

 

=[ 

 

 

− 

] 

] 

] 

Try It #1 

AB 

 

A= [ 

 

 

 

 

− ] 

B= [ 

 

− 

− 

 

] Finding Scalar Multiples of a Matrix 

 

 

Th 

scalar multiple 

 

Th 

. Th Table 2 

Lab A 

Lab B 

 

 

 

Table 2


627 

 

 

C 

 

=[ 

] 

C 

 

C 

 

=[ 

 

 

 

 

 

 

 

[ ] 

 

 

] [ + 

 

 

] [ = ] 

 

[ 

 

 

Th 

 

C 

 

=[ 

] 

 

] [ = 

] ThAB 

 

scalar multiplication 

s fi 

A = 

[ a a 

a 

a 

] 

cA 

cA=c 

[ a a 

a 

a 

] 

= 

[ ca ca 

ca 

ca 

] 

ABCab 

 

aA+B=aA+aB 

 

a+bA=aA+bA 

 

 

Example 6 Multiplying the Matrix by a Scalar 

A 

A = [ ] 

Solution A 

A = [ ] 

= [ ċ ċ ċ ċ ] 

=[ ]

628 


Try It #2 

B, fi−B 

A= [ ] 

Example 7 Finding the Sum of Scalar Multiples 

A +B 

 

− 

A=[ − 

− 

Solution , fiAB 

 

ċ − 

A 

ċ 

=[ ċ 

− ċ 

] 

 

ċ ċ − 

 

− 

 

=[ − 

] 

− 

 

− ċ 

B 

ċ 

=[ ċ 

− ċ 

 

ċ ċ − 

− 

] [ B = − 

] ] 

 

 

− 

 

− 

 

 

=[ − 

 

− 

A +B 

 

− − 

 

A +B =[ − 

] [ + 

 

− 

] − − 

 

− −+ + 

 

=[ + −− 

+ 

] 

 

+ + −− 

 

− 

 

=[ − 

] 

− 

Finding the Product of Two Matrices 

] 

 

fi 

Am ×rBr ×n 

ABm ×nABA 

Bfi 

A ⋅ B 

× × 

 

ABfiTh 

 

iABiAjB 

ABA×B×AB 

a 2 × 

A = 

[ a a b 

 

a 

b 

b 

 

 

a 

a 

a 

] [ B = 

b 

b 

b 

 

b 

b 

b 

]


629 

e fiAB 

1. AB A B 

b 

 

a 

a 

a 

 

[ 

b 

 

b 

] =a 

⋅b 

+a 

⋅b 

+a 

⋅b 

 

2. AB AB 

b 

 

a 

a 

a 

 

[ 

b 

 

b 

] =a 

⋅b 

+a 

⋅b 

+a 

⋅b 

 

3. AB AB 

b 

 

a 

a 

a 

 

[ 

b 

 

b 

] =a 

⋅b 

+a 

⋅b 

+a 

⋅b 

 

ABABA 

BAB 

AB= [ a ⋅b 

+a 

⋅b 

+a 

⋅b a 

⋅b 

+a 

⋅b 

+a 

⋅b 

a 

⋅b 

+a 

⋅b 

+a 

⋅b 

 

 

 

 

 

 

a 

⋅b 

+a 

⋅b 

+a 

⋅b 

a 

⋅b 

+a 

⋅b 

+a 

⋅b 

a 

⋅b 

+a 

⋅b 

+a 

⋅b 

] 

properties of matrix multiplication 

ABC 

 

ABC=ABC 

CA + B = CA + CB, 

 

A + BC = AC + BC. 

 

Example 8 Multiplying Two Matrices 

AB 

= [ 

 

] B = [ ] 

Solution A×B 

×ThTh× 

 

AB= [ ] [ ] 

= [ 

+ + 

+ + ] 

 

= [ ] 

Example 9 

Multiplying Two Matrices 

AB 

a. AB b. BA 

A =[ − 

] B = [ 

] 

 

− 

−

630 


Solution 

a. A×B× 

ABTh× 

AB 

AB=[ − − 

][ 

 

− 

 

] 

= [ 

−+−+ −−++ 

 

+−+ −++ ] 

 

=[ − ] 

b. ThB×A×Th 

× 

− 

BA=[ 

 

− 

 

] [ − 

] 

−+− +− 

 

+− 

=[ 

−−+ 

−+ −+ 

 

 

−+ + + 

− 

 

 

=[ 

− 

− 

 

Analysis Notice that the products AB and BA are not equal. 

AB=[ − − 

] [ ≠ 

− − 

] =BA 

 

This illustrates the fact that matrix multiplication is not commutative. 

] 

] 

Q & A… 

Is it possible for AB to be defined but not BA? 

A×B×AB 

fiBA 

fi 

Example 10 

Using Matrices in Real-World Problems 

Table 3, 

 

Wildcats Mud Cats 

 

 

 

Table 3 

Table 4. 

Goals 

Balls 

Jerseys 

Table 4


631 

. Th 

 

 

E =[ 

] 

Th 

C = 

 

] 

 

CE= [ 

 

 

 

=++++ 

 

= 

Th 

How To… 

 

1. ABC 

2. 

3. fifi 

 

Example 11 

Using a Calculator to Perform Matrix Operations 

AB−C 

 

− 

A=[ − 

− 

− 

] B = [ 

 

− 

− 

− − 

− − − 

] [ C = 

− 

] 

 

− − 

Solution AAB 

BCC 

 

A×B−C 

Th 

[ 

] 

− 

 

− 

 

− 

 

− 

 

Dimensions of a Matrix (http://openstaxcollege.org/l/matrixdimen) 

Matrix Addition and Subtraction (http://openstaxcollege.org/l/matrixaddsub) 

Matrix Operations (http://openstaxcollege.org/l/matrixoper) 

Matrix Multiplication (http://openstaxcollege.org/l/matrixmult)



VERBAL 

1. 

 

 

3. ABBA 

 

5. e? Th 

AB=BA 

 

2. 

 

4. 

 

 

 

ALGEBRAIC 

 

fi 

A=[ 

 

] B = [ ] [ C = 

 

 

 

 

] [ D = 

] E = 

[ 

 

 

 

] F = [ 

] 

 

 

 

 

 

6. A+B 7. C+D 8. A+C 9. B−E 10. C+ 11. D−B 

 

A=[ 

 

] B = [ 

 

 

 

 

 

] C = 

[ 

 

 

 

 

] D = [ 

] 

12. A 13. B 14. −B 15. −C 16. 

C 

17. D 

 

A=[ − 

 

 

] B = [ − ] [ C = 

 

 

− 

 

 

] D = [ 

] 

 

− 

 

− 

18. AB 19. BC 20. CA 21. BD 22. DC 23. CB 

 

 

A=[ − 

] B = [ − 

− ] C =[ 

 

 

 

] D = [ − − 

 

] [ E = 

− − 

] 

24. A + B − C 25. A + D 26. C + B 27. D + E 28. C − D 29. D − E


 

A =A ċA 

A=[ − 

 

 

− 

 

− 

] B = 

[ − ] [ C = 

] 

30. AB 31. BA 32. CA 33. BC 34. A 35. B 

36. C 37. B A 38. A B 39. AB 40. BA 

 

A =A ċA 

A=[ 

 

] B = [ − − − ] [ C = 

 

 

 

− 

] D = [ 

] 

 

− 

− 

 

41. AB 42. BA 43. BD 44. DC 45. D 46. A 

47. D 48. ABC 49. ABC 

TECHNOLOGY 

 

 

 

− 

A=[ − 

 

] B = [ 

 

 

− 

 

 

] [ C = 

] 

50. AB 51. BA 52. CA 53. BC 54. ABC 

EXTENSIONS 

 

 

= [ 

 

] 55. B 56. B 57. B 58. B 

59. B n B B

634 




Write the augmented matrix of a system of equations. 

Write the system of equations from an augmented matrix. 

Perform row operations on a matrix. 

Solve a system of linear equations using matrices. 

7. 6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION 

Figure 1 German mathematician Carl Friedrich Gauss (1777–1855). 

 

fifi 

ff 

 

fiSystems of Linear Equations: Two Variables 

 

Writing the Augmented Matrix of a System of Equations 

 

ffi 

ffi 

augmented matrix 

× 

 

x +y = 

 

x −y = 

 

[ − ∣ ] 

ffi. Thcoefficient matrix 

[ − ]

SECTION 7.6 SOLVING SYSTEMS WITH GAUSSIAN ELIMINATION 

635 

 

 

 

 

ffi 

 

[ 

x −y −z = 

[ 

x +y = 

x −z = 

] 

 

− − 

 

− 

] 

 

− − 

 

 

∣ 

− 

xfi 

yz 

ax+by+cz=d 

ffi 

How To… 

 

1. ffixe fi 

2. ffiy 

3. zffi 

4. 

Example 1 Writing the Augmented Matrix for a System of Equations 

 

 

x +y −z = 

 

x −y +z = 

 

x −y +z = 

Solution Thffi 

 

− 

[ − 

 

∣ 

] 

− 

Try It #1 

 

 

x −y = 

 

x+y= 

Writing a System of Equations from an Augmented Matrix 

 

 

fi

636 


Example 2 Writing a System of Equations from an Augmented Matrix Form 

 

[ 

] 

 

− − − 

− − 

 

∣ 

− 

Solution xyz 

 

− − 

[ − − 

 

− 

∣ 

] → 

 

x −y −z =− 

x −y −z = 

 

 

− −x +y +z = 

Try It #2 

 

[ 

] 

 

− 

− 

 

 

∣ 

 

− 

Performing Row Operations on a Matrix 

row operations 

 

 

row-echelon formmain diagonal 

 

 

a b 

[ 

 

d 

] 

 

row-equivalent 

 

1. leading 

2. 

3. 

4. 

ffi 

o fi 

1. R i 

↔R j 

 

2. cR i 

 

3. R i 

+cR j 

 

 

 

fi 

fi


637 

Gaussian elimination 

Gaussian eliminationTh 

A 

 

a 

a 

a 

A= a 

 

[ 

a 

a 

] =[ ft A 

 

b 

b 

 

b 

a 

a 

a 

] 

 

 

 

 

How To… 

 

1. The fiffi 

2. e fie fi 

3. 

4. 

5. 

6. 

 

7. 

Example 3 

Solving a 2 × 2 System by Gaussian Elimination 

 

 

x +y = 

 

x −y = 

 

Solution 

 

[ − 

] 

 

. Th 

∣ 

R 

↔R 

→ [ − ∣ 

] 

fiTh 

− 

−R 

+R 

=R 

→ [ − ∣ 

] 

 

 

 

 

R 

=R 

→ [ − ∣ 

] 

. Thy =y =e fi 

 

 

Th( 

) 

x −= 

 

x =

638 


Try It #3 

 

 

x +y = 

x−=− 

Example 4 

Using Gaussian Elimination to Solve a System of Equations 

× 

 

x +y = 

 

x +y = 

Solution 

[ ∣ ] 

. The fi 

 

 

 

R 

=R 

→[ 

∣ 

] 

− 

−R 

+R 

=R 

→[ 

∣ 

] 

Th=. Th 

Example 5 

 

 

 

Solving a Dependent System 

x +y = 

x +y = 

Solution 

A =[ ∣ ] 

 

− 

R 

+R 

=R 

→ [ ∣ ] 

R 

↔R 

→[ ∣ ] 

Thy =Thfi 

fio fiy 

 

x +y = 

 

y =−x 

 

( x− 

x ) 

y =− 

x 

Example 6 Performing Row Operations on a 3×3 Augmented Matrix to Obtain Row-Echelon Form 

 

[ 

] 

 

− 

− 

∣ 

− 

Solution ThfiTh−w


639 

Th 

− 

−R 

+R 

=R 

→ [ 

 

 

− 

 

∣ 

 

] 

− 

 

 

− 

R 

+R 

=R 

→ 

 

[ − 

 

 

∣ 

 

] 

− 

 

 

− 

 

R 

+R 

=R 

→ [ − 

 

 

∣ 

 

] 

 

Th 

 

− 

 

R 

=R 

→[ 

− 

 

 

∣ 

− 

 

] 

 

Try It #4 

 

x −y +z = 

 

−x+y =− 

 

x−y+z= 

Solving a System of Linear Equations Using Matrices 

 

 

Th 

 

Example 7 

Solving a System of Linear Equations Using Matrices 

 

 

x −y +z = 

 

x +y −z =− 

 

x −y −z = 

Solution 

 

− 

− 

 

 

∣ 

− 

− − 

 

− 

−R 

+R 

=R 

→ [ 

 

− 

 

 

∣ 

− 

− − 

ThR 

R 

 

− 

R 

R 

 

→[ 

− − 

− − 

[ 

] 

] −R 

+R 

=R 

→ [ 

] 

] 

 

− 

 

− 

 

 

∣ − 

 

− −

640 


 

 

− 

−R 

+R 

=R 

→[ 

− 

 

 

∣ 

− ] − R 

=R 

→[ 

 

− 

 

− 

 

 

∣ 

− 

 

 

Th 

 

x −y +z = 

 

y −z =− 

 

z = 

− 

Example 8 

Solving a Dependent System of Linear Equations Using Matrices 

 

 

−x−y +z =− 

 

x +y = 

 

y −z = 

Solution 

− − 

[ 

− 

 

 

 

∣ 

− 

−. Th 

 

− − − 

−R 

→ 

 

[ 

 

 

∣ 

] 

− 

 

 

R 

↔R 

→ 

− 

[ − 

 

∣ 

 

 

 

−R 

+R 

=R 

→ 

− 

[ − 

 

∣ 

− 

− 

 

R 

+R 

=R 

→[ 

 

 

− 

 

∣ 

 

Th 

 

x +y −z = 

 

y −z = 

 

= 

=fifi 

yfiz 

x 

 

x +y −z = 

 

y = 

 

x +z−z = 

 

x +z = 

 

] 

z = −x 

 

 

] 

] 

] 

]


641 

zyx 

 

y −z = 

 

z = −x 

 

 

 

y −( −x 

) = 

 

y = −x 

 

Th( 

 

x−x 

−x ) 

Try It #5 

 

x +y −z = 

 

x +y +z = 

x+y−z= 

Q & A… 

Can any system of linear equations be solved by Gaussian elimination? 

 

How To… 

 

1. ABC 

2. ref( 

Example 9 

 

 

 

 

Solving Systems of Equations with Matrices Using a Calculator 

x +y +z =− 

−x +y −z =− 

−x−y +z = 

Solution 

 

− 

 

− − 

 

∣ 

− 

− − − 

A 

− 

A=[ 

 

− − − 

] 

 

− − 

ref(A 

 

A 

 

[ 

 

] 

x + y + z =− 

 

− → y + 

z =− 

− 

z =− 

( 

− 

− 

) 

[ 

]

642 


Example 10 

Applying 2 × 2 Matrices to Finance 

 

Th 

Solution x = 

y = 

 

x +y = 

 

x +y = 

 

[ 

∣ ] 

− 

[ ∣ ] 

 

 

y = 

 

y = 

−= 

Th 

Example 11 Applying 3 × 3 Matrices to Finance 

 

ThTh 

 

Solution xy 

z. Th 

 

x +y +z = 

 

x +y +z = 

 

x −z = 

 

 

 

[ 

 

∣ 

 

] − 

 

 

 

−R 

+R 

=R 

 

→[ 

 

∣ 

] − 

 

 

−R 

+R 

=R 

→ 

 

[ 

 

 

∣ 

 

− − − 

 

 

 

 

R 

=R 

→[ 

 

∣ 

− − − 

[ 

 

 

 

R 

+R 

=R 

→ 

 

∣ 

− 

 

] − 

Th− z =−z = 

 

] 

]


643 

Thy + z =z = 

 

 

y + 

= 

 

y += 

 

y = 

The fix +y +z =y =z = 

 

x ++= 

 

x = 

Th 

Try It #6 

 

t 10%. Th 

 

 

 

 

Solve a System of Two Equations Using an Augmented Matrix (http://openstaxcollege.org/l/system2augmat) 

Solve a System of Three Equations Using an Augmented Matrix (http://openstaxcollege.org/l/system3augmat) 

Augmented Matrices on the Calculator (http://openstaxcollege.org/l/augmatcalc)



VERBAL 

1. 

 

 

3. 

o diff 

 

2. 

 

 

4. 

 

 

[ − ∣ ] 

5. 

 

ALGEBRAIC 

 

6. x−y= 

x+y= 

9. x+y+z= 

x+y= 

x+y+z=− 

7. y= 

x−y= 

10. x+y+z= 

x−y+z=− 

x+y=− 

8. x+y+z= 

−x+y+z= 

x+z= 

 

 

∣ ] 

11. [ − − ∣ ] 12. [ 

 

14. [ 

 

 

− 

 

∣ 

 

] 15. [ 

] 

 

− 

 

 

 

∣ 

 

− − 

 

16. 

[ ∣ 

 

 

 

] 

[ 13. 

 

− − 

 

 

∣ 

− 

 

− ∣ − ] 

] 17. 

[ ∣ ] 18. 

[ ∣ ] 19. 

[ − 

 

20. 

[ − ∣ − ] 

21. x−y=− 

x+y= 

22. x+y=− 

x+y=− 

23. x+y= 

x+y= 

24. −x−y=− 

x−y=− 

25. −x+y= 

x+y= 

26. x+y= 

−x−y=− 

27. −x+y= 

x−y=− 

28. x+y= 

x+y= 

32. 

x− 

y=− 

 

x+ 

y= 

29. x−y= 

x+y= 

33. 

[ 

 

 

∣ 

 

] 34. 

30. −x−y= 

−x−y= 

 

 

[ ∣ 

 

 

 

− 

] 35. 

31. 

x− 

y= 

 

x+ 

y= 

 

 

[ ∣ 

]


36. 

 

− − 37. −x+y−z= 

[ − 

 

∣ 

 

] x+y−z= 

− 

x−y+z=− 

39. x+y+z= 

−x−y−z=− 

x+y+z= 

40. x+y−z= 

−x−y+z=− 

x+y−z= 

38. x+y−z=− 

x−y−z= 

x+y+z= 

41. x+y−z= 

−x−y+z=− 

x+y−z= 

42. x+y= 

x+z= 

−y−z=− 

45. − 

x+ 

y+ 

z=− 

 

 

x− 

y+ 

z= 

 

x+ 

y+ 

z= 

 

43. x+y+z= 

x+z= 

−y+z= 

46. − 

x− 

y+ 

z=− 

 

 

x+ 

y− 

z= 

 

− 

x+ 

y+ 

z=− 

 

44. 

x− 

z=− 

 

 

x+ 

y= 

 

 

y− 

z= 

 

EXTENSIONS 

 

47. x− 

+y − 

+z − 

= 

x+y+z= 

x+ 

+y+z− 

= 

48. x− 

−y+ 

+z=− 

x+ 

+y+ 

−z= 

x+y− z− 

= 

49. x− 

−y− 

+z=− 

x+ 

+y+ 

+z+ 

= 

x+y+z= 

50. x− 

+y+ 

−z= 51. x− 

−y− 

+z=− 

x+ 

−y− 

+z= 

x+ 

+y+ 

+z+ 

= 

x− 

+y+ 

+z= 

x+y+z= 


 

52. 

a fle fl 

a fl 

 

53. 

. Th 

 

 

h fl 

54. 

 

ft 

ft 

55. 

ft


56. 

 

ft 

 

58. Thm fl 

 

e fl 

 

 

 

y fl 

60. 

s. Th 

 

Th 

 

 

57. 

. Th 

 

 

 

fift 

 

59. e fl 

 

 

 

les. Th 

 

 

 

 

 

61.

SECTION 7.7 SOLVING SYSTEMS WITH INVERSES 

647 



Find the inverse of a matrix. 

Solve a system of linear equations using an inverse matrix. 

7.7 SOLVING SYSTEMS WITH INVERSES 

ffThfi 

 

 

Th 

fift 

 

Finding the Inverse of a Matrix 

aa − aa − =a − a =( a ) a = − = 

( 

) =ThA 

A − Th 

I n 

Th 

×× 

I 

=[ ] 

 

I =[ 

 

 

AI=IA=A 

] 

 

 

AA − =I 

 

A − A =I 

 

AA − =A − A =A 

A − fi× 

×× 

the identity matrix and multiplicative inverse 

identity matrixI n 

 

I 

=[ 

] I 

=[ 

 

] 

× × 

An ×nBn ×nAB=BA=I n 

B =A − multiplicative inverse 

of a matrixA

648 


Example 1 Showing That the Identity Matrix Acts as a 1 

AAI=IA=A 

 

A =[ − ] 

Solution A 

A 

AI=[ − ] 

[ ] = [ ċ+ċ ċ+ċ 

 

−ċ+ċ −ċ+ċ ] = [ − ] 

AI=[ ] [ − ] = [ 

ċ+ċ− ċ+ċ 

 

ċ+ċ− ċ+ċ ] 

=[ − ] 

How To… 

 

1. An ×nBn ×nAB 

2. AB=n fiBABA=IB =A − A =B − 

Example 2 

Showing That Matrix A Is the Multiplicative Inverse of Matrix B 

 

A =[ − − ] B = [ − − ] 

Solution ABBA 

AB=[ − − ] [ − − ] 

 

= [ −+ −+ 

 

 

 

−−− −−− ] 

 

=[ ] 

BA=[ − − ] [ − − ] 

= [ −−− −−− 

 

+− −+− ] 

 

=[ ] 

AB 

Try It #1 

 

A=[ −− ] B= [ − − ] 

Finding the Multiplicative Inverse Using Matrix Multiplication 

fi 

fi


649 

Example 3 

Finding the Multiplicative Inverse Using Matrix Multiplication 

o fi 

A =[ − − ] 

Solution A 

[ − c d ] =[ ] 

 

− ] [ a b 

[ − − ] [ a b 

c d ] = [ a −c b −d 

a −c b −d ] 

fi 

 

a −c = R 

 

a −c = R 

 

−R 

+R 

→R 

c 

 

a −c = 

 

+c =− 

 

c =− 

a 

 

a −−= 

 

a += 

 

a =− 

 

 

b −d = R 

 

b −d = R 

 

−R 

+R 

=R 

d 

 

b −d = 

 

+d = 

 

d = 

b 

 

b −= 

 

b −= 

 

b = 

A − =[ − − ] 

Finding the Multiplicative Inverse by Augmenting with the Identity 

fiA 

IIA − 

 

A =[ ] 

A

650 


[ ∣ ] 

A 

1. 

2. − 

3. − 

4. 

[ ∣ ] 

[ ∣ − ] 

[ − ∣ − − ] 

5. − 

ThA − 

 

[ − ∣ − − ] 

[ ∣ − − ] 

A − =[ − − ] 

Finding the Multiplicative Inverse of 2 × 2 Matrices Using a Formula 

fi× 

 

A× 

A = [ a b 

c d ] 

A 

 

A − = 

ad−bc [ d −b 

−c a ] 

ad−bc≠ad−bc=A 

Example 4 Using the Formula to Find the Multiplicative Inverse of Matrix A 

o fi 

A =[ − − ] 

Solution 

 

A − = 

 

 

−−− [ − − ] 

 

= 

−+ [ − − ] 

=[ − − ] 

Analysis We can check that our formula works by using one of the other methods to calculate the inverse. Let’s augment 

A with the identity. 

[ − − ∣ ] 

Perform row operations with the goal of turning A into the identity.


651 

1. Multiply row by −and add to row 

[ − ∣ − ] 

2. Multiply row by and add to row 

[ ∣ − − ] 

So, we have verified our original solution. 

A − =[ − − ] 

Try It #2 

o fiA 

A − =[ − ] 

Example 5 

Finding the Inverse of the Matrix, If It Exists 

 

A = [ ] 

Solution 

1. 

[ ∣ ] 

[ ∣ ] 

2. − 

[ ∣ − ] 

3. Th. Th 

Finding the Multiplicative Inverse of 3 × 3 Matrices 

×fi× 

 

× 

 

A =[ 

 

] 

A 

 

A 

 

A I =[ 

 

∣ 

 

 

 

] A 

Aft 

ftfi 

 

How To… 

×, fi 

1. 

2. ft 

3. 

4. AA − =IA − A =I

652 


Example 6 Finding the Inverse of a 3 × 3 Matrix 

×A, fi 

 

A =[ 

 

] 

Solution AA 

ThA 

 

[ 

 

∣ 

 

 

 

] 

 

[ 

∣ 

 

 

 

] 

 

 

− 

−R 

+R 

=R 

→[ 

 

∣ 

 

 

 

] 

 

 

−R 

+R 

=R 

 

→[ 

∣ 

 

− 

 

] − 

 

− 

R 

↔R 

→ [ ∣ 

 

 

− 

] 

 

−R 

+R 

=R 

→ [ 

− 

 

∣ 

 

 

− 

] − 

 

−R 

+R 

=R →[ 

− 

 

∣ 

 

 

− 

 

] − − 

 

− 

A − =B 

 

=[ 

− 

] − − 

Analysis To prove that B = A −1 , let’s multiply the two matrices together to see if the product equals the identity, if 

A A − = I and A − A = I. 

R 

R 

 

− 

] 

 

[ ] ] 

] [ 

 

] 

 

 

 

] 

 

AA =[ − 

 

 

 

− 

 

 

− − 

−+−+ ++− ++− 

=[ 

−+−+ 

++− ++− 

] 

 

−+−+ ++− ++− 

 

=[ 

 

 

− 

A − A 

 

=[ 

− 

 

− − 

−++ −++ −++ 

 

=[ 

 

−++ −++ −++ 

] 

 

+−+− +−+− +−+− 

 

=[


653 

Try It #3 

× 

] 

− 

A=[ 

 

− − 

 

− 

Solving a System of Linear Equations Using the Inverse of a Matrix 

fiX 

B 

fi 

 

AX=B 

AffiX 

BThAX=B 

 

 

a 

x +b 

y =c 

 

 

a 

x +b 

y =c 

 

ffi 

A = [ a b ] a 

b 

 

Th 

 

 

 

AX=B 

X = 

[ x 

y ] 

B = 

[ c c ] 

[ a b ] a 

b 

[ x 

c ] 

y ] = [ c 

− =( 

) = 

ax=bx 

a 

 

ax=b 

 

 

 

 

 

( 

a ) ax=( a ) b 

a − ax=a − b 

a − ax=a − b 

x =a − b 

x =a − b 

Thfffi 

 

 

××

654 


solving a system of equations using the inverse of a matrix 

ffiAXB 

 

AX=B 

A 

 

 

 

A − AX=A − B 

A − AX=A − B 

IX=A − B 

X =A − B 

Q & A… 

If the coefficient matrix does not have an inverse, does that mean the system has no solution? 

ffi 

fi 

Example 7 Solving a 2 × 2 System Using the Inverse of a Matrix 

 

 

x +y = 

 

x +y = 

Solution ffi 

A=[ ] X = [ x y] B = [ ] 

 

[ ] [ x y] = [ ] 

A − 

 

A − = 

ad−bc [ 

d −b 

−c a ] 

 

= 

− 

[ − − ] 

 

= 

 

[ − − ] 

 

A − =[ − − ] 

A − 

 

[ − 

A − AX=A − B 

− ] 

[ ] [ x y] = 

[ − ] [ 

 

− 

 

[ ] 

[ x y] = [ 

+− 

−+ ] 

 

] 

 

Th− 

 

[ x 

] 

y ] = [ −


655 

Q & A… 

Can we solve for X by finding the product B A −1 ? 

A − B ≠BA − 

 

A − AX=A − B 

 

A − A=A − B 

 

IX=A − B 

 

X =A − B 

fiA − A − A 

B 

Example 8 Solving a 3 × 3 System Using the Inverse of a Matrix 

 

 

x +y +z = 

 

−x −y −z =− 

 

−x−y −z =− 

Solution AX=B 

x 

[ 

 

− − 

− 

] [ y 

 

] [ = 

 

 

− ] 

− − − z − 

l fiA 

 

[ 

− 

− 

−∣ 

 

− − − 

] 

 

 

 

[ 

− 

− 

− 

∣ 

 

 

] 

− − − 

 

 

 

[ 

 

 

 

 

 

∣ 

 

 

] 

 

− − − 

 

 

− 

 

[ 

 

 

] 

 

∣ 

 

 

 

− 

[ − 

] 

− 

 

∣ 

 

 

 

[ 

 

 

 

 

 

− 

 

∣ 

 

− 

 

− 

 

]

656 


 

 

− 

 

 

− − 

[ ∣ ] 

 

− − 

[ 

∣ − 

− 

] 

 

] 

 

− − 

A =[ − − − 

 

 

A − A − AX=A − B 

 

 

 

[ 

Th 

] [ 

x 

] [ ] [ = 

 

− − 

− − 

 

 

− − − 

y 

 

− − − z 

] [ 

] 

 

− − 

− − 

 

 

 

− 

− 

−+− 

A − B =[ 

 

−−+ 

 

+− 

] [ = 

] 

 

Try It #4 

ffi 

 

x −y +z = 

 

−x+y −z = 

 

y−z=− 

How To… 

 

1. ffiAB 

2. 

3. ffiffi 

 

Example 9 Using a Calculator to Solve a System of Equations with Matrix Inverses 

 

 

x +y +z = 

 

x +y +z =− 

 

x +y +z =− 

Solution ffiA 

B 

 

A=[ 

 

] B= 

 

[ − 

] 

 

−


657 

X 

 

 

A − ×B 

 

 

 

− [ − ] 

 

 

The Identity Matrix (http://openstaxcollege.org/l/identmatrix) 

Determining Inverse Matrices (http://openstaxcollege.org/l/inversematrix) 

Using a Matrix Equation to Solve a System of Equations (http://openstaxcollege.org/l/matrixsystem)



VERBAL 

1. 

AB≠BA 

 

 

A − A=AA − 

3. × 

 

5. 

 

× 

2. × 

 

 

4. 

 

ALGEBRAIC 

AB 

6. A=[ 

8. A=[ 

 

 

] 

 

− ] B= [ ] 7. A= [ ] B= [ − − 

 

− 

] B= [ 

 

 

 

10. 

 

A=[ − 

] B= 

[ 

 

− 

 

− 

 

 

12. A=[ 

] B= [ 

 

− − 

− 

] 

− − 

] 

9. A= [ − 

 

] [ 11. A= 

 

− ] B=[ − − − − ] 

, fi 

] B= 

[ 

 

− 

− − 

] 

 

− 

13. [ − ] 14. [ − ] 15. [ − ] 16. [ − − − ] 

17. [ ] 18. [ ] 19. [ − ] [ 20. 

21. [ 

 

− 

 

 

] 22. [ 

 

− 

− 

− − − 

] 23. [ 

 

− 

 

− 

] 24. [ 

] 

 

 

− 

 

] 

 

− 

− − 

 

 

25. [ 

 

] 

26. [ 

 

 

 

]


× 

27. x−y=− 

x+y=− 

28. x+y=− 

x−y= 

29. x−y= 

−x+y=− 

30. x−y=− 

x+y= 

31. −x−y= 

x+y=− 

32. −x+y= 

−x+y= 

 

33. 

x− 

y= 

 

− 

x+ 

y= 

34. 

x+ 

y=− 

 

 

x− 

y=− 

 

× 

35. x−y+z= 

x+y= 

x−y−z= 

36. x+y+z= 

x−y+z=− 

−x+y+z= 

37. x−y−z= 

−x+y+z=− 

x+y+z= 

38. x−y+z=− 

x+y−z= 

x+y+z= 

39. x−y+z=− 

x+y−z= 

y−z= 

40. 

x− 

y+z=− 

 

 

x−y+ 

z=− 

 

x+y− 

z= 

41. 

x− 

y+ 

z= 

 

− 

x− 

y+ 

z= 

− 

x− 

y+ 

z= 

42. x+y+z=− 

x−y+z= 

y+z=− 

TECHNOLOGY 

 

43. x−y=− 

−x+y= 

45. x−y −z= 

x+y −z=− 

y−z=− 

44. − 

x− 

y=− 

 

 

x+ 

y= 

 

46. x−y+z=− 

x−y=− 

x+y+z= 

EXTENSIONS 

 

47. 

[ 

51. 

 

 

 

] 

 

 

 

 

 

[ ] 

 

 

 

 

 

 

 

 

 

 

 

[ 

 

− 

48. 

− 

[ 

 

49. 

 

 

 

 

− 

] − 

− − 

] 50. 

 

[ ]



Th 

 

52. 

e diff 

 

 

54. o diff 

s. Th 

 

 

 

 

56. 

ies. Th 

 

t $0.75. Th 

 

58. 

 

 

 

60. 

. Th 

 

t. Th 

 

ft ft 

 

 

53. 

 

 

55. 

 

 

 

 

 

57. 

e diff 

ts. Th 

 

 

 

 

 

 

59. 

 

 

 

 

61.

SECTION 7.8 SOLVING SYSTEMS WITH CRAMER'S RULE 

661 



Evaluate 2 × 2 determinants. 

Use Cramer’s Rule to solve a system of equations in two variables. 

Evaluate 3 × 3 determinants. 

Use Cramer’s Rule to solve a system of three equations in three variables. 

Know the properties of determinants. 

7.8 SOLVING SYSTEMS WITH CRAMER'S RULE 

 

 

 

 

Evaluating the Determinant of a 2 × 2 Matrix 

 

 

 

 

Th 

 

fi 

find the determinant of a 2 × 2 matrix 

determinant× 

A= [ 

a b 

c d ] 

fi 

A=∣ a b 

c d∣ =ad−cb 

ThA 

∣ A ∣ 

Example 1 

Finding the Determinant of a 2 × 2 Matrix 

 

Solution 

 

 

A=[ − ] 

 

A=∣ − ∣ 

=−− 

= 

Using Cramer’s Rule to Solve a System of Two Equations in Two Variables 

fi 

 

Introduction à l'Analyse des lignes Courbes algébriques 

ffifi

662 


 

fifi 

 

 

 

a 

x+b 

y=c 

 

a 

x+b 

y=c 

 

x 

ffiyffiy 

y 

b 

a 

x+b 

b 

y=b 

c 

R 

b 

 

 

−b 

a 

x−b 

b 

y =−b 

c 

R 

−b 

 

 

b 

a 

x−b 

a 

x=b 

c 

−b 

c 

 

x 

 

 

b 

a 

x−b 

a 

x=b 

c 

−b 

c 

 

xb 

a 

−b 

a 

=b 

c 

−b 

c 

 

 

yx 

x= bc [ 

 

−b 

c 

c b ] b 

a 

−b 

a 

= 

c 

b 

 

 

[ a b ] 

a 

b 

 

 

a 

a 

x+a 

b 

y=a 

c 

R 

a 

 

 

−a 

a 

x−a 

b 

y=−a 

c 

R 

−a 

 

 

a 

b 

y−a 

b 

y=a 

c 

−a 

c 

 

y 

 

a 

b 

y−a 

b 

y=a 

c 

−a 

c 

 

 

ya 

b 

−a 

b 

=a 

c 

−a 

c 

 

 

 

y= ac 

−a 

c 

 

_ 

a 

b 

−a 

b 

=a c 

−a 

c 

[ a c a 

b 

−a 

b 

= 

a 

c ] 

 

[ a b 

a 

b 

 

xyffi 

xy 

 

D 

D x 

x 

x= D _ x D 

D y 

y 

 

D y 

 

 

y=_ 

D 

Th 

xy


663 

Cramer’s Rule for 2 × 2 systems 

Cramer’s Rule 

 

 

 

a 

x+b 

y=c 

 

 

a 

x+b 

y=c 

 

Th 

x= D [ c b ] _ x 

 

D = 

c 

b 

 

D 

_ 

 

[ a b ] y 

[ a c 

D≠y= 

_ 

a 

b 

D = 

a 

c ] 

_ 

 

[ a b ] D≠ 

a 

b 

 

xxyy 

 

Example 2 

Using Cramer’s Rule to Solve a 2 × 2 System 

× 

 

x +y = 

 

x −y = 

Solution x 

x= D x 

∣ D = −∣ _ 

∣ = 

−∣ −− 

−− =− 

− = 

y 

y= D y 

 

∣ D = ∣ _ 

 

∣ = 

−∣ − 

−− =− 

=− 

Th− 

Try It #1 

× 

x+y=− 

 

−x+y=− 

Evaluating the Determinant of a 3 × 3 Matrix 

×fi× 

×fi× 

Thdown 

up Th 

 

× 

 

a 

b 

c 

= a 

 

[ 

b 

c 

 

a 

b 

c 

]

664 


 

1. A 

 

a 

b 

c 

 

a 

A=∣ 

 

b 

c 

 

a 

b 

 

∣ a 

b 

∣ 

a 

b 

c 

a 

b 

 

2. 

 

3. ftfi 

 

 

a 

b 

c 

 

 

a 

A=∣ 

 

b 

c 

 

a 

b 

 

∣ a 

b 

∣ 

a 

b 

c 

a 

b 

 

Th 

∣ A ∣=a 

b 

c 

+b 

c 

a 

+c 

a 

b 

−a 

b 

c 

−b 

c 

a 

−c 

a 

b 

 

Example 3 

Finding the Determinant of a 3 × 3 Matrix 

× 

] 

 

 

=[ − 

 

e fi. Th 

 

 

 

∣ 

∣=∣ 

− 

 

 

∣ − 

∣ 

 

 

=−++−−−− 

 

=+++−− 

 

= 

Try It #2 

× 

 

− 

 

A=∣ 

∣ 

− 

Q & A… 

Can we use the same method to find the determinant of a larger matrix? 

×× 

ft 

Using Cramer’s Rule to Solve a System of Three Equations in Three Variables 

fi× 

 

×× 

 

fio fi


665 

× 

 

a 

x + b 

y + c 

z =d 

 

 

a 

x + b 

y + c 

z =d 

 

 

a 

x + b 

y + c 

z =d 

 

x= D D 

_ x 

 

D y= 

y 

 

_ 

 

D z=D _ z 

 

D D≠ 

 

a 

 

b 

c 

d 

a D=∣ 

 

b 

c 

 

b 

c 

 

∣D x 

d a 

b 

c 

 

=∣ 

 

b 

c 

 

a 

d 

c 

 

∣D y 

a 

d 

b 

c 

 

=∣ 

 

d 

c 

 

a 

b 

d 

 

∣D z 

a 

a 

d 

c 

 

=∣ 

 

b 

d 

∣ 

a 

b 

d 

 

D x 

x 

D y 

yD z 

 

z 

Example 4 

Solving a 3 × 3 System Using Cramer’s Rule 

× 

 

x +y −z = 

 

x −y +z =− 

 

x +y −z = 

Solution 

 

 

− 

 

D=∣ =∣ − 

∣D 

− 

− − 

− 

x 

 

− 

∣D y 

 

− 

=∣ =∣ − 

∣D 

 

 

− − 

− 

z ∣ 

 

 

 

 

Th− 

x= D _ x 

 

D =− 

− = 

D y 

 

y=_ 

 

D =− 

− = 

z= D _ z 

 

D = 

− =− 

Try It #3 

× 

x−y+z= 

x+y+z= 

x−y+z= 

Example 5 Using Cramer’s Rule to Solve an Inconsistent System 

 

x −y = 

x −y = 

Solution y fiDD x 

D y 

 

D=∣ − −∣ =−−−=

666 


fi 

 

1. − 

2. 

 

−x +y =− 

 

x −y = 

 

=− 

=−Th 

Figure 1 

y 

 

 

 

 

 

– – – – – – 

 

 

 

 

 

y = x 

y = x − 

x 

– 

– 

– 

– 

Figure 1 

Example 6 Use Cramer’s Rule to Solve a Dependent System 

fi 

x −y +z = 

x +y −z = 

x −y +z = 

Solution s fit fie fi 

 

∣ 

− 

 

 

− 

 

− 

∣ 

 

∣ 

− 

− 

+−−+−−−−−−−= 

fi 

o fi 

1. − 

 

 

 

−x +y −x = 

x −y +z = 

= 

2. = 

 

Figure 2


667 

x − y +z = 

x−y+z= 

x+y+z= 

Understanding Properties of Determinants 

Figure 2 

Th 

 

properties of determinants 

1. 

 

2. 

3. 

4. 

5. ThA − A 

6. 

Example 7 

Illustrating Properties of Determinants 

 

Solution 

 

] 

 

 

 

 

A=[ 

− 

Ae fi 

 

 

 

A=[ 

 

 

 

 

∣ 

 

] 

− 

 

 

A=−++−−+ 

 

=− 

 

A=[ − − ] A=−−−=−=− 

 

B=[ − − ] B=−−−=−=

668 


 

 

 

 

A=∣ 

 

 

 

∣ 

] − − 

 

 

A=+−++−− 

 

=−++−−= 

. Th 

A=[ ] A=−= 

A − A 

A=[ ] A=−=− 

 

A − = [ 

− 

− ] A − =−( − 

) − ( 

 

) =− 

 

 

. Th 

A=[ ] A=−=− 

 

B=[ 

] B=−=− 

Example 8 

Using Cramer’s Rule and Determinant Properties to Solve a System 

× 

x +y +z = 

x +y +z =− 

x +y +z = 

Solution 

 

 

 

 

D=∣ 

 

∣ 

 

 

fio fi 

1. 

 

−x−y−x=− 

 

x+y+z= 

 

=− 

 

 

Solve a System of Two Equations Using Cramer's Rule (http://openstaxcollege.org/l/system2cramer) 

Solve a Systems of Three Equations using Cramer's Rule (http://openstaxcollege.org/l/system3cramer)



VERBAL 

1. 

 

3. 

 

ALGEBRAIC 

, fi 

2. 

 

× 

 

4. Th×A 

 

 

 

5. ∣ ∣ ∣ 6. − −∣ ∣ 7. 

− ∣ ∣ 8. − − ∣ 

 

− 

9. ∣ −∣ ∣ 10. −∣ ∣ 11. ∣ ∣ 12. − ∣ 

13. ∣ 

− 

 

− 

 

∣ 14. ∣ − 

− ∣ 15. ∣ 

− 

 

 

 

 

∣ 

− 

∣ 

 

− 

16. 

∣ 

− 

∣ 

 

 

 

17. 

∣ 

 

∣ 

 

− 

18. − 

∣ 

− 

∣ 

− 

 

 

19. − 

− ∣ 

− − 

∣ 

 

− 

20. − 

− 

∣ 

− 

∣ 

 

− 

21. ∣ 

− − 

∣ 

− 

22. − 

 

 

∣ 

− 

∣ 

 

23. 

− 

− 

−∣ 

 

∣ 

24. 

− 

 

 

− ∣ 

 

 

25. x−y=− 

x+y= 

26. x−y= 

−x+y= 

27. x−y= 

−x+y=− 

28. x+y= 

x−y= 

29. x+y= 

x−y=− 

30. x−y= 

−x+y=− 

31. x−y=− 

x+y=− 

32. x−y= 

−x+y= 

33. x+y= 

−x−y=− 

34. x−y=− 

−x+y= 

 

35. x+y−z=− 

x+y+z= 

−x−y+z=− 

36. −x+y−z=− 

x−y−z=− 

x−y+z= 

37. x+y−z=− 

−x−y+z= 

y+z= 

38. x−y+z= 

x−z=− 

x+y−z=− 

39. x−y+z= 

−x+y=− 

x+y+z= 

40. x+y−z= 

−x−y+z= 

x−y+z= 

41. x−y+z= 

−x+y+z= 

x+y−z=− 

42. −x−y−z=− 

x−y+z= 

x−y−z=−


43. x−y+z= 

−x+y−z=− 

x+y+z= 

44. x−y+z= 

−x+y−z=− 

x+y−z=− 

TECHNOLOGY 

 

∣ ∣ ∣ 

 

 

45. 

 

 

 

 

46. 

∣ 

47. 

 

 

 

 

− 

 

 

 

− ∣ 

− 

 

∣ 

− 

 

∣ 

 

48. 

 

∣ 

 


Th 

 

49. 

 

50. 

 

 

51. o 106. Th 

. Th 

 

52. o 216. Th 

s 112. Th 

 

Th 

 

53. 

 

 

 

55. 

 

 

 

 

 

57. 

 

 

 

 

 

 

 

 

54. 

 

 

t. Th 

 

 

 

56. 

 

 

 

 

58. 

 

. Th


59. 

. Th 

 

 

 

61. 

 

 

 

 

63. ies. Th 

 

 

. Th 

 

 

 

65. 

 

 

 

. Th 

 

 

 

60. 

 

 

 

 

62. . Th 

 

 

et. Th 

 

 

 

64. 

 

 

Th 

 

 

 

 

 

 

t y 

 

. ThTable 1 

Fat (g) Protein (g) Carbohydrates (g) 

 

 

 

Table 1 

66. 

f mix. Th 

 

 

 

67. 

 

 

 

68.



Key Terms 

addition method 

 

 

augmented matrix 

break-even point fi 

coefficient matrix 

column 

consistent system 

 

cost function 

Cramer’s Rule 

dependent system 

 

determinant 

 

entry 

feasible region 

 

Gaussian elimination 

identity matrix 

 

inconsistent system 

 

independent system xy 

main diagonal 

matrix 

multiplicative inverse of a matrix 

nonlinear inequality 

partial fraction decomposition 

diff 

partial fractions diff 

 

profit function fiPx=Rx−Cx 

revenue function R=xpx=p= 

row 

row operations 

 

row-echelon form ft 

 

row-equivalent AB 

 

scalar multiple 

solution set 

substitution method


system of linear equations 

system of nonlinear equations 

system of nonlinear inequalities 

 

Key Equations 

× I 

 

=[ ] 

 

 

× I =[ 

 

] 

× 

A − = 

ad−bc 

[ −b 

−c a ] ad−bc≠ 

Key Concepts 

7.1 Systems of Linear Equations: Two Variables 

 

 

Th 

Example 1 

 

 

 

Example 2 

 

Example 3 

 

Example 4 

ft 


 

Example 8 

Th 

Example 9 

 

fiExample 10Example 11 

7.2 Systems of Linear Equations: Three Variables 

xyzExample 1 

 

Th 

Example 2 

diff 

Example 3 

ft 

Example 4


ft 

Example 5 

 

 

7.3 Systems of Nonlinear Equations and Inequalities: Two Variables 

Th 

 

Example 1 

Th 

 

Example 2 

Thfi 

 

 

Example 3 

>< 

Example 4 

 

Example 5 

7.4 Partial Fractions 

Px 

Qx 

A 

a 

x+b + 

B 

 

a 

x+b 

 

 

 

Example 1 

Th Px 

Qx 

Example 2 

Th Px 

Qx 

A x 

+ 

Bx+C 

Example 

3 

ax +bx+c 

Px 

Qx Qx 

 

Ax+B 

__ 

ax +bx+c + 

A 

x+B 

 

__ 

ax +bx+c ++ 

A n 

x+B n 

 

__ 

ax +bx+c n Example 4 

 

7.5 Matrices and Matrix Operations 

 

Th× 

Example 1 

 


Example 6 

ftExample 7 

fi 

 

ThABA 

BABExample 8Example 9


ftExample 10 

ftExample 11 

7.6 Solving Systems with Gaussian Elimination 

Example 1 

Example 2 

 


Example 6 

 


Example 9 


7.7 Solving Systems with Inverses 

AI=IA=AExample 1 

AA − =A − =IExample 2 

×Example 3 

ThExample 4 

Example 5 

× 

Example 6 

AX=BAA − AX=A − BExample 7 

Example 8 

Example 9 

7.8 Solving Systems with Cramer's Rule 

Th [ a b 

c d ] ad−bcExample 1 

x= D D 

_ x 

 

D y= 

y 

 

_ 

Example 2 

D 

× 

Example 3 

 

x= D D 

_ x 

 

D y= 

y 

 

_ 

 

D z=D _ z 

Example 4 

D 

 


 

 

○ 

 

○ 

 

○ 

 

○ 

○ ThA − A 

 

○ 

Example 7Example 8



SYSTEMS OF LINEAR EQUATIONS: TWO VARIABLES 

 

1. x−y= 

x+y=−− 

2. x−y= 

−x+y= 

 

3. x+y=− 

x−y=− 

4. 

x+ 

y= 

 

 

x− 

y=− 

 

5. x+y= 

x+y= 

 

6. x+y=− 

x+y= 

7. x+y= 

x+y= 

8. x+y= 

x−y= 

 

9. 

Cx=x+ 

Rx=x 

10. Cx=x+x 

. Th 

ft 

 

 

SYSTEMS OF LINEAR EQUATIONS: THREE VARIABLES 

 

11. x−y= 

−y+x= 

x+z= 

12. x+y−z= 

x−y+z= 

x+y+z= 

13. x+y+z= 

x+y+z= 

x+y= 

14. x−y+z=− 

x+y+z=− 

x+y−z= 

17. x−y+z= 

x+y−z= 

x−y−z= 

15. x+y−z=− 

x−y+z= 

−x+y+z=− 

18. x−y−z= 

x+y−z= 

y−z= 

16. x+z=− 

x−y= 

y−z=− 

 

19. o 61. Th 

 

 

20. . Th 

f $8,070.00. Th


SYSTEMS OF NONLINEAR EQUATIONS AND INEQUALITIES: TWO VARIABLES 

 

21. y=x − 

y=x− 

22. y=x − 

y=x+ 

23. x +y = 

y=x− 

24. x +y = 

y=x + 

25. x +y = 

y−x = 

 

26. y>x − 27. 

x +y < 

 

28. x +y +x< 

y>−x − 

29. x −x+y −x< 

y


 

56. x−y=− 

−x+y= 

59. x+y+z= 

−x−y−z=− 

x+y+z= 

57. x−y= 

−x+y= 

60. −x+y−z= 

y+z=− 

−x+y+z= 

58. −x−y= 

−x−y= 

SOLVING SYSTEMS WITH INVERSES 

, fi 

61. [ − − ] [ 62. 

 

− 

 

− 

] 

 

− 

[ 63. 

 

− 

− − 

] [ 64. 

 

] 

, fi 

65. x−y=− 

−x+y= 

66. x−y=− 

−x+y= 

67. x+y−z=− 

x−y−z=− 

x+z=− 

68. −x−y+z= 

−x+y+z=− 

−y+z=− 

 

69. 

 

 

 

 

70. 

ies. Th 

 

$1. Th 

 

SOLVING SYSTEMS WITH CRAMER'S RULE 

, fi 

 

− 

71. [ ] 72. [ − − 

− ] [ 73. 

] 

√ 

] [ — 74. 

— √ 

√ 

— 

 

75. x−y= 

−x−y=− 

76. x−y= 

−x+y= 

77. −x+y= 

−x+y= 

78. x+y+z= 

x+y+z= 

x−y+z= 

79. x−y+z=− 

 

x−y−z= 

 

x−y−z= 

 

80. 

x− 

y− 

z=− 

 

x− 

y− 

z=− 

 

x− 

y− 

z=−


679 


 

1. −x−y= 

x+y=− 

 

 

2. 

x− 

y= 

 

x−y= 

3. − 

x−y= 

x+y= 

4. x−y= 

−x+y=− 

5. x−y−z= 

x−y+z=− 

 

x+y−z= 

 

6. x+z= 

x+y+z= 

x+y+z= 

7. x−y−z= 

x+y+z= 

x−y−z= 

8. y=x +x− 

y=x− 

9. y +x = 

y −x = 

 

10. y 

y


 

23. x−y= 

x−y= 

24. x+y+z=− 

x−y+z= 

x−y−z= 

 

25. x−y=− 

−x+y= 

26. 

x− 

y+ 

z=− 

 

x− 

y− 

z= 

 

x− 

y− 

z= 

 

27. x−y= 

x+y= 

28. x+y−z=− 

x−y+z=− 

x−y+z=− 

 

29. 

 

Cx=x +x+Rx=x +x 

 

fi 

fi 

30.

8 

Analytic Geometry 

a 

b 

Figure 1 (a) Greek philosopher Aristotle (384–322 BCE) (b) German mathematician and astronomer Johannes Kepler (1571–1630) 


8.1 The Ellipse 

8.2 The Hyperbola 

8.3 The Parabola 

8.4 Rotation of Axes 

8.5 Conic Sections in Polar Coordinates 

Introduction 

Th 

 

d diff 

 

 

 

 

 

 

fi 

fifi 

h fi 

681

682 

CHAPTER 8 ANALYTIC GEOMETRY 



Write equations of ellipses in standard form. 

Graph ellipses centered at the origin. 

Graph ellipses not centered at the origin. 

Solve applied problems involving ellipses. 

8.1 THE ELLIPSE 

Figure 1 The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr) 

 

ThFigure 1 

whispering chamber 

 

 

Writing Equations of Ellipses in Standard Form 

conicTh 

Figure 2 

 

Figure 2 

 

Thffi 

Th

SECTION 8.1 THE ELLIPSE 683 

ellipsex y 

o fih fifocusfoci 

 

 

fi 

g. ThFigure 3 

 

Figure 3 

Thmajor axis 

minoraxisvertexvertices 

Thcenter of an ellipseTh 

Th 

Figure 4 

y 

 

 

 

 

 

 

 

 

x 

Figure 4 

 

Th 

xy

684 


Deriving the Equation of an Ellipse Centered at the Origin 

−ccTh 

xyxyFigure 5 

x, y 

y 

a, 

d 1 

d 2 

c, 

c, ) 

a, 

x 

Figure 5 

a−caa −−c=a +cThc 

aa −c . Th 

 

a+c+a−c=a 

xyfi 

d 

=−cxy 

d 

=cxy 

fid 

+d 

xy 

aad 

+d 

=a 

. Th 

d 

+d 

=√ —— 

x−−c +y− +√ — x−c +y− =a 

√ — x+c +y +√ — x−c +y =a 

 

√ — x+c +y =a −√ — x−c +y 

 

x+c +y = a − √ — x−c +y 

x +cx+c +y =a −a √ — x−c +y +x−c +y 

 

x +cx+c +y =a −a √ — x−c +y +x −cx+c +y 

 

cx=a −a √ — x−c +y −cx 

cx−a =−a √ — x−c +y 

cx−a =−a √ — x−c +y 

 

 

[ cx−a ] =a [ √ — x−c +y ] 

 

c x −a cx+a =a ( x −cx+c +y ) 

 

c x −a cx+a =a x −a cx+a c +a y 

a 

a x −c x +a y =a −a c 

 

x a −c +a y =a a −c 

 

x b +a y =a b 

b =a −c 

x b 

a b y 

+a a b b 

=a a b 

a b 

x 

a +y b = 

 

Th x 

_ 

 

a +y b =Thfi 

a >bb >a


Writing Equations of Ellipses Centered at the Origin in Standard Form 

 

 

 

 

Th 

 

ThThfi 

 

 

 

standard forms of the equation of an ellipse with center (0, 0) 

Thx 

x _ 

 

a +y _ 

 

b 

 

= 

a>b 

a 

±a 

b 

±b 

±cc =a −b Figure 6a. 

Thy 

x _ 

 

b +y _ 

 

a 

 

= 

a>b 

a 

±a 

b 

±b 

±cc =a −b Figure 6b. 

c =a −b 

fi 

 

y 

y 

b 

a 

c 

−a 

−c 

 

 

 

c 

a 

x 

−b 

 

 

 

 

 

b 

x 

−c 

−b 

(a) 

Figure 6 (a) Horizontal ellipse with center (0, 0) (b) Vertical ellipse with center (0, 0) 

−a 

(b)

686 


How To… 

 

1. xy 

 

a. ±a±c 

x x _ 

 

a +y _ 

 

b = 

b. ±a±c 

y x _ 

 

b +y _ 

 

a = 

2. c =a −b b 

3. a b 

Example 1 

Writing the Equation of an Ellipse Centered at the Origin in Standard Form 

±± 

Solution Thxx. Th 

 

x 

a +y _ 

 

b = 

Th±a =a = 

Th±c =c = 

c =a −b b 

c =a −b 

=−b c a 

b = b 

a =b = 

x _ 

 

+y _ 

 

= 

Try It #1 

√ — 

Q & A… 

Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex? 

 

±a±a±c±c 

acc =a −b o fib 

Writing Equations of Ellipses Not Centered at the Origin 

h 

khkTh 

xx−hyy−k


standard forms of the equation of an ellipse with center (h, k) 

Thhkx 

_ 

x −h 

a +y _ 

−k 

b = 

 

a >b 

a 

h±ak 

b 

hk ±b 

h±ckc =a −b Figure 7a 

Thhky 

_ 

x −h 

b +y _ 

−k 

a = 

 

a >b 

a 

h k±a 

b 

h±bk 

h, k±cc =a −b Figure 7b 

h, k 

c = a − b . 

 

h −a, k 

h −c 

, k 

y 

h, k + b 

h, k 

 

 

 

h + c 

, k 

h + a, k 

x 

h −b, k 

y 

h, k + a 

 

 

hk 

 

 

h, k + c 

h + b, k 

x 

h, k −b 

hk −a 

(a) 

(b) 

Figure 7 (a) Horizontal ellipse with center (h, k) (b) Vertical ellipse with center (h, k) 

hk −c 

How To… 

 

1. xy 

a. yx 

_ 

x −h 

a +y _ 

−k 

b = 

b. xy 

_ 

x −h 

b +y _ 

−k 

a =

688 


2. hk 

3. a a 

4. c hk 

5. b c =a −b 

6. hka b 

Example 2 

Writing the Equation of an Ellipse Centered at a Point Other Than the Origin 

−−−−−− 

Solution xyTh 

 

 

_ 

x −h 

b +y _ 

−k 

a = 

hk. Th−−− 

 

 

hk=( −+− 

−+ ) 

 

=−− 

fia Thaafi 

y 

 

a =−− 

 

a = 

 

a = 

a = 

fic Thhk ±chk −c=−−hk +c=−k =− 

c 

 

k +c = 

 

−+c = 

 

c = 

c = 

b c =a −b 

 

c =a −b 

 

=−b 

 

b = 

hka b 

 

_ x+ +_ y+ = 

 

Try It #2 

−−√ — 

+√ — 

Graphing Ellipses Centered at the Origin 

 

_ 

x 

_ 

 

a +y b =a >bx _ 

 

_ 

 

b +y a 

a >b 

=


How To… 

 

1. 

a. _ 

x 

_ 

 

a +y b=a >b 

x 

±a 

±b 

±c 

b. _ 

x 

_ 

 

b +y a=a >b 

y 

±a 

±b 

±c 

2. cc =a −b 

3. 

Example 3 

Graphing an Ellipse Centered at the Origin 

x _ + 

y _ 

= 

Solution >yTh 

x _ 

 

b +y _ 

 

a =b =a = 

 

±a=±√ — =± 

±b=±√ — =± 

±cc =a −b c 

 

c =±√ — a −b 

 

=±√ — − 

 

=±√ — 

 

=± 

Th± 

 

Figure 8 

y 

 

 

 

 

 

x 

 

 

Figure 8

690 


Try It #3 

x 

+y _ 

= 

 

Example 4 

Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form 

x +y =Th 

 

Solution 

 

x +y = 

 

x 

+y _ 

= 

 

 

_ x 

+y _ 

= 

>xTh 

_ 

x 

_ 

 

=b = 

a +y b =a 

 

±a=±√ — =± 

±b=±√ — =± 

±cc =a −b c 

 

c =±√ — a −b 

 

=±√ — − 

 

=±√ — 

Th±√ — 

 

Figure 9 

y 

 

 

 

 

x 

− 

Figure 9 

Try It #4 

x +y =Th 

 

Graphing Ellipses Not Centered at the Origin 

fi 

hk x _ 

−h 

a +y _ 

−k 

b =a >b 

_ 

x −h 

b +y _ 

−k 

a =a >b


How To… 

hk 

1. 

 

a. _ 

x −h 

a +y _ 

−k 

b =a >b 

hk 

x 

h±ak 

hk ±b 

h±ck 

b. _ 

x −h 

b +y _ 

−k 

a =a >b 

hk 

y 

hk ±a 

h±bk 

hk ±c 

2. cc =a −b 

3. 

Example 5 Graphing an Ellipse Centered at (h, k) 

_ x+ + y _ − = 

 

 

Solution >y 

Th_ 

x −h 

b +y _ 

−k 

a =b =a = 

hk=− 

hk ±a=−±√ — =−±−− 

h±bk=−±√ — =−±− 

hk ±cc =a −b c 

 

c =±√ — a −b 

 

=±√ — − 

 

=±√ — 

Th−−√ — −+√ — 

 

y 

− 

−+√ 

− 

− 

 

−−√ 

− 

x 

Figure 10

692 


Try It #5 

_ x− +_ y− = 

 

 

How To… 

hk 

1. ax +by +cx+dy+e = 

2. 

 

3. ffix y 

4. 

m 

x−h +m 

y−k =m 

m 

m 

m 

 

5. 

Example 6 

Graphing an Ellipse Centered at ( h, k) by First Writing It in Standard Form 

x +y −x +y += 

 

Solution 

 

x +y −x +y += 

 

 

x −x+y +y=− 

ffi 

 

x −x+y +y=− 

 

 

x −x ++y +y +=−++ 

 

 

x− +y+ = 

 

 

_ x− +_ y+ = 

 

> 

xTh_ 

x −h 

a +y _ 

−k 

b =a =b = 

 

hk=− 

h±ak=±√ — −=±−−− 

hk ±b=−±√ — =−±− 

h±ckc =a −b c 

 

c =±√ — a −b 

 

=±√ — − 

 

=±√ — 

Th−√ — −+√ — − 

 

Figure 11


Try It #6 

−√5, − +√5, − 

 

 

x 

 

− 

 

− 

− 

 

 

− 

Figure 11 

 

 

x +y −x+y+= 

Solving Applied Problems Involving Ellipses 

 

fl 

 

 

Th 

fl Figure 12 

 

 

Example 7 

Figure 12 Sound waves are reflected between foci in an elliptical room, called a whispering chamber. 

Locating the Foci of a Whispering Chamber 

Th 

Figure 13 

a. 

 

b. 

 

 

 

Figure 13

694 


Solution 

 

a. 

x _ 

 

_ 

 

a +y b=a >ba 

b 

 

aa =a =a = 

bb =b =b = 

Th_ 

x 

+ _ 

y 

= 

b. ±c 

c =a −b c 

c =a −b 

c =− 

c =±√ — − 

c =±√ — 

c ≈± 

Th±. Th= 

Try It #7 

 

a. 

 

b. 

 

 

Conic Sections: The Ellipse (http://openstaxcollege.org/l/conicellipse) 

Graph an Ellipse with Center at the Origin (http://openstaxcollege.org/l/grphellorigin) 

Graph an Ellipse with Center Not at the Origin (http://openstaxcollege.org/l/grphellnot)



VERBAL 

1. 2. 

3. 

 

5. 

 

y 

4. 

 

 

ALGEBRAIC 

 

6. x +y= 7. x +y = 8. x −y = 

9. x +y = 10. x −x+y −y+= 

 

 

11. _ x + _ 

y = 12. 

 

_ x 

+ y _ 

 

= 

13. x +y = 

14. x +y = 15. x− 

+y− = 16. x− _ 

+_ y+ = 

 

17. _ x+ +_ y− = 18. x− + y− = 19. x −x+y −y+= 

20. x −x+y −y+= 21. x −x+y −y+= 

22. x +x+y −y+= 23. x +x+y −y+= 

24. x +x+y −y+= 25. x +x+y +y+= 

26. x +x+y +y+= 

, fi 

27. _ x+ + _ y+ = 28. _ x+ + _ y− = 29. x +y = 

 

30. x +y +x+y= 31. x +y +x = 

GRAPHICAL 

 

32. _ x 

+y _ 

 

= 

33. _ x 

+y _ 

= 

34. x +y = 

35. x +y = 36. _ x− +_ y− = 37. _ x+ +_ y− = 

 

38. _ x + 

_ y+ = 39. x −x+y −y−= 

 

40. x −x+y −y+= 41. x +x+y −y+= 

42. x +x+y −y−= 43. x +x+y −y+= 

44. x +x+y −y+= 45. x +x+y +y+= 

 

46. x- 

y 

47. 

x-y−

696 CHAPTER 8 ANALYTIC GEOMETRY 

48. 

x-y 

 

49. +√ — 

50. +√ — 51. −−+√ — 

 

52. 

55. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y 

 

 

 

 

 

 

 

 

 

 

y 

 

 

x 

x 

53. y 

54. 

 

56. 

 

 

 

 

 

 

 

 

 

y 

EXTENSIONS 

, fi. Th=a ċb ċπ 

57. _ x− +_ y− = 58. _ x+ +_ y− = 59. _ x+ +_ y− = 

 

60. x −x+y −y+= 61. x −x+y −y+= 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 

 

x 

 

 

 

 

 

 

 

 

 

 

 

 

y 

 

 

 

x 


62. t fi 

 

64. 

e). Th 

 

 

 

66. 

et. Th 

 

 

 

68. 

 

 

 

63. t fi 

 

h 

65. . Th 

 

 

 

 

67.

SECTION 8.2 THE HYPERBOLA 697 



Locate a hyperbola’s vertices and foci. 

Write equations of hyperbolas in standard form. 

Graph hyperbolas centered at the origin. 

Graph hyperbolas not centered at the origin. 

Solve applied problems involving hyperbolas. 

8.2 THE HYPERBOLA 

 

Th 

 

Figure 1 

 

 

 

 

Figure 1 

A shock wave intersecting the ground forms a portion of a conic and results in a sonic boom. 

ft 

fiflTh 

Thfi 

 

Locating the Vertices and Foci of a Hyperbola 

hyperbola 

Th 

Figure 2 

Figure 2 A hyperbola 

fi 

xyffxy

698 


fiTh 

fidifferencefisum 

Thtransverse axis 

Th 

conjugate axisThcenter of a 

hyperbola 

asymptotes 

Thcentral rectangle 

 

Figure 3 

y 

 

 

 

 

 

 

 

x 

 

 

 

 

Figure 3 

Key features of the hyperbola 

 

xy 

 

Deriving the Equation of an Ellipse Centered at the Origin 

−ccThxy 

ffxyFigure 4 

y 

xy 

d 

d 

x 

–c 

c 

–a 

a 

Figure 4 

a−caa −−c=a +cThc 

ac −a. Th 

 

a+c−c−a=a


xyfi 

d 

=−cxy 

 

d 

=cxy 

fid 

−d 

xyff 

aad 

−d 

=a 

Th 

 

d 

−d 

=√ —— 

x−−c +y− −√ — x−c +y− =a 

√ — x +c +y −√ — x −c +y =a 

 

√ — x +c +y =a +√ — x −c +y 

 

x +c +y =a +√ — x −c +y 

x +cx +c +y =a +a √ — x −c +y +x −c +y 

 

x +cx +c +y =a +a √ — x −c +y +x −cx +c +y 

 

cx =a +a √ — x −c +y −cx 

cx −a =a √ — x −c +y 

cx −a =a √ — x −c +y 

cx−a =a ( √ — x−c +y ) 

 

 

 

c x −a cx+a =a x −cx+c +y 

c x −a cx+a =a x −a cx+a c +a y a 

a +c x =a x +a c +a y 

c x −a x −a y =a c −a 

x c −a −a y =a c −a 

x b −a y =a b b =c −a 

x b 

a b y 

−a a b b 

=a a b 

a b 

_ x 

_ 

 

a −y b = 

Thfi±a±b 

standard forms of the equation of a hyperbola with center (0, 0) 

Thx 

_ x 

_ 

 

a −y b = 

 

a 

±a 

b 

±b 

cc =a +b 

±c 

y=± b __ 

a x

700 


Figure 5a 

Thy 

y _ 

 

_ 

 

a −x b = 

 

a 

±a 

b 

±b 

cc =a +b 

±c 

y =±_ 

a b x 

Figure 5b 

c =a +b 

 

y 

y 

y = − b 

a x 

b 

y = b 

a x 

y = − a 

b x 

c 

a 

y = a 

b x 

−c 

−a 

 

a 

c 

x 

−b 

 

b 

x 

−b 

−c 

−a 

(a) 

Figure 5 

(a) Horizontal hyperbola with center (0, 0) (b) Vertical hyperbola with center (0, 0) 

(b) 

How To… 

 

1. xya 

ffifi 

 

 

a. _ 

x 

_ 

 

a −y b =xTh 

±a±c 

b. _ 

y 

_ 

 

a −x b =yTh 

±a±c 

2. aa =√ — a 

3. cc =√ — a +b


Example 1 

Locating a Hyperbola’s Vertices and Foci 

_ 

y −x _ 

= 

Solution Th_ 

y 

_ 

 

a −x b =yTh 

yo fix =y 

 

 

 

 

 

Th±cc 

 

=_ 

y −x _ 

=_ 

y − _ 

=_ 

y y = 

y =±√ — =± 

c =√ — a +b =√ — +=√ — = 

Th± 

Try It #1 

_ x − 

_ 

y = 

Writing Equations of Hyperbolas in Standard Form 

 

 

fi 

Thfi 

 

Hyperbolas Centered at the Origin 

 

c =a +b b =c −a Th 

 

How To… 

 

1. xy 

a. ±a±c 

x_ 

x 

_ 

 

a −y b = 

b. ±a±c 

y_ 

y 

_ 

 

a −x b = 

2. b b =c −a 

3. a b

702 


Example 2 

Finding the Equation of a Hyperbola Centered at (0, 0) Given its Foci and Vertices 

±±√ — 

Solution ThxTh_ 

x 

_ 

 

a −y b = 

Th±a =a = 

Th±√ — c =√ — c = 

b 

 

b =c −a 

b =− c a 

b = 

a =b =_ 

x 

_ 

 

a −y b =Th 

_ 

x 

−y _ 

=Figure 

6 

 

 

 

 

y 

 

 

 

 

 

 

x 

 

 

Figure 6 

Try It #2 

√ — 

Hyperbolas Not Centered at the Origin 

h 

khkTh 

xx−hyy−k 

standard forms of the equation of a hyperbola with center (h, k) 

Thhkx 

_ 

x −h 

a −y _ 

−k 

b = 

 

a 

h±ak 

b 

hk ±b 

cc =a +b 

h±ck


ThTh 

abTh± b 

a hk 

point-slope formulay =± b 

a x−h+k 

Figure 7a 

Thhky 

y _ 

−k 

a −x _ 

−h 

b = 

 

a 

hk ±a 

b 

h±bk 

cc =a +b 

hk ±c 

y =± a x−h+kFigure 7b 

b 

y 

y = − b x − h+k 

a 

h, k + b 

y 

h, k + c 

h, k + a 

y = a x − h+k 

b 

h − b, k 

h + b, k 

h − c, k 

h − a, k 

h, k 

h + c, k 

h + a, k 

x 

h, k 

h, k − a 

y = − a x − h+k 

b 

x 

h, k − b 

h, k − c 

y = 

a 

b x − h+k 

 

 

Figure 7 (a) Horizontal hyperbola with center (h, k) (b) Vertical hyperbola with center (h, k) 

hk 

c =a +b 

o fi 

How To… 

hk 

1. xy 

a. yx 

_ 

x −h 

a −y _ 

−k 

b = 

b. xy 

y _ 

−k 

a −x _ 

−h 

b = 

2. hk

704 


3. a a 

4. c hk 

5. b b =c −a 

6. hka b 

Example 3 

Finding the Equation of a Hyperbola Centered at (h, k) Given its Foci and Vertices 

−−−− 

− 

Solution yx 

 

 

_ 

x −h 

a −y _ 

−k 

b = 

hkTh−− 

 

 

hk=( + 

−+− ) =− 

fia Thafia fi 

x 

 

a =|−| 

 

a = 

 

a = 

 

a = 

fic Thh±ckh−ck=−−h+ck=− 

xc−h = 

 

h +c = 

 

+c = 

 

c = 

 

c = 

b b =c −a 

 

b =c −a 

 

=− 

 

= 

hka b 

 

_ x− −_ y+ = 

 

Try It #3 

−− 

Graphing Hyperbolas Centered at the Origin 

 

 

_ 

x 

_ 

y 

_ 

 

a −x b = 

_ 

 

a −y b =


How To… 

 

1. 

2. fi 

 

a. _ 

x 

_ 

 

a −y b = 

x 

±a 

±b 

±c 

y =±_ 

a b x 

b. _ 

y 

_ 

 

a −x b = 

y 

±a 

±b 

±c 

y =± a 

b x 

3. c =±√ — a +b 

4. 

 

Example 4 

Graphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form 

_ y −_ x = 

 

 

Solution Th_ 

y 

_ 

 

a −x b 

y 

=Th 

Th±a=±√ — =± 

Th±b=±√ — =± 

Th±cc =±√ — a +b c 

Th± 

c=±√ — a +b =±√ — +=±√ — =± 

Thy =± 

__ a b x =± 

x =± 

x 

 

 

Th 

Figure 8

706 


y 

y = − x 

 

 

 

y = x 

 

− 

y 

 

 

 

 

 

x 

− 

 

 

x 

 

− 

−− 

− 

 

Try It #4 

Figure 8 

_ 

x 

−y _ 

= 

 

 

Graphing Hyperbolas Not Centered at the Origin 

hk 

x _ 

−h 

a −y _ 

−k 

b = 

y _ 

−k 

a −x _ 

−h 

b = 

 

 

How To… 

hk 

1. 

 

2. fi 

 

a. _ 

x −h 

a −y _ 

−k 

b = 

x 

hk 

h±ak 

hk ±b 

h±ck 

y =± b 

a x−h+k


b. y −k 

a −x −h 

b = 

y 

hk 

hk ±a 

h±bk 

hk ±c 

y =±_ 

a 

b x−h+k 

3. c =±√ — a +b 

4. 

 

Example 5 

Graphing a Hyperbola Centered at (h, k) Given an Equation in General Form 

x −y −x −y −= 

 

Solution 

 

 

x −x−y +y= 

ffi 

 

x −x−y +y= 

 

 

x −x +−y +y +=+− 

 

 

x− −y+ = 

 

 

_ x− − y _ + = 

 

Th_ 

x−h 

a −y−k _ 

 

b =a =b =a = 

b =. Thx 

hk=− 

h±ak=±−−−− 

hk ±b=−±− 

h±ckc =±√ — a +b c 

c=±√ — +=±√ — =±√ — 

Th−√ — −+√ — − 

Thy =± b 

a x−h+k =± 

x−− 

 

Figure 9

708 


y 

 

x 

−− 

− 

− 

− 

Figure 9 

Try It #5 

_ y+ −_ x− = 

 

 

Solving Applied Problems Involving Hyperbolas 

fi 

Thffi 

ft 

ffi 

Figure 10 

 

Figure 10 Cooling towers at the Drax power station in North Yorkshire, United Kingdom (credit: Les Haines, Flickr) 

Thfi 

Example 6fi


Example 6 

Solving Applied Problems Involving Hyperbolas 

ThFigure 11ThTh 

 

 

 

 

 

Figure 11 Project design for a natural draft cooling tower 

 

fi 

fi 

Solution 

_ 

x 

_ 

 

a −y b = 

t fia b 

fia aTh 

a =. Tha =a = 

b xy 

fixy 

yx 

ThyTh 

 

 

_ 

x 

_ 

 

a −y b = 

b = 

y 

_ 

_ 

x 

a − 

 

b 

 

= 

 

_ 

 

− 

a xy 

≈ 

Th 

_ x 

− y y _ 

 

=x _ 

 

− 

_ 

 

=

710 


Try It #6 

Figure 12 

 

e fid fi 

 

 

 

 

Figure 12 

 

Conic Sections: The Hyperbola Part 1 of 2 (http://openstaxcollege.org/l/hyperbola1) 

Conic Sections: The Hyperbola Part 2 of 2 (http://openstaxcollege.org/l/hyperbola2) 

Graph a Hyperbola with Center at Origin (http://openstaxcollege.org/l/hyperbolaorigin) 

Graph a Hyperbola with Center not at Origin (http://openstaxcollege.org/l/hbnotorigin)



VERBAL 

1. 2. 

 

3. 4. 

 

5. 

 

ALGEBRAIC 

 

 

6. y +x= 7. x 

−y 

= 

8. y +x =x 9. x −y = 

10. −x +x+y +y−= 

 

 

11. x 

x −y = 

12. 

 

 

−y 

= 

13. y 

−x 

= 

14. y −x = 

15. x− 

−y − 

= 16. y − 

−x+ = 

17. x− 

−y + 

= 18. x −x−y −y+= 

19. −x −x+y −y+= 20. x −x−y −y+= 

21. −x +x+y −y+= 22. −x +x+y −y+= 

23. x +x−y −y+= 24. −x +x+y +y+= 

25. x +x−y +y−= 

, fi 

26. y 

−x = 

27. x− 

−y + 

= 

28. y − 

−x+ 

= 29. x −x−y +y−= 

30. y +y−x +x+= 

GRAPHICAL 

 

31. x 

−y 

= 

33. y 

−x 

= 

32. x 

−y 

= 

34. x −y = 

35. y + 

−x− 

= 36. x− 

−y+ 

= 

37. y − 

−x− 

= 38. −x −x+y −y−=


39. x −x−y −y−= 40. −x +x+y −y+= 

41. x +x−y −y−= 42. x +x−y −y−= 

43. −x +x+y −y−= 44. x +x−y +y+= 

, fi 

45. − 46. − 

− 

47. 48. −√ — 

49. +√ — 50. +√ — 

, fi 

51. 

52. 

y 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

 

 

 

 

x 

 

 

 

 

 

x 

53. 

 

y 

54. 

y 

 

− 

 

 

− 

x 

 

 

 

−− 

 

 

55. 

y 

 

 

x 

–– 

–– 

 

–


EXTENSIONS 

yx 

 

56. _ x − 

_ y = 

57. _ y − 

_ x = 

58. _ x− − y _ + = 

 

59. −x −x+y −y−= 60. x −x−y −y+= 


 

 

61. Thy=x 

y=−x 

 

62. Thy=x 

y=−x 

 

63. Thy= 

x 

y=− 

x 

 

64. Thy= 

x 

y=− 

x 

 

65. Thy= 

x 

y=− 

x 

 

 

x 

fl 

66. Th 

y=x− 

 

 

 

y=−x+ 

68. Th 

y=x+ 

 

 

y=−x− 

70. Th 

y=x− 

 

 

y=−x+ 

67. Th 

y=x− 

 

 

y=−x+ 

69. Th 

y= 

x− 

 

 

y=− 

x+

714 




Graph parabolas with vertices at the origin. 

Write equations of parabolas in standard form. 

Graph parabolas with vertices not at the origin. 

Solve applied problems involving parabolas. 

8.3 THE PARABOLA 

Figure 1 The Olympic torch concludes its journey around the world when it is used to light the 

Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force) 

Th 

flTh 

fi 

Figure 1 

e fl 

flTh 

 

fl 

 

ffi 

Graphing Parabolas with Vertices at the Origin 

The Ellipse 

. ThparabolaFigure 2 

Figure 2 Parabola

SECTION 8.3 THE PARABOLA 715 

fi 

xyfidirectrixfi 

focus 

Quadratic Functions 

Figure 3 

. Th 

Thlatus rectumTh 

fidP 

P 

y 

 

 

 

 

 

 

 

x 

Figure 3 Key features of the parabola 

 

 

y 

y = −p 

p 

 

d 

xy 

d 

x−p 

x 

Figure 4 

xypy =−pFigure 4 

dxyx−pffyd =y +pTh 

pxyd 

 

 

d =√ — x− +y−p 

=√ — x +y−p 

dy 

xypxyx−p 

 

√ — x +y−p =y +p 

 

 

x +y−p =y+p 

 

 

 

x +y −py+p =y +py+p 

x −py=py 

x =py 

Thy =pxxx =py 

yTh

716 


standard forms of parabolas with vertex (0, 0) 

Table 1Figure 5 

Axis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum 

x y =px p x=−p p±p 

y x =py p y=−p ±pp 

Table 1 

y 

y =px 

p > 

y =px 

p < 

y 

p|p| 

p|p| 

 

p 

p−|p| 

x 

p 

p−|p| 

 

x 

x =−p 

(a) 

(b) 

x = −p 

x =py 

p > 

y 

y 

 

y =−p 

x 

−|p|, p 

p 

|p|, p 

−|p|, p 

p 

|p|, p 

 

(c) 

x 

y = −p 

(d) 

x =py 

p < 

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 

the x-axis, the parabola opens left. (c) When p < 0 and the axis of symmetry is the y-axis, the parabola opens up. 

(d) When p < 0 and the axis of symmetry is the y-axis, the parabola opens down. 

ThFigure 5 

 

 

 

Figure 6 

y 

 

y =x 

 

x 

− 

x=− 

Figure 6


How To… 

 

1. y =pxx =py 

2. fi 

 

a. y =px 

xy = 

p xpp >p < 

 

p p 

p x =−p 

p p±px =p 

b. x =py 

yx = 

p ypp >p < 

 

p p 

p y =−p 

p ±pp 

3. 

Example 1 

Graphing a Parabola with Vertex (0, 0) and the x-axis as the Axis of Symmetry 

y =x 

Solution Thy =pxThx 

 

=pp =p > 

p= 

x =−p=− 

x x = 6 

± 

Figure 7 

y 

x= − 

 

 

 

 

 

 

 

– – – – – 

– 

 

– 

– 

– 

– 

Figure 7 

– 

x

718 


Try It #1 

y =−x 

Example 2 

Graphing a Parabola with Vertex (0, 0) and the y-axis as the Axis of Symmetry 

x =−y 

Solution Thx =pyThy 

 

−=pp =− p < 

 

p=( − 

) 

y =−p = 

 

y = 

( ±− 

) 

 

y 

y = 

 

x 

−− 

− 

− 

Figure 8 

x = −y 

Try It #2 

x =y 

Writing Equations of Parabolas in Standard Form 

 

 

How To… 

 

1. xy 

a. px 

y =px 

b. py 

x =py 

2. p 

3. 

Example 3 

Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix 

( − 

) x = 

 

Solution Thpy =px 

pp =−_ =− 

py =px=−x 

Thy =−x


Try It #3 

( 

) y=− 

 

Graphing Parabolas with Vertices Not at the Origin 

h 

khkTh 

xx−hyy−k 

hky−k =px−h 

xx−h =py−k 

yTh 

 

standard forms of parabolas with vertex (h, k) 

Table 2Figure 9hk 

Axis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum 

y = k y−k =px−h h+pk x=h −p h+pk ±p 

x = h x−h =py−k hk +p y=k −p h±pk +p 

Table 2 

y 

y−k =px −h 

p > 

y 

y−k =px −h 

p < 

h+p k +p 

h + p k + p| 

y=k 

y=k 

h, k 

h+p k 

h+p k −p 

h+p k 

h + p k −p| 

hk 

x=h −p 

x 

x=h −p 

x 

(a) 

(b) 

y 

x−h =py −h 

p > 

y 

x−h =py −k 

p < 

x=h 

hk +p 

y = k − p 

x=h 

h, k 

h−p k +p h−p| k +p h+p| k +p 

h+p k +p 

hk +p 

x 

y=k −p h, k 

(c) 

x 

(d) 

Figure 9 (a) When p > 0, the parabola opens right. (b) When p < 0, the parabola opens left. 

(c) When p > 0, the parabola opens up. (d) When p < 0, the parabola opens down.

720 


How To… 

hk 

1. y−k =px−hx−h =py−k 

2. fi 

 

a. y−k =px−h 

hkhk 

ky =k 

p x−hpp > 

p 

p < 

hkp hk +p 

kp y =k −p 

hkp h±pk +p 

3. 

Example 4 

Graphing a Parabola with Vertex ( h, k) and Axis of Symmetry Parallel to the x-axis 

y− =−x+ 

 

Solution Thy−k =px−hTh 

x 

hk=− 

y =k = 

−=pp =−p < 

h+pk=−+−=− 

x =h −p =−−−= 

h+pk ±p=−+−±−−−− 

 

Figure 10 

− 

y 

y− = −x + 

− 

− 

y= 

x 

−− 

x= 

Figure 10


Try It #4 

y+ =x− 

 

Example 5 Graphing a Parabola from an Equation Given in General Form 

x −x −y −= 

 

Solution Th 

x−h =p y−Thy 

x 

 

x −x −y −= 

 

x −x =y + 

 

x −x +=y ++ 

 

x− =y + 

 

x− =y+ 

 

x− =ċċy+ 

 

hk=− 

x =h = 

p =p > 

hk +p=−+=− 

y =k −p =−−=− 

h±pk +p=±−+−−− 

 

Figure 11 

y x− = y+ 

−− 

− 

− 

x 

− 

y= − 

x= 

Figure 11 

Try It #5 

x+ =−y− 

 

Solving Applied Problems Involving Parabolas 

 

 

fl 

flFigure 12Th

722 


 

 

 

 

 

Figure 12 

 

Thffi 

d fi 

Example 6 

Solving Applied Problems Involving Parabolas 

fiFigure 13Thfl 

 

fl 

a. 

 

b. 

 

 

 

Solution 

 

Figure 13 Cross-section of a travel-sized solar fire starter 

a. Thx =py 

p >0. Th. Thp = 

x =py 

x =y p 

x =y 

b. Th 

=x 

 

x =y 

=y x 

y ≈ y 

Th


Try It #6 

Th 

. Thfl 

a. 

x 

 

b. o fi 

 

Conic Sections: The Parabola Part 1 of 2 (http://openstaxcollege.org/l/parabola1) 

Conic Sections: The Parabola Part 2 of 2 (http://openstaxcollege.org/l/parabola2) 

Parabola with Vertical Axis (http://openstaxcollege.org/l/parabolavertcal) 

Parabola with Horizontal Axis (http://openstaxcollege.org/l/parabolahoriz)



VERBAL 

1. 2. 

p 

 

3. 

p 

 

 

5. 

 

 

4. e eff 

 

p 

ALGEBRAIC 

 

 

6. y =−x 7. y=x 8. x −y = 

9. y− =x− 10. y +x−y−= 

VF 

d 

11. x=y 12. y= 

x 13. y=−x 14. x= 

y 15. x=y 16. x= 

y 

17. x− =y− 18. y− = 

x+ 

19. y− =x+ 

20. x+ =y+ 21. x+ =y+ 22. y+ =x+ 

23. y +x−y+= 24. x −x−y+= 25. x −x−y+= 

26. y −x+y−= 27. x −x+y−= 28. y −y+x−= 

29. y −x−y+= 30. x +x+y−= 

GRAPHICAL 

 

31. x= 

y 32. y=x 33. y= 

x 

34. y=−x 35. y− =− 

x+ 

36. −x+ =y+ 

37. −y+ =x− 38. y −y−x+= 39. x +x+y+= 

40. x +x−y+= 41. y −x+y+= 42. x +x+y+= 

43. y +y−x+= 44. −x +x−y−= 

, fi 

45. y=− 46. x=− 

47. x=−√ — 

+√ — 

49. √ — −√ — x=√ — 

−√ — 

48. −x=− 

( − 

) 

50. y= 

( 

)


 

51. 52. 

y 

 

 

 

− 

 

 

x 

53. 

 

− 

y 

54. 

 

 

− 

 

 

− 

 

 

− 

y 

x 

x 

 

 

55. y 

 

 

 

 

x


EXTENSIONS 

 

56. V− 57. V−−− 

58. V− 59. V−−− 

60. V−( − 

) ( − 

) 


61. Th 

 

 

x =y 

 

63. 

 

. Th 

 

12 f 

 

65. 

 

 

 

67. 

 

 

 

69. 

y=−x +xx 

y 

 

 

62. 

 

 

64. 

 

 

66. 

 

 

 

68. 

 

 

 

20 f 

70. 

y=−x +x

SECTION 8.4 ROTATION OF AXIS 727 



Identify nondegenerate conic sections given their general form equations. 

Use rotation of axes formulas. 

Write equations of rotated conics in standard form. 

Identify conics without rotating axes. 

8.4 ROTATION OF AXIS 

 

xfiTh 

 

 

 

Figure 1 

Diagonal Slice Horizontal Slice Deep Vertical Slice Vertical Slice 

Ellipse 

Circle Hyperbola Parabola 

Figure 1 The nondegenerate conic sections 

nondegenerate conic sections 

degenerate conic sectionsFigure 2 

x

728 


Intersecting Lines Single Line Single Point 

Figure 2 Degenerate conic sections 

Identifying Nondegenerate Conics in General Form 

 

Th 

ffi 

 

Ax +Bxy+Cy +Dx+Ey+F = 

ABCffi 

 

xy 

xyB 

Conic Sections 

 

 

 

 

 

 

 

 

 

Example 

x +y = 

x +y = 

x −y = 

x =yy =x 

x +y = 

x−y+= 

x−x−= 

x +y = 

x +y =− 

Table 1


general form of conic sections 

nondegenerate conic section 

ABC 


Table 2ffB =ACTh 

 

Conic Sections 

 

 

 

 

Example 

Ax +Cy +Dx+Ey+F =A ≠CAC> 

Ax +Cy +Dx+Ey+F =A =C 

Ax −Cy +Dx+Ey+F =−Ax +Cy +Dx+Ey+F = 

AC 

Ax +Dx+Ey+F =Cy +Dx+Ey+F = 

Table 2 

How To… 

 

1. Ax +Bxy+Cy +Dx+Ey+F = 

2. AC 

a. AC 

b. AC 

c. AC 

d. AC 

Example 1 

Identifying a Conic from Its General Form 

 

a. x −y +x +y −= 

b. y +x +y −= 

c. x +y −x −y −= 

d. −x −y +x +y += 

Solution 

a. 

 

 

 

 


x +xy+−y +x +y +−= 

A=C =−ACs. Th 

b. 


x +xy+y +x +y +−= 

A=C =A

730 


 

 

 

 

c. 


x +xy+y +−x+−y+−= 

A=C =A =C 

d. 


−x +xy+−y +x +y += 

A=−C =−AC>A ≠C 

Try It #1 

 

a. y −x +x−y−= b. x +y +x+y−= 

Finding a New Representation of the Given Equation after Rotating through a Given Angle 

xyx 

y-xyxy 

θxy 

xyx′y′fi 

x′y′Figure 3 

y 

y’ 

x’ 

θ 

x 

Figure 3 The graph of the rotated ellipse 

 

x +y −xy−= 

xyx′y′Figure 4 

y 

y' 

x' 

θ 

θ 

θ 

θ 

x 

− θ 

θ 

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.


ThxyijThi′j′ 

θangle of rotationFigure 5 

 

 

i′=θi+θ j 

j′=−θi+θ j 

y' 

j' 

y 

j 

x' 

θ 

i' 

θ 

− θ 

θ 

θ 

i 

x 

Figure 5 Relationship between the old and new coordinate planes. 

u 

 

u =x′i′+y′j′ 

u =x′iθ +jθ+y′−iθ +jθ 

u =ix'θ +jx'θ −iy'θ +jy'θ 

u =ix'θ −iy'θ +jxθ +jy'θ 

u =x'θ −y'θi+x'θ +y'θj 

u =x′i′+y′j′xy 

x=x′θ −y′θ y =x′θ +y′θ 

equations of rotation 

xy 

θx 

x′y′xyx′y′ 

x=x′θ −y′θ y =x′θ +y′θ 

How To… 

, fift 

1. x yx =x′θ −y′θy =x′θ +y′θ 

2. xy 

3. x′y′ 

Example 2 Finding a New Representation of an Equation after Rotating through a Given Angle 

x −xy+y −=ftθ = 

Solution xyx =x′θ −y′θy =x′θ +y′θ

732 


θ = 

 

 

 

 

 

 

 

x =x′−y′ 

x =x′ 

( _ 

x = x′−y′ _ 

 

√ — 

_ 

√ — ) 

√ — ) −y′ ( 

y =x′+y′ 

y =x′ 

( _ 

y = x′+y′ _ 

 

√ — 

_ 

√ — ) 

√ — ) +y′ ( 

x =x′θ−y′θy =x′θ +y′θx −xy+y −= 

 

 

/ 

 

( 

x′−y′ _ 

 

√ — ) 

− ( 

x′−y′ _ 

 

√ — ) 

( 

x′+y′ _ 

 

√ — ) 

+ ( 

x′+y′ _ 

 

 

√ — ) 

−= 

x′−y′x′−y′ __ 

 

/ −x′−y′x′+y′ __ 

 

+ / 

x′+y′x′+y′ __ 

−= 

/ 

 

x′ −x′y′+y′ − x′ −y′ 

_ 

+x′ +x′y′+y′ −= 

 

 

 

x′ +y′ − x′ −y′ 

_ 

= 

 

 

( x′ +y′ − x′ −y′ 

_ 

 

) = 

x′ +y′ −x′ −y′ = 

x′ +y′ −x′ +y′ = 

 

x′y′ 

_ x′ + 

_ y′ = 

_ 

 

 

 

_ x′ + 

y′ _ 

= 

 

ThFigure 6 

y’ 

 

 

 

x’ 

 

 

 

 

 

 

 

θ= 

 

 

x 

 

 

 

Figure 6


Writing Equations of Rotated Conics in Standard Form 

fi 


x′y′ 

x′y′θfi 

θ= A −C 

B 

 

 

 


ABCB ≠xy 

θθ= A −C 

 

B 

θ>θ θ 

θ

734 


e fi ———— 

θ = √ 

 

__________ −θ √ 

− ————— 

__ 

 

= √ 

__ 

_ 

 

= − __ 

 

_ 

 

= √ 

_____ − 

ċ __ = √ 

__ 

= √ 

__ 

 

θ =_ 

 

√ — 

———— 

θ = √ 

 

__________ +θ √ 

+ ————— 

__ 

 

= √ 

__ 

_ 

 

= + __ 

 

_ 

 

= √ 

_____ + 

ċ __ = √ 

_ 

 

= √ __ 

 

θ =_ 

 

√ — 

θθx =x′θ −y′θy =x′θ +y′θ 

 

x =x′θ −y′θ 

 

 

 

 

x =x′ 

( _ 

x = x′−y′ _ 

 

√ — 

_ 

√ — ) 

√ — ) −y′ ( 

y =x′θ +y′θ 

 

y =x′ 

( _ 

_ 

√ — ) 

√ — ) +y′ ( 

 

y = x′+y′ _ 

 

√ — 

xy 

 

 

( 

x′−y′ _ 

 

√ — ) 

− ( 

x′−y′ _ 

 

√ — ) 

= 

√ — ) 

( 

x′+y′ _ 

 

√ — ) 

+ ( 

x′+y′ _ 

 

 

 

 

 

 

( x′−y′x′−y′ __ ) −( x′−y′x′+y′ __ ) +( x′+y′x′+y′ __ ) = 

 

x′ −x′y′+y′ −x′ +x′y′−y′ +x′ +x′y′+y′ = 

x′ −x′y′+y′ −x′ −x′y′+y′ +x′ +x′y′+y′ = 

x′ +y′ = 

 

 

x′ + 

y′ = 

 

x′y′ 

_ x′ + 

_ y′ = 

Figure 8 

y 

 

 

 

 

 

 

 

 

 

x 

 

Figure 8 

Try It #2 

x −√ — xy+y =x′y′x′y′


Example 4 

Graphing an Equation That Has No x´y´ Terms 

x′y′ 

 

x +xy−y = 

Solution e fiθ 

 

x +xy−y =⇒A =B =C =− 

 

θ= A −C 

B 

 

 

θ= −− 

 

 

θ= 

θ= Figure 9 

 

y 

 

θ 

 

θ 

 

x 

 

Th 

 

 

 

Figure 9 

θ= 

= _ 

+ =h 

+=h 

=h 

 

h = 

e fiθθ 

 

———— 

θ = 

√ −θ √ 

− ————— 

 

= √ 

_ 

_ 

 

= − _ 

 

_ 

 

= √ 

 

ċ = _ 

√ — 

———— 

 

θ = 

√ +θ √ 

+ ————— 

 

= √ 

_ 

_ 

 

= + _ 

 

_ 

 

= √ 

ċ = _ 

√ — 

e fixy 

 

x =x′θ −y′θ 

 

 

x =x′ 

( 

 

_ 

√ — ) −y′ ( 

x = x′−y′ _ 

 

√ — 

_ 

√ — )

736 


 

 

 

 

y =x′θ +y′θ 

y =x′ 

( 

 

_ 

√ — ) +y′ ( 

y = x′+y′ _ 

 

√ — 

_ 

√ — ) 

x = x′−y′ _ 

 

√ — y =x′+y′ _ 

 

√ — x +xy−y = 

 

 

 

( 

x′−y′ _ 

 

√ — ) 

+ ( 

x′−y′ _ 

 

√ — ) 

= 

√ — ) 

( 

x′+y′ _ 

 

√ — ) 

− ( 

x′+y′ _ 

 

( ) x′−y′ +x′−y′ x′+y′ −x′+y′ = 

( ) x′ −x′y′+y′ +x′ +x′y′−y′ −x′ +x′y′+y′ = 

 

 

 

 

( ) x′ −x′y′+y′ +x′ +x′y′−y′ −x′ −x′y′−y′ = 

 

( ) x′ −y′ = 

x′ −y′ = 

 

Figure 10_ x′ − 

_ y′ = 

 

y’ 

 

 

 

 

y 

_ x′ − 

_ y′ = 

 

x’ 

 

 

 

 

 

 

 

 

 

 

x 

 

 

 

 

Identifying Conics without Rotating Axes 

Figure 10 

 

 

 


 

 

A′ x′ +B′x′y′+C′y′ +D′x′+E′y′+F′= 

B −AC=B′ −A′C′Thft 

ThB −ACft


using the discriminant to identify a conic 


A′x′ +B′x′y′+C′y′ +D′x′+E′y′+F′=B −AC=B′ −A′C′ 

ThAx +Bxy+Cy +Dx+Ey+F = 

 

B −AC 

< 

= 

> 

Example 5 Identifying the Conic without Rotating Axes 

 

a. x +√ — xy+y −= 

b. x +√ — xy+y −= 

Solution 

a. ABC 

 

 

 

 

 

 

 

 

 

 

A B 

 

x +√ — 

xy+ y −= 

B −AC=( √ — ) − 

=− 

=− 

=−< 

Thx +√ — xy+y −= 

b. AB 

A B 

e fi 

C 

 

 

x +√ — 

xy+ y −= 

 

 

C 

B −AC=( √ — ) − 

=− 

=− 

=−< 

Thx +√ — xy+y −= 

Try It #3 

 

a. x −xy+y −= 

b. x −xy+y −= 

 

Introduction to Conic Sections (http://openstaxcollege.org/l/introconic)



VERBAL 

1. t effxy 

 

3. 

Ax +Bxy+Cy +Dx+Ey+F= 

B −AC 

2. 

Ax +By +Cx+Dy+E=AB= 

 

4. ax +x+y −= 

a 

5. Ax +Bxy+Cy +Dx+Ey+F=θ θ= A−C 

B 

 

ALGEBRAIC 

 

6. x +y +x+y−= 7. x −x+y−= 

8. x −y +x−y−= 9. x −y +x−= 

10. y −x+y+= 11. x +y −x−y+= 

12. x +xy+y −y−= 13. x +xy+y −y−= 

14. −x +√ — xy−y += 15. x +√ — xy+y −x−= 

16. −x +√ — xy+y −y+= 17. x +√ — xy+y −x+= 

, fift 

18. x +xy+y −=θ= 19. x −xy+y −=θ= 

20. x +xy−=θ= 21. −x +xy+=θ= 

22. x +√ — xy+y +y+=θ= 

θxy 

xy 

23. x +√ — xy+y +y−= 24. x +√ — xy+y +y−= 

25. x −√ — xy+y +y−= 26. −x −√ — xy−y −x= 

27. x +xy+y +x−y+= 28. x +xy+y +x−= 

29. x +xy+y −x+= 30. x −√ — xy+y −= 

GRAPHICAL 

 

 

31. y=−x θ=− 32. x=y θ= 33. _ x + 

_ y =θ= 

 

34. _ 

y 

+x _ 

=θ= 

35. y −x =θ= 36. y=_ x θ= 

 

37. x=y− θ= 38. _ x + 

_ y =θ=


x′y′x′y′ 

39. xy= 40. x +xy+y −= 

41. x −xy+y −= 42. x −√ — xy+y −= 

43. x +√ — xy+y −= 44. x +√ — xy+y −= 

45. x +√ — xy+y −= 46. x +xy+y −x+y= 

47. x +xy+y −x+y= 48. x −√ — xy+y −= 

49. x −xy+y −√ — x−√ — y= 

xyTh 

 

50. x −√ — xy+y +x−y= 51. x −xy+y +x−y= 

52. x −√ — xy+y +x−y= 53. x +√ — xy+y +x−y= 

54. x +xy+y +x−= 55. x +xy+y +x−y= 

k 

56. x +kxy+y +x+y−= 

k 

58. x +kxy+y −x+y+= 

k 

57. x +kxy+y +x−y+= 

k 

59. kx +xy+y −x+y+= 

k 

60. x +xy+ky +x+y+= 

k

740 




Identify a conic in polar form. 

Graph the polar equations of conics. 

Define conics in terms of a focus and a directrix. 

8.5 CONIC SECTIONS IN POLAR COORDINATES 

Figure 1 Planets orbiting the sun follow elliptical paths. (credit: NASA Blueshift, Flickr) 

 

 

ft 

 

 

 

 

 

fi 

Th 

fi 

 

Identifying a Conic in Polar Form 

fi 

x =+y Figure 2 

Pr, θ 

D 

 

r 

θ 

F 

 

x=+y 

Figure 2

SECTION 8.5 CONIC SECTIONS IN POLAR COORDINATES 741 

The Parabolafififi 

fifi 

Prθ 

FfiDfiefi 

eccentricityfi 

ThPe = PF 

PD fi 

PPFP 

De 

e 

≤e < 

e = 

e > 

fifix =±peθ 

Thpolar equationrθ 

the polar equation for a conic 

x =±ppeccentricity 

epolar equation 

ep 

r =_ 

± e θ 

y =±pp 

e 

ep 

r =_ 

± e θ 

How To… 

 

1. 

 

2. effi 

3. e 

4. x =py =pep 

xy 

Example 1 

Identifying a Conic Given the Polar Form 

 

 

a. r = 

b. r = 

+θ 

 

c. r = 

+θ 

 

−θ 

Solution 

Thfi 

 

c c 

a. 

 

 

( 

r= 

 

+θ ċ ) 

( 

_ 

) 

= 

__ 

=__ 

 

( 

) ( 

) + ( 

) θ + 

_ 

θ

742 


θy =pe = 

 

Th 

 

=ep 

 

= 

p 

( 

) = ( 

 

) 

p 

 

=p 

e a. The = 

c. 

r = 

_ 

−θ ċ 

x = 

= 

( _ 

) 

( 

_ 

 

) 

_ 

 

( _ 

) 

r = 

( _ 

) −( 

_ 

 

) θ 

__ 

 

_ 

r= 

_ 

−θ 

y =−pe = 

Th 

 

 

=ep 

 

 

=p 

 

 

=p 

e =. The =y =− 

=−


Try It #1 

 

r= 

−θ 

Graphing the Polar Equations of Conics 

 

Th 

 

fiThfi 

Th 

eThθr 

θ π 

ππ 

 

Example 2 Graphing a Parabola in Polar Form 

 

r = 

 

+θ 

Solution 

 

 

 

( _ 

r =_ 

 

) 

 

+θ = __ 

 

( _ 

) +( 

_ 

 

) θ 

 

 

 

r=_ 

 

+θ 

e =Thθ 

x =p 

 

 

=ep 

 

 

=p 

 

 

=p 

Thx = 

 


A B C D 

θ π 

π π 

 

 

 

r= 

+θ 

 

 

≈ 

 

≈ fi 

≈ 

Table 1 

B 

F 

A 

 

 

D 

r 

x = 

 

Figure 3

744 


Analysis We can check our result with a graphing utility. See Figure 4. 

 

Example 3 

Graphing a Hyperbola in Polar Form 

Figure 4 

 

r = 

−θ 

 

Solution 

 

 

 

( _ 

r =_ 

 

) 

 

−θ = 

( _ 

) −( 

_ 

 

) θ 

__ 

 

 

r=_ 

 

−_ θ 

e = e >Thθ 

 

y =−p 

 

=ep 

 

=( 

) p 

 

( 

) =p 

 

 

=p 

Thy =− 

 


A B C D 

θ π 

π π 

 

 

 

r= 

−θ 

 

− 

= 

Table 2 

C 

D 

r 

A 

 

B 

Figure 5


Example 4 

Graphing an Ellipse in Polar Form 

 

r = 

−θ 

 

Solution 

 

 

 

r = 

 

−θ = 

( _ 

 

( _ 

) 

 

_ 

) θ 

__ 

) − ( 

 

 

r=_ 

 

− 

θ 

e = e

746 


Analysis 

We can check our result with a graphing utility. 

See Figure 7. 

10 

Figure 7 r = 

_ 

graphed on a viewing window 

5 − 4 cos θ 

of [−3, 12, 1] by [ −4, 4, 1], θ min = 0 and θ max = 2π. 

Try It #2 

 

r= 

−θ 

 

Defining Conics in Terms of a Focus and a Directrix 

 

 

How To… 

 

1. y 

x 

2. p 

3. ffi 

4. p 

Example 5 

Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix 

e =y =− 

Solution Thy =−p 

y =−−< 

ep 

 

r = 

−eθ 

e =|−|==p 

Th 

 

 

r = 

−θ 

 

 

r = 

−θ 

Example 6 Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and 

Directrix 

e =_ x = 

 

Solution x =px => 

 

ep 

 

r = 

+ e θ 

e = 

||==p


Th 

 

( _ 

r = 

) 

 

+_ θ 

 

_ 

r = 

 

+_ θ 

 

_ 

r = 

 

( _ 

) +_ θ 

 

_ 

r = 

 

_ +_ θ 

 

 

r =_ 

ċ +θ 

 

r = 

+θ 

 

 

Try It #3 

e=x=− 

Example 7 

Converting a Conic in Polar Form to Rectangular Form 

 

r = 

−θ 

Solution r =√ — x +y x =rθy =rθ 

 

 

r = 

−θ 

 

 

r ċ− θ) = 

−θ ċ−θ 

r −r θ = 

r =+rθ r 

r =+r θ) 

x +y =+y r =√ — x +y y = r θ 

x +y =+y +y 

x −y = 

Try It #4 

 

r= 

+θ 

 

Polar Equations of Conic Sections (http://openstaxcollege.org/l/determineconic) 

Graphing Polar Equations of Conics - 1 (http://openstaxcollege.org/l/graphconic1) 

Graphing Polar Equations of Conics - 2 (http://openstaxcollege.org/l/graphconic2)



VERBAL 

1. 

 

3. 

θ 

 

5. 

 

2. 

 

4. 

 

 

ALGEBRAIC 

 

 

6. r = 

−θ 

 

7. r = 

−θ 

 

8. r = 

−θ 

 

9. r = 

+θ 

 

10. r = 

+θ 

 

11. r = 

 

+θ 

 

12. r = 

−θ 

 

13. r = 

+θ 

14. r−θ= 15. r+θ= 16. r−θ= 17. r+θ= 

 

 

18. r = 

+θ 

 

19. r = 

−θ 

 

20. r = 

−θ 

 

21. r = 

+θ 

 

22. r = 23. r = 

+θ −θ 

 

24. r = 

+θ 

 

25. r = 

−θ 

θ 

26. r+θ= 27. r−θ= 28. r−θ= 29. r = 

−+θ 

θ 

30. r = 

+θ 

 

 

 

31. r = 

+θ 

 

32. r = 

+θ 

 

33. r = 

−θ 

 

34. r = 

+θ 

 

35. r = 

−θ 

 

36. r = 

−θ 

 

37. r =_ 

 

− θ 

 

38. r = 

+θ 

39. r+θ= 40. r−θ= 41. r−θ= 42. r−θ= 

fi 

 

43. x = e = 

44. x =−e = 45. y =e = 

 

46. y =− e = 

47. x =e = 48. x =− e = 

 

49. x =− 

e = 

 

52. x =− e = 

 

55. x =− e = 

 

50. y = 

e = 

 

53. x =− e = 

 

51. y = e = 

 

54. y =e = 

EXTENSIONS 

Rotation of Axesxy 

rθ 

56. xy= 57. x 2 +xy+y 2 = 58. x 2 +xy+y 2 = 

59. x 2 +xy+y 2 = 60. xy+y =



Key Terms 

angle of rotation θ> θ 

θ< θ θ= θ = 

center of a hyperbola 

center of an ellipse 

conic section 

conjugate axis 

degenerate conic sections 

 

directrix 

 

eccentricity P F D 

e = PF e 

PD 

ellipse x, y o fi 

foci 

focus (of a parabola) a fi 

focus (of an ellipse) fi 

x, y 

hyperbola x, y x, y 

 

latus rectum 

 

major axis 

minor axis 

nondegenerate conic section 

 

parabola x, yfifi 

 

polar equation rθ 

transverse axis 

Key Equations 

x 2 

a 2+y 

b 2= a > b 

x 2 

b 2+y 

a 2= a > b 

h,k (x − h)2 

+ (y − k)2 

= a > b 

a 2 b 2 

h,k (x − h)2 

+ (y − k)2 = a > b 

b 2 a 2 

x- x 2 

a 2−y 

b 2= 

y- y 2 

a 2−x 

b 2=


h, kx (x − h)2 

− (y − k)2 

= 

a 2 b 2 

(h,k), y- (y − k)2 

− (x − h)2 

= 

a 2 b 2 

x 

y 2 =px 

y 

x 2 =py 

h, kx y−k 2 =p(x − h) 

(h,k), y- (x − h) 2 =p(y − k) 

 

 

 

 

Ax 2 +Bxy+Cy 2 +Dx+Ey+ F = 

x =x θ −y θ 

y =x θ +y θ 

θθ= A − B 

Key Concepts 

8.1 The Ellipse 

x, y 

 

 


 

 


 


 

Example 7 

8.2 The Hyperbola 

x, y ffx, y 

 

ThExample 1 

 


 


 

Example 6


8.3 The Parabola 

x, y 

 

Thx 

p >p < Example 1 

Thy 

p >p p < Example 4 

Thh, ky 

p >p



THE ELLIPSE 

Th 

 

1. x 

 

+y 

= 

2. (x − 

+(y + 

= 

3. x +y +x −y += 4. x +y −x +y += 

 

5. x 

 

+y 

 

= 

6. (x − 

+(y + 

= 

7. x +y +x +y −= 8. x 2 +y 2 −x +y += 

o fi 

9. − 10. −−− 

11. 

 

 

 

THE HYPERBOLA 

Th 

 

12. x 

 

−y 

= 

13. (y + 

−(x − 

= 

14. y 2 −x 2 +y −x += 15. x −y −x −y −= 

 

16. x 

 

−y 

 

= 

17. (y − 

−(x + 

= 

18. x −y +x +y −= 19. y −x −y −= 

, fi 

20. − 21. 

THE PARABOLA 

Th 

 

22. y =x 23. (x + = 

(y − 24. y −y −x −= 25. x 2 + x − y += 

 

26. x +y = 27. (y − = 

(x + 28. x −x −y += 29. y +y +x +=


 

30. − x = 31. ( _ 

) y=_ 

32. 

 

 

 

ROTATION OF AXES 

 

33. x +xy+y +x −y −= 34. x +xy+y +x −y += 

35. x +xy+y +x −y += 

θ xy 

xy 

36. x +xy−y −= 37. x −xy+y −= 

x'y'x'y' 

38. x −xy+y −x −y += 39. x −xy+y −= 

40. x +xy−y −x +y += 

CONIC SECTIONS IN POLAR COORDINATES 

 

 

 

41. r= 

−θ 

 

42. r= 

+θ 

 

43. r= 

+θ 

 

44. r= 

−θ 

 

 

 

45. r= 

−θ 

 

46. r= 

+θ 

 

47. r= 

+θ 

 

48. r= 

−θ 

fi 

 

49. x = e = 50. y =− e =



 

1. x 2 

 

+y 2 

 

= 

2. y 2 +x 2 −y +x −= 

 

3. (x − 

+(y − 

= 4. x 2 +y 2 +x −y −= 

5. 

 

6. 

 

 

 

 

7. x 2 

 

−y 2 

 

= 

8. y 2 −x 2 +y += 

 

 

9. (x − 

−(y + 

= 10. y 2 −x 2 +y −x −= 

11. 

 

 

 

12. y 2 +x = 13. x 2 −x − y += 

 

14. (x − =−(y + 15. y 2 +x −y += 

16. 

y =− 

17. 

 

 

 

 

 

θ xy 

18. x 2 −xy+y 2 = 19. x 2 +xy+y 2 +x −y = 

x'y'x'y' 

20. x 2 +√ — xy+y 2 = 21. x 2 +xy+y 2 −x = 

 

 

22. r= 

23. r= 

−θ 

+θ 

 

 

 

24. r= 

−θ 

 

25. r= 

+θ 

26. 

e = x =

9 

Sequences, Probability and 

Counting Theory 

Figure 1 (credit: Robert S. Donovan, Flickr.) 


9.1 Sequences and Their Notations 

9.2 Arithmetic Sequences 

9.3 Geometric Sequences 

9.4 Series and Their Notations 

9.5 Counting Principles 

9.6 Binomial Theorem 

9.7 Probability 

Introduction 

 

 

Th 

fi 

 

ThftTh 

 

 

 

 

 

755

756 

CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY 



Write the terms of a sequence defined by an explicit formula. 

Write the terms of a sequence defined by a recursive formula. 

Use factorial notation. 

9.1 SEQUENCES AND THEIR NOTATIONS 

Th 

Thfi 

Table 1 

Day 

Hits 

Table 1 

fi 

fi 

 

Writing the Terms of a Sequence Defined by an Explicit Formula 

sequence 

. Th 

 

fitermTh 

fit fi 

fi 

ffifi 

fi 

fi 

fififi 

finn 

n 

 

↓ ↓ ↓ ↓ ↓ ↓ 

 

n 

Thfi = = =Thn 

n 

abcn. Th 

 

a n 

= n 

n 

fi 

fifi 

n 

 

a 

= 

 

=

SECTION 9.1 SEQUENCES AND THEIR NOTATIONS 757 

Th 

Th 

 

Thfifin 

Table 2 

n n 

nth term of the sequence, a n 

n 

Table 2 

 

Figure 1Th 

 

 

a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1 

 

 

 

n 

fiinfinite sequence. Thfi 

e fi 

n 

Thfinite sequencefi 

sequence 

sequencefinite sequence 

finThtermsTha 

 

 

a 

a 

a 

a n 

 

a 

fia 

a 

 

Thnth term of the sequenceexplicit 

formulafinfi 

infinite sequence 

n 

Q & A… 

Does a sequence always have to begin with a 1 

? 

fia 

a

758 


How To… 

e fin 

1. n n =o fie fia 

 

2. o fia 

n = 

3. fin 

Example 1 

Writing the Terms of a Sequence Defined by an Explicit Formula 

e fit fifia n 

=−n + 

Solution n =n 

n = a 

=−+= 

n = a 

=−+= 

n = a 

=−+=− 

n = a 

=−+=− 

n = a 

=−+=− 

The fit fi−−− 

Analysis The sequence values can be listed in a table. A table, such as Table 3, is a convenient way to input the function 

into a graphing utility. 

n 

a n 

− − − 

Table 3 

Figure 2 

 

a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n 

 

 

 

 

 

 

 

 

 

Figure 2 

Try It #1 

t fi t n 

=n−


Investigating Alternating Sequences 

Th 

fi 

n 

−− 

fi 

Th 

 

How To… 

e fin 

1. nn =fifia 

Th 

− n 

2. o fia 

n = 

3. fin 

Example 2 

Writing the Terms of an Alternating Sequence Defined by an Explicit Formula 

e fit fi 

 

Solution n =n = 

a n 

=_ 

−n n 

n + 

 

 

 

 

n = a 

= − 

+ =− 

 

n = a 

= − 

+ = 

 

n = a 

= − 

+ =− 

 

n = a 

= − 

+ = 

 

 

n = a 

= − 

+ =− 

 

The fit fi{ − 

 

− 

 

− 

} 

Analysis The graph of this function, shown in Figure 3, looks different from the ones we have seen previously in this 

section because the terms of the sequence alternate between positive and negative values. 

 

a n 

 

 

 

 

 

 

 

 

 

 

 

n 

 

− 

 

− 

 

 

 

− 

Figure 3

760 


Q & A… 

In Example 2, does the (−1) to the power of n account for the oscillations of signs? 

nn +n − 

 

Try It #2 

t fi 

n 

a n 

= 

− n 

 

Investigating Piecewise Explicit Formulas 

Th 

fi 

ff 

How To… 

e fin 

1. n = 

2. o fie fia 

n = 

3. n = 

4. o fia 

n = 

5. fin 

Example 3 

Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula 

e fi 

 

n 

a n 

= n 

 

 

__ n n 

Solution n =n =n n 

_ n n 

 

a 

= = n 

a 

= = n 

 

a 

= 

= n 

 

a 

= = n 

a 

= = n 

 

a 

= 

= n 

 

The fi 

Analysis Every third point on the graph shown in Figure 4 stands out from the two nearby points. This occurs because 

the sequence was defined by a piecewise function. 

a n 

 

 

 

 

 

 

 

 

 

 

n 

Figure 4


Try It #3 

 

a n 

= 

n 

{ n 

 

n 

n 

Finding an Explicit Formula 

Thfi 

n 

fifiTh 

fi 

 

How To… 

e fi, fi 

1. 

2. 

3. 

4. a n 

nn =n =n = 

Example 4 

Writing an Explicit Formula for the nth Term of a Sequence 

n 

a.{ − 

− 

− 

} 

b.{ − 

− 

− 

− 

− 

} 

c.e e e e e 

Solution 

a. Th− n Th 

n +1. Thn + 

 

b. Th 

a n 

= −n n+ n+ 

 

{ − 

− 

− 

− 

− 

} 

Th 

{ − 

 

− 

− 

− 

− 

− 

− } Th 

n 

n + 

 

a n 

=− 

 

 

n+ 

c. Then = e n + 

a n 

=e n + 

Try It #4 

n 

−−

762 


Try It #5 

n 

Try It #6 

{ − 

n 

− 

− 

− 

− 

} 

{ 

e 

_ 

e 

e e } 

Writing the Terms of a Sequence Defined by a Recursive Formula 

 

fl 

Th 

Th 

 

ff 

Th 

recursive formula 

fi 

fia n 

 

 

 

a 

= 

 

a n 

=a n− 

−n ≥ 

n fie fi 

 

a 

= 

 

a 

=a 

−=−= 

 

a 

=a 

−=−= 

 

a 

=a 

−=−= 

e fi 

Thfifi 

 

 

a 

= 

 

a 

= 

 

a n 

=a n− 

+a n− 

n ≥ 

fi 

 

 

a 

=a 

+a 

=+= 

recursive formula 

recursive formulafi


Q & A… 

Must the first two terms always be given in a recursive formula? 

Thfifi 

. The fifi 

How To… 

e fie fin 

1. a 

. The fi 

2. o fia 

a n− 

 

3. o fia 

 

4. n 

Example 5 


e fit fifi 

 

a 

= 

 

a n 

=a n− 

−n ≥ 

Solution Thfia n− 

 

 

n = a 

= 

n = a 

=a 

−=−=−= 

n = a 

=a 

−=−=−= 

n = a 

=a 

−=−=−=− 

n = a 

=a 

−=−−=−−=− 

The fit fiFigure 5. 

a n 

 

− 

− 

− 

− 

− 

 

n 

 

− 

− 

Try It #7 

Figure 5 

e fit fifi 

 

a 

= 

 

a n 

a n− 

+n≥ 

How To… 

e fin 

1. a 

 

2. a 

 

3. o fi 

4. n

764 


Example 6 


e fifi 

 

a 

= 

 

a 

= 

 

a n 

=a n− 

+a n− 

n ≥ 

Solution Thfia n− 

a n− 

 

 

n = a 

=a 

+a 

=+= 

n = a 

=a 

+a 

=+= 

n = a 

=a 

+a 

=+= 

n = a 

=a 

+a 

=+= 

The fi Figure 6 

 

 

 

 

 

a n 

 

 

 

 

n 

Figure 6 

 

Try It #8 

e fifi 

 

a 

= 

 

a 

= 

 

a 

= 

 

a n 

= a a n− +a n− 

n ≥ 

n− 

Using Factorial Notation 

Thn factorialn 

n 

 

=ċċċ= 

 

=ċċċċ= 

a n 

=n+ 

n 

 

a 

=+=== 

Thnnn−ċ


n factorial 

n factorialfiThn 

nfin 

 

= 

 

= 

 

n=nn−n−⋯n ≥ 

Thfi= 

Q & A… 

Can factorials always be found using a calculator? 

 

 

Example 7 Writing the Terms of a Sequence Using Factorials 

n 

e fit fifia n 

= 

 

n+ 

Solution n =n = 

n = 

 

a 

= 

 

+ = 

= 

 

= 

 

n = 

 

a 

= 

 

+ = 

= 

 

= 

n = 

 

a 

= 

 

+ = 

= 

 

= 

 

n = 

 

a 

= 

 

+ = 

= 

 

 

= 

n = 

 

a 

= 

 

+ = 

= 

 

 

= 

 

 

The fit fi{ 

 

 

 

 

 

} 

Analysis Figure 7 shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the 

factorial term in the denominator results in the denominator becoming much larger than the numerator as n increases. 

This means the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero. 

Try It #9 

 

 

 

 

 

 

 

 

 

 

 

a n 

 

 

 

 

 

 

 

Figure 7 

t fi a n 

= n+ 

n 

 

 

 

 

 

n 

 

Finding Terms in a Sequence (http://openstaxcollege.org/l/findingterms)

766 CHAPTER 9 SEQUENCES, PROBABILITY AND COUNTING THEORY 


VERBAL 

1. 

 

 

3. 

 

 

5. 

 

 

2. 

4. a n 

 

 

n 

 

ALGEBRAIC 

e fi 

6. a n 

= n − 

 

7. a n 

=− 

 

n+ 8. a n =−−n− 9. a n 

= n 

 

n 

10. a n 

= n+ n 11. a =ċ n −n− 12. a n 

=− ċ − n− n 

13. a n 

= 

 

n+ 

14. a n 

=− n + 15. a n 

=−( ċ−n− ) 

e fi 

 

16. a n 

= −n −n 

n− n 

18. a n 

= n+ n 

_ n 

n 

17. a n 

= _ n 

n+ n≤ 

n − n> 

19. a n 

= −ċn− n 

ċ− n− n 

20. a n 

= n − n≤n> 

_ 

n − 


 

34. −−−−− 35. −−− 36. 

37. 38. 

 

 

 

 

39. 40. ( 

) 

41. 

 

e fi 

43. a n 

= n 

n 

44. a =ċn 

n 

ċn 

45. a = n 

n 

n −n− 

 

42. 

 

46. a = ċn 

n 

nn− 

GRAPHICAL 

e fit fi 

47. a n 

= −n 

n 

+n 

48. a n 

= +n _ 

 

n n 

+n n 

50. a n 

=a n 

=a n− 

+ 51. a n 

= n+ 

n− 

49. a 

=a n 

=−a n− 

+ 

e fit fi 

52. 

 

a n 

 

 

 

 

 

 

 

 

 

 

 

n 

53. a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n 

54. a n 

 

 

 

 

 

 

 

 

 

 

 

 

n 

e fit fi 

55. 

56. 

a n a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n 

 

 

 

 

n


TECHNOLOGY 

fi 

a 

[ENTER] 

 

[2ND] ANSa n− 

[ENTER] 

[ENTER] 

o fi 

57. t fia 

= 

a n 

= 

a + 

n− 

>Frac 

 

59. t fi 

a 

=a n 

= a n − − + 

61. a 

=a n 

=na n− 

58. 

a 

=a n 

=a n− 

+ 

60. 

a 

=a n 

= a + _ 

n− a n− 

 

te a fifi 

 

[2ND] LIST 

OPS“seq(”[ENTER] 

“Expr:”[X,T, θ, n]n 

“Variable:” 

“start:”n 

“end:”n 

[ENTER] 

[ENTER] 

 

 

[2ND] LIST 

OPS“seq(”[ENTER] 

“Expr”, “Variable”,“start”, “end” 

 

[ENTER] 

 

fi 

 

62. t fi 

a n 

=− 

n+ _ 

 

63. 

a n 

= n −n +n− 

n 

 

64. t fi 

65. 

a n 

= nċ−n− a n 

= n +n− 

66. a n 

= n 

n 

EXTENSIONS 

67. a n 

=−−n 

a n 

=− 

68. a n 

= n +n+ 

n+ 

 

69. 

−−−− 

 

 

71. 

70. 

a n 

= n+ 

n− b n 

=n +n +n

SECTION 9.2 ARITHMETIC SEQUENCES 769 



Find the common difference for an arithmetic sequence. 

Write terms of an arithmetic sequence. 

Use a recursive formula for an arithmetic sequence. 

Use an explicit formula for an arithmetic sequence. 

9.2 ARITHMETIC SEQUENCES 

ftTh 

Th 

 

ft 

fiTh 

fiThftfift 

ftftfi 

fi 

Finding Common Differences 

Tharithmetic sequence 

common difference 

ff− 

− − − − − 

 

Thff 

o fi 

+ + + + 

 

 

arithmetic sequence 

arithmetic sequenceff 

Thcommon differencea 

fi 

dff 

a n 

=a 

a 

+da 

+da 

+d 

Example 1 

Finding Common Differences 

, fiff 

a. b. − 

Solution ff 

a. Thn diff 

−= −= −= −= 

b. Thn diff. Thn diff 

−−= −= −= −=

770 


Analysis ThFigure 1. We can see from the graphs that, although both 

sequences show growth, a is not linear whereas b is linear. Arithmetic sequences have a constant rate of change so their 

graphs will always be points on a line. 

a n 

 

a n 

 

 

 

 

 

 

 

 

 

 

− 

 

n 

 

− 

 

 

 

 

n 

Figure 1 

Q & A… 

If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the 

common difference? 

 

o fiff 

Try It #1 

, fiff 

 

Try It #2 

, fiff 

 

Writing Terms of Arithmetic Sequences 

fifi 

ffThfiff 

nd 

a n 

=a 

+n−d 

How To… 

e fiff, fie fi 

1. ffe fio fi 

2. ffo fi 

3. fi 

4. 

Example 2 


e fit fia 

=d =− 

Solution −fifi 

 

The fit fi


Analysis As expected, the graph of the sequence consists of points on a line as shown in Figure 2. 

a n 

 

 

 

 

Figure 2 

 

 

 

n 

Try It #3 

t fia 

=d= 

How To… 

y fi, fi 

1. a 

a n 

na n 

=a 

+n−dd 

2. a 

nda n 

=a 

+n−d 

Example 3 

a 

=a 

=, fia 

 


Solution Thffd 

 

+d+d+d 

a 

+d =+d 

n fiffd 

 

a n 

=a 

+n−d 

 

a 

=a 

+d 

a 

=+d a 

d 

=+d a 

 

d = n diff 

e fiftff 

 

a 

=a 

+= 

Analysis Notice that the common difference is added to the first term once to find the second term, twice to find the third 

term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to 

the first term nine times or by using the equation a n 

= a 1 

+ (n − 1)d. 

Try It #4 

a 

=a 

= a 

 

Using Recursive Formulas for Arithmetic Sequences 

fiTh 

fi

772 


ffff 

e fi 

a n 

=a n− 

+d n ≥ 

recursive formula for an arithmetic sequence 

Thff 

a n 

=a n− 

+d n ≥ 

How To… 

 

1. o fiff 

2. ff 

Example 4 

Writing a Recursive Formula for an Arithmetic Sequence 

 

−− 

Solution Thfi−Thfffi 

 

d=−−−= 

ff 

 

a 

=− 

 

a n 

=a n− 

+n ≥ 

Analysis We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as 

shown in Figure 3. The growth pattern of the sequence shows the constant difference of units. 

 

 

 

 

− 

− 

a n 

Figure 3 

 

n 

Q & A… 

Do we have to subtract the first term from the second term to find the common difference? 

 

fiftf fiff 

Try It #5


Using Explicit Formulas for Arithmetic Sequences 

 

Thff 

 

a n 

=a 

+dn− 

fiyfffi 

 

− 

− 

− 

− 

 

 

Thff−−fiy 

−−−=+=fiy 

ThFigure4 

 

 

 

 

 

 

 

a n 

Figure 4 

 

n 

y =mx+ba n 

yn 

xmb 

− 

a n 

=−n + 

fi 

a n 

=−n−fia n 

=−n + 

explicit formula for an arithmetic sequence 

n 

a n 

=a 

+d n − 

How To… 

e fi 

1. ffa 

−a 

 

2. ffe fia n 

=a 

+dn− 

Example 5 Writing the nth Term Explicit Formula for an Arithmetic Sequence

774 


Solution Thffe fi 

 

d =a 

−a 

 

 

=− 

= 

Thfffffi 

 

 

a n 

=+n− 

 

a n 

=n − 

Analysis The graph of this sequence, represented in Figure 5, shows a slope of and a vertical intercept of −. 

 

 

 

 

 

 

 

 

a n 

Figure 5 

 

n 

Try It #6 

 

 

Finding the Number of Terms in a Finite Arithmetic Sequence 

fifi 

fffffi 

e fi 

How To… 

e fif a fi, fi 

1. ffd 

2. ffe fia n 

=a 

+dn− 

3. a n 

n 

Example 6 

Finding the Number of Terms in a Finite Arithmetic Sequence 

e fi 

−− 

Solution Thffe fi 

 

−=−


Thff−ffn 

 

 

a n 

=a 

+dn− 

 

a n 

=+−n− 

 

a n 

=−n 

−a n 

n 

 

−=−n 

 

=n 

Th 

Try It #7 

e fi 

 

Solving Application Problems with Arithmetic Sequences 

fta 

a 

 

ff 

 

a n 

=a 

+dn 

Example 7 

Solving Application Problems with Arithmetic Sequences 

fi 

a. 

b. 

Solution 

 

 

 

a. Thdiff 

Anft 

 

A n 

=+n 

b. 

−= 

Try It #8 

ftfi 

 

A 

=+= 

Th 

 

ftn 

 

 

Arithmetic Sequences (http://openstaxcollege.org/l/arithmeticseq)



VERBAL 

1. 2. n diff 

 

3. 

 

5. 

y diff 

ALGEBRAIC 

4. in diff 

 

 

, fiff 

6. 7. { 

 

} 

o fiff 

8. 9. 

fififi 

 

10. a 

=−d=− 11. a 

=d= 

 

e fit fi 

12. a 

=a 

=− 13. a 

=−a 

=− 

fififi 

 

14. n diff 

t 

16. n diff 

t 

18. n diff 

t 

15. n diff 

t 

17. n diff 

t 

 

19. a 

 

a 

=a 

= 

21. a 

 

a 

=a 

= 

23. a 

 

a 

=a 

= 

20. a 

 

a 

=a 

= 

22. a 

 

a 

=a 

= 

, fifi 

24. a 

=a 

=−a 

25. a 

=−a 

=−a 

 

e fit fi 

26. a 

=a n 

=a n− 

− 27. a 

=−a n 

=a n− 

−


 

28. a= 29. a= 30. a=− 

31. a= 32. a=−− 33. a= 

34. a=−−− 35. a={ 

 

} 

36. a= { − 

− 

− } 

37. a={ 

− 

− } 

e fit fi 

38. a= 39. a= 

40. a= 

e fit fi 

41. a=−n 42. a= 

n− 

 

 

43. a= 44. a= 45. a=− 

46. a=−−− 47. a= 48. a=−−− 

49. a= 50. a={ 

− 

− } 

52. a= { −− 

− 

} 

51. a= { 

 

} 

, fin fi 

53. a=−−− 54. a= 55. a={ 

 

} 

GRAPHICAL 

 

56. 

 

 

 

 

 

 

 

 

 

 

 

− 

− 

− 

− 

− 

− 

− 

− 

− 

− 

− 

− 

a n 57. 

 

− 

− 

 

 

 

 

 

n 

 

 

 

 

 

 

 

 

 

a n 

 

 

 

 

 

 

 

 

− 

− 

 

 

 

 

 

 

 

 

n


e fi 

58. a 

=d= 59. a 

=a n 

=a n− 

− 60. a n 

=−+n 

TECHNOLOGY 

a n 

=n − 

 

MODE 

 

SEQ] 

 

DOT] 

 

ENTER 

Y= 

n n= 

unun=n − 

un un= 

2NDWINDOWTBLSET 

 

= 

 

Δ= 

 

 

2NDGRAPHTABLE] 

61. 

un 

62. n= 

un 

63. WINDOWn=n= 

x=x=y=− 

y=14. ThGRAPH 

 

a n 

= n + 

 

 

64. 

unTABLE] 

65. 

WINDOW] 

 

EXTENSIONS 

66. 

 

68. 

bbb 

70. 

 

72. 

{ _ 

_ _ 

} 

67. 

 

69. 

a−ba+b−a+b, 

71. { _ _ _ 

 

} 

 

73. 

 

 

74.

SECTION 9.3 GEOMETRIC SEQUENCES 779 



Find the common ratio for a geometric sequence. 

List the terms of a geometric sequence. 

Use a recursive formula for a geometric sequence. 

Use an explicit formula for a geometric sequence. 

9.3 GEOMETRIC SEQUENCES 

fffl 

fi 

 

ftftft 

 

 

Finding Common Ratios 

Thgeometric sequence 

common ratioTh 

 

 

 

a n 

definition of a geometric sequence 

geometric sequenceTh 

common ratioTh 

a 

r 

 

a 

a 

ra 

r a 

r 

How To… 

 

1. 

2. 

Example 1 

Finding Common Ratios 

, fi 

a. b. 

Solution 

a. _ = 

_ = 

_ = 

_ = 

Th. Th 

b. _ = _ 

= _ _ =_ 

 

Th

780 


Analysis The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have 

the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so 

this interpretation holds, but (b) is not. 

a n 

a n 

 

 

 

 

 

 

 

 

 

 

 

 

n 

 

a 

 

b 

n 

Figure 1 

Q & A… 

If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the 

common ratio? 

 

o fi 

Try It #1 

, fi 

Try It #2 

, fi 

Writing Terms of Geometric Sequences 

 

 

 

fi 

fiTh 

fifi 

a 

=−r =fi−ċ− 

−ċ− 

 

a 

=− 

 

a 

=−ċ=− 

 

a 

=−ċ=− 

 

a 

=−ċ=− 

The fi−−−− 

How To… 

e fi, fie fi 

1. a 

o fia 

 

2. a n 

=a 

o fia 

a 

o fia 

fi 

3.


Example 2 

Writing the Terms of a Geometric Sequence 

e fia 

=r =− 

Solution a 

−o fia 

a 

o fia 

 

 

a 

= 

 

a 

=−a 

=− 

 

a 

=−a 

= 

 

a 

=−a 

=− 

The fi−− 

Try It #3 

t fia 

=r= 

 

Using Recursive Formulas for Geometric Sequences 

fi 

. Th 

 

recursive formula for a geometric sequence 

Thrd fia 

 

a n 

=r a n− 

n ≥ 

How To… 

e fi 

1. 

2. 

3. 

Example 3 

Using Recursive Formulas for Geometric Sequences 

 

 

 

Solution The fi. The fi 

 

r =_ = 

fia 

 

 

a n 

=ra n− 

 

a n 

=a n− 

n ≥ 

 

a 

= 

Analysis The sequence of data points follows an exponential pattern. The common ratio is also the base of an 

exponential function as shown in Figure 2. 

a n 

 

 

 

 

 

 

Figure 2 

n

782 


Q & A… 

Do we have to divide the second term by the first term to find the common ratio? 

 

e fiftf fi 

Try It #4 

 

{ _ 

_ _ 

} 

Using Explicit Formulas for Geometric Sequences 

 

o fi 

a n 

=a 

r n − 

Th 

 

a n 

= n − 

Th Figure 3 

 

 

 

 

 

 

 

 

 

 

a n 

 

 

n 

Figure 3 

explicit formula for a geometric sequence 

n 

a n 

=a 

r n − 

Example 4 

Writing Terms of Geometric Sequences Using the Explicit Formula 

a 

=a 

=, fia 

 

Solution Thr 

rr r


 

a n 

=a 

r n − 

a 

=r a 

r 

=r a 

=r 

r = 

e fi 

 

 

 

a 

=a 

= 

= 

Analysis The common ratio is multiplied by the first term once to find the second term, twice to find the third term, 

three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common 

ratio nine times or by multiplying by the common ratio raised to the ninth power. 

Try It #5 

a 

=a 

= a 

 

Example 5 

Writing an Explicit Formula for the nth Term of a Geometric Sequence 

n 

 

Solution The fi. The fi 

 

 

= 

The fi 

 

 

a n 

=a 

r n− 

a n 

=ċ n − 

ThFigure4 

 

 

 

 

 

 

a n 

 

n 

Figure 4 

Try It #6

784 


Solving Application Problems with Geometric Sequences 

a 

a 

 

 

a n 

=a 

r n 

Example 6 

Solving Application Problems with Geometric Sequences 

 

 

a. 

b. 

Solution 

 

a. ThTh 

 

Pnft 

 

P n 

=ċ n 

 

b. 

−= 

ftn 

 

P 

=ċ ≈ 

Th 

Try It #7 

Th 

 

a. 

b. 

 

Geometric Sequences (http://openstaxcollege.org/l/geometricseq) 

Determine the Type of Sequence (http://openstaxcollege.org/l/sequencetype) 

Find the Formula for a Sequence (http://openstaxcollege.org/l/sequenceformula)



VERBAL 

1. 2. 

 

3. 

 

5. 

y diff 

ALGEBRAIC 

, fi 

4. e diff 

 

6. 7. −−− 8. −− 

− 

− 

− 

 

, fi 

9. −−−−− 10. 11. − 

− 

 

− 

 

12. 13. 

fififi 

14. a 

=r= 15. a 

=r= 

 

e fit fi 

16. a 

=a 

= 17. a 

=a 

= 

fififi 

18. Th 

 

19. Th − 

 

 

 

20. a n 

=−−a 

21. a n 

={ − 

− 

 

} a 

 

e fit fi 

22. a 

=−a n 

=− 

a n− 

23. a 

=a n 

=a n− 

 

24. a n 

=−− 25. a n 

=−−−− 

26. a n 

= 27. a n 

=−− 

28. a n 

= 29. a n 

={ 

 

 

 

} 

30. a n 

={ − 

− 

 

} 

31. a = n { 

− 

 

− 

} 

e fit fi 

32. a n 

=−ċ n− 33. a n 

=ċ( − 

 

) n−


 

34. a n 

=−−−− 35. a n 

= 

36. a n 

=−−−− 37. a n 

=−− 

38. a n 

=−−−− 39. a n 

={ −− 

− 

− } 

40. a n 

={ 

 

 

} 

41. a = n { − 

− 

} 

, fifi 

42. a 

=a n 

=−a n− 

a 

43. a n 

=−( − 

) n− a 

 

, fin fi 

44. a n 

=−− 45. a n 

={ 

 

 

} 

GRAPHICAL 

 

46. 

a n 

 

 

 

 

 

 

− 

 

 

 

− 

− 

− 

− 

− − 

n 

47. 

 

 

a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

− 

− 

− 

− 

 

e fit fi 

48. a 

=r= 

49. a =a =a 50. a 

n n− n 

=ċ n− 

EXTENSIONS 

51. 

 

52. 

 

n 

53. 

bbb 

55. 

 

57. 

a n 

=−( 

) n− 

59. 

 

 

 

54. 

a−ba−ba−b 

56. 

{ 

 

 

 

 

} 

58. 

 

 

60.

SECTION 9.4 SERIES AND THEIR NOTATIONS 787 



Use summation notation. 

Use the formula for the sum of the first n terms of an arithmetic series. 

Use the formula for the sum of the first n terms of a geometric series. 

Use the formula for the sum of an infinite geometric series. 

Solve annuity problems. 

9.4 SERIES AND THEIR NOTATIONS 

ThTh 

 

 

 

Using Summation Notation 

fi 

Thseries 

 

 

+++++ 

nth partial sumfifiTh 

S n 

 

 

S 

= 

 

S 

=+= 

 

S 

=++= 

 

S 

=+++= 

Summation notationft 

fi 

fi. 

index of summationThlower 

limit of summationfiTh 

upper limit of summation 

→ 

→ 

k ← k 

← 

 

∑ 

k = 

fia k 

=kk = 

k =k 

 

a 

== 

 

a 

== 

 

a 

== 

 

a 

== 

 

a 

== 

n fi 

 

 

∑ 

k =++++= 

k =

788 


summation notation 

The finseriessummation notation 

n 

∑ 

a k 

 

k = 

a k 

k =k =n 

kindex of summationlower limit of summationnupper limit of summation 

Q & A… 

Does the lower limit of summation have to be 1? 

Th 

 

How To… 

 

1. 

2. 

3. k 

4. o fi 

Example 1 

∑ k 

 

 

k = 

Using Summation Notation 

Solution fi 

k k =k =fik = 

k o fi 

 

 

 

 

∑ 

k 

= + + + + 

k = 

=++++ 

= 

Try It #1 

 

∑ k− 

k= 

Using the Formula for Arithmetic Series 

 

ff dTh 

arithmetic sequencefin 

 

 

S n 

=a 

+a 

+d+a 

+d++a n 

−d+a n 

 

 

 

S n 

=a n 

+a n 

−d+a n 

−d++a 

+d+a 

 

fi n 

e fin


 

S n 

=a 

+a 

+d+a 

+d++a n 

−d+a n 

 

+S n 

=a n 

+a n 

−d+a n 

−d++a 

+d+a 

 

S n 

=a 

+a n 

+a 

+a n 

++a 

+a n 

 

n 

 

S n 

=na 

+a n 

 

o fie fin 

 

S n 

= n a 

+a n 

formula for the sum of the first n terms of an arithmetic series 

arithmetic seriesThfin 

 

S n 

= n a 

+a n 

How To… 

, fie fin 

1. a 

a n 

 

2. n 

3. a 

a n 

nS n 

= n a 

+a 

_ n 

 

 

4. o fiS n 

 

Example 2 

Finding the First n Terms of an Arithmetic Series 

 

a. +++++++++ 

b. ++++− 

 

c. ∑ k− 

Solution 

k= 

a. a 

=a n 

= 

o fin = 

a 

a n 

n 

 

 

 

 

S n 

= n a 

+a n 

S 

= + 

= 

b. a 

=a n 

=− 

o fin 

 

a n 

=a 

+n−d 

 

−=+n−− 

 

−=n−− 

 

=n − 

 

=n 

a 

a n 

n 

S n 

= n a 

+a n 

S 

= − 

=−

790 


 

 

c. a 

k = 

a k 

=k − 

a 

=−=− 

n =o fia 

k = 

 

a k 

=k − 

 

a 

=−= 

a 

a n 

n 

 

S n 

= na +a _ n 

 

 

 

Try It #2 

o fi 

S 

=__ 

−+ 

= 

++++++++++ 

Try It #3 

o fi 

Try It #4 

o fi 

++++ 

 

∑ 

k= 

−k 

Example 3 

Solving Application Problems with Arithmetic Series 

ft 

ft 

Solution Tha 

= 

d = 

 

ftn =S 

fia 

 

 

 

a n 

=a 

+dn− 

 

 

 

 

 

a 

= 

+ 

−= 

 

S n 

= na +a _ n 

( 

+ 

) 

 

= 

__ 

 

= 

Try It #5 

ft


Using the Formula for Geometric Series 

 

geometric series 

re fin 

 

S n 

=a 

+ra 

+r a 

++r n a 

 

fin 

r 

 

rS n 

=ra 

+r a 

+r a 

++r n a 

 

 

S n 

=a 

+ra 

+r a 

++r n a 

 

 

−rS n 

=−ra 

+r a 

+r a 

++r n a 

 

 

−rS n 

=a 

−r n a 

fi 

S n 

−r 

 

S n 

= a −rn 

−r r ≠ 

formula for the sum of the first n terms of a geometric series 

geometric seriesThfin 

 

S n 

= a −r n 

 

−r r ≠ 

How To… 

, fie fin 

1. a 

rn 

2. a 

rnS n 

= a −r n 

−r 

3. o fiS n 

 

Example 4 

Finding the First n Terms of a Geometric Series 

o fi 

a. S 

+−++ 

 

b. ∑ ċ 

k 

Solution 

k= 

a. a 

=n = 

n fire fi 

r= − 

=− 

 

a 

rn 

S n 

= a −rn 

_ −r 

 

( − ( − 

S 

= 

−( − 

) 

) ) 

__ 

≈

792 


 

b. a 

k = 

a 

=ċ = 

r =. Thn = 

a 

rn 

S n 

= a −r n 

 

−r 

 

S 

= − 

− = 

Try It #6 

o fi 

S 

+++ 

Try It #7 

o fi 

 

∑ 

k= 

k 

Example 5 

Solving an Application Problem with a Geometric Series 

 

 

Solution Tha 

=n =r = 

a 

rno fi 

S n 

= a −rn 

−r 

 

S 

= − 

 

− ≈ 

 

Try It #8 

 

 

Using the Formula for the Sum of an Infinite Geometric Series 

Thfi 

fifininfinite seriesfi 

fi++++ 

∞ 

Th∑ kfi 

k = 

Th 

fifidiverges 

Determining Whether the Sum of an Infinite Geometric Series is Defined 

fififiTh 

 

+++++


Thr =nr n 

ff 

fiThfi−

794 


fir = 

r n n 

 

( _ 

) = 

 

 

( _ 

) = 

 

 

( _ 

) = 

Thr n n 

 

 

( _ 

) 

= 

 

 

( _ 

) 

= 

 

 

 

( _ 

) 

= 

 

 

nr n nr n r n 

−r n a 

Th 

fi 

formula for the sum of an infinite geometric series 

Thfi−


 

 

a 

=r =− o 

fi 

a 

S =_ 

 

−r 

 

S =__ 

= 

−( − 

) 

d. Thr > 1. Th 

Example 8 

Finding an Equivalent Fraction for a Repeating Decimal 

_ 

Solution _ = 

 

_ =+++ 

fi 

 

 

_ =+ + 

 

fi 

fi 

 

 

Try It #12 

 

Try It #13 

 

Try It #14 

 

a 

S =_ 

 

−r = 

 

− = 

= 

 

+ 

+ 

+ 

∞ 

∑ 

k = 

k + 

∞ 

∑ 

k = 

( − 

) k 

Solving Annuity Problems 

 

annuity 

 

 

fi 

 

 

a 

= 

r =100.5% = 

 

n n 

n =a 

=r =n = 

 

 

S 

= − 

 

− ≈ 

 

=

796 


How To… 

, fi 

1. a 

 

2. n 

3. r 

a. 

b. o fir 

4. a 

rne fin 

S n 

= a rn 

_ r 

5. o fiS n 

ftn 

Example 9 

Solving an Annuity Problem 

Th 

ft 

 

Solution Tha 

= 

n =firfi 

 

 

r =+ 

= 

a 

=r =n =fin 

o fi 

 

S 

= − 

 

− ≈ 

ft 

Try It #15 

Th 

 

 

 

Arithmetic Series (http://openstaxcollege.org/l/arithmeticser) 

Geometric Series (http://openstaxcollege.org/l/geometricser) 

Summation Notation (http://openstaxcollege.org/l/sumnotation)



VERBAL 

1. n 2. e diff 

 

3. 4. 

ies diff n 

5. 

ALGEBRAIC 

 

6. Thm +mm=m= 7. Thn=n=n 

8. Thk−k=−k= 9. Ther 4 fi 

 

 

10. +++++++++ 11. ++++ 

12. 

++ 

+++ 

e fin 

13. 

++ 

++ 

14. ++++ 15. ++++ 

 

 

16. +++++++ 17. ++++ 

18. − 

+ 

− 

++ 

 

n 

 

19. +++ _ + _ 

 

20. ∑ ċ 

n− 

n= 

 

21. ∑ ċ a− 

a= 

fi 

 

22. ++++ 23. ++++ 24. ∑ m− 

∞ 

25. ∑ −( − 

) k− 

k= 

∞ 

m=


GRAPHICAL 

 

ft 

 

26. 

 

∞ 

∑ 

k = 

( 

) k 

27. 

 

28. 29. S n 

 

 

 

NUMERIC 

, fi 

 

30. ∑ a 

a= 

 

31. ∑ nn− 

n= 

 

32. ∑ k 

k= 

 

33. ∑ k 

e fino fi 

34. −+−++++ 35. + 

++ 

++ 

+ 

 

36. −++++ 37. ∑ ( k 

− 

) 

finfi 

38. S 

−−−− 39. S 

−+− 

k = 

k= 

 

40. ∑ k− 

k= 

 

41. ∑ −ċ( 

n= ) n− 

, fifi 

42. +++ 

 

∞ 

45. ∑ ċ n− 

n= 

43. −− 

− 

− 

 

k= 

 

∞ 

44. ∑ ċ( 

) k− 

 

 

46. 47. 

 

 

48. 

 

49. 

 

EXTENSIONS 

50. Th−k k=x 

x 

52. n 

n 

∑ 

k= 

k− 

51. a k 

 

 

∑ 

k= 

a k 

= 

53. 

−−−−−


54. _ 

n. Th 

 

 

56. 

 

. Th 

 

ff 

 

fter fi 

 

55. Th s fi 

 

 

57. 

 

 

ff 

 


58. 

 

 

 

ft 

60. 

 

ft 

59. g 6 f 

 

 

ft 

61. 

 

 

 

 

 

62. 

. Th

800 




Solve counting problems using the Addition Principle. 

Solve counting problems using the Multiplication Principle. 

Solve counting problems using permutations involving n distinct objects. 

Solve counting problems using combinations. 

Find the number of subsets of a given set. 

Solve counting problems using permutations involving n non-distinct objects. 

9.5 COUNTING PRINCIPLES 

 

 

Th 

Th 

ff 

Th 

 

 

Using the Addition Principle 

ThffTh 

ThAddition Principle 

fi 

Figure 1 

Figure 1 

the Addition Principle 

Addition Principlem 

ne fim +n 

Example 1 

Using the Addition Principle 

Th

SECTION 9.5 COUNTING PRINCIPLES 801 

Solution fi 

 

+ + + + + + 

↓ ↓ ↓ ↓ ↓ ↓ ↓ 

+ + + + + + 

Th 

Try It #1 

 

 

Using the Multiplication Principle 

ThMultiplication Principle 

n a fi 

Figure 2 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2 

Th 

1. 

2. 

3. , fi 

4. , fi 

5. 

6. 

7. 

8. 

9. , fi 

10. , fi 

11. 

12. 

o fi 

 

×× 

× × =

802 


the Multiplication Principle 

Multiplication Principlemn 

ftfim ×nTh 

Fundamental Counting Principle 

Example 2 

Using the Multiplication Principle 

 

fifi 

fi 

Solution fififi 

 

×× 

× × = 

Thfi 

Try It #2 

ffTh 

 

 

Finding the Number of Permutations of n Distinct Objects 

Th 

 

permutation 

Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle 

ftTh 

fi 

 

 

× × 

The fie fi 

× × 

fte fin fi 

× × 

ftfi 

e fi 

× × = 

Th 

How To… 

n 

1. e fi 

2. 

3. e fi 

4.


Example 3 

Finding the Number of Permutations Using the Multiplication Principle 

 

a. 

b. 

 

c. 

Solution 

a. 

 

× × 

 

Thfifi 

e fi 

× × = 

o fi 

b. 

 

× × 

 

fifiTh 

 

× × = 

o fis fi 

c. 

× × × × × × × × 

Thfi 

 

× × × × × × × × = 

Th 

Analysis Note that in part c, we found there were ways for people to line up. The number of permutations of n 

distinct objects can always be found by n!. 

Try It #3 

fi 

 

Try It #4 

fi 

 

Try It #5 

fi

804 


Finding the Number of Permutations of n Distinct Objects Using a Formula 

 

 

nr 

P nr n 

P r 

 

Pn, rfinnn−r 

n−r 

Th 

fi 

Th 

××= 

 

= 

== 

Thffi 

. Th 

n 

Pn, r= 

 

n− 

n 

n−nnn 

n 

n 

formula for permutations of n distinct objects 

nr 

n 

Pn, r= 

 

n−r 

How To… 

 

1. n 

2. r 

3. nr 

4. 

Example 4 

Finding the Number of Permutations Using the Formula 

 

 

Solution n =r = 

n 

 

Pn, r= 

 

n−r 

 

 

P= 

 

− = 

= 

Th 

Analysis We can also use a calculator to find permutations. For this problem, we would enter 12, press the n 

P r 

function], 

enter 9], and then press the equal sign. The n 

P r 

function] may be located under the MATH] menu with probability 

commands. 

Q & A… 

Could we have solved Example 4 using the Multiplication Principle? 

ċċċċċċċċo fi


Try It #6 

fi 

 

Try It #7 

fi 

 

Find the Number of Combinations Using the Formula 

Th 

 

combinations. rn 

C nrCn, r n 

C r 

 

 

n 

C n, r= 

 

rn−r 

 

 

fi 

 

Th==Th 

 

 

ThnotTh 

 

formula for combinations of n distinct objects 

nr 

n 

C n, r= 

 

rn−r 

How To… 

 

1. n 

2. r 

3. nr 

4. 

Example 5 

Finding the Number of Combinations Using the Formula 

ffs fi 

a 

b. 

Solution 

 

 

a. 

 

C= 

 

− = 

b. 

 

C= 

 

− =

806 


Analysis We can also use a graphing calculator to find combinations. Enter 5, then press n 

C r 

, enter 3, and then press 

the equal sign. The n 

C r 

function may be located under the MATH menu with probability commands. 

Q & A… 

Is it a coincidence that parts ( a) and ( b) in Example 5 have the same answers? 

rn notn−r. ThCn, r=Cn, n −r 

Try It #8 

ffs 10 fle 3 fl 

Finding the Number of Subsets of a Set 

r 

ff 

ff 

ThC= 

ThC= 

 

C+C+C+C+C+C= 

Th. Th 

n 

n 

 

 

formula for the number of subsets of a set 

n n 

Example 6 

Finding the Number of Subsets of a Set 

ffff 

 

Solution n = 

 

n = 

 

Th 

Try It #9 

= 

diff 

 

Finding the Number of Permutations of n Non-Distinct Objects 

 

 

 

Th


n 

 

r 

r 

r k 

 

 

fi 

Th 

 

= 

Th 

formula for finding the number of permutations of n non-distinct objects 

nr 

r 

r 

r k 

 

 

n 

 

 

r 

r 

r k 

 

Example 7 

Finding the Number of Permutations of n Non-Distinct Objects 

 

Solution Thn =r 

=r 

= 

Th 

 

= 

Try It #10 

 

 

Combinations (http://openstaxcollege.org/l/combinations) 

Permutations (http://openstaxcollege.org/l/permutations)



VERBAL 

nAmB 

AB 

1. 

AB 

 

3. 

 

 

 

 

2. 

AB 

4. nts diff 

r 

n 

5. r 

n 

r 

 

 

NUMERIC 

Th 

 

6. A=−−− 

 

A 

8. 

 

10. 

 

12. A 

B 

A=bcd B=aeiou 

14. 

 

7. B=−−−− 

 

A 

9. 

 

 

11. 

 

13. 

 

 

15. P 16. P 17. P 18. P 19. P 

20. C 21. C 22. C 23. C 24. C 

, fi 

25. 26. abcz 

27. 28. Th 

 

29. Th 

 

, fi 

30. Th 31. Th 

32. Th 33. Th 

 

34. Th


EXTENSIONS 

35. ThS 

 

SHint 

 

t 0.) 

36. Th 

n 

 

n 

 

37. Cn, rPn,r 38. A 

A 

39. 

 

 


40. 

 

 

a. 

 

b. 

 

c. 

 

42. 

 

 

y diff 

 

44. 

 

 

 

 

46. 

 

 

48. 

 

 

 

 

 

50. 

 

diff 

52. ff 

 

 

 

 

 

54. 

 

 

41. ffs 6 diff 

d 8 diff 

3 p 

 

 

43. ff 

 

 

y diff 

 

45. s in 12 diff 

 

 

47. 

 

49. 

s—diff 

 

 

51. ff 

 

 

 

 

53.

810 




Apply the Binomial Theorem. 

9.6 BINOMIAL THEOREM 

 

 

o fix+y n n 

Identifying Binomial Coefficients 

Counting Principlesfix+y n 

fiffi( _ n r 

) 

Cn, r 

( _ n n 

r 

)=Cn, r= 

 

rn−r 

Th( _ n r 

)binomial coefficientffi( 

 

) =C= 

binomial coefficients 

nrn ≥rbinomial coefficient 

( _ n n 

r 

)=Cn, r= 

 

rn−r 

Q & A… 

Is a binomial coefficient always a whole number? 

ffi 

 

Example 1 

ffi 

Finding Binomial Coefficients 

a. ( 

) b ( 

) c. ( 

Solution 

) 

ffi n 

C r 

 

a. ( 

) = 

 

− =ċċ 

= 

( _ n n 

r 

)=Cn, r= 

rn−r 

 

b. ( 

c. ( 

) = 

 

− =ċċ 

= 

) = 

 

− =ċċ 

= 

Analysis 

 

 

( _ n r 

)=( 

n_ 

n − r )

SECTION 9.6 BINOMIAL THEOREM 811 

Try It #1 

ffi 

a. ( 

) b. ( 

) 

Using the Binomial Theorem 

x+y n binomial expansion 

ffix+y x+yfiftTh 

fifi 

 

 

x+y =x +xy+y 

 

 

x+y =x +x y +xy +y 

x+y =x +x y +x y +xy +y 

xy 

. Thn 

ffiffi 

Thffi 

( n _ 

) ( n _ 

) ( n _ 

) ( n _ 

n ) 

ThBinomial Theorem 

n 

 

x+y =∑ 

n ( n _ k ) xn−k y k 

 

k = 

n_ 

=x n +( n _ 

) xn− y +( n _ 

) xn− y ++( n − ) xy n − +y n 

ffix +y 

 

 

x+y =x +y 

 

x+y =x +xy+y 

 

 

x+y =x +x y +xy +y 

x+y =x +x y +x y +xy +y 

x+y 

Pascal’s Triangle 

Exponent 

x+y = x+y 

 

Pattern 

+ 

# of Terms 

 

x+y = x +xy + y 

 

+ 

 

x+y = x +x y + xy + y 

 

+ 

 

x+y = x +x y + x y + xy + y 

 

+ 

 

n 

n+ 

n+ 

+ 

+ 

+ 

+ 

+ 

xy xy xy xy xy 

x 

y 

Figure 1

812 


Figure1 

Thn +x+y n 

Thn 

Thxn 

Thyn 

Th 

x+y n =+= 

x 

x xx = 

y =yy 

x x yx y x y xy y 

Th 

x+y =x +x y +x y +x y +xy +y 

ffiThffiffi 

Figure2 

+ = 

 

Pascal’s Triangle 

 

 

 

 

 

 

+ = 

Figure 2 

fl 

+finfl 

 

ffi 

 

 

 

→ x+y = 

→ x+y =x +y 

→ x+y =x +xy+y 

→ x+y =x +x y +xy +y 

→ x+y =x +x y +x y +xy +y 

→ x+y =x +x y +x y +x y +xy +y 

the Binomial Theorem 

Binomial Theorem 

n 

 

x+y =∑ 

n ( n k ) x n −k y k 

k = 

 

n_ 

=x n +( n _ 

) x n − y +( n _ 

) x n − y ++( n − ) xy n − +y n

SECTION 9.6 BINOMIAL THEOREM 813 

How To… 

 

1. n 

2. k =k =nl Th 

3. 

Example 2 

Expanding a Binomial 

 

a x+y 

b (x −y 

Solution 

a. n =k =k = 

 

 

 

 

x+y =( _ 

) x y +( _ 

) x y +( _ 

) x y +( _ 

) x y +( _ 

) x y +( _ 

) x y 

x+y =x +x y +x y +x y +xy +y 

b. n =k =k =x 

x−yy 

x −y =( _ 

) x −y +( _ 

) x −y +( _ 

) x −y +( _ 

) x −y +( _ 

) x −y 

x −y =x −x y +x y −xy +y 

Analysis Notice the alternating signs in part b. This happens because (−y) raised to odd powers is negative, but (−y) 

raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign. 

Try It #2 

 

a. x−y b. xy 

Using the Binomial Theorem to Find a Single Term 

x+y 

 

o fifi 

ffix+y 

x+y =x +( _ 

) x y +( _ 

) x y +( _ 

) x y +( _ 

) xy +y 

Th( _ 

) x y. Th( _ 

) x y 

( n _ 

r )x n−r y r 

the (r + 1)th term of a binomial expansion 

r+x+y n 

( n _ 

r )x n −r y r

814 


How To… 

fi 

1. n 

2. r+ 

3. r 

4. rr+ 

Example 3 

Writing a Given Term of a Binomial Expansion 

x+y 

Solution r +=r = 

Try It #3 

( n _ 

r )x n−r y r 

( _ 

) x− y =x y 

x −y 

 

The Binomial Theorem (http://openstaxcollege.org/l/binomialtheorem) 

Binomial Theorem Example (http://openstaxcollege.org/l/btexample)



VERBAL 

1. 

 

2. 

 

 

3. l Th 4. 

Th 

ALGEBRAIC 

ffi 

5. ( _ 

) 

6. ( _ 

) 

7. ( _ 

) 

8. ( _ 

) 

9. ( _ 

) 

10. ( _ 

) 

11. ( _ 

) 

12. ( _ 

) 

l Th 

13. a−b 14. a+ 15. a+b 16. x+y 17. x+y 

18. x−y 19. x−y 20. ( _ x 

+y ) 21. x − +y − 22. √ — x −√ — y 

l The fi 

23. a+b 24. x− 25. a−b 26. x−y 27. a+b 

28. a+b 29. x −√ — y 

, fi 

30. Thx−y 31. Thx−y 

32. Thx−y 33. Th+y 

34. Tha+b 35. Th x−y 

36. Thx− 37. Tha−b 

38. Th( x − 

) 

 

 

39. Th( y _ 

+_ x 

) 

GRAPHICAL 

Thf x=x+ Thfi 

 

40. f 

xf 

x 

 

42. f 

xf 

x 

 

41. f 

xf 

x 

 

43. f 

xf 

x 

 

44. f 

xf 

x 

t fi


EXTENSIONS 

45. x+y n 

( n _ 

k ) an−k b k k 

n ( n _ 

 

k ) = ( _ 

) 

46. a+b n a n−k b k 

 

47. x+b 

bk 

49. 

l Th 

a. x −x+ 

b. √ — a +√ — a − 

c. x +y −z 

d. x −√ — y 

48. ( 

n_ 

k − 1 ) + ( n _ 

 

k ) 

( n _ 

k ) 

pp≥ 

p=pp−

SECTION 9.7 PROBABILITY 817 



Construct probability models. 

Compute probabilities of equally likely outcomes. 

Compute probabilities of the union of two events. 

Use the complement rule to find probabilities. 

Compute probability using counting theory. 

9.7 PROBABILITY 

Figure 1 An example of a “spaghetti model,” which can be used to predict possible paths of a tropical storm. [33] 

 

Figure 1Th 

Th 

 

Constructing Probability Models 

experiment 

ThoutcomesTh 

sample spaceTh 

event 

ThprobabilityThpfi 

≤p ≤. probability model 

 

ffleTable 1 

Outcome 

ffle 

ffle 

Table 1 

Probability 

 

 

Th 

The fi

818 


How To… 

 

1. 

2. 

3. 

Example 1 

Constructing a Probability Model 

 

Solution Th 

. Th 

 

Th 

 

 

Outcome 

Probability 

 

 

 

 

 

 

 

 

 

 

 

 

Table 2 

Q & A… 

Do probabilities always have to be expressed as fractions? 

 

 

Try It #1 

 

Computing Probabilities of Equally Likely Outcomes 

SS 

fi 

S 

fiTh 

S 

= 

 

computing the probability of an event with equally likely outcomes 

ThES 

PE= E 

S = n(E) ____ 

n(S) 

ES≤PE≤ 

Example 2 

Computing the Probability of an Event with Equally Likely Outcomes 

 

Solution ThTh 

o fi 

PE= 

=


Try It #2 

 

Computing the Probability of the Union of Two Events 

ftfi 

fi 

Thunion of two eventsEFE ∪F 

 

PE∪F=PE+PF−PE∩F 

Figure 2o fib 

 

 

 

 

 

 

Figure 2 

Th 

= 

Th 

bb 

= 

 

bfi 

bb 

 

+ 

− 

= 

 

Thb 

 

probability of the union of two events 

ThEF E ∪FE 

FEEF 

E ∩F 


Example 3 

Computing the Probability of the Union of Two Events 

 

Solution 

 

Th 

 

 

Th 

 

PH= 

P= 

PH∩= 

 


Th 

 

= 

+ 

− 

 

=

820 


Try It #3 

 

Computing the Probability of Mutually Exclusive Events 

Figure 2 

dThd 

mutually exclusive events 

 

PE ∪F=PE+PF 

EFTh 

 

= 

d 

fi 

d 

PE∪F=PE+PF 

Thd 

 

= 

+ 

 

= 

 

probability of the union of mutually exclusive events 

ThE F 

PE∪ F=PE+PF 

How To… 

 

1. e fi 

2. e fi 

3. 

4. 

5. 

Example 4 

Computing the Probability of the Union of Mutually Exclusive Events 

 

Solution Th 

Th 

 

 

 

 

+ 

= 

 

Try It #4

SECTION 9.7 PROBABILITY 

821 

 

 

Thcomplement of an eventEE′ 

E 

WW 

fi 

 

PE′=−PE 

Th. Th 

 

 

 

 

− = 

the complement rule 

Thcomplement of an event 

PE′=−PE 

Example 5 

Using the Complement Rule to Calculate Probabilities 

 

a. 

b. 

Solution ThfiTh 

fi 

× 

 

 

 

 

 

 

Table 3 

a. 

. Th 

 

= 

 

b. 

fi 

 

PE′=−PE 

=− 

 

 

= 

 

 

Try It #5

822 


Computing Probability Using Counting Theory 

 

fi 

Th 

 

fi 

 

Th 

CTh 

C. Th 

 

____ 

 

 

=C _ 

 

C 

 

 

Example 6 

Computing Probability Using Counting Theory 

= 

 

= 

 

a. 

b. 

c. 

Solution 

 

 

a. 

ThCThC 

 

C 

C = 

 

= 

 

 

b. 

ThCTh 

C 

ThCċC 

 

CC 

C =ċ 

= 

 

 

c. ft 

 

 

ThC 

C 

 

 

C 

C = 

 

 

 

 

ThCċC 

 

CC 

C =ċ 

=


 

o fi 

 

 

 

 

+ 

 

= 

 

o fi 

 

− 

= 

 

 

Try It #6 

 

 

a. 

b. 

c. 

 

Introduction to Probability (http://openstaxcollege.org/l/introprob) 

Determining Probability (http://openstaxcollege.org/l/determineprob)



VERBAL 

1. 2. 

 

 

3. 4. e diff 

 

 

5. union of two sets 

 

union 

of two events 

diff 

NUMERIC 

Figure 3o fi 

6. 

 

 

7. 

 

 

8. 

9. 

 

 

 

 

10. 

11. 

12. 

Figure 3 

13. 

 

14. 15. 

16. 17. 

 

18. 19. 

20. 21. 

 

22. 23. 

24. 

 

25. 

 

 

 

26. 27. 28. 29. 

30. 31. 32. 

 

33. 

 

34.


35. 

 

37. 

 

39. 

 

36. 

n 9. 

38. 

40. 

 

41. 42. 

 

 

 

43. 44. 

45. 46. 

 

 

47. 48. 

49. 50. 

EXTENSIONS 

 

ft 

 

 

51. 

 

52. 

 

53. 

5 

54. 

55. 

3 w 

 


 

 

56. 

 

 

58. 

 

 

57. et fi 

 

 

59. et fi 

 

 

60. e in fi



Key Terms 

Addition Principle mn 

m +n 

annuity 

arithmetic sequence e diff 

arithmetic series 

binomial coefficient rn 

Cnr( _ 

n r ) 

binomial expansion x+y n 

Binomial Theorem 

combination 

common difference e diff 

common ratio 

complement of an event E 

diverge 

event 

experiment 

explicit formula 

finite sequence n 

n 

Fundamental Counting Principle mnft 

m ×n 

geometric sequence 

geometric series 

index of summation 

 

infinite sequence 

infinite series 

lower limit of summation 

Multiplication Principle mnft 

m ×n 

mutually exclusive events 

n factorial n 

nth partial sum n 

nth term of a sequence 

outcomes 

permutation 

probability 

probability model


recursive formula 

sample space 

sequence 

series 

summation notation 

 

term 

union of two events 

upper limit of summation 

Key Equations 

 

 

 

n 

n 

n 

n 

e fin 

e fin 

fi−


Key Concepts 

9.1 Sequences and Their Notations 

 


Example 3 

nExample 4 

 

 


 

ThnnExample 7 

9.2 Arithmetic Sequences 

e diff 

Thn diff 

Thdiff 

Example 1 

Thdiff 


diffda n 

=a n− 

+dn ≥ 

Example 4 

 

diffda n 

=a 

+dn−Example 5 

Example 6 

a n 

=a 

+dnExample 7 

9.3 Geometric Sequences 

 

Th 

ThExample 1 

Th 


ra n 

=ra n− 

n ≥ 

Example 3 

ra n 

=a 

r n − Example 5 

a n 

=a 

r n Example 6 

9.4 Series and Their Notations 

Th 

 

Example 1 

Th 

Th nExample 2Example 3 

Th 

Th nExample 4Example 5 

Th −



ts. Th 

Example 9 

9.5 Counting Principles 

mn 

m +nExample 1 

mnft 

m ×nExample 2 

n 

nrPn, r 

Pn, rExample 3 

Example 4 

 

nrC n, r 

Example 5. 

n n Example 6 

 

Example 7 

9.6 Binomial Theorem 

r ) C n, rExample 1 

Thl ThExample 2 

Example 3 

( n _ 

9.7 Probability 

 

 

ThExample 1 

 

 

Example 2 

 

Example 3 

 

Example 4 

The diff 

Example 5 

 

Example 6



SEQUENCES AND THEIR NOTATION 

1. 

a 

=a n 

=a n− 

+n 

3. 

a n 

= n + 

 

2. 

− 

 

4. 

n 

a n 

= 

 

nn+ 

ARITHMETIC SEQUENCES 

5. _ _ _ _ 

 

n diff 

7. a 

= 

n diffd=− 

fi 

9. 

−− 

11. 

_ _ _ _ 

 

6. 

n diff 

8. a 

= 

a 

=− 

10. 

−_ −−_ 

 

12. 

 

GEOMETRIC SEQUENCES 

13. 

 

15. a 

= 

a 

= t fi 

17. t fi 

a 

=a n 

=ċa n− 

 

19. 

−_ −_ 

− _ 

− _ 

 

14. 

 

16. a 

=− 

r=_ 

 

18. 

_ _ _ 

 

20. 

−−_ −_ − 

 

 

SERIES AND THEIR NOTATION 

21. 

_ m+m=m= 

 

23. n 

 

 

22. 

 

24. 

n diff. Th


25. n 26. Th 

S 

 

Table 1 

 

 

Year Membership Fees 

 

 

 

 

 

 

Table 1 

27. 

∞ 

∑ 

k− 

ċ ( − 

) k= 

29. 

 

 

ft 

28. _ 

 

 

 

 

 

 

 

30. Th 

 

 

 

 

 

 

COUNTING PRINCIPLES 

31. 

−− 

 

33. 

 

32. 

 

 

34. 

 

 

 

35. P 36. 

 

 

37. C 38. ff 

 

 

ff 

39. 

 

41. 

 

40. 

 

 

 

 

 

42.


BINOMIAL THEOREM 

43. ( _ 

) 

45. l Th 

a+b 

44. l Th( x+_ y ) 

46. a −b 

 

PROBABILITY 

 

47. 48. 

49. 50. 

 

51. 

 

52. 

53. 

 

 

 

 

54. 

 

56. 

 

55. 

 

57.


833 


1. 

a=−a n 

= +an 

 

3. 

n diff 

5. 

−− _ −− _ 

 

 

7. −−− _ − _ 

 

 

9. 

− _ 

_ − _ 

 

11. 

k − 

kk=−k= 

13. 

 

∑ −ċ− 

k− 

k= 

15. 

. Th 

 

 

 

 

17. 

m 5 diffs, 3 diff 

 

 

 

19. 

 

 

21. 

 

 

2. 

a n 

= n −n− n 

 

4. a 

=− 

diffd=− 

_ 

 

6. 

 

8. 

−−−− 

10. 

−_ _ − 

_ 

12. 

 

n. Th 

 

14. 

∞ 

∑ 

k= 

 

_ ċ ( − _ 

) k− 

16. 

 

 

 

 

18. 

 

 

 

20. 

 

 

22. l Th( _ x−_ y ) 

23. ( x − 

)


Figure 1 

 

 

 

 

 

 

 

Figure 1 

24. 

 

e t 

25. 

 

26. 27. 

 

28. 

 

29. 

y fl

Try It Answers 

Chapter 1 

Section 1.1 

1. a. b. __ c.− 

__ 

2. a. b. _ 

c.− 3. a. 

b. c. d. 

e. 4. a. 

b. c. d. 

e. 

5. 

a. − 

N W I Q Q' 6. a. b. c. 

d. e. 

b. × × × 

c. √ — × × × × 

d. √ — × 

e. × 

7. a. 

 

b. 

c. d. 

 

 

e. 

 

8. Constants Variables 9. a. b. c. 

d. 

a. πrr + h π r, h 

10. a. b. c. 

b. L + W L, W 

π 

d. e. 

c. y +y 4 y 11. 

12. a.−y−z−y + zb. −c. pq−p+q d. r−s+ 

t 

13. A=P+rt 

Section 1.2 

1. a.k b.( y ) c.t 2. a.s b.− c.ef 

3. a.y) b.t c.−g 4. a. b. c. d. 

 

 

5. a. 

−t b. 

f c. 

k 6. a.t − = 

t b. 

 

 

7. a.g h b. t c.−y d. 

 

a b e. r 

s 

8. a. b 

c b. u c.− 

w d.q 

p e. 

c d 

9. a. v 

u b. 

x c. e _ 

 

f 

_ 

s 

d.r 

e. f. h 

10. a.× 

b.× c.× d. × − e. × − 

11. a. b.− c. − 

d. 12. a.−× b.× − c. × 

d. −× e. ≈ × 13. × 

× × 

Section 1.3 

1. a. b. c. d. 2. ∣x ∣ ∣ y ∣ √ — yz 

xy? Th 

3. ∣ x ∣ 4. x√— 

y 

y 

5. b √ — ab 6. √ — 7. 8. √ — 9. −√ — 

10. a.− b. c. √ 11. ( √ — ) = = 12. x y 13. x 

Section 1.4 

1. Th−x 

− 2. x +x −x− 3. −x −x +x − 

4. x −x −x + x+ 5. x + x− 

6. x − x+ 7. x − 8. x + xy−x −y+ 

Section 1.5 

1. b −ax+ 2. x−x− 3. a.x+x+ 

b. x−x+ 4. x− 5. y+y− 

6. a+ba −ab+b 7. x−x +x+ 

8. a− − a− 

Section 1.6 

 

1. 

x+ 2. x+x+ 

 

x+x+ 

3. 

x− 

4. 

x+x− 

5. x −y xy 

Chapter 2 

Section 2.1 

2. x 3. √ — =√ — 

y 

– – 

 

 

 

 

 

– – – – 

– 

– 

y 

 

x 

4. ( − 

) 

 

1. 

x y = 1__ 

y 

x + 2 

2 (x, y) 

− y = − += 

− 

 

− y = − += ( − 

 

) 

 

 

y = += 

 

− 

y = += ( – – – – – 

– 

) 

– 

y = += 

x 

A-1

A-2 

TRY IT ANSWERS 

Section 2.2 

1. x=− 2. x=− 3. x= 4. x= 

 

5. x = − 

− 

− 

 

6. x = 

7. m = − 8. y =x − 9. x + y= 

 

10. y= 

11. y 

 

 

Section 2.3 

12. y =x + 

1. 2. C =x + 3. 

4. L=W= 5. 250 ft 

Section 2.4 

1. √ — −=+i √ — 2. 

3. −i−+i=−i 

4. 

−i 

5. +i 

6. −−i 7. − 

Section 2.5 

 

 

 

 

 

– – – – – – 

– 

– 

– 

– 

i 

 

1. x−x+=x=x=− 2. x−x+= 

x=x=− 3. x+x−=x=−x= 

4. x+x+=x=− 

x=− 

 

5. x=x=− 

x=− 6. x=±√ — 7. x=±√ — 8. x=− 

 

x= 9. 

 

Section 2.6 

1. 

2. 3. − 4. x=x= __ x=− 

 

5. x=− 6. x=− 

 

− 7. x=−x= 8. x=−−ii 

 

9. x=x= 10. x=− 

Section 2.7 

1. − 2. −∞−∪∞ 3. x< 4. x≥ − 

5. ∞ 6. [ − 

8. ( − 

 

∞ ) 7. < x ≤ 

) 9. ∣ x−∣≤ 

y 

10. k≤k≥ 

 

 

 

 

−∞∪∞ 

 

 

 

 

 

 

– – – 

– 

– 

 

 

 

 

 

– – – – – – 

– 

 

 

 

x 

x 

r 

Chapter 3 

Section 3.1 

1. a. b. 

 

2. w =f d 3. 4. g= 5. m= 

6. y=f x= ___ √ 

 

— x 7. g= 8. x=x= 

 

9. a. 

b. 

c. 

10. a. 

b. . Th 

e 100 diff 

fi 

11. 

12. 

Section 3.2 

1. − 2. −∞∞ 3. ( −∞ __ 

 

) ∪ ( __ 

∞ ) 

5. a.– 

4. [ − 

__ ∞ ) 

– 

b. x |x≤––≤x< c. –∞–∪– 

6. = = 

7. −∞−∞ 

8. 

y 

 

 

 

 

 

 

– – – – – – – 

– 

– 

– 

– 

– 

Section 3.3 

 

1. ___________ 

− 

= _____ 

= 2. __ 

 

3. a + 4. Th− 

−80). Th 

−∞−∪∞− 

− 

 

– – 

– – 

– 

– 

– 

– 

– 

– 

f x 

Section 3.4 

 

 

 

 

 

− 

1. a. fgx=fxgx=x−x −= x −x −x+ 

f−gx=fx−gx=x−−x −=x−x 

b. 

x 

x

TRY IT ANSWERS A-3 

2. aGr 

r 

GaF 

3. fg =f=g f=g = 4. g f=g = 

5. a. b. 6. −∪∞ 

7. gx=√ — 

+ x hx=____ 

 

− x f= h ∘ g 

Section 3.5 

1. bt=ht+=−t +t+ 

2. 

y 

3. 

5. a. 

– – – – 

– – – – – 

– – – 

 

 

 

 

 

– 

– 

– 

hx 

 

 

 

 

 

– 

– 

 

 

 

y 

– 

– 

– 

– 

gx 

fx 

 

 

 

 

x 

hxx – + 

 

 

 

 

x 

 

x 

Thf x 

g x. Th 

 

ft. Thft 

 

 

b. 

 

4. gx= 

____ 

x– + 

– – – 

y 

 

 

 

 

– 

– 

– 

6. a.gx=−f x b. hx=f −x 

x − x − 

g(x) − − − − h(x) 

7. 

– – – – – 

– 

– 

– 

– 

– 

y fx = x 

hx = f− x= − x 

 

 

 

 

 

 

x 

gx = −f x = −x 

 

 

hx=f−x 

fx 

8. 

9. x 

g(x) 

10. g(x) = x− 

11. gx=f ( __ 

x ) 

gx= √ 

x 

Section 3.6 

1. p| p−|≤ 

2. f x= −| x+|+ 3. x= −x = 

Section 3.7 

1. h= 2. 3. 4. Th 

f − −∞−f − ∞ 

5. a.f= 

b. f − = 

 

 

x 

 

6. a. b. 7. x=y+ 8. f − x=−x 

f∞f − −∞ 

9. y 

y=x 

 

 

 

 

 

fx 

 

 

 

– – – – 

– 

– 

Chapter 4 

Section 4.1 

f – x 

 

1. m= − − = 

− =− __ 

m< 

2. m= − 

 

− = 

= 

3. y=−x+ 4. H(x)=x+ 

5. 

7. 

– – – – 

– – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

y = x 

y 

y = x + 

 

 

 

 

x 

x 

x 

y = x 

6. 

−−− 

8. 

9. a. f x=x 

b. gx=− 

__ x 

10. y=− 

__ x+ 

Section 4.2 

1. Cx=x+25,000; Thy 

 

2. a. b. 3. 

Section 4.3 

1. 2. 

Chapter 5 

Section 5.1 

1. Th− 

hx= − 

__ 

x+ +h− 

h−≈ 

2. gx=x −x+gx=x− +

A-4 


3. Ths. Thfx≥ 

[ 

∞ ) 4. yx 

5. 

 

Section 5.2 

1. fxfx=x 

Th 

2. x fx 

x→±∞fx→−∞ 

3. Ths 6. Thm i−x 

Th − 4. x→∞fx→−∞ 

x→−∞fx→−∞ 

5. Th 

x x 

fxx 

fx 

6. yx− 

7. Thx 

8. Th 

x 

 

9. x 

−y 

 

Section 5.3 

1. yx− 

2. Th−− 

 

3. 

 

– – – – 

– 

– 

– 

– 

– 

y 

4. f 

f 

f 

 

x=x= 

x 

5. fx=− 

 

x− x+ x− 6. Th 

− 

 

Section 5.4 

 

1. x −x+− 

x+ 2. 

3. x −x+ 

Section 5.5 

x −x +x−+ 

 

x+ 

1. f −=− 2. −− 

3. 4. − 

 

5. f x=− 

 

x + 

 

x −x+ 6. 

 

 

7. 

Section 5.6 

1. x→±∞f x→x→ 

f x→∞e nxy 

2. 

y 

Th 

hift 

 

 

 

 

x→fx→∞ 

 

 

x→±∞fx→− 

3. __ 

 

– – – – 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

y = – 

x = 

x 

Th 

 

fx=______ 

 

x− − 

4. x=x= 

5. x=x= 

x= 6. x=x=– 

y= 7. 

 

 

f x=______ 

 

x− − 

 

8. 

= 

___________ 

−x− x− 

_______________ 

= −x −x+ 

x−x− 

______________ 

= −x +x− 

x −x+ 

–– – – 

Section 5.7 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x →±∞f x→− 

y =– 

 

 

x =x → 

f x→∞ 

x 

 

 

y ( − ) 

y=_ 

 

x= 

x=y( _ 

) 

x– 

x – 

 

x 

 

 

 

1. f − fx=f ( − x+ _____ 

) = ( x+ _____ 

) −=x−+=x 

ff − x=fx−=__________ 

x−+ 

=x __ 

=x 

2. f − x= x − 3. f − x= √ — x − 

4. f − x=______ 

x − 

x 5. f− x=______ 

x+ 

x− 

Section 5.8 

1. 

____ 

2. __ 

3. x=


Chapter 6 

Section 6.1 

1. gx= x jx= −x 

2. 3. 

 

4. Nt= t 

5. f x= x 6. f x=√ — √ — x 

x 

7. y≈· x 8. 9. 

10. e − ≈ 11. 12. 

× − 

 

 

 

Section 6.2 

1. 

2. 

fx 

fx = 4 x 

 

 

 

 

– 

– – – – – – 

 

– 

 

 

 

– 

 

fx 

– – – – – – 

– 

 

Th−∞∞ 

∞ 

y= 

x 

f(x) = 2 x – + 3 

 

Th−∞∞ 

∞ 

y= 

 

y = 3 

 

x 

3. x≈− 

4. 

( _ 

 

) =− 5. = 6. ≈ 

7. The diff 8. 

 

 

Section 6.4 

1. ∞ 2. (∞) 

3. 

fx 

4. 

5. 

– – 

 

 

 

 

––– 

– – – 

– 

– 

y 

 

 

 

 

 

f (x) = (x) 

 

x = 0 

x= − x= 

 

 

f (x) = (x + ) 

 

y= x 

x 

– – – – 

– 

 

– 

– 

– 

– 

 

 

− 

− 

 

 

y= x 

 

 

x 

x 

Th∞ 

−∞∞ 

x= 

Th−∞ 

−∞∞ 

x=– 

y fx= x+ Th∞ 

x= 

−∞∞ 

x= 

4. 

 

 

 

 

– 

fx 

– – – – – – 

– 

fx= ( x 

 

y = 0 

 

 

Th−∞∞ 

∞ 

y= 

x 

6. 

x= 

y 

 

 

y= x 

Th 

∞ 

 

fx= x 

−∞∞ 

 

x 

 

x= 

5. 

g(x) = 1.25 –x 

 

 

 

– 

––– – – – 

gx 

 

 

y = 0 

x 

 

 

Th−∞∞ 

∞ 

y= 

6. fx=− 

__ e x −−∞∞ 

 

−∞y= 

Section 6.3 

1. a. 

= = 

b. 

= = 2. a. = 

 

= b. = 

= 

c. − =_ 

( _ 

) =− 

3. 

=_ √ — = _ 

= 

 

7. 

8. 

y 

x = 2 

 

 

 

 

––– 

– – – 

 

– 

– 

– 

– 

 

 

 

 

y 

– – – – – 

 

– 

– 

x 

Th∞ 

−∞∞ 

x= 

x 

Th−∞ 

−∞∞ 

x= 

9. x≈ 10. x= 

11. fx=x+−

A-6 


Section 6.5 

1. b 

+ b 

+ b 

+ b 

k= b 

+ b 

k 

2. 

x+− 

x−− 

x− 3. x 

4. −x 5. 

6. x+y−z 

7. 

__ x 

8. __ 

x−+x+−x+−x− 

9. ( ⋅ ____ 

⋅ ) ( __ 

) 

10. ( x− √ — ___________ x 

x− ) 

11. ________ 

x x+ 

x+ ( _______ 

x x+ 

x+ ) 

12. Th 13. ____ 

 

 

14. _____ 

 

≈ _____ 

= 

Section 6.6 

1. x= − 2. x= − 3. x=__ 

 

4. 5. x= 

6. t = ( __ 

 

) ( __ 

 

) 

 

( _ 

) 

 

7. t= 

( _ 

√ — ) =− __ 

8. x= 9. x=e 

 

10. x=e − 11. x≈ 12. x= x=− 

13. t=× 

______ ≈ 

Section 6.7 

1. f t=A 

e −t 2. 

3. f (t) = A 

e ( ____ 

 

)t 4. 5. 

6. y=e x 7. y=e x 

Section 6.8 

1. a. Tht fi 

y= x b. 

ft 

2. a. Tht fi 

y=+x b. 

 

3. a. Tht fi 

_____________________ 

 

y= 

−x 

+e 

b. 

c. 

 

Chapter 7 

Section 7.1 

1. 

2. Th 

− 

y 

 

 

 

 

 

 

 

− 

– –– 

– – – 

– 

 

– 

– 

– 

– − 

3. −− 4. −− 5. − 6. 

7. Th 

fixx + 

8. 

Section 7.2 

1. − 2. 3. fi 

xx−−x + 

Section 7.3 

1. ( − _ 

 

4. 

_ 

) 2. − 3. −−−− 

y 

 

+ 

 

 

−− 

 

– –– 

– – 

– 

 

– 

− 

– 

Section 7.4 

 

1. _ 

 

x− − _ 

x− 2. _ 

x− − _ 

x− 

x− 

4. _ 

 

x −x+ + __ x+ 

x −x+ 

Section 7.5 

1. A+B= [ 

 

 

 

− 

2. B=[ − − − − 

] 

Section 7.6 

] 

+ [ 

 

 

 

− 

− 

] 

= [ 

 

3. _ 

x− +x− _ 

 

x + 

] = [ 

 

+ +− 

+ 

 

+ 

+− −+ 

1. [ − ∣ ] 2. x−y+ z= 3. 

x−y+z= 

y+z=− 

4. [ − 

 

∣ 

 

5. 

] 

 

6. 

Section 7.7 

1. AB=[ 

 

 

 

− − 

] [ − − ] = [ 

BA=[ − − 

 

2. A =[ − 

 

 

 

− 

 

 

 

 

 

− 0 ] 

 

−+ −+ 

 

 

−1−+− −−+−] 

= [ 

 

] 

] [ 

− − 

] = [ −+−− 

−+−− 

= [ 

 

] 

] 

3. A− = [ 

 

 

+− 

 

− 

− 

+− ] 

] 

4. X= [ 

 

 

]


Section 7.8 

1. − 2. − 3. ( −_ _ 

) 

Chapter 8 

Section 8.1 

1. x +_ 

y 

= 2. x− _ 

 

+y− _ 

 

= 

y 

3. 

 

± 

 

 

± − 

 

±√ — 

– –– 

– – 

– 

– 

– − 

– 

4. _ 

x 

+y _ 

 

= 

 

± 

± 

±√ — 

5. 

− 

( −√ — ) 

( +√ — ) 

 

6. x− 

+y+ 

=−− 

−−( −−√ — ) 

( −+√ — x 

) 7. a. 

+ y 

= b. Th 

 

 

Section 8.2 

y 

 

 

 

 

 

 

 

− 

– –– 

– – – 

– 

 

– 

– 

– 

– − 

– 

− 

 

+ 

 

 

−− 

 

 

– –– 

– – 

– 

 

– 

− 

– 

1. ±( ±√ — ) 2. y −x 

= 

3. y− 

+x− 

= 

4. ± 

± 

± 

y 

 

y=± _ x 

− 

− 

y 

− 

 

 

x 

5. −− 

−−−( −−√ — ) 

( −+√ — )y=± x −− 

 

 

−− 

− 

y 

− 

− 

6. Th 

_ 

x 

− y _ 

 

= _ x 

y 

− _ 

 

= 

Section 8.3 

1. − 

x= 

 

−± 

2. 

y= − 

 

± 

3. x =y 

4. − 

y= − 

− 

x= 

 

− 

5. − 

x= − 

−− 

− 

y= 

 

−− 

−− 

−− 

− 

x=− 

− 

6. a. y =x b. Th 

Section 8.4 

1. a. b. 2. x' _ 

 

+y' _ 

 

= 

3. a. b. 

x 

y =−x 

− 

y 

− 

−− 

 

x= 

y y + =−x − 

y=− 

 

y y + =−x − 

y=− 

− 

x= 

y 

 

− 

− 

x 

y= 

x 

x 

x + =−y − 

x

A-8 


Section 8.5 

Section 9.7 

b. x +x y+xy +y 3. x y 

1. e= 

x=− 

1. 

Outcome 

y 

 

2. 

3. r= 

 

 

 

−θ 

4. −x+ x −y = 

− − 

 

− 

− 

 

x 

 

 

 

− 

 

Chapter 9 

Section 9.1 

1. The fit fi 2. The fit fi 

{ −−_ −_ 

} 3. The fi 

4. a n 

=− n+ n 5. a n 

=− _ n n 

6. a n 

=e n− 7. 8. { _ _ 

} 

9. The fit fi{ _ 

} 

Section 9.2 

1. Th. Thff− 

2. Th−≠− 

3. 4. a 

= 5. a 

=a n 

=a n − 1 

+ 

n≥ 6. a n 

=−n 7. Th 

8. ThT n 

=+n 

 

Section 9.3 

1. Th_ ≠_ 

 

2. Th. Th_ 

3. { _ 

_ 

} 4. a 

=a n 

=_ a n − 

n≥ 

5. a 

= 6. a n 

=−− n − 7. a. P n 

=3 ∙ 1a n 

b. Th 

Section 9.4 

1. 2. 3. 4. − 5. 

6. ≈ 7. 8. 9. Th 

fi 10. Thfi 

fi 11. Thfifi 

12. 13. Th 14. − 

_ 

 

15. 

Section 9.5 

1. 2. Th 3. 

4. 5. 6. P= 7. P= 

8. C= 9. 10. 

Section 9.6 

1. a. b. 2. a. x −x y+x y −x y +xy −y 

Probability 

2. 

_ 

3. 

_ 4. 

 

5. _ 6. a. _ 

b. 

 

_ 

 

 

_ 

c. _

Odd Answers 

CHAPTER 1 

Section 1.1 

1. . Th 

 

3. Th 

 

d diffff 

 

5. − 7. − 9. − 

11. 13. − 15. 17. 19. 21. 

23. − 25. 27. 29. − 31. − 33. ± 

35. 37. 39. −y− 41. −b+ 

43. z− 45. y+ 47. −b+ 49. _ x 

51. x 53. −+ 

55. 

 

57. g+−= 59. 

61. 63. 65. 67. 

Section 1.2 

1. 

 

×× × 

3. 

5. 7. 9. 

11. 13. 15. 

 

17. 19. 

21. ×− 23. 

25. a 27. b c 29. ab d 31. m 33. q 

p 

35. y 

x 37. 39. a 41. c y 

 

b 43. 

z 

45. 47 × 49. 

51. m 53. a 

55. n 

a c 57. 

a b c 

59. 

Section 1.3 

1. 

t. Th 

3. Th 

5. 7. 9. 11. √ — 

13. _____ √— 15. 17. √— 19. √ — 21. √ — 

23. √ — 25. 

27. √— 29. __________ −+√— 

31. √ 

— 33. √ — 35. x 37. √ — p 

39. m √ — m 41. b√ — a 43. x 

45. y √ — 

47. ______ √— d 

d 49. +√ — ____________ x √— 

51. −w√ — w 

−x 

53. √— x−√ — x 55. n √ — 57. m √— m 59. 

d 

61. √ 

— ______ x 63. z √ 

— 65. 67 _________ −√— − 

69. ______ √— mnc a cmn 71. x+√ — ___________ √— 73. ____ √— 

Section 1.4 

1. Th 

 

Th 

 

3. 

5. 7. 9. 11. x +x+ 

13. w +w+ 15. b − b +b −b+ 

17. x −x− 19. b −b + 

21. v −v+ 23. n −n +n− 

25. y −y+ 27. p +p+ 

29. y −y+ 31. c − 33. n − 

35. −m + 37. q − 39. t +t −t −t+ 

41. y −y −y+ 43. p −p −p+ 

45. a −b 47. t −tu+u 49. t +x +t−tx−x 

51. r +rd−d 53. x −x−m 

55. t −t +t+ 57. a +a c−ac −c 

Section 1.5 

1. Th 

x 

−y 

x −y =x+yx−y 3. 

x 

 

5. m 7. m 9. y 11. a−a+ 

13. n − n+ 15. p+p− 

17. h+h− 19. d−d− 

21. t+t − 23. x+x− 

25. p+p− 27. d+d− 

29. b+cb−c 31. n+ 33. y+ 

35. p− 37. x+x −x+ 

39. a+a −a+ 41. x−x +x+ 

43. r+sr −rs+s 45. c+ − −c− 

47. x+ x+ 49. z− − z− 

51. x−x+ 53. x+x− 

55. x+ x− 57. z +a z+az−a 

 

59. 

 

x+x−x+ 

Section 1.6 

1. 

 

3. 

 

 

B-1

B-2 

ODD ANSWERS 

5. y+ y+ 7. b+ 9. x+ x+ 11. a+ a− 

13. n− n− 15. c− c+ 17. 19. d − d − 

21. t+ t+ 23. x− x+ 25. p+ p+ 27. d+ d+ 

29. b+ 

b− 31. y− 33. x+y 

35. a− 

 

y+ xy 

a −a− 

37. y −y+ 

y −y− 39. z +z+ 

41. x+xy+y 

 

z −z− x+xy+y+ 

43. b+a 

ab 45. +ab 47. a−b 49. c +c− b 

c +c+ 

51. x+ 

x− 53. x+ x− 55. 

y+ 57. 

Chapter 1 Review Exercises 

1. − 3. 5. y= 7. m 9. 

11. 13. 15. a 17. x 

y 

19. a 

21. × 23. 25. √ — 27. √— 

29. √— 31. √— 33. − 35. x +x + 

37. x −x+ 39. k −k− 41. x +x +x+ 

43. a +ab−b 45. p 47. a 

49. a−a+ 51. x+ 53. h−k 

55. p+p −p+ 57. q−pq +pq+p 

59. p+ −p− 61. x+ 

x− 63. 

 

67. x+y 

69. 

xy 

 

Chapter 1 Practice Test 

1. 3. x= 5. 7. 

9. 11. x 13. 15. √— x 17. √— 

19. q −q −q 21. n −n +n− 

23. x+x− 25. c−c +c+ 

27. z− z− 29. a+b 

b 

CHAPTER 2 

Section 2.1 

65. m+ 

m− 

1. 

xy 

3. y 

y 5. x 

y 7. x 

y− 9. x 

y( 

) 11. y=−x 13. y=−x 

 

15. y=x− 

17. d=√— 19. d=√ — = 

21. d≈ 23. ( − 

) 25. − 27. 29. y= 

31. 

y 

 

 

 

− 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

35. 

x − 

y 

y 

 

 

 

 

 

− 

 

 

x 

– – – – – – 

– 

– 

– 

– 

– 

– 

39. 

– – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

x 

 

 

− 

x 

33. − 

37. x − 

y 

 

 

 

 

 

− 

– – – – – – 

– 

 

– 

– 

– 

– 

– 

41. 

– – – – – 

y 

y 

 

 

 

 

 

 

 

– 

– 

 

– 

– 

– 

– 

– 

43. d= 45. d= 47. − 49. x=y=− 

51. x=y= 53. x=−y= 

55. −= 57. 59. 

 

 

61. 63. 54 ft 

Section 2.2 

1. 3. Th 

x 5. 

 

7. x= 

9. x= 

11. x= 13. x= 15. x=− 

 

17. x≠−x=− 19. x≠x= 

21. x≠x=− 

 

23. y=− ___ 

x+ 

 

25. y=− ___ 

x+ 

27. y= 

x+ 

29. y=−x− 31. y= 33. y=− 35. x+y= 

37. y 

39. y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

x 

x 

 

 

x

ODD ANSWERS B-3 

41. m=− 

43. m= 

45. m 

=− 

m = 

 

47. y=x−yy 

=−y 

=− 

49. y=−x+y y 

=−y 

= 

51. y=− A 

B x+C 53. Th− 

B 

Th−− 

Th 

3. Th− 

 

55. 30 ft 57. 59. 

Section 2.3 

1. 

 

3. −x 5. v+ 

7. 9. +m 11. 

13. +P 15. 17. −x 19. 

21. 

23. 25. B=+x 

27. 29. R= 31. r= 

 

33. W= P−L 

=− 

= 

pq 

35. f= 

p+q = 

+ = 

37. m=− 

 

39. h= 

A 41. = =160 ft 

b 

+b 

43. 45. A= 47. 49. h= V 

πr 

51. r= √ V 53. C=π 

πh 

Section 2.4 

1. 

3. ii− 

5. −+i 7. +i 9. − 

+ 

i 

11. 

i 

13. 

i 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

 

r 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

 

15. −i 17. −+i 19. −i 21. +i 

23. −+i 25. −−i 27. 29. − 

i 

31. −i 33. 

+ 

i 35. i 37. +i√— 

 

39. 41. − 43. i 45. ( √— 

+ 

i ) =− 

47. i 49. 51. −i 53. −i 55. 

− 

i 

Section 2.5 

1. 

3. 

a ∙ b= 

ff 

a=b= 

r 

5. xx = 

ax+b =d 

7. x=x= 9. x=− 

x=− 11. x=x=− 

 

13. x=− 

x= 

15. x=− 17. x=x=− 

 

19. x=−x= 21. x=x=− 23. x=x=− 

25. x=−x= 27. x=±√ — 29. z= 

z=− 

 

31. x= ±√— 

33. 35. 

 

37. 39. x= −±√— 

41. x= ±√— 43. x=−±√— 

45. x≈x≈ 47. x≈−x≈ 

49. ax +bx+c= 

 

x + b 

a x=−c 

a 

x + a b x+b 

 

a =−c 

a + b 

a 

 

 

( x+ b 

a ) = b −ac 

a 

 

=±√ 

x+ b 

a b−ac 

a 

 

 

x= −b±√— b −ac 

 

a 

51. xx+= 53. x= 

55. Th 

x−x −x+=300. Thx 

57. 

Section 2.6 

1. 

 

 

3. 

− 

. Th 

 

− 5. 

 

 

7. x= 9. x= 11. x=x= 

13. x=−− 15. y= 

− 

 

17. m=− 

19. x= 

± i 21. x= 23. t= 25. x= 

 

27. x=− 29. x=− 

31. x=− 

 

 

33. x=− 35. x=− 37. x=−− 

39. x=− 41. x= 43. 

45. x=−− 47. 49. 

Section 2.7 

1. 

 

3. −∞∞

B-4 

ODD ANSWERS 

5. x= 

 

y = x−e left 

y 

7. ( −∞ __ 

] 9. [ − ____ 

∞ ) 11. −∞ 13. ( −∞− ___ 

] 

15. −∞∞ 17. ( −∞− 

) ∪∞ 

19. −∞−∪+∞ 21. 23. − 

25. 27. − 

29. x>−x>− 

x>−−+∞ 

31. x



1. y= 

x+ 

x 

y 

– – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

3. − 

– – – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

 

 

− 

5. −∞ 7. x=− 9. x≠−x=− 

 

11. x= ±√— 13. − 15. y=− 

x− 

 

17. y= 

x− 19. i 21. 

− 

i 23. x=− 

 

25. x= 

±√— 27. 29. x= 

− 

CHAPTER 3 

Section 3.1 

1. 

 

3. 

 

 

5. 

 

 

 

7. 9. 11. 13. 

15. 17. 19. 

21. 23. 25. 

27. f−=−f=−f−a=−a−−fa=−a+ 

fa+h=a+h− 29. f−=√ — +f= 

f−a=√ — +a +−fa=−√ — −a −fa+h= 

√ — −a−h + 31. f−=f=− 

f−a=∣ −a−∣−∣ −a+∣−fa=−∣ a−∣+∣ a+∣ 

fa+h=∣ a+h−∣−∣ a+h+∣ 

33. gx−ga _ 

x−a 

=x + a+ x ≠ a 35. a. f −= b. x= 

37. a. f = b. x=− 39. a. r=− 

__ t 

b. f−= c. t= 41. 43. 

45. 47. 49. 

51. 53. a. f = b. f x=−x=− 

55. 57. 

59. 61. 63. 

65. 67. f x=x= 

69. f−=f−=f=f=f= 

71. f−=f−=f=f=f= 

x 

73. f−= 

__ f−= __ 

f=f=f= 75. 

77. Th 79. Th 

 

− 

y 

y 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

81. Th 

− 

y 

 

 

 

 

 

 

– – 

– 

– 

– 

– 

– 

 

85. Th 

− 

y 

 

 

 

 

 

– – 

– 

– 

– 

– 

– 

x 

x 

x 

 

 

 

 

 

– 

 

– 

– 

 

– 

– 

– 

– 

83. Th 

0, 10 

y 

 

 

 

 

 

– 

x 

 

87. Th 

− 

y 

 

 

 

 

 

– – 

– 

 

– 

– 

– 

– 

89. a. g=b. Th 

f 100 s 91. a. 

fts 200 ft. 

b. Thfts 350 ft. 

Section 3.2 

1. Th 

 

3. Thxfx= √ — x 

 

−∞∞ 

x 

fx= √ — x 

∞ 5. 

 

xy 

 

−∞∞ 

7. −∞∞ 9. −∞ 

11. −∞∞ 13. −∞∞ 15. ( −∞− 

 

) ∪ ( − 

 

∞ ) 

17. −∞−∪−∪∞ 19. −∞−∪−∪∞ 

x 

x

B-6 

ODD ANSWERS 

21. −∞ 23. ∞ 25. −∞−∪−∪∞ 

27. 29. − 

31. − 33. −∞∞ 

35. [ −− 

 

] ∪[ 

] [ −− 

 

] ∪[ 

] 

37. −∞∞ 

39. −∞∞ 41. −∞∞ 

y 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

43. −∞∞ 

y 

– 

– 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

 

 

 

x 

x 

– 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

45. −∞∞ 

y 

– – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

47. f−=f−=f−=f= 

49. f−=−f=f=f= 

51. f −=−f=f=f= 

53. −∞∪∞ 

55. 

y 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

– – – – – 

– 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

– 

– 

x 

 

x 

 

Th Th 

 

 

 

57. 59. fx= 

√ — x− 

61. a. Th b. Th 

c. Th 

Section 3.3 

1. 

3. Th 

 

5. b+ 7. 9. x+h 

 

11. − 

_________ 

+h 13. h +h+ 15. x+h− 

17. 19. −∞−∪∞ 

 

− 21. −∞∪ 

∪∞ 

 

 

 

 

 

 

x 

 

x 

23. − 

25. 

−− 

27. a.− b. − 

29. − 31. 33. ≈− 35. 

−−∞∞ 

37. −−−− 

−∞ 39. − 

−−−−∞ 

−−−−∞ 

41. 43. b= 45. ≈ 

47. ≈− 

Section 3.4 

1. g 

f 

g 

3. f x= x+gx= x− 

Thf (gx=f x−=x−+=x 

g ( f x=g x+=x+−=x f∘g=g∘f 

5. f+gx=x+−∞∞ 

f−gx=x +x−−∞∞ 

fgx=−x −x +x +x−∞∞ 

( f _ 

g ) x=x +x 

−x −∞− √— ∪−√ — √ — ∪√ — ∞ 

7. f+gx= x +x + 

−∞∪∞ 

x 

f−gx= x +x − 

−∞∪∞ 

x 

fgx=x+−∞∪∞ 

( f _ 

g ) x=x +x −∞∪∞ 

9. f+gx=x +√ — x−∞ 

f−gx=x −√ — x−∞ 

fgx=x √ — x−∞ 

( f _ 

g ) x= x ∞ 11. a.fg= 

√ — x− 

b.fgx=x −x+ c.gfx=x − 

d.g ∘ gx=x− e.f ∘ f−= 

13. fgx=√ — x ++gfx=x+√ — x + 

√ 

15. fgx= — 

x+ √ 

x 

; gfx= 

— x + 

x 

 

17. fgx= 

__ x x≠gfx=x−x≠ 

 

 

19. fghx= 

x+ + 

 

21. a.g∘f x=− 

 

√ — −x b. ( −∞ __ 

) 

23. a. ∪∞x=−b.∞c.∞ 25. ∞ 

27. fx=x gx=x− 

29. fx= x ; gx=x+ 

31. fx= √ — 

x ; gx= 

x− 

33. fx= √ — x ; gx= x− 

x+ 

35. fx=√ — x ; gx=x+


37. fx= √ — x gx=x− 

 

39. fx=x gx= 

x− 

41. fx=√ — x ; gx= x− x+ 

43. 45. 47. 49. 51. 53. 

55. 57. 59. 61. 63. 65. 

67. 69. 71. 73. fg=gf=− 

75. fg= 

gf= 77. fgx=x +x+ 

79. g∘gx=x+ 81. f∘gx=g∘f x= 

83. −∞∞ 85. 87. f∘g =g∘f = 

89. f∘g =g∘f = 91. 

93. At=π ( √ — t+) A=π ( √ — ) =π 

95. A=π 

97. a.NTt=t +t− b. ≈ 

Section 3.5 

1. 

 

3. 

 

 

5. f 

−xxfx 

f−x= fx 

 

f−x= −fx 

 

7. gx= ∣ x−∣− 

 

9. gx= 

 

x++ 11. Thfx+ 

f 

13. Thfx− 

f 15. Thfx+ 

f 17. Thfx− 

f 19. Th 

fx+− hift 

f 21. −∞− 

−∞ 23. ∞ 

25. y 27. 

y 

– – – – – – – – 

 

 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

 

h 

 

x 

– – – – 

 

 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

 

 

f 

 

x 

29. y 31. gx= fx− 

– – – – – 

 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

k 

 

 

 

hx= fx+ 

33. fx= ∣ x−∣− 

35. fx= √ — x+− 

37. fx= x− 

39. fx= ∣ x+∣− 

x 41. fx= − √ — x 

43. fx= −x+ + 

45. fx= √ — −x + 

47. 49. 

51. 53. Th 

g 

x 

f 55. Thg 

 

f 

57. Thg 

 

f 59. Thg 

f 61. Thg 

y 

f 63. gx= ∣ −x ∣ 

 

65. gx= 

 

x+ − 67. gx= 

 

x− + 

69. hift 

 

 

hift 

y 

– – – – 

 

 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

73. 

 

 

 

y 

– – – – 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

g 

 

 

m 

 

 

 

 

 

 

x 

x 

71. 

 

hift 

 

 

hift 

y 

– – – – – 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

75. Th 

 

hift 

 

y 

– – – – – 

 

 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

h 

 

 

 

 

p 

 

x 

x

B-8 

ODD ANSWERS 

77. Th 

fx= √ — x hift 

 

 

y 

y 

 

 

 

 

 

 

 

 

a 

 

 

 

 

 

 

 

 

 

 

 

 

x 

 

Section 3.6 

79. 

– – – – – – – 

81. 

– – – – – – – 

y 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

g 

 

g 

 

 

1. 

∣ A ∣ =B 

A 

B 

A 

−B 3. 

x 

 

x 5. Thx 

∣ x−∣= 7. ∣ x−∣≥ 

9. Thx 11. − 

13. −− 15. − 

17. 

y 19. 

y 

21. 

– 

– 

– – – – 

 

 

 

 

 

– 

y 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

x 

x 

23. 

– – – – 

– – – – – 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

 

 

x 

x 

x 

x 

25. 

– – – – – 

29. 

– – – – 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

y 

 

 

 

 

 

 

– 

– 

– 

– 

– 

– 

– 

33. − 

y 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

 

f 

 

 

 

x 

x 

35. 

 

 

27. 

– – – – – 

31. 

y 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– – – – – – 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

 

x 

 

x 

 

37. Tha 

yTh 

yx= 

39. ∣ p− ∣ ≤ 41. ∣ x−∣≤ 

SECTION 3.7 

1. 

 

y 

 

y 

3. f x= x 

5. y=f − x 

7. f − x=x− 9. f − x=−x 11. f − x=− 

_ x 

x− 

13. f x−∞f − x=√ — x − 

15. f x∞f − x=√ — x+ 

17. fgx=xgfx=x 19. 

21. 23. 25. 27. 

29. y 

31. 33. 

35. − 37. 39. 

 

41. 

 

f 

 

– x 

f 

 

f −1 (x) 

–– – – 

– 

– 

– 

 

 

 

 

x 

x 

x


43. f − x=+ x 

– – – – 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

f 

 

f – 


 

 

 

45. f − x = x − 

 

47. td=_ d t= 

Th 

 

 

x 

 

1. 3. 5. f−=−f=− 

f−a=−a −a; −fa=a −a; 

fa +h=−a −ah −h +a +h 

7. 9. 11. 

13. 

y 

15. 17. − 

 

 

19. −+a−a 

−+a 

 

23. 

– – – – 

– – – – 

 

– 

– 

– 

– 

 

 

 

y 

– 

– 

– 

– 

 

 

 

 

 

 

 

 

x 

x 

=−a +a≠ 

21. −∞−∪(−∪(∞ 

25. 

27. ∞ 

−∞ 

29. −−∞−∞ 

31. −− 

33. 

35. f∘g x=−x, g∘f x=−−x 

37. f∘g x= √ 

x 

+ g∘f x=_ 

 

√ — x+ 

 

 

 

x 39. f∘g x= 

+ =_ 

+x 

 

 

+x ( −∞− 

) ∪ ( − 

) ∪(∞) 

+ 

x 

41. f∘g x= 

√ — x ∞) 

43. gx= x − 

x + 

fx=√ — x 

45. 

y 

47. 

y 

– – – – – 

 

 

 

 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

x 

– – – – – 

 

 

 

 

 

 

 

 

– 

– 

– 

 

 

 

 

 

 

x 

49. 

53. 

– – – – 

– – – – – 

67. 

– – – 

75. 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

– 

– 

y 

 

 

y 

 

 

 

 

 

– 

– 

 

– 

– 

– 

– 


 

 

 

 

 

 

 

 

 

x 

x 

x 

51. 

– – – 

 

y 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

– 

 

55. fx=∣ x −∣ 

57. 59. 

61. 

63. fx= 

∣ x +∣+ 

65. fx=−∣ x−∣+ 

69. f − x= x − 

71. f − x=√ — x − 

73. Th 

y 

 

 

 

 

 

– – – – – 

– 

– 

– 

– 

– 

 

1. 3. − 5. Th 

 

7. a −a 9. −a+b+b≠a 11. √ — 

13. 

15. 17. 

y 

19. f − x= 

x+ 

 

 

 

 

21. −∞−∞ 

 

23. − 25. f= 

– – – – – 

 

– 

– 

– 

– 

– 

– 

CHAPTER 4 

Section 4.1 

 

 

 

 

 

 

x 

27. fx= { 

|x| x ≤ 

x > 

29. x = 31. 

33. f − x=− x − 

−x _ 

 

1. 

3. dt = −t 

5. Thaa 

ya 

xa. Th 

 

x 

x

B-10 

ODD ANSWERS 

7. 9. 11. 13. 15. 

17. 19. 21. 

23. 25. 27. − 29. 

_ x − 

 

31. y = x− 33. y = − 

 

x+ 35. y = −x− 

 

37. 39. 41. yx 

43. 

 

y−x( ) 

47. m=−m=− 

49. m=−m= 

45. yx− 

51. m=− m=− 

53. y =x− 55. y=− 

__ t+ 57. 59. y=− 

__ x+ 

61. y =x− 63. y =− 65. 67. 69. 

71. 

– – – – – 

75. 

– – – – – 

79. 

– – – – – 

83. 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

x 

x 

x 

x 

73. 

– – – – – 

77. 

– – – – – 

81. 

– – – – – 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

 

 

 

y 

– 

– 

– 

– 

– 

– 

– 

 

 

 

85. y = 

87. x = − 

89. gx = −x+ 

91. fx = x− 

93. gx = − 

__ 

x+ 

95. fx = x− 

97. fx = −x+ 

x 

x 

x 

99. 

 

 

 

 

 

 

y 

−−−−− 

− 

− 

− 

− 

− 

− 

– – 

– 

– 

– 

 

 

 

x 

x 

101. aa = b = 

b.qp = p− 

103. y 105. x = − 

__ 107. x = a 

 

 

109. y=____ 

d 

x−____ 

ad 

 

c−a c−a 

 

111. y=x− 

113. x< 

x> 

115. 

 

117. Th 

ts. Th 

 

119. Th− 

 

121. 

Section 4.2 

1. 

3. 

5. 

7. 9. 11. 

13. Pt=t + 15. − 

 

17. ft 

19. Wt=t+ 21. −15, 0) Th 

x 

 

23. 

25. Ct=−t 27. 

 

 

29. 31. y =−t + 

33. fi 35. y =t − 

37. fi 

39. 41. 43. 

45. a. b. c. d. 

e. Pt=+t f. 47. a. Cx=x + 

b. The fl 

c. 49. a. Pt=t + 

b. 51. a. Rt= −t + 

b. c. 

53. 55. 

57.


Section 4.3 

1. ft 

3. 

5. Th 

 

7. 

9. 

11. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13. 15. 17. 

 

19. 

21. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

23. 

25. y=x+r= 27. y=−x+ 

r=− 29. y=−x+r=− 

31. y=x−r= 33. −−− 

−(−− 35. 

fi 

37. y=x+ 39. y=x− 

41. y=−x+ 


1. 3. 5. y=−x+ 7. 

9. y=x− 11. 13. 15. −− 

17. m−m=− 19. y=−x+ 

21. 

 

 

 

 

 

 

y 

– – – – – – – 

 

– 

– 

– 

– 

– 

35. 37. 

y 

y 

 

 

 

 

 

 

 

 

 

Population 

 

 

 

 

 

 

 

 

Year 


23. 

25. 

27. y=−x+ 

29. a. b. 

c. Pt= t+ 

x 31. 33. y= 

1. 3. 5. y=−x− 7. y=−x− 

9. 11. 13. −− 

15. y=−x+ 

17. =− 

y= 

y 

 

 

 

 

 

 

 

 

– – – – 

 

– 

– 

– 

x 

 

x 

 

 

 

 

 

 

 

x 

39. 

41. y=−x+ 

r=− 

43. 

45. 

19. 21. 

23. y=x+ 

25. a. b. 

 

 

c.y=−t+ 

27. 29. 

y 

31. y=x+ 

 

 

33. r= 

 

 

 

 

 

 

x

B-12 

ODD ANSWERS 

CHAPTER 5 

Section 5.1 

1. 

3. a= 

5. 

7. gx=x+ − −− 

9. fx=( x+ 

) − 

( − 

− 

) 

11. kx= x− −− 

13. fx= ( x− 

) − 

( 

− 

) 

15. − 

 

x= 

 

17. − 

− 

x=− 

 

19. − 

−x=− 

21. −∞∞∞ 23. −∞∞ 

−∞ 25. −∞∞−∞ 

27. fx=x +x+ 29. fx=x −x+ 

31. fx=− 

__ 

x + 

__ 

x+ __ 

33. fx=x −x+ 

35. −) 37. ( 

x= 

− 

) 

+√ — − √ — x= 

 

 

 

y 

( +√— 

)( − √— 

) 

 

 

 

 

 

− − 

− 

 

− 

− 

− 

− 

39. ( 

− ) 

x= 

 

( +√— 

)( − √— 

y 

 

 

 

 

 

−−−−− 

− 

 

− 

− 

− 

− 

x 

) 

x 

 

 

 

 

 

− − 

− 

 

− 

− 

− 

− 

41. fx=x +x+ 

43. fx=−x −x− 

y 

45. fx=− 

__ 

x − x+ 

47. fx=x +x+ 

49. fx=−x +x 

51. Th 

 

h. Th 

 

53. Thhift 

 

55. Th 

 

57. −∞∞ 

−∞ 

59. −∞∞ ∞ 61. fx=x + 

63. fx=−x − 65. fx=x +x− 67. 

69. 71. Th 

 

73. 75. 

 

x 

Section 5.2 

1. Th 

. Th 

3. x 

fxx 

fx 5. Th 

7. 9. 

11. 13. − 15. 

− 17. x→∞fx→∞x→−∞fx→∞ 

19. x→−∞fx→−∞x→∞fx→−∞ 

21. x→−∞fx→−∞x→∞fx→−∞ 

23. x→∞fx→∞x→−∞fx→−∞ 

25. yt− 

27. y−x− 

29. yx− 

31. 33. 35. 37. 39. 

41. 

43. 

45. 

47. x→−∞fx→∞ 

x→∞fx→∞ 

x 

fx 

 

 

 

− 

 

− 

51. y 

x 

x→−∞fx→∞ 

x→∞fx→∞ 

y 

 

 

 

 

 

−−−−−− − 

− 

− 

− 

− 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

 

x 

x 

49. x→−∞fx→∞ 

x→∞fx→−∞ 

x 

 

 

− 

− 

fx 

− 

− 

 

 

53. y 

x 

x→−∞fx→−∞ 

x→∞fx→∞ 

y 

 

 

 

 

 

−− 

− 

− 

− 

− 

− 

 

55. y 57. y− 

x− x− 

x→−∞fx→−∞ x→−∞fx→∞ 

x→∞fx→∞ 

x→∞fx→∞ 

y 

y 

 

 

 

 

 

−−−−−− 

− 

− 

− 

− 

− 

 

x 

x


59. yx 

− 

x→−∞fx→−∞ 

x→∞fx→∞ 

61. fx=x − 

63. f x=x −x +x 

65. f x=x + 

67. Vm=m +m +m+ 

69. Vx=x −x +x 

Section 5.3 

 

 

 

 

 

y 

−−−−− 

− 

− 

− 

− 

− 

 

1. x 

x 

fx= 3. a b 

 

a b 5. Th 

7. −− 9. − 

11. − 13. − 

15. −− 17. −( 

_ ) 

19. − 21. ( √ — )( −√ — ) 

23. −− 

25. f =−f = 

27. f =f =− 

29. f =f =− 

31. − 

 

33. − 

35. − 

37. 

 

− 

39. 

41. 

 

 

 

43. x1, 0 

−4, 0 

y 

(0, 4); ax→−∞g x→ 

−∞x→∞g x→∞ 

 

 

 

 

 

gx 

−−−−− 

− 

 

− 

47. x0, 0−2, 0 

 

y(0, 0); ax→−∞ 

n x→∞x→∞ 

n x→−∞ 

45. x3, 0 

2, 0 

y 

(0, −x→−∞k x→ 

−∞x→∞k x→∞ 

kx 

x 

 

 

−−−−−− 

− 

 

− 

− 

− 

− 

− 

− 

− 

− 

− 

 

 

 

 

 

nx 

−−−−− 

− 

 

− 

− 

− 

− 

x 

x 

x 

49. f x=− 

x−x+x+ 51. f x= 

x+ x− 

53. −− 55. − 

57. f x=− 

x+x−x− 

59. f x= 

x− x− x+ 61. f x=−x− x− 

63. f x=−x+x+x− 

65. f x=− 

x− x−x+ 

67. −−− 

69. : −− 71. − 

73. f x=x− x+ 75. f x=x −x +x 

77. f x=x −x +x+ 

79. f x= π x +x +x+ 

Section 5.4 

1. Th 

 

3. x++ 

 

x− x+ 

5. x+x+ 7. x− 

 

x− 9. x−+ 

 

x− 

x+ 

 

11. x−+ 

 

x+ x− 

13. x −x+x −x+ 

 

15. x +x++ 

 

x− 17. x −x+− 

x+ 

19. x −x+− 

x+ 21. x +x+ 

23. x −x+− x+ 25. x −x+ 

 

27. x +x+ 

x− 29. x −x+ 31. x −x + 

33. x −x +x− 35. x −x +x− 

37. x −x+ 39. x−x − 41. 

x−x +x +x+ 43. 

45. x−x +x+ 47. x−x +x+ 

49. x +x+− 51. 

x +x+ 53. x −x + 

x−− 55. x −x +x −x +x −x+ 

 

57. x −x +x−+ 

 

x+ 59. ++i 

 

x−i 

61. +−i 

x+i 

63. x +ix−+ −i 

x−i 65. x + 67. x+ 

69. x+ 71. x− 73. x − 

Section 5.5 

1. Th 

3. 

5. 

 

7. − 9. 11. 

13. − 15. − 17. − 19. − 

 

21. − 

√— − √ — 23. −− 25. −− 

 

27. −− 

29. 

+√— 

−√— 

31. 

33. −− 35. 

− 

− 

37. −−√ — −√ — 39. − 

− 

41. +i−i 

43. − 

+i−i 

45. − 

 

+i−i

B-14 

ODD ANSWERS 

47. 

 

f x 

 

 

 

 

 

−−−−−− − 

− 

− 

− 

− 

 

51. 

f x 

 

 

 

 

 

−−−−−−− 

− 

− 

− 

− 

− 

x 

x 

 

55. 

f x 

 

 

 

 

 

−−−−−− 

− 

− 

− 

− 

− 

 

x 

49. 

f x 

 

 

 

 

 

−−−−−− 

− 

− 

− 

− 

− 

 

53. 

f x 

 

 

 

 

 

−−−−−− 

− 

− 

− 

− 

− 

 

57. ± 

±±± 

 

59. ±± 

± 

± 

 

61. 

− 

 

63. 

− 

65. 

 

67. fx= 

x +x −x− 

69. fx=− 

__ x −x 

71. 73. 

75. 77. 

79. 

Section 5.6 

1. Th 

3. Th 

5. es. Th 

 

 

7. x=− 9. x=−− 

11. x=− 

y= 

x=− 13. 

 

x=−y= 

x=− 15. x=− 

y=x=− 

17. x=−y= 

x=− 

19. x= 

y=− 

 

x= 21. 

 

23. xy( 

) 

25. x→− 

+ f x→−∞x→− 

− f x→∞ 

x→±∞f x→ 

x 

x 

27. x→ + f x→−∞ x→ − f x→∞; 

x→±∞f x→− 29. 

x→− 

+ f x→∞ x→− 

− f x→−∞ x→− 

− f x→∞ 

x→− 

+ f x→−∞x→±∞f x→ 

 

31. y=x+ 33. y=x 

35. 

x= 

y= y 

37. x= 

y= 

y 

 

y= 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

 

x= 

39. 

x=− 

y=( 

) ( − 

) px 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

x= − 

 

43. 

x=− 

 

y= 

( − 

) ( ) fx 

 

 

 

 

 

 

−−−−−− − 

− 

x= − 

− 

− 

− 

− 

y= 

x 

x 

x= 

y= 

x 

 

47. 

x= 

y=x+−( 

) 

( 

) hx 

 

 

 

 

 

y=x+ 

x= 

−−−−− 

 

− 

− 

− 

− 

− 

x 

 

 

 

 

 

−−−− − 

y= − 

− 

− 

− 

− 

 

x= 

41. x= 

y= 

sx 

x= 

 

 

 

 

 

 

 

 

 

 

 

−−−−− − 

− 

y= 

 

45. x=− 

y= 

− 

ax 

 

 

x=− 

 

 

−−−−− − 

− 

− 

− 

− 

x 

x 

y= 

x 

 

49. x=− 

y= 

−( − 

) wx 

x= 

 

x= − 

 

 

 

 

 

−−−−−− − 

− 

− 

− 

− 

y= 

x


51. fx=________ 

x − x− x − 53. +x− fx=x 

x +x+ 

55. fx= − x+ 

⋅x x− 

57. fx= 

x+ x − x− 

59. fx=− x− x − 61. fx= 

∙ +x− x x− 

x− 

63. fx=−____________ 

 

x+x− 

65. x=y= 

x 

y − − 

x 

y 

67. x=−y= 

x − − − − − 

y − − 

x 

y 

69. x=−y= 

x −0 −0 −0 − − 

y 

x 

y 

71. ( 

∞ ) 

fx 

 

 

 

 

 

–– – – – – 

– 

– 

– 

– 

 

x 

73. −∞∪∞ 

fx 

 

 

 

 

 

–– – – – – 

– 

– 

– 

– 

 

75. () 77. () 79. (−) 

+t 

81. Ct= 

+t 

83. ft 85. 

87. 

Section 5.7 

x 

 

1. diffict oxy 

3. 

5. f − x=√ — x + 7. f − x=√ — x+ − 

9. f − x=√ — −x 11. f − x= ± √ x− 

√ 

— 

13. f − x= x− 

15. f− x= −x 

 

17. f − x= 

______ −x 

∞ 19. f− x= 

__________ x− + 

√ 

21. f − x=−x 23. f − x= 

______ x+ 

x 

— 

27. f − x= 

______ x− x+ 29. f− x=√ — x+− 

25. f − x= 

______ x+ 

−x 

31. f − x=√ — x− 

y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

35. f − x= √ 

— x− 

y 

 

 

 

 

 

 

− −−−− − − 

− 

− 

− 

− 

− 

39. 

 

 

 

 

 

y 

−−−−− 

− 

− 

− 

− 

− 

 

 

 

 

 

43. −∪ ∞ 

f x 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

47. − 

y 

 

 

 

 

 

−−−− – 

− 

− 

− 

− 

− 

51. f − x= 

x−b 

a 

55. f − x= cx−b 

a−x 

 

 

 

 

 

x 

x 

x 

x 

x 

33. f − x=√ — x − 

y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

37. f − x=√ — x+ − 

y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

41. −∪∞ 

f x 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

 

 

 

 

45. −∞−ċ− 

f x 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

49. −−− 

y 

 

 

 

 

 

−−−− – 

− 

− 

− 

− 

− 

 

 

 

 

53. f − x= √ x −b 

a 

57. th= √ −h 

 

59. r A= √ A 

π , ≈ 61. lT= ( __ 

π) T ≈ 3.26 ft 

63. rA= √ A+π 

−2, 3.99 ft 65. rV= √ V ≈ 5.64 ft 

π 

π 

x 

x 

x 

x 

x

B-16 

ODD ANSWERS 

Section 5.8 

1. Th 

3. 5. y=x 

7. y=x 9. y=x 11. y= 

x 13. y= 

x 

15. y= 17. y=xzw 19. 

√ 

— y=x√ — z 

x 

21. y= xz 

xz 

w 

23. y= 

25. y= 

√ — wt 

27. y= 29. y= 31. y= 33. y= 

35. y= 37. y= 39. y= 

41. y= 

x y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

45. y= 

__ 

x 

 

 

 

 

 

y 

−−−−− 

− 

− 

− 

− 

− 

 

 

 

 

 

 

 


1. f x = x − − 

− 

−− 

fx 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

 

x 

x 

x 

43. y= 

__ 

√— x 

y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

 

47. ≈ 

49. ≈ 

51. 

53. 

55. ≈ 

57. ≈ 

59. ≈ 

3. f x= 

x+ + 

5. 

 

 

7. 

 

9. 

 

11. x→−∞f x→−∞ 

x→∞f x→∞ 

13. −− 

− 

15. 17. 

 

19. x + 

 

21. x −x+− x+ 

23. x −x−x+x −x− 

25. { −− 

 

} 27. { 

29. 

} 

x 

31. −( − 

 

) 

x=y= 

y = 

 

 

 

 

 

y 

−−−−− 

− 

− 

− 

− 

− 

 

x = 

x 

33. − 

( 

) x=− 

y= 

y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

− 

x = − 

y = 

x 

 

x = 

35. y=x− 37. f − x=√ — x + 39. f − x=√ — x+− 

41. f − x= x+ − 

x≥− 

 

43. y= 45. y= 

47. ≈ 


1. − 3. x→−∞ 

f x→∞ x→∞f x→∞ 5. f x=x− 

7. 

 

9. − 

 

 

 

11. x +x +x++ x− 13. { −− 

} 

15. −− 

 

 

17. f x=− 

x− x−x+ 19. 

21. −( 

) 23. f − x=x− +x≥ 

x=−y= 

y 

 

 

 

 

 

−−−−− 

− 

− 

− 

− 

x = − 

− 

CHAPTER 6 

Section 6.1 

 

x = 

y = 

x 

25. f − x= x+ 

x− 

27. y= 

1. 

 

3. 

 

t. Th 

 

nominal 5. 

7. 

 

9. 

11. ft 

13.


t infl 

 

 

15. 

17. 

19. f x= x 21. f x=( 

__ 

) − 

( 

__ 

) x ≈ x 

23. 25. 27. 29. 

31. 33. P=At⋅( + n) r 35. 

37. 39. 

41. 

43. 45. f −=− 47. f −≈− 

49. f ≈ 51. y=⋅ x 53. y≈⋅ x 

55. y≈⋅ x 

57. = At−a a( + ___ r 

) −a 

a = 

a 

a[ ( + ___ r 

) −] 

= 

a 

 

=( + r 

) − 

In=( + n) r − 

59. ff x=a⋅( b) x 

b> 1. Thn> 

fx=a⋅( b) x =ab − x =ae n − x =ae −n x =ae −nx 

61. 63. 65. 

67. 

Section 6.2 

1. 

x. Th 

 

 

3. gx= −x y 

 

5. gx) =− x +y 

 

7. gx=( 

) x y 

 

9. y− 

y 

 

 

 

 

gx=− x 

x 

−−−−− 

− 

− 

− 

− 

− 

g−x=− −x 

11. 

 

 

 

 

 

y 

−−−−− − 

− 

− 

− 

− 

gx= x 

hx= x 

f x= x 

 

13. 15. 17. 19. 21. 

23. y 

25. y 

− −fx= − 

− 

− 

− 

fx=− x + 

 

 

 

 

fx= x 

 

 

 

−f x= x − 

 

 

x 

−−−−− 

− 

−−−−− 

− 

x 

− 

− 

− 

− 

x 

27. 

hx= 

 

 

−−−−− 

hx 

 

 

 

 

 

 

 

 

 

x 

29. x→∞fx→−∞ 

x→−∞f x→−1 

31. x → ∞fx→ 

x→ −∞fx→∞ 

33. fx= x − 

35. fx= x − 

37. fx= −x 

39. y=− x + 

41. y=− x + 

43. g≈ 

45. h−=− 47. x≈− 49. x≈− 

51. Thgx) =( b) x y 

fx=b x b> 

fx=b ( x 

b) x yf −x 

53. Thgxhx 

fxn 

b>fx=b ( x 

b ) b x 

n 

fx−n 

Section 6.3 

1. 

b 

b. Th 

yb y b 

x 

d txby 

3. 

 

b y =x 

x 5. Th 

b 

e e 

(x) 

(x) 

7. a c =b 9. x y = 11. b =a 13. a = 

15. e n =w 17. c 

k=d 19. 

y=x 

21. n 

= 23. y 

( 

) =x 25. h=k 

27. x= 

29. x= 31. x= 33. x= 

 

35. x=e 37. 39. 41. 43. 

 

45. 47. − 49. − 51. 53. 

55. ≈ 57. ≈ 59. 

x=x= 

fx=x). Thn 

n= 

n =n 

Thx=f x=x 

61. x 

x=x=e 

x=e . Th 

x=e = 63. = e 

 

 

65.

B-18 

ODD ANSWERS 

Section 6.4 

1. 

y=xab 

ba 

3. hift 

y 

ff 5. 

 

 

7. ( −∞, 

) −∞∞ 

9. ( − 

∞ ) −∞∞ 

11. ∞x= 

13. ( − 

 

∞ ) x=− 

 

15. −∞x=− 

17. ( 

∞ ) x= 

 

x→( 

) + fx→−∞s x → ∞fx → ∞ 

19. −∞x=− 

x→−+fx→−∞x→∞fx→∞ 

21. ∞−∞∞x= 

x( 

) y 

23. −∞, 0); ra−∞∞ 

x=x−e y 

25. ∞−∞∞x= 

xe y 

27. 29. 31. 33. 

35. y 37. 

 

 

 

 

 

− − 

− 

− 

− 

− 

41. 

fx= x 

 

gx= x 

 

fx 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

45. 

gx 

 

 

 

 

 

−−−−−− − 

− 

− 

− 

− 

x 

 

 

 

x 

x 

y 

39. 

 

 

 

 

f x= e x 

 

−−−−− − 

− 

− 

− 

− 

43. 

x 

 

gx= x 

y 

 

 

 

 

 

−−−−−− −− − 

− 

− 

− 

− 

47. f x= 

−x− 

49. f x= 

x+ 

51. x= 

53. x≈ 

55. x≈− 

 

x 

57. Thfx= xgx=− x 

 

b≠ b 

x=−_ 

 

b x 

59. 

x+ 

x− > 

fx= x+ 

x− > 

x−∞− 

∞Th−∞−∪∞ 

Section 6.5 

1. 

 

_ 

. Th b 

x n = n x b 

3. b 

+ b 

+ b 

x+ b 

y 

5. b 

− b 

7. −k 9. xy 

11. b 

13. b 

15. x+y−z 

17. x− y 

19. 

23. ( xz 

√ — y 

) 

 

x+ 

y 

25. = 

a 

21. x 

27. 

= 29. 

b 

 

( 

) =a−b 31. 

b b− 

33. ≈ 35. ≈ 37. ≈− 

39. x= 

 

x+− 

x−= 

( 

_____ x+ x− ) = 

x 

= 

_____ x+ 

x− 

= 

_____ x+ 

x− − 

= 

_____ x+ 

x− −x− _______ 

 

x− 

= 

_____________ 

x+−x+ 

x− 

= 

_____ x− 

x− 

x= 

 

+− 

−= 

 

− 

x= 

41. bnn 1. Th 

b 

n= n n 

n 

b = 

n 

b 

Section 6.6 

1. 

 

 

 

3. Th 

 

 

 

. Th 

 

5. x=− 

7. n=− 9. b= 

 

11. x= 13. 15. p=( 

) − 

( 

17. k=− 

 

) − 

19. x= 

 

21. x=


( 

) − 

23. x= 

 

25. 27. x= 

 

29. − = 

31. n= 33. k= 

35. x=−e 

 

37. n= 39. 41. 

43. x=± 

45. x= 47. x= 49. x= 

 

51. x= 

53. x= e ≈ 

y 

y 

 

− 

− 

− 

− 

− 

− 

− 

 

55. x=− 

y 

 

 

 

 

 

−−−−−−−− – 

− 

− 

− 

59. 

 

 

 

y 

−−−− − 

− 

− 

− 

− 

− 

− 

− 

63. x= 

≈ 

y 

 

 

 

 

 

 

− 

− 

− 

 

 

 

 

x 

 

65. 

y 

x 

x 

 

 

 

 

 

−− − 

− 

− 

− 

− 

 

57. x= e+ 

≈ 

y 

 

− − 

− 

− 

− 

− 

− 

x 

 

61. x= 

≈ 

y 

 

 

 

 

 

−−−−− − 

− 

− 

− 

− 

x 

x 

 

 

 

 

 

 

 

fx=e x 

 

 

− x 

− 

x 

67. 

y 

 

 

 

 

 

 

 

Section 6.7 

x 

69. ≈ 71. ≈ 

73. ≈ 

75. ≈− 

77. 

79. t=( ( y 

_ 

A) 

k 

) 

k) 

81. t=( ( − T−T 

s 

 

T 

−T s) 

_ 

1. 

s. Th 

 

3. 

 

Th 

 

5. 

 

 

 

t diff 

 

 

orders of magnitude 

7. f ≈ 

9. 11. f x= x 

13. 

15. 

y 

fx 

 

 

 

 

 

 

 

 

 

 

 

17. 

 

x 

 

 

 

 

 

 

 

 

 

 

−−−−−−−−− − 

− 

 

 

 

 

Pt 

19. 21. 

23. 

 

 

 

 

 

 

 

t 

x

B-20 

ODD ANSWERS 

25. M= 

( S 

S ) 

 

M= ( S 

S ) 

M __ = 

( S 

S ) 

S 

M __ =S 

27. y=b x 

bb≠ 1. Th 

y=b x 

y=xb 

e y =e xb 

y=e xb 

29. A=e −t A≈ 31. 

33. f t=e −t ; 

35. r≈− 

37. f t=e t ftP ≈ 

39. f t=e t 

41. 43. Tt=e −t +t 

45. 113 47. 

x=x≈ 

49. ≈ 51. N≈ 53. 

Section 6.8 

1. 

 

 

3. 

t fi 

 

 

. Th 

 

 

 

5. y 

 

7. 9. 11. P = 

13. p≈ 15. y 17. 

19. 

21. y 

31. 

 

 

 

 

 

 

 

 

 

 

 

 

 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 

 

x 

23. 

25. 

27. f x= x 

29. y 

 

 

 

 

 

 

 

 

 

33. f x=e −x 

35. f x=x≈ 

37. y=+x 

39. y 

 

 

 

 

 

 

 

 

 

x 

x 

41. y 

 

 

 

 

 

 

 

 

 

 

 

51. y 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

53. y 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 

43. f ≈ 

45. f x=x≈ 

 

47. f x= 

+e −x 

49. 

 

 

 

55. f x=x≈ 57. f x= x 

g x= 

y =x 

_ 59. f − x= a−( c x 

−) 

 

 

b 


1. 

3. y = x 5. 7. 

 

y 

9. 

 

 

y 

11. gx= −x y 

13. x = 

15. a 

b= − 17. x = 19. =− 

 

x 

x 

 

 

 

 

 

 

 

 

 

 

−−−−−−−− 

 

− 

x


21. e − =− 

23. 

gx 

 

−−−−− 

− 

 

− 

− 

− 

− 

− 

− 

− 

x 

25. x>− 

x =− 

x→− + 

f x→−∞ 

x→∞f x→∞ 

27. 

xy 

29. ( xy) z 

31. y 

 

33. +b+ b+−b− 

35. 

( 

v w 

) 

√ u 

37. x = 39. x =− 41. 43. 

 

45. x = 47. a=e − 49. x =± 

__ 

 

51. 53. f − x= √ — x − 

55. f t= t f ≈g 57. 

59. 

61. 

y 

63. y = x ; y =e –x 

65. 

67. 

y =−x 

y 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

 

 

 

 

−−−−− 

 

− 

x 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. 3. 

5. y 7. a = 9. x = 

 

y 

11. ≈− 

fx= −x f−x= −x 

13. x< 

x = 

x→ − f x→−∞ 

x→−∞f x→∞ 

15. t 

 

17. y+z+ x− 

_______ 

 

+ 

19. x =__ 

≈ 21. a= 

________ + 

23. 25. x = 27. x =± 

____ √— 

 

29. f t=e −t ; 

31. Tt=e −t +T≈°F 

x 

x 

33. 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CHAPTER 7 

Section 7.1 

x 

35. 

y = x 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

37. 

 

y = 

 

+e −x 

y 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. 

3. 

 

fi 5. 

xy 

7. 9. 11. − 13. − 

 

15. ( − 

 

) 17. 19. ( 

 

) 

21. − 23. ( − __ 

___ ) 25. 

27. ( − __ 

33. ( 

__ 

) 

 

29. ( x x+ ) 31. − 

) 35. ( 

) 37. xx− 

39. ( − __ 

__ 

) 41. 

43. 

45. 

47. − 49. − 51. ( A+B 

 

A−B ) 

 

53. ( − 

 

A−B A A−B ) 55. ( _ 

EC−BF 

AE−BD DC−AF _ 

BD−AE ) 

x 

x

B-22 

ODD ANSWERS 

57. Thn a pfi 59. 

61. Th 63. 

65. 67. 

69. 

71. 73. 

75. 

77. 

 

Section 7.2 

1. 

3. . Th 

 

 

x+y−z=−x−y+z=−x+y+z= 

5. 

 

 

7. 9. 

11. − 13. ( − ____ 

____ 

____ 

) 15. ( 

) 

17. − 19. ( x−x _______ 

+x ______ 

) 21. ( − ___ 

___ 

− ) 

23. 25. 27. ( __ 

− __ 

− __ 

) 

29. 31. − 33. 

35. −−− 37. 39. ( 

 

 

) 

41. ( 

 

 

) 43. 45. 

47. ( 

 

 

) 49. − 51. 

53. 55. 

 

57. Th 

59. 

61. Th 

63. 

 

65. 

 

67. 

69. e 19.3%, fi 

 

Section 7.3 

1. 

s. A 

f a pd a cir 

 

3. . Th 

e a fsider a sys 

 

 

 

5. 

CxRx 

Cx


5. x=−B 

A= 

x=− 

B− 

 

7. 

x+ − 9. 

x− x+ + 

x+ 

11. 

 

x− + 13. 

x− x+ + 

x− 

15 

 

x+ + 

x− 17. 

x+ + 

x− 

 

19. 

x− − 21. 

x− x− + 

x− 

23. − 

 

x+ + 

x+ 25. − 

x− − 

x− 

27. 

x − 

x+ + 

x+ 

29. 

x + 

 

x − 

x+ + 

x+ 

31. x+ 

 

x +x+ + 33. −x x+ x +x+ + 

x− 

35. x− 

 

x +x+ + 37. 

x+ x +x+ + 

x− 

39. 

 

x −x+ + 41. − 

x+ x +x+ + 

x− 

43. 

x + 

x+ −___________ 

x 

x −x+ 45. x+ 

x + + x+ 

x + 

47. x+ 

x+ + x+ 

x+ 49. 

x +x+ − x 

x +x+ 

51. 

x − x 

x + + −x 

x + 

53. − 

x − 

 

x + 

x− − 

x− 

 

55. 

x+ − 

x+ + 

x+ 

57. 

 

x− − 

x+ + 

x+ − 

x− 

59. − 

x − 

x+ + 

x+ + 

x− 

Section 7.5 

1. 

f diff 

s a 2× 

s a 2× [ ] + [ ] 

3. Am × nB 

n × m 5. 

AB A B 

AB BA 

B A BATh 

 

 

 

7. 

[ 9. ] [ − 

] 

11. 

 

 

13. 

[ 

15. ] [ − − − − 

 

 

− − − − ] 

] 19. [ ] 

 

− 

23. 

[ − − ] 

 

17. 

 

[ 

 

 

 

21. [ 

 

− − ] 

− 

25. 

− 

 

− − 

27. 

[ − 

− 

 

− 

29. ] [ 

] 

 

− 

31. [ − ] 

33. 

35. [ 

− ] 37. [ 

− ] 

39. [ ] 41. [ − −− ] 

43. [ − 

− − 

− − − 

] 45. 

[ − 

] 

− 

 

− − 

47. 

[ − 

49. ] [ − 

− ] 

 

− 

51. 

[ − 

 

 

53. ] [ 

] 

− 

 

 

 

55. 

[ 

57. ] [ ] 

 

 

59. B n = 

[ 

{ 

 

n 

 

] 

 

 

[ ] n 

 

Section 7.6 

1. 

d a v 

 

3. 

n a m 

ThR 

=R 

−R 

 

R 

=R 

−R 

. Th 

5. 

 

7. [ − 

] 9. 

[ 

 

 

] 

− 

11. −x+y= 

x−y= 

15. x+y−z= 

y+z= 

x+y − z=− 

23. 25. ( 

 

) 

31. ( ____ 

− 

13. x+y= 

−x−y+z= 

x+y+z= 

17. 

19. −− 

21. 

27. ( x 

x+ ) 29. 

___ 

) 33. − 35. ( 

 

 

 

)

B-24 

ODD ANSWERS 

37. ( ___ 

___ 

− ___ 

) 39. ( xy 

− x − 

y ) 

41. ( x−x __ 

− ) 43. − 45. − 

47. 49. ( −z+ 

__ z−__ 

z ) 

51. 53. 

55. 57. 

59. 

61. 

Section 7.7 

1. A − AA − =I 

A − A − =I 

r a 2 × 3. adbc 

ad−bc= 

5. [ ] . Th 

 

 

A − = 

− − 

[ 

] 

− ] = [ 

7. AB=BA= [ ] = 9. AB=BA= [ ] = 

 

 

11. 

AB=BA=[ = 13. ] 

 

 

[ − ] 

15. 

[ − ] 17. Th 19. 

[ − ] 

21. − 

 

 

[ 

− 

− 

 

] 23. 

− [ 

− 

 

 

− 

] 

− − 

− 

25. [ 

 

− − 

27. − 29. 

] 

− 

31. ( 

− 

) 33. ( − __ 

− ___ 

) 35. ( 

 

) 

 

37. − 39. ( − 

__ 

41. ( _ 

 

−__ 

−__ 

 

 

) 

− ___ 

− _ 

) 43. ( − 

 

) 

− − 

45. ( 

 

− 

) 47. 

 

[ 

 

− 

 

− 

 

 

− 

− 

49. 

 

[ 

 

− 

 

− 

 

] 

− − − 

 

 

 

 

51. [ 

 

 

] 53. 

 

 

 

 

 

− − − − − 

55. 

57. 

59. 

61. 

] 

Section 7.8 

1. 

 

3. Th 5. − 

7. 9. − 11. 13. − 15. 17. − 

19. 21. 23. − 25. 27. ( 

29. 31. ( −− __ 

) 

) 

33. 35. 

37. − 39. ( 

) 41. 

43. 45. 47. 49. 

51. 53. 

55. 57. 

59. 

61. 63. 

 

65. 67. 

 


1. 3. − 5. − 7. 

9. 11. − 13. 

15. −− 17. ( xx ___ 

x ____ 

) 19. 

21. − 23. 25. 

27. y 

29. y 

 

 

 

 

 

−−−−− 

− 

 

− 

− 

− 

− 

x 

 

 

 

 

 

−−− 

− 

 

− 

− 

− 

− 

 

31. − 

x+ + 33. 

x+ x+ − x+ 

 

35. 

x− + −x+ 

x +x+ 37. x− 

x − + x+ 

x − 

 

39. [ − − − ] 41. 

43. 

] 

47. [ 

 

 

 

− 

45. 

[ − 

 

 

 

− 

− 

− 

 

49. 

51. x−z= 

y+z=− 

− 

 

 

53. 

[ − 

 

 

 

55. ] [ − 

 

 

] 

− 

− 

57. 59. 

61. 

[ ] 63. 65. − 

67. − 69. 

71. 73. 75. ( __ 

) 77. xx+ 79. ( − 

] 

 

x 

 

)



1. 3. 5. ( 

 

 

) 

7. ( xx ____ 

−x ____ 

) 

9. −√ — −√ — −√ — √ — √ — −√ — √ — √ — 

11. 

y 

 

 

 

 

 

−−−−− 

− 

 

− 

− 

− 

− 

x 

 

13. 

x+ − x+ 

x+ 

15. [ − ] 

− 

17. [ − ] 

19. − 

 

 

− 

21. 

[ − 

− 

 

 

− 

] 23. x 

− 

25. 27. ( ____ 

) 

CHAPTER 8 

Section 8.1 

29. 

1. 

s a co 

3. e a cir 5. 

xy 

7. x 

+y = 9. x 

( + y 

( = 

 

) 

) 

11. x 

+y =− 

−√ — −√ — 

y 

13. x 

+ 

 

 

( =− 

 

) 

( 

) ( − 

) ( √— 

) 

( − √— 

) 15. x− 

+y − 

= 

−− 

+√ — −√ — 17. x + 

+y − 

= 

−− 

−−−+√ — −−√ — 

19. x − 

+y − 

=− 

+√ — 

−√ — 21. x − 

( √ — ) + y − 

( √ — ) = 

+√ — −√ — 

+√ — −√ — − 

23. _______ x + 

+y _______ − 

=− 

−−−+√ — 

−−√ — 

25. _______ x + 

+y _______ + 

=− 

−−−−−− 

−+√ — −−−√ — − 

27. −−+√ — −−−√ — 29. 

31. −−− 

33. 

−− 

√ — −√ — 

y 

 

 

 

 

 

−−−−− 

− 

 

− 

− 

− 

− 

37. − 

−−− 

− 

s a cir. Th 

 

 

y 

 

 

 

 

−−−− 

− 

− 

− 

− 

x 

x 

35. ( __ 

( − __ 

) ( 

( √— 

) 

 

) ( − __ 

) 

 

) 

) ( − √— 

 

 

 

y 

−−.− 

− 

 

−. 

− 

39. 

−− 

+√ — −√ — 

y 

 

 

 

 

 

−−−−− 

− 

 

− 

− 

− 

− 

41. − 43. − 

−−−− −−−− 

−+√ — −−√ — −+√ — −−√ — 

y 

y 

 

 

 

 

−−−− 

− 

− 

− 

45. −− 

−−−− 

−−−− 

 

 

−−−−− 

− 

 

− 

− 

− 

− 

y 

x 

x 

 

 

 

−−.− 

− 

−. 

− 

47. x 

+y 

= 

49. x − 

+y − 

= 

51. x + 

+y − 

= 

53. x 

+y 

= 

55. x + 

+y − 

= 

57. =π 59. =√ — π 

x 

x 

x

B-26 

ODD ANSWERS 

61. =π 63. x 

y 

h + 

 

= 

 

h 

65. x 

+ y 

= 

67. 

Section 8.2 

1. ts in a pe diff 

s a p 

3. Th 

5. Th 

7. x 

 

−y = 

9. x 

 

−y = 

11. x 

 

−y =− 

√ — −√ — y= 

xy=− 

x 

13. y 

 

 

−x =−√— 

−√ — y= 

xy=− 

x 

15. x − 

−y − 

=− 

−y= 

x −+y=− x −+ 

 

17. x − 

−y + 

=−−− 

+√ — −−√ — −y=x−y=−x− 

19. x + 

−y − 

=− 

−+√ — −−√ — y=x+y=−x 

33. 

− 

35. 

− 

− 

− 

y 

 

 

 

− 

− 

 

− − − − 

− − 

37. 

− 

 

− 

− 

 

− 

− 

 

 

 

− 

− 

− 

− 

y 

y 

 

x 

 

− 

− 

 

 

− 

− 

 

 

x 

 

x 

 

 

− 

21. y − 

−x − 

=+√— 

−√ — y= 

x −+y=− x −+ 

 

23. _______ y + 

−x _______ + 

=−−− 

−−+√ — −−−√ — 

y 

39. 

−− 

− 

− 

− 

− 

− 

− 

−− 

− 

 

− 

x 

y= ___ 

x+−y=− ___ 

x+− 

25. x + 

−y − 

=− 

−+√ — −−√ — y= x ++ 

 

y=− __ x ++ 

27. y= __ 

x−−y=− __ x −− 

 

29. y= 

x −+y=− 

x−+ 

31. 

y 

− 

− 

− − − − 

 

 

− 

− 

 

x 

 

41. 

y 

43. 

 

 

−− − 

x 

− − 

− − 

−− − − 

45. x 

−y 

= 

47. x − 

−y − 

= 

− 

 

 

 

− 

− 

− 

y 

 

 

 

 

x 

 

 

− 

 

− 

49. x − 

−y − 

= 51. y 

−x 

= 

53. y 

−x + 

= 55. x + 

−y + 

=


57. yx=√ — x + 59. yx=+√ — x +x+ 31. y 

33. 

y 

yx=−√ — x + 

yx=−√ — x = − 

x +x+ 

 

 

 

y 

y 

 

 

 

 

 

 

 

 

x 

 

 

 

−−−− 

− 

 

 

 

− 

x 

 

 

− − − − 

− 

x 

x 

−−−−− 

− 

 

−−−−− 

− 

 

− 

− 

y = − 

− 

− 

− 

− 

− 

− 

− 

− 

− 

− 

35. 

61. x 

−y 

= 

63. x 

−y 

= 

y 37. 

y 

 

 

y 

y 

x− x = 

 

 

x 

 

− − − − 

 

 

 

 

 

− 

 

 

 

x 

x 

− 

− − − − − − − 

x 

− 

− − − 

− 

− 

− 

− 

− 

− 

39. 

y 41. 

y 

− 

65. x 

− y 

= 

67. x 

 

− 

− y 

= 

y= 

x 

 

− − − − x= − 

y 

69. x − 

−y 

= 

− 

x 

−− 

− − 

 

− 

− − 

 

x 

− 

− 

−−−−− 

− 

− 

− 

− 

− 

43. y 

45. x =−y 

Section 8.3 

 

47. y − =√ — x − 

1. 

49. y +√ — =−√ — x −√ — 

 

m a fixd a fix 

x= 

51. x =y 

3. Th 

 

53. y − 

5. Th 

= x + 

x 

 

7. y=x 9. y − =x − 

− 55. y −√ — 

11. y = 

xVF ( 

) dx=− − 

=√ — x +√ — 

 

 

− 

57. y =−x 

13. x =− 

yVF ( − − 

) dy= 

 

 

59. y + =x + 

15. y = 

xVF ( 

) dx=− − 

61. 

 

 

63. 65. 

17. x − =y −VFdy= 

67. x =−y − 69. 

19. y− =x+V−F( − 

) dx=− 

 

Section 8.4 

21. x + =y +V−−F−dy=− 

23. y − =−x +V−F−dx= 

1. xyes a r 

25. x − = 

y +V−F ( − 

) dy=− 

 

3. Ths a h 5. 

xy 

27. x − =−y −VF( 

) dy= 

 

7. AB= 9. AB=−< 

29. y − = 

x−VF ( 

) dx= 11. AB=> 13. B −AC= 

 

15. B −AC= 17. B −AC=−< 

19. x′ +y′ −= 21. x′ +x′y′−y′ += 

23. θ=x′ −y′ +√ — x′+y′−= 

25. θ=x′ +y′ +x′−√ — y′−= 

27. θ≈x′ +x′−y′+= 

29. θ=x′ −y′ −√ — x′+√ — y′+=

B-28 

ODD ANSWERS 

31. ____ √— 

x′+y′ = __ x′−y′ 

 

y 

− 

− 

 

 

− 

− 

35. ________ x′+y′ 

−x′−y′ ________ 

= 

y 

− 

39. 

43. 

− 

− 

 

 

− 

− 

θ = 

− 

− 

−− 

 

 

 

 

 

 

− 

− 

 

y 

 

x 

 

 

−−−− 

− 

 

− 

− 

47. y 

θ = 

 

− 

− 

− 

− 

51. θ= 

y 

y' 

 

 

 

 

−−−−− 

− 

 

− 

− 

55. θ≈ 

y 

y' 

 

 

 

 

x' 

−−−−− 

− 

 

− 

− 

 

x' 

 

 

x 

x 

x 

x 

x 

x 

33. x′−y′ 

+x′+y′ 

= 

 

 

 

−−−−− 

− 

 

− 

− 

37. √— 

41. 

49. 

y 

x′− 

y′= 

( __ 

____ x′+√— 

y′− ) 

y 

 

 

− 

− 

− 

θ = 

− 

− 

θ = y 

45. 

θ = 

− 

− 

− 

 

 

− 

− 

y 

 

− 

53. θ= 

y 

y' 

 

 

− 

− 

 

 

 

y 

 

θ = 

 

x 

 

 

 

 

x' 

−−−−− 

− 

 

− 

− 

57. −√ —



1. x 

+y =− 

−√ — −√ — 3. x + 

+y − 

= 

−−−−−−−+√ — 

−−√ — 

5. 

−− 

√ — −√ — 

 

 

 

−−−− 

− 

− 

− 

y 

x 

7. −− 

−−−− 

−−−−+√ — 

−−−√ — 

y 

 

 

 

−−−− 

− 

− 

− 

9. x 

+y = 11. 

 

13. y + 

−x − 

=− 

−−+√ — −−√ — 

15. x − 

− y + 

( √ — ) =−− 

−−−− 

17. y 

 

− 

 

−− 

 

 

 

 

 

− 

−−−− 

− 

− 

− 

− 

 

−− 

x 

19. 

 

− 

 

 

 

 

y 

 

 

−−−− 

− 

− 

− 

− 

 

− 

x 

 

 

x 

21. x − 

−y − 

= 23. x + = y − 

 

−( − 

) y= 

 

25. x + =y +−−( −− 

) 

y=− 

 

27. 

29. 

y 

 

y 

x= 

x = − 

 

 

 

−−−−− 

− 

 

− 

− 

− 

x 

− − 

 

 

31. x − = ( 

) y − 

33. B −AC= 35. B −AC=−< 

37. θ=x′ +y′ −= 

− 

 

 

− 

− 

− 

 

 

 

 

x 

39. θ= 41. e= 

y 

 

− 

43. e= 

 

 

 

45. 

47. 

y 

y 

 

 

 

 

 

 

 

 

x 

−−−− 

− 

y= − − 

− − − − 

− 

− 

− 

 

49. r= 

+θ − 


 

 

 

1. x 

+y =− 

−√ — −√ — 

3. 

−− 

+√ — −√ — 

y 

 

 

 

−−− 

− 

− 

− 

x 

5. x − 

+y − 

= 

7. x 

 

−y = 

− 

√ — −√ — 

y=± 

x 

9. −−−−+√ — − 

−√ — −y=± x −− 

 

y 

−− − 

 

 

x 

− 

 

− 

− 

− 

− 

− 

 

− 

11. y − 

−x − 

= 13. x − = y + 

 

−( − ___ 

) y=− 

 

15. 

y 

17. 

 

19. θ≈ 

− 

− 

− 

− 

 

 

 

 

− 

− 

 

−−−− − 

 

− 

− 

− 

 

 

− 

− 

x 

 

x=− 

 

x 

x

B-30 

ODD ANSWERS 

21. x′ −x′+y′= 

23. e= 

47. 

a n 

49. 

a n 

y 

 

⎛ 

 

 

 

⎝ ⎞ 

 

 

⎭ 

 

 

 

 

 

 

45. − 

 

55. Th 

57. Th 

x 

 

 

−−−−−− 

− 

 

 

 

− 

n 

−−−−−− −−−−−− 

− 

− 

− 

 

− 

− 

− 

n 

− 

− 

− 

− 

− 

− 

− 

− 

25. 

y 

− 

− 

 

 

y= 

51. a n 

53. a n 

= n− 

 

⎛ ⎝ ⎞ 

 

55. a 

 

 

=a n 

=a n− 

− 

⎭ 

 

x 

 

− − − − 

 

57. t fi 

 

− 

 

 

n 

 

 

 

 

− 

−−−−− 

 

− 

− 

59. t fi 

− 

− 

− 

− 

− 

− 

 

61. a 

= 

− 

63. 

65. 

67. a n 

=−s a t 

CHAPTER 9 

−=−−nnld a n 

−=−−nn= 

Section 9.1 

1. 

en a fini 

y a f 

a n 

=− 

69. a 

=a 

=a n 

=a n− 

−a n− 

71. n + 

n − =n +n 

+nn − 

 

n − 

y a f =nn +n +=n +n +n 

3. 

 

5. f a p 

 

Section 9.2 

1. 

efi y a co 3. 

 

 

e diff 

s a co 

13 ∙ 12 ∙ 11 ∙ 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1. 

diff 5. 

7. −− 

−− 9. e a co. The diff 

 

 

 

 

 

11. −− 

 

13. 

 

 

15. r a s 7. Th 

 

− 

− 17. 

 

 

 

 

 

diff 9. Th 

 

−≠− 11. 

 

 

13. −−−− 

19. −−−−−−−− 

15. a 

21. a n 

=n + 23. a n 

= 

n n− 

n 25. a n 

=( − 

) − 

= 17. a 

= 19. a 

= 21. a 

= 

n 23. a 

= 25. a 

=− 27. −−−−− 

27. t fi−− 29. t fi 

29. a 

=a n 

=a n− 

+n≥ 31. a 

=a n 

=a n− 

+n≥ 

−− 

 

31. 

 

 

 

 

 

 

33. a 

=a n 

=a n− 

+n≥ 35. a 

= 

a =a + 

n n− 

n≥ 

33. 

 

35. a=−a =a +n 

37. a 

= 

a =a − 

n n− 

n≥ 39. a=a =a +a = 

n n− 

n n− 

41. t fi 43. a n 

=+n 

37. a 

=a n 

=a n− 

+ 39. 41. 

45. a n 

=−+n 47. a n 

=n 49. a n 

=+n 

43. 

 

 

 

51. a n 

= 

n− 

 

53. Th


59. 

 

 

− 

− 

− 

− 

− 

− 

− 

− 

65. 

a n 

 

− 

n 

− 

− 

− 

 

a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

−−−−−− 

− 

n 

61. 

63. 

a n 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

−−−−−− 

− 

67. 

a n 

=n 

a n 

=+n 

69. a 

=−a+b 

71. Th 

a 

=− 73. 

 

a 

= 

a n 

=a n− 

−−−a 

=− 

Section 9.3 

1. 

3. m in a s 

 

5. 

e a co 

 

 

er difff a 

 

 

7. Th− 9. Th 

Th 11. Thic. Th 

− 13. Thic. Th 

 

15. 

 

 

17. 

19. a 

=− 

21. a 

=− 23. 

 

25. a=−a n 

= 

a 27. a=a =−a n− n n− 

29. a 

= 

a = n 

a 31. a= 

n− 

a =−a n n− 

33. −− 

 

35. a n 

= n− 

37. a n 

= 0.8 ∙ (− n− 39. a n 

=−( __ 

) n− 

41. a n 

= 3 ∙ ( − __ 

 

) n− 43. a 

= 

 

45. Th 

47. Tht a g 

49. 

a n 

 

 

 

 

 

− 

− 

 

 

 

 

 

n 

51. 

a 

=a n 

=a n− 

 

a 

=a n 

=a n− 

53. a 

=b 

55. Th 

a 

≈ 

n 

57. a 

=− 

 

59. h a de 

a n 

= 400 ∙ 0.5 n− 

a 

= 

Section 9.4 

1. n nf a 

3. s in a 

5. s a s 

n a co 

 

 

 

7. ∑ n 9. ∑ 11. ∑ k+ 13. S 

= 

n= 

k= 

k= 

 

 

15. S 

= 17. ∑ 8 ∙ 0.5 k− 

( −( 

19. S 

= 

− 

 

 

) ) 

k= 

 

= 

≈ 

21. S 

= − 

− = 

≈ 

 

23. Th S= 

− 

25. Th S= − 

−( − 

) 

27. y 

 

 

 

 

 

 

 

 

 

 

 

 

29. S: Th 

S n 

 

g 1. Th 

∞ 

∑ ( 

k= ) k 

 

x 

 

 

S= 

−( 

) = 

31. 33. 35. S 

= 

37. S 

= 

 

39. S 

= 41. S 

=− 

43. S=− 

 

45. S= 47. 49. 

51. a k 

=−k 53. 55. r= 

 

57. 59. 61. 

Section 9.5 

1. Thm + nAB 

3. Th 

. Th 

 

. Th 

lem. Th 

5. 

n 

Cn, r= 

 

7. += 9. ++= 

n−rr 

11. = 13. = 15. P= 

17. P= 19. P= 21. C= 

23. C= 25. = 27. = 

29. = 31. = 

33. 35. 

 

37. r=r=r= 

Cnr=Pnr=r=r=Cnr=Pnr=n

B-32 

ODD ANSWERS 

39. ×= 

41. −+−= 43. ××= 33. 

 

45. ××= 47. P= 

1 2 3 4 5 6 

49. C×C×C= 51. = 

53. 1 

 

 

= 

 

2 

 

 

Section 9.6 

3 

 

 

1. 

Cnr ( 

n r n 

) 

=Cnr= 

rn−r 4 

 

 

n 

3. Thl Th x +y =∑ ( n 

k= k ) 

x n−k y k 5 

 

 

 

5. 7. 9. 11. 

6 

 

 

13. a −a b +ab −b 15. a +a b +ab +b 

17. x +x y +x y +x y +xy +y 35. 

37. 39. 

41. 

43. 

 

19. x −x y +x y −x y +xy −y 

21. 

 

x + 

x y + 

x y + 

xy + 

y 23. 45. 

a +a b +a b 47. C 

C = 

49. CC 

 

C = 

 

25. a −a b +a b 

51. CC 

 

C ≈ 53. CC 

 

C ≈ 

27. a +a b +a b 

29. x −x √ — y +x y 31. −x y 

55. +−= 

33. y 35. x y 

57. CC 

 

37. a b 39. y x 

 

C = 

59. 

41. f 

x=x +x 43. f 

x=x +x +x +x 

CC 

 

C = 

y 

y 

f x 

f x 

 

 

 

 


 

 

 

 

1. 3. 

 

 

 

 

5. Thic. Thn diffd= 

 

 

. 

x 

 

7. −− 9. a 

−−−−− 

 

=−a n 

=a n− 

+ 

− 

 

− 

x 

11. a 

− 

−−−−−−−−− 

n 

= 

− 

n+ 13. r= 15. 

 

− 

− 

17. 19. a 

− 

− 

n 

=− 

∙ ( 

) n− 

− 

 

− 

21. ∑ ( 

45. x y − 

m= m+ ) 23. S = 25. S ≈ 

 

 

− 

− 

27. S= 

47. k − 

− 

29. 31. 33. = 

35. P= 37. C= 

49. Thx +y −z 

l Th 39. =× 41. = 

43. 

 

45. a 

Section 9.7 

+a b+a b 

47. 

1 2 3 4 5 6 

1. 

 

1 

3. 

2 

5. Thunion of two events 

3 

 

4 

 

5 

ABAB 

ABh. The diff 

6 

 

7. 

 

49. 

9. 51. 

53. 

55. −C 

C ≈ 

 

11. 

13. 

15. 

17. 

19. 

 

21. 

23. 

25. 

27. 

29. 57. CC 

 

 

31. 

C ≈



1. −−− 3. Thic. Th 

n diffd=. 5. a 

=−a n 

=a n− 

− 

 

a 

=− 7. Thic. Th 

 

r= 

 

. 9. a=a =− n 

∙ a n− 

11. ∑ ( − 

k=− k ) 

13. S 

=− 15. 

17. ××××= 

19. C= 21. 

 

= 23. x 

25. 

 

27. 

29. CC 

 

C ≈

Index 

A 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B 

 

 

 

 

 

l Th 

 

 

C 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n diff 

 

 

 

 

 

 

 

 

 

 

 

 

 

e Th 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

diff 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F 

 

r Th 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l Th 

 

G 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C-1

C-2 

INDEX 

H 

 

 

 

 

 

 

 

 

 

 

 

hift 

 

 

 

 

 

I 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ue Th 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J 

 

L 

 

 

 

 

 

 

 

 

 

n Th 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 

n 

 

 

 

 

 

 

 

 

n 

n 

n 

O 

 

 

 

 

 

 

 

 

 

 

 

P 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n 

 

 

 

 

 

 

fi 

 

 

n Th 

 

Q 

 

 

 

 

 

 

 

 

 

 

R 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o Th 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

der Th

INDEX C-3 

 

 

 

 

 

 

 

 

 

 

S 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T 

 

 

 

 

 

 

 

 

 

U 

 

 

 

 

V 

 

 

 

 

 

 

 

 

 

 

 

 

hift 

 

 

 

 

 

 

W 

 

X 

x 

x 

x 

Y 

y 

y 

y 

 

Z

College Algebra, 2015a

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