Mathspace for Florida B.E.S.T Algebra 1

Mathspace for Florida

Algebra 1

B.E.S.T. 2022 edition

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Contents

1

Equations and Inequalities in

One Variable 2

1.01 Interpreting algebraic expressions 4

1.02 Equations with variables on one side 9

1.03 Equations with variables on both sides 14

1.04 Literal equations 19

1.05 Absolute value equations 22

1.06 Multi-step inequalities 28

1.07 Compound inequalities 35

1.08 Honors: Absolute value inequalities - solidifylesson 40

2

Equations and Inequalities in

Two Variables 46

2.01 Slope-intercept form 48

2.02 Standard form 60

2.03 Point-slope and connecting forms 68

2.04 Parallel and perpendicular lines 78

2.05 Linear inequalities in two variables 84

3

Systems of Linear Equations and

Inequalities96

3.01 Solving systems of equations by graphing 98

3.02 Solving systems of equations by substitution 106

3.03 Solving systems of equations by elimination 111

3.04 Systems of linear inequalities 116

Contents

iii

4

Functions and Linear relationships 126

4.01 Evaluating functions 128

4.02 Domain and range 132

4.03 Average rate of change 140

4.04 Characteristics of functions from a graph 146

4.05 Linear relationships 153

4.06 Transforming linear functions 162

4.07 Characteristics of linear functions 172

4.08 Comparing linear and nonlinear functions 183

4.09 Linear absolute value functions 189

4.10 Transforming absolute value functions 194

5

Exponential Functions 204

5.01 Operations with numerical radicals 206

5.02 Rational exponents 213

5.03 Exponential relationships 220

5.04 Characteristics of exponential functions 228

5.05 Exponential growth and decay 237

5.06 Percent growth and decay 249

5.07 Simple interest 259

5.08 Compound interest 264

6

Polynomials and Factoring 270

6.01 Adding and subtracting polynomials 272

6.02 Multiplying polynomials 276

6.03 Multiplying special products 280

6.04 Dividing polynomials by a monomial 284

6.05 Factoring GCF 287

6.06 Factoring by grouping 291

6.07 Factoring trinomials where a = 1 295

6.08 Factoring trinomials where a > 1 300

6.09 Factoring special products 305

iv

Mathspace Florida B.E.S.T - Algebra 1

7

Quadratic Functions 310

7.01 Quadratic relationships 312

7.02 Key features of quadratic graphs 321

7.03 Quadratic functions in factored form 334

7.04 Quadratic functions in vertex form 344

7.05 Quadratic functions in standard form 355

7.06 Modeling with quadratic functions 366

7.07 Linear, quadratic, and exponential models 374

7.08 Honors: Combining linear and quadratic functions 384

8

Solving Quadratic Equations 392

8.01 Solving quadratic equations using graphs and tables 394

8.02 Solving quadratic equations by factoring 403

8.03 Solving quadratic equations using square roots 409

8.04 Solving quadratic equations by completing the square 414

8.05 The quadratic formula and the discriminant 419

8.06 Solving quadratic equations using appropriate methods 425

9

Descriptive Statistics 432

9.01 Populations and sampling 434

9.02 Classifying data 440

9.03 Displaying and interpreting univariate data 445

9.04 Two-way frequency tables 457

9.045 Honors: Two-way relative frequency tables 466

9.05 Associations in bivariate data 478

9.06 Scatter plots and lines of fit 483

9.07 Analyzing lines of fit using residuals 490

Contents

v

1

Equations and

Inequalities in

One Variable

Chapter outline

1.01 Interpreting algebraic expressions 4

1.02 Equations with variables on one side 9

1.03 Equations with variables on both sides 14

1.04 Literal equations 19

1.05 Absolute value equations 22

1.06 Multi-step inequalities 28

1.07 Compound inequalities 35

1.08 Honors: Absolute value inequalities 40

1.01 Interpreting

algebraic expressions

Concept summary

An expression is a mathematical statement that contains one or more numbers and variables

joined together by operators and grouping symbols. An expression does not contain an equal sign

or inequality symbol.

An algebraic expression is an expression which includes at least one variable.

Algebraic operation

A mathematical process, such as addition, subtraction, multiplication, or division

Example: +, −, ×, and ÷

Variable

A symbol, usually a letter, used to represent an unknown value.

Example: Expression: 3x − y + 2 Variables: x and y

Term

One part of an expression. Terms are separated by addition or subtraction.

Example: Expression: 3x − y + 2 Terms: 3x, −y, and 2

Coefficient

The number or constant that multiplies a variable in an algebraic term. If no number is

specified, the coefficient is 1.

Example: Term: 3x Coefficient: 3

Constant term

A term that has a fixed value and does not contain a variable. Sometimes just called a constant.

Example: Expression: 3x − y + 2 Constant: 2

4


Like terms are terms that have the same variables with the same exponents. An example is shown

below:

Worked examples

Example 1

3x + 5x 2 − 2x − 8

Terms: 3x, 5x 2 , −2x, and −8

Like terms: 3x and −2x

Consider the algebraic expression: 15y 2 + 7y 3 − 3y 2 − 5

Identify:

• The constant term

• The like terms

• The coefficient of y 3

Approach

The constant term is the numeric term without any variable factor, the like terms are the algebraic

terms that have the same algebraic factors, and the coefficient of y 3 is the numeric factor of the

term containing a factor of y 3 .

Solution

The constant term is −5, as it is the term without any variable part.

The like terms are 15y 2 and −3y 2 , as they have the same variables with the same exponents.

The coefficient of y 3 is 7, as this is the numeric factor of the term 7y 3 .

Reflection

Notice that we include the sign with all terms. For example, one of the like terms is −3y 2 , which has

a coefficient of −3.

Example 2

Vincenzo runs a removalist company that charges $37.50 per hour plus a one-off truck hire fee of

$150.00.

Write an expression that models how much he charges for a job that lasts a hours.

Chapter 1 Equations and Inequalities in One Variable 5

Approach

We need to look at the two values that affect the price of the job; the cost per hour of $37.50 and

the truck hire fee of $150.00. Once we determine how to write the cost per job in terms of a hours,

we need to write an expression that is the sum of these two values.

Solution

Cost: 37.5a + 150

Example 3

Bernie and Paula are painting the walls of their house. Bernie can paint x square meters an hour

while Paula can paint y square meters an hour. Bernie spends 4 hours painting while Paula spends

6 hours over the weekend. Together they paint a total of 4x + 6y square meters over the weekend.

Determine what 6y represents in the expression.

Approach

We can look at the units that are included in the term 6y. Paula paints for 6 hours and y is the

number of square meters she can paint per hour. We can use this to help us determine what

6y represents.

Solution

The number of square meters Paula painted over the weekend.

What do you remember?

1 Match the following terms with their definitions:

a Constant i Symbol that represents an unknown number

b Variable ii A purely numeric term in an algebraic expression

c Coefficient iii The value that indicates how many of a variable in a term

d Algebraic term iv Term including a variable

2 Write the expression 9x in words.

3 State whether the following expressions are like terms:

a 11p and 3p b 2p and 15 c 9p and 6q d 12 and 8

e 5h and 5hk f 4h 4 and 3h g 8h 3 k 4 and 9k 4 h 3 h 6h k and 7k h

6


4 For the following algebraic expressions:

i State the number of terms.

ii What is the numerical coefficient of the first term?

iii What is the constant term?

iv Does it contain like terms?

a 2x + 4 b 7y + 3 + 5x

c 3x + 2y − 8x + 9 d 8p + 5

Let’s practice

5 Consider the algebraic expression 3x + 10 + 4x.

a

b

c

d

State the number of terms.

What is the numerical coefficient of the third term?

What is the constant term?

Does it contain like terms?

6 Dylan purchased 3 pens, 4 pencils and a single note pad. The total cost of all these items

was 3x + 4y + 5 dollars.

a What does the variable y represent? b What does the variable x represent?

c

What does 5 represent?

7 Deborah earns $25 per hour of work and is paid double for every hour worked on the

weekend. At the end of a week, Deborah calculates her pay for that week to be 25x + 50y

dollars.

a

b

What represents the number of hours Deborah worked during the weekdays?

What represents the number of hours Deborah worked during the weekends?

8 Mohamad and Valentina run a clothes business. Mohamad is able to sew 5 shirts an hour,

while Valentina is able to sew 3 shirts an hour. The expression 6( 5 + 3 ) represents the total

number of shirts the couple is able to produce in a six-hour period.

What does 5 + 3 represent?

9 Brad and Patricia are making paper cranes to decorate their room. Brad can make m an hour

while Patricia can make n an hour. Brad spends 6 hours making cranes while Patricia spends

5 hours over the weekend.

Together they make a total of 6m + 5n paper cranes over the weekend.

a What does 6m represent? b What does 5n represent?

10 The width of a rectangular football field is 7 yards more than half of its length. If L represents

the length of the field, write an expression for the width.


11 Roxy is 5 inches taller than Jane, while Katrina is 9 inches shorter than Jane. Let k be Jane’s

height in inches.

a

b

Write an algebraic expression for Roxy’s height.

Write an algebraic expression for Katrina’s height.

12 Valerie is with a mobile phone provider that charges $0.26 per minute plus a connection fee

of $0.45. Write an expression that models how much he will be charged for a call that lasts

a minutes.

Let’s extend our thinking

13 For each of the following, determine if the statement is true or false. Explain your reasoning.

a

b

An expression must include a variable.

Any expression with more than one term can be simplified.

14 Tobias has two times as many books as Isabelle does, and Isabelle has four less than triple

the number of books that William does. If William has n books, explain the method used to

write an expression for the number of books that Tobias has.

8


1.02 Equations with

variables on one side

Concept summary

An equation is a mathematical relation statement where two equivalent expressions and values

are seperated by an equal sign. The solutions to an equation are the values of the variable( s ) that

make the equation true. Equivalent equations are equations that have the same solutions.

One particular type of equation is a linear equation.

Linear equation

An equation that contains a variable term with an exponent of 1, and no variable terms with

exponents other than 1.

Example: 2x + 3 = 5

Equations are often used to solve mathematical and real world problems. To solve equations we

use a variety of inverse operations and the properties of equality.

Inverse operation

Two operations that, when performed on any value in either order, always result in the original

value; inverse operations “undo” each other.

Example: Operation: Multiply by 2 Inverse: Divide by 2

The following are the properties of equality:

Reflexive property of equality

a = a

Symmetric property of equality

If a = b, then b = a

Transitive property of equality

If a = b and b = c, then a = c


Addition property of equality

If a = b, then a + c = b + c

Subtraction property of equality

If a = b, then a − c = b − c

Multiplication property of equality

If a = b, then ac = bc

Division property of equality

If a = b and c ≠ 0, then a ÷ c = b ÷ c

Substitution property of equality

If a = b then b may be substituted for a in any expression containing a.

The following is another important property:

Distibutive property

a( b + c ) = ab + ac

Worked examples

Example 1

Solve the following equation:

Solution

Subtract 3 from both sides of the equation

x = 4 Multiply both sides of the equation by 2

10


Example 2

Solve the following equation:

Solution

3x + 4x + 7 = 3 Multiply the entire equation by 3

7x + 7 = 3

Combine like terms

7x = -4 Subtract 7 from both sides of the equation

Divide both sides of the equation by 7

Example 3

Yolanda works at a restaurant 5 nights a week and receives tips. On the first three nights, the total

tips she received was $32, $27, and $26. She earned twice as much in tips on the fourth night

compared to the fifth night. The average amount of tips received per night for the week was $29.

If the amount she received on the fifth night was $f, determine how much she received that night.

Approach

The average is equal to the sum of values divided by the number of values. We can use this to

build an equation in terms of f about the average tips Yolanda received. Then we can solve the

equation for f.

Solution

Writing an equation in terms of f

Combining like terms in numerator

Yolanda received $20 on the fifth night.

Multiply both sides of equation by 5


Divide both sides by 3



1 Solve the following equations:

a 4x = 8 b 5m = −15 c 3p = 12 d 2k = −18


a 8m + 9 = 65 b −10 + 3k = 5 c 7 − 8t = 15 d −2d + 10 = 24

Let’s practice


a b c


a −2( −3h − 5 ) = 34 b 9x + 6x + 7 − 3x = 55

c 4x + 2 + 9x − 12 + 6x + 7 = 130


a

b

c

6 The product of 6 and the sum of x and 4 is equal to 42. Solve for the value of x.

7 Consider the given triangle which has a

perimeter of 171 cm.

Solve for the value of x.

8x cm

5x + 76 cm

6x + 19 cm

8 Consider the given

quadrilateraL which has a

perimeter of 393 cm.

Solve for the value of x.

8x − 30 cm

2x + 24 cm

9x + 29 cm

5x − 14 cm

12


9 Ursula is constructing a tabletop from a rectangular piece of board. The tabletop is to have

a perimeter of 68 inches, and its length is to be 5 inches shorter than double the width.

Solve for the width of the tabletop.

10 Emma and Beth do some fundraising for their tennis team and together they raised $664.

Emma raised $292 more than Beth.

Solve for the amount of money raised by Beth.

11 Yvonne is the youngest child in her family. She has three older brothers of ages 23, 20 and

16, and a sister who is eight years older than Yvonne is. The average age of the five children

is 19 years.

If Yvonne is k years old, solve for the value of k.


12 Stormie tries to guess how many people are at a concert, but she guesses 800 too many.

Pierce guesses 300 too few. The average of their guesses is 4250.

a

b

Describe a method to construct the equation to solve for the exact number of people at

the concert.

Solve for the exact number of people at the concert.

13 Athena and Emilio want to go karting. It costs 50 cents per lap of the course.

a

b

c

Write an equation to solve for the number of laps Athena and Emilio can afford if they

have $12.

Solve for the number of laps they can afford.

How does that equation change if they have to put a deposit of $1 on each cart used?

14 Bassam is changing km/h into mi/h for a project on peregrine falcons. He knows that the top

speed of the peregrine falcon is 389 km/h. He knows that there is 1.6 km to 1 mi. To convert

389 km/h to mi/h, Bassam multiplies 389 by

What error does Bassam make?

15 Determine whether each statement is true or false. Explain your thinking.

a

b

A known solution can be checked by substituting it into the equation.

No two equations have the same solution.


1.03 Equations with

variables on both sides

Concept summary

The first step in solving equations with variables on both sides is usually to move all variable terms

to one side of the equation, by applying the properties of equality to variable terms.

A fully simplified equation in one variable will take one of the following three forms, corresponding

to how many solutions the equation has:

• x = a, where a is a number ( a unique solution )

• a = a, where a is a number ( infinitely many solutions )

• a = b, where a and b are different numbers ( no solutions )

An equation of the second form, which is true for any possible value of the variable( s ), is

sometimes called an identity.

Worked examples

Example 1

Determine how many solutions the following equations have:

a 3( −8 + x ) = 3( −8 + x )

Solution

3( −8 + x ) = 3( −8 + x )

−24 + 3x = −24 + 3x Distribute the 3

−24 = −24

Subtract 3x from both sides

Since the equation is of the form a = a, where a is a number, there are infinitely many solutions.

14


b

Solution

9 + x = x + 5 Multiply both sides by 9

9 = 5 Subtract x from both sides

Since the equation is of the form a = b, where a and b are different numbers, there are

no solutions.

c 5( 7 + x ) = 2x + 85

Solution

5( 7 + x ) = 2x + 85

35 + 5x = 2x + 85 Distribute the 5

35 + 3x = 85 Subtract 2x from both sides

This equation will simplify to one of the form x = a, where a is a number, so there is a unique solution.

Reflection

We did not have to solve the equation to determine that it has a unique solution - we only needed

to check that the variable terms did not cancel each other out, like they did in part ( a ) and part ( b ).

Example 2

Solve the following equation: 4( x − 9 ) = x + 6

Solution

4( x − 9 ) = x + 6

4x − 36 = x + 6 Distribute the 4

3x − 36 = 6

Subtract x from both sides

3x = 42

Add 36 to both sides

x = 14 Divide both sides by 3

Reflection

We can check if x = 14 is correct by substituting 14 in for x into each side and checking that they

are equal:

LHS = 4( 14 − 9 ) = 4 ⋅ 5 = 20

RHS = 14 + 6 = 20


Example 3

Right now, Bianca’s father is 48 years older than Bianca. 2 years ago, her father was 5 times older

than her. Solve for y, Bianca’s current age.

Approach

We want to write expressions that represent Bianca’s age and her father’s age. Then we relate

them with an equation and solve for y.

Bianca’s father is currently y + 48 years old.

Two years ago, Bianca was y − 2 years old. Her father was y + 48 − 2 = y + 46 years old.

Her father’s age was five times her age at this time, which produces our equation:

y + 46 = 5( y − 2 )

Solution

y + 46 = 5( y − 2 )

y + 46 = 5y − 10 Distribute the 5

y + 56 = 5y

Add 10 to both sides

56 = 4y Subtract y from both sides

14 = y Divide both sides by 4

Bianca is currently 14 years old.



a m + m + 3 + 12 = 13 + m + m + m b x + x + x + x + 2 + 5 = x + 4 + x + x

c p + p − 3 + 5 = p − p − p + 1 − 5 d −k − k − k − 6 = −k − k − k − k + 6

2 Determine the number of solutions of the following equations:

a 7( 7 + x ) = 2x + 28 b 10( 5 + x ) = 10( x + 5 )

c d −31 − 3( x − 9 ) = 2( x − 2 ) − 5x

3 For each of the following relations:

i Write an equation for the relation, using n to represent the unknown number.

ii Determine the number of solutions to the equation.

a

b

c

Three more than three times a number is equal to six more than four times the number.

Six more than two times a number is equal to five more than two times the number.

Two more than four times a number is equal to two added to quadruple the number.

16


Let’s practice


a 3f − 8 = f b 10r + 4 = 6r

c 4w + 24 = w + 15 d −72 − 9p = −32 − p


a 4( x − 15 ) = x b 5( x − 16 ) = x − 16

c 3( x − 3 ) = 2( x + 1) d 2( 3x − 4 ) = 2( x + 4 )

e 3y = −13 − 4( 2y + 5 ) f x − 6 − 7( x + 3 ) = 21 − 9( x − 18 )


a

b

c

d

7 Consider the given rectangle with a perimeter of 126 + 3y centimeters.

4y + 8

4y + 3

Find the value of y.

8 A Payroll Officer has been told to distribute a bonus to the employees of a company worth

15% of the company’s net income. Since the bonus is an expense to the company, it must be

subtracted from the income to determine the net income. If the company has an income of

$120 000 before the bonus, then the Payroll Officer must solve:

for B. Find the bonus, B.

B = 0.15( 120 000 − B )

9 Right now, Skye’s father is 36 years older than Skye. 4 years ago, her father was 4 times as

old as her. Solve for y, Skye’s current age.

10 Xanthe, a tennis player, has won 69 out of 96 matches in her career. Find x, the number of

matches she must win in a row to raise her win percentage to 75%.



11 Three consecutive integers are such that the sum of the first and twice the second is 15

more than twice the third.

a Let x be the smallest of the integers. Find x.

b

c

State the three consecutive integers.

Are there any other sets of three consecutive integers that could fulfill the requirements?

Explain how you know.

12 Determine whether the following statements are true or false. Explain your thinking.

a

b

Any equation with variables on both sides must have multiple solutions.

Two equations will always have the same solution if one is a multiple of the other.

18


1.04 Literal equations

Concept summary

Some equations involve more than one variable. These are usually still referred to as just

equations, but sometimes they are given other names.

Literal equation

An equation that involves two or more variables

Example: x = 5y + 3z

We can use the properties of equality to isolate any variable in a literal equation.

Formula

A type of literal equation that describes a relationship between certain quantites

Example: A = l ⋅ w where A is area, l is length, and w is width.

The same variable might be used to represent different quantities acorss different formulas.

For example, in the formula for the area of a rectangle w is used to represent the width of the

rectangle, however in another context w might be used to represent a weight or other value.

Worked examples

Example 1

Solve for x in the following equation:

y = 5( 1 + x )

Approach

We need to rearrange the equation to isolate x. We can use the properties of equality and inverse

operations to solve literal equations for a variable, just as we would for linear equations.

Solution

y = 5( 1 + x )

Divide both sides of the equation by 5




Reflection

When rearranging an equation, we reverse the operations acting on the variable we want to isolate

in the reverse order of operations. Whatever is done to one side of the equation, must be done to

the other to keep the equation balanced.

Example 2

Given the formula for Ohm’s law:

V = IR

where V is voltage, I is current and R is resistance.

Write the formula for current.

Approach

The formula for current is Ohm's law with I isolated. We use inverse operations and properties of

equality to get the solution.

Solution

V = IR

Divide both sides of the equation by R



1 What benefits come from rewriting literal equations?

2 Solve for x in the following equations:

a y = x + z b c y = −8( 7 + x )

d e f kx = m + nx

3 Solve for m in the following equations:

a b y = 9mx − 5 c x = −2k( n + m ) d x = 3k( n + m )

4 Solve for

5 Solve for R : V = IR − E

20


Let’s practice

6 The formula for the force of an object is F = ma, where F is the force, m is the mass of the

object, and a is the object’s acceleration. Write an equation that could be used to solve for

the mass of the object.

7 The formula for the perimeter of a rectangle is P = 2l + 2w, where P is the perimeter, l is

the length, and w is the width. Write an equation that could be used to find the length of the

rectangle.

8 Betsy knows the formula to convert Celsius into Fahrenheit is She wants to

convert a temperature from Fahrenheit to Celcius. Find an equation that she could use to do

this conversion.

9 The displacement of an object is given by Write an equation that could be

used to find v 0 .

10 The bend allowance for sheet metal is given by the formula Write an

equation that could be used to solve for the thickness of the sheet, T.

11 The formula to the distance traveled in a time period is d = rt, where d is the distance

traveled, r is the rate of travel, and t is the time spent traveling.

a

b

Write an equation that could be used to solve for the rate of travel.

Floyd is riding his bike 5 miles to Jue’s house. Jue is expecting Floyd in 15 minutes.

Find the average speed Floyd must ride in order to get to Jue’s on time. Give your

answer in mi/h.


12 Determine the similarities and differences of solving 3x + y = z and 3x + 8 = 2. Explain your

thinking.

13 Rufino has submitted the following work to write the equation for the mass of an object, m,

given the kinetic energy, KE, and velocity, v:

Determine whether Rufino is correct by explaining each step of his working if the step is

correct, otherwise fix his error.


1.05 Absolute value

equations

Concept summary

The absolute value of a number is the distance of that number from 0 on a number line. Distance is

expressed as a positive value, and so the absolute value of a number is always positive. Absolute

value is sometimes called magnitude.

To denote the absolute value of a quantity, we use a single vertical bar on either side. The absolute

value of x can be thought of as:

• If x ≥ 0, then | x | = x

• If x < 0, then | x | = -x

Absolute value equation

An equation that contains a variable expression inside absolute value bars.

Example: | x | = 4

Extraneous solution

A solution of an equation that emerges from the process of solving the problem, but is not a

valid solution to the problem.

An absolute value equation can be solved by applying arithmetic operations as usual, but

sometimes one or more solutions to an absolute value equation will be extraneous solutions

depending on the context.

Worked examples

Example 1

Solve each absolute value equation.

a |x | = 6

Solution

The solutions to this equation will be any values of x that have an absolute value of 6.

The solutions are x = 6 and x = −6.

22


Reflection

Notice that this equation has different solutions from the equation | x − 6 | = 0 which has only a

single solution of x = 6. This shows that adding or subtracting terms in or out of the absolute value

bars changes the equation, similar to parentheses.

b |2x | = 4

Solution

The solutions to this equation will be any values of x that make 2x have an absolute value of 4.

The solutions are x = 2 and x = −2.

Reflection

Notice that this equation has the same solutions as the equation 2|x | = 4. This demonstrates that

multiplying or dividing a constant term in or out of the absolute value bars does not change the

equation.

c |3x | + 4 = 2x + 9

Approach

This equation is harder to solve because it involves x-terms both inside and outside absolute value

bars. One way to work around this is to split up the problem into two cases: when 3x ≥ 0, and when

3x < 0.

Solution

When 3x ≥ 0, we know that | 3x | = 3x, so we can write and solve the equation:

|3x | + 4 = 2x + 9

3x + 4 = 2x + 9 Take the case where 3x ≥ 0

x + 4 = 9

x = 5


Subtract 4 from both sides

When 3x < 0, we know that | 3x | = −3x, we can write and solve the equation:

|3x | + 4 = 2x + 9

−3x + 4 = 2x + 9 Take the case where 3x < 0

4 = 5x + 9 Add 3x to both sides

−5 = 5x


−1 = x Divide both sides by 5

Using two cases, we have found two solutions: x = 5 and x = −1.


We can substitute each solution into the starting equation to check that they are valid.

|3( 5 ) | + 4 = 2( 5 ) + 9

15 + 4 = 10 + 9

19 = 19

and

|3( −1) | + 4 = 2( −1) + 9

3 + 4 = −2 + 9

7 = 7

Reflection

If substituting either solution into the starting equation resulted in an untrue equation, then that

solution would be extraneous.

Example 2

Clay wants to find the point in a 100 yard dash where his distance in yards from the middle of the

race is equal to 40 less than twice the number of yards he has run so far.

Find the number of yards that Clay can have run to make this equality true.

Approach

The varying factor in this context is the number of yards that Clay has run. We can represent this

with the variable y.

Clay’s distance from the middle of the race can be described as the difference between the

number of yards he has run, y, and 50 yards. We can write this as the absolute value expression

| y − 50|.

Since y represents how far Clay has run in the race, we can express “40 less than twice the yards

he has run so far” as the expression 2y − 40.

Putting these two expressions together, we get the equation that Clay wants to solve:

Solution

|y − 50 | = 2y − 40

To solve this equation, we can use the two cases method.

Taking the case where y − 50 ≥ 0, we have:

24


Taking the case where y − 50 < 0, we have:

Solving using the two cases method gives us the possible solutions y = −10 and y = 30. We can

see that if we substitute y = −10 into our starting equation we will get

|−10 − 50 | = 2( −10 ) − 40

which is not true. This tells us that y = −10 is an extraneous solution.

We can also see that if we substitute y = 30 into our starting equation we will get

which is true.

|30 − 50 | = 2( 30 ) − 40

So Clay must run exactly 30 yards so that his distance from the middle of the race is equal to

40 less than twice the number of yards he has run.

Reflection

Notice also that the solution y = −10 is made non-valid by the context, since Clay cannot run a

negative number of yards.


1 Solve each of the following equations:

a |x | = 8 b 9 | x | = 4 c −3 | x | = −12

d |x + 3 | = 7 e |1 − x | = 3 f

g |2x | − 5 = 4 h |−3.9x | + 3 = 22.5 i |3x + 6 | = 9

j −3 | x + 5 | = −9 k 7 − | 4x | = 5 l |4x − 8 | + 1 = 13

m |2x + 9 | = x + 6 n |3x | + 2 = 2x + 7 o −2 | x + 3 | = −14

p q −6 | 4x − 1| = −12 r

2 Solve 3 | x + 2 | − 1 = 3x + 11 and state how many solutions it has.


Let’s practice

3 For each of the following, solve the absolute value equation and then check each answer to

determine which solutions are extraneous and which are valid:

a |x + 6 | = 2x b |3x + 2 | = 4x + 5

c |x − 1| = 5x + 10 d −2 | x | = 6 − 2x

4 Write 5x − 2 = 3 or 5x − 2 = −3 as an equivalent absolute value equation.

5 Write an absolute value equation that represents each of the following:

a All real numbers x that are 8 units away from 0.

b All real numbers x that are 5 units away from −2.

6 The minimum and maximum lengths of AA batteries, b in millimeters, can be represented

with the absolute value equation | b − 49.9 | = 0.6.

Find the minimum and maximum lengths in millimeters of a AA battery.

7 Baby clothing is rated using a TOG rating for what temperatures it is designed for to avoid

the baby being too cold or overheating. A sleep sack with rating 2.5 TOG is designed for

65°F, plus or minus 4°F.

Write an absolute value equation that represents the minimum and maximum temperatures

for that sleep sack.

8 Compare and contrast the process for solving each equation and state their solutions.

• Equation A: | 3x | + 4 = 5

• Equation B: | 3x + 4 | = 5

9 Compare and contrast the process for solving each equation and state their solutions.

Assume a, b > 0.

• Equation A: | ax | = b

• Equation B: a | x | = b

10 Determine the value( s ) of b for which the equation | x | = b has only one solution.

26



11 Describe why the absolute value of a number cannot be negative. Give an example of an

absolute value equation with no solution.

12 Clem and Finn are waiting in line at a petting zoo. Clem is 6 spots in line away from Finn.

Consider an absolute value equation that represents Clem’s possible position, compared to

Finn’s position.

a

b

c

Describe the positions for Finn where there are two possible positions for Clem.

Describe the positions for Finn where there is only one possible position for Clem.

Describe the positions for Finn where there is no possible position for Clem.

13 Roxanne performed the following steps to solve the equation 4 | x | = −2:

1 4 | x | = −2

2 4x = ±2

3

Show that neither of these solutions is valid and explain where her error was.

14 Kaydence is running to meet her friend at the local market. She knows that her average

speed is 7.5 mph ( 0.125 miles per minute ). She tells her friend that she will be there

in 30 minutes, plus or minus 6 minutes depending on the route she ends up taking.

Remember that:

Write and solve an equation for the minimum and maximum distances for her run.


1.06 Multi-step

inequalities

Concept summary

Some mathematical relations compare two non-equivalent expressions - these are known as

inequalities.

We can solve inequalities by using various properties to isolate the variable, in a similar way to

solving equations:

Asymmetric property of inequality

If a > b, then b < a

Transitive property of inequality

If a > b and b > c, then a > c

Addition property of inequality

If a > b, then a + c > b + c

Subtraction property of inequality

If a > b, then a − c > b − c

Multiplication property of inequality

If a > b and c > 0, then a ⋅ c > b ⋅ c

If a > b and c < 0, then a ⋅ c < b ⋅ c

Division property of inequality

If a > b and c > 0, then a ÷ c > b ÷ c.

If a > b and c < 0, then a ÷ c < b ÷ c.

When multiplying or dividing an inequality by a negative value the inequality symbol is reversed.

28


Linear inequality

An inequality that contains a variable term with an exponent of 1, and no variable terms with

exponents other than 1

Example: 3x − 5 ≥ 4

Solving an inequality using the above properties of inequalities results in a solution set.

Solution set

The set of all values that make the inequality or equation true

Example: 3 ≤ x

We can represent solutions to inequalities algebraically, by using numbers, letters, and/or symbols,

or graphically, by using a coordinate plane or number line.

An algebraic solution to an inequality can be represented as an inequality, such as 3 ≤ x, or in other

ways, including interval notation and set-builder notation.

Interval

A set of numbers that lie between two values

Interval notation

A way to represent a solution set or interval as a pair of numbers using a combination of

square brackets and parentheses.

Example: Inequality solution 3 ≤ x

Interval notation [3, ∞ )

We use square brackets if the endpoint is included and parentheses if the endpoint is not included.

We always use parenthese for infinity. We can join two sets together using the union symbol ∪. We

may see x ∈ which says “x is in”.

Set-builder notation

A shorthand used to write sets. The set {x ∈ |x > 0} is read as “the set of all real values of x

such that x is greater than zero”. Sometimes a colon : is used instead of a vertical line | .

Example: Inequality solution 3 ≤ x Set-builder notation {x ∈ |x ≥ 3}

• Inequality notation: −3 ≤ x < 4

• Interval notation: [−3, 4 )

• Set-builder notation: {x ∈ | − 3 ≤ x < 4}

−5 −4 −3 −2 −1 0 1 2 3 4 5


• Inequality notation: x ≥ −3

• Interval notation: [−3, ∞ )

• Set-builder notation: {x ∈ | x ≥ −3

−5 −4 −3 −2 −1 0 1 2 3 4 5

Two inequalities that have the same set of solutions are called equivalent inequalities.

Worked examples

Example 1

Consider the inequality

a Solve the inequality

Approach

We want to isolate x on one side of the inequality and a number on the other.

Solution

Reflection

Multiply both sides of the inequality by 2

Add 8 to both sides.

Divide both sides by −3. Reverse the direction of the

inequality symbol since we are dividing by a negative

number.

Solving an inequality is similar to solving an equation. However, we need to reverse the direction of

the inequality when multiplying or dividing by a negative number.

b Plot the inequality on a number line.

Solution

Plot the solution set of the inequality x ≥ −6. Note that since we include −6 the point should be filled.

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Reflection

What if the solution was x > −6?

End points included in the interval are filled points.

End points not included in the interval are unfilled points.

30


Example 2

Oprah charges $37.72 to style hair, as well as $6 per foil. Pauline would like a style and foils,

but has no more than $95.86 to spend.

a Write an inequality that represents the number of foils Pauline could get.

Approach

Pauline would like a style which is $37.72 and some foils at $6 each. If the number of foils Pauline

can get is N, then we can write an expression that represents the cost to style hair as 6N + $37.72.

Pauline has no more than $95.86 to spend. Remember that “no more than” means “less than or

equal to.”

Solution

6N + 37.72 ≤ 95.86

b Write the solution set to the inequality.

Approach

Solve the inequality and then write the solution set.

Solution

37.72 + 6N ≤ 95.86

6N ≤ 58.14 Subtract 37.72 from both sides of the inequality

N ≤ 9.69 Divide both sides by 6

Inequality notation: N ≤ 9.69

Reflection

We could also write this in interval notation as ( −∞, 9.69].

c Determine the solution set in the context of the question.

Approach

We need to work out what the solution means in context.

Solution

Pauline cannot get part of a foil, so she can get a maximum of 9 foils and a minimum of 0 foils.

In context, the solution set contains the following numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 because

Pauline can get any number between 0 and 9 foils.


Reflection

Sometimes the context restricts the set of solutions.


1 Explain the differences between the inequalities: 0 ≤ x < 4 and 3 ≤ x ≤ 7.

2 Plot the following inequalities on a number line:

a x > −9 b x ≤ 15

3 Describe the range of values that satisfy each inequality.

a x ≥ 29 b x < −29

4 Which of the following inequalities represents the solution for x in 10 ≤ 6 − 4x?

A x ≤ −4 B x ≥ −1 C x ≤ 1 D x ≥ 4

5 Which of the following number lines represent the solution for 4x − 7 < 5?

A

−5 −4 −3 −2 −1 0 1 2 3 4 5

B

−5 −4 −3 −2 −1 0 1 2 3 4 5

C

−5 −4 −3 −2 −1 0 1 2 3 4 5

D

−5 −4 −3 −2 −1 0 1 2 3 4 5

6 Which of the following inequalities represents the solution for r in “5 more than 2r is less

than 39”?

A r > 17 B r < 17 C r > 22 D r < 22

7 To ride the ferris wheel in Fantasy World, a child must be at least 47 inches tall and less than

64 inches tall. Which of the following number lines represent these conditions?

A

44 46 48 50 52 54 56 58 60 62 64 66

B

44 46 48 50 52 54 56 58 60 62 64 66

C

44 46 48 50 52 54 56 58 60 62 64 66

D

44 46 48 50 52 54 56 58 60 62 64 66

Let’s practice

8 Consider the inequality: 5( x + 3 ) ≤ 35.

a Solve for x.

b State whether the following values satisfy the inequality:

i x = −4 ii x = 8 iii x = 4 iv x = 0

32


9 For the following inequalities:

i Solve for x. ii Plot the solutions on a number line.

a 3x − 7 < 8 b 4 < 6x − 2 c −6x − 7 ≤ 5 d 2( x − 3 ) < −16

10 Consider the inequality: 4 − 2x < 3x − 2.

a Solve for x.

b

State whether the following values satisfy the inequality:

i ii x = 1 iii iv x = 2

11 Solve the following inequalities:

a 5x − 40 ≥ 50 b −8 − m > 3 c −8( x + 8 ) ≥ −40

d e f

12 Consider the situation: “3 less than 3 groups of p is no more than 24”.

a Solve the inequality. b Find the largest value p can take.

13 Skye has a budget for school stationary of $43, but has already spent $21.14 on books and

folders. Let p represent the amount that Skye can spend on other stationery. Solve for p.

14 When breeding certain types of fish it is recommended that the number of female fish is

more than double the number of male fish.

a

b

Write the inequality for the recommended relationship. Let f be the number of female fish

and m be the number of male fish.

State whether the following combinations align with the recommendation:

i f = 10, m = 8 ii f = 23, m = 9

iii f = 13, m = 10 iv f = 7, m = 4

15 James is saving up to buy a laptop that is selling for $550. He has $410 in his bank account

and expects a nice sum of money for his birthday next month.

a

b

Write the inequality that models the situation in which James can afford the laptop. Let x

represents the amount he is to receive for his birthday.

Plot the solution to the inequality on a numberline.

16 To get a grade of C, Uther must obtain a total score of at least 300 over his four exams. So

far he has taken the first three exams and achieved scores of 68, 60, and 86.

a

b

Solve for x, the score on his last exam to get a C or better.

Describe the solution regarding Uther’s score.



17 Ryan wants to save up enough money so that he can buy a new sports equipment set, which

costs $40.00. Ryan has $22.10 that he saved from his birthday. In order to make more

money, he plans to wash neighbors’ windows for $2 per window.

a

b

Let w be the number of windows that Ryan washes. Solve for w, correct to two decimal

places.

State whether each statement is correct. Explain your thinking.

i

ii

iii

iv

Ryan must wash more than 9 windows to be able to afford the equipment.

Ryan must wash at least 8 windows to be able to afford the equipment.

If Ryan washed 8 windows, and 95% of another window, he could afford the equipment.

The number of windows Ryan must wash to be able to afford the equipment must be

greater than or equal to 9.

18 Rochelle tried to solve the following inequality but made a mistake in her work:

Step 0: −4 − 2x > 10

Step 1: −2x > 14

Step 2: x > −7

Determine which step is incorrect and explain the error.

19 Percy tried to plot the solution to the inequality 4x + 28 ≥ −8 on a number line, however, his

answer is incorrect.

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Identify the errors and explain how to rectify them.

34


1.07 Compound

inequalities

Concept summary

A compound inequality is a conjuction of two or more inequalities. The set of solutions for a

compound inequality are the values which make all of the inequalities true.

• We use “and” to indicate that a value must

satisfy both inequalities in order to be in the

solution set. For example:

−5 −4 −3 −2 −1 0 1 2 3 4 5

x < 3 and x ≥ −2

We can also write this compound inequality

more simply as −2 ≤ x < 3

• We use “or” to indicate that a value need only

satisfy at least one inequality in order to be in

the solution set. For example:

−5 −4 −3 −2 −1 0 1 2 3 4 5

x > 3 or x ≤ −2

Worked examples

Example 1

Write a compound inequality to represent the solution set shown on the number line.

−5 −4 −3 −2 −1 0 1 2 3 4 5

Approach

We want to describe the compound inequality then write the solution to the compound inequality

algebraically.

Description: x is less than −1 or x is greater than or equal to 2

Solution

Compound inequality: x < −1 or x ≥ 2


Example 2

Find the solution set of the compound inequality x ≥ −5 and x < 3 using set-builder notation and

plot the solution on a number line.

Approach

1. Plot x ≥ −5 on a number line.

2. Plot x < 3 on a number line.

3. Find the solution of the compound inequality by comparing the number lines.

4. Represent the solution graphically.

5. Represent the solution using set-builder notation.

Solution

Plotting x ≥ −5:

The minimum value, −5, is included in the interval and

is represented by a filled circle.

Plotting x < 3:

The maximum value, 3, is not included in the interval

and is represented by a unfilled circle.

Finding the solution to the compound inequality:

We want the solution to both x ≥ −5 and x < 3. The solution to the compound inequality will be the

values greater than or equal to −5 and less than 3.

Solution represented graphically on a number line:

−6 −4 −2 0 2 4 6

−6 −4 −2 0 2 4 6

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Solution represented algebraically using set-builder notation: {x ∈ | − 5 ≤ x < 3}

Reflection

We can check our solution( s ) are correct by taking a number in our solution set and seeing if

it works in both inequalities.

For example, we can see that 0 falls within the region shown on the number line,

and −5 ≤ 0 < 3 as required.

Example 3

The formula for converting temperatures from Celsius to Fahrenheit is:

During a recent year, the average temperatures in Tampa, Florida ranged from 59° to 95° Fahrenheit.

Write a compound inequality to solve for the corresponding range of values of C, the temperature

in Florida in degrees Celcius.

36


Approach

The degrees in Fahrenheit must be between 59 and 95 inclusive, so the minimum is 59 and

the maximum is 95. The middle part of our compound inequality is going to be the formula for

converting temperatures.

Once we have the compound inequality we can solve it.

Solution

Subtract 32 from each section of the inequality

Multiply each section of the inequality by 5

Divide each section of the inequality by 9

The average temperatures in Tampa ranged from 15° to 35° Celcius.

Reflection

When solving a compound inequality, whatever is done to one part of the inequality must be done

to all parts.


1 For each pair of inequalities, plot the solution to the compound inequality that is Inequality 1

and Inequality 2 on a number line.

a Inequality 1: x ≥ 2 b Inequality 1: x > 3

Inequality 2: x ≤ 7 Inequality 2: x ≤ 7

2 For each pair of inequalities, plot the solution to the compound inequality that is Inequality 1

or Inequality 2 on a number line.

a Inequality 1: x ≤ 3 b Inequality 1: x < 8

Inequality 2: x > 8 Inequality 2: x > 3

3 For the following compound inequalities:

i

ii

iii

Rewrite the compound inequality as a pair of inequalities joined by either and or or.

Plot the solution to the compound inequality on a number line.

Write the solution in interval notation.

a −5 < x < 8 b −7 < x ≤ 5

4 Which of the following sets of inequalities represents the solution for x in 1 ≥ x − 4 > −6?

A x > −2 and x ≥ 5 B x > −2 or x ≥ 5

C x > −2 and x ≤ 5 D x > −2 or x ≤ 5


5 Freshwater temperature can affect the development of various fish species including salmon.

The optimum temperature for salmon ranges from 33°F to 68°F.

Which of the following sets of inequalities represent the temperatures where salmon will

thrive?

A T > 33 or T < 68 B T > 33 and T < 68

C T < 33 and T > 68 D T < 33 or T > 68

6 Which of the following number lines represents the solution for m + 2 < −5 or m − 2 ≥ −5?

A

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

B

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

C

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

D

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

7 Which of the following number lines represents the solution for 9 − 2p > −6 and 12 < 3p?

A

3 4 5 6 7 8 9

B

3 4 5 6 7 8 9

C

3 4 5 6 7 8 9

D

3 4 5 6 7 8 9

Let’s practice

8 For each pair of inequalities, state the solution to the compound inequality that is Inequality

1 and Inequality 2.

a Inequality 1: −3x + 2 ≤ −13 b Inequality 1: 4x + 3 ≥ −17

Inequality 2: −4x − 2 ≥ 26 Inequality 2: 3x − 4 < 5

9 For each pair of inequalities, state the solution to the compound inequality that is Inequality

1 or Inequality 2.

a Inequality 1: 5x − 7 < −22 b Inequality 1: −6x + 8 < 26

Inequality 2: −6x + 8 < −28 Inequality 2: 5x − 4 ≤ 26

10 For the following compound inequalities:

i Solve for x. ii Write the solution in interval notation.

a 16 < 3x + 7 ≤ 37 b −52 < −5x + 3 ≤ −22

c d 8 < 2x − 5( x − 7 ) < 41

e

38


11 For each of the following scenarios, state whether the compound inequality would use and

or or:

a

Francesco solves a compound inequality and expresses the solution in interval notation

as ( −∞, ∞ ).

b Zenaida solves a compound inequality and find that there are no possible values for x.

12 The velocity, in feet per second, of a tennis ball t seconds after it is projected directly upward

is v = 85 − 32t.

Solve for the range of time t at which the ball will be faster than 37 ft/sec but slower than

69 ft/sec.

13 The formula for converting temperatures from Fahrenheit to Celsius is

During a recent year, the temperatures in Moscow ranged from −20°C to 30°C.

Solve for the corresponding range of values of F.

14 To get a final grade of B, a student must have an average mark on five tests that is greater

than or equal to 80 and less than 90. Maria’s grades on the first four tests were 96, 87, 78

and 94.

Solve for the range of grades x that she could score on the fifth test that would result in her

getting a final grade of B.


15 Dora solved a compound inequality and expressed their solution as the interval ( 5, 13].

a

b

Rewrite this solution as a pair of inequalities joined by either and or or.

Write a compound inequality which Dora could have solved, using at least two steps, to

produce this solution.

16 Carlton tried to solve the following compound inequality, but made a mistake in his work:

Step 0: −8 ≤ 10 − 3x < 4

Step 1: −18 ≤

−3x < −6

Step 2: 6 ≤ 3x < 18

Step 3: 2 ≤ x < 6

Determine which step is incorrect and explain the error.

17 Skyler tried to plot the solution to the compound inequality 4x − 1 ≤ 11 and

on a number line, however their answer is incorrect.

−5 −4 −3 −2 −1 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Identify the error( s ) and explain how to rectify them.


1.08 Honors: Absolute

value inequalities

Concept summary

The absolute value of a number is a measure of the size of a number, and is equal to its

distance from zero (0), which is always a non-negative value. Absolute value is sometimes

called “magnitude”.

An absolute value inequality is an inequality containing the absolute value of one more

variable expressions.

Inequality

A mathematical relation that compares two non-equivalent expressions

Solutions to absolute value inequalities usually involve multiple inequalities joined by one of the

keywords “and” or “or”. Solutions with two overlapping regions joined by “and” can be rewritten as

a single compound inequality:

| x | ≥ 1 has solutions of x ≤ −1 or x ≥ 1 or in interval notation (−∞, −1] ∪ [−1, ∞).

| x | < 3 has solutions of x > −3 and x < 3 which is equivalent to −3 < x < 3 or in interval

notation (−3, 3).

In general, for an algebraic expression p(x) and k > 0, we have:

• | p(x)| < k can be written as −k < p(x) < k

• | p(x)| ≤ k can be written as −k ≤ p(x) ≤ k

• | p(x)| > k can be written as p(x) < −k or p(x) > k

• | p(x)| ≥ k can be written as p(x) ≤ −k or p(x) ≥ k

Solutions to absolute value inequalities can also be represented graphically using number lines:

−4 −3 −2 −1 0 1 2 3 4

| x | ≥ 1

−4 −3 −2 −1 0 1 2 3 4

| x | < 3

40


Worked examples

Example 1

Consider the inequality | x | > 2:

a Represent the inequality | x | > 2 on a number line.

Approach

This inequality represents values of x which are “more than 2 units away from 0”. To plot this,

we will need to use two regions. Also note that this inequality doesn’t include the endpoints,

so we will use unfilled points to show this.

Solution

−5 −4 −3 −2 −1 0 1 2 3 4 5

Reflection

We can give a quick check of the answer by thinking about whether this is an “and”-type inequality

or an “or”-type inequality.

In this case, the inequality | x | > 2 has a solution of x < −2 or x > 2. Our solution on the numberline

has two distinct parts, which matches what we expect for an “or”-type inequality.

b Rewrite the solution to | x | > 2 in interval notation.

Approach

We can use either the number line or x < −2 or x > 2 to help write this in interval notation.

Solution

The x < −2 part can be written as (−∞, −2).

The x > 2 part can be written as (−2, −∞).

Since this is an “or”, not “and”, we will find the union of these two sets to give:

(−∞, −2) ∪ (−2, −∞)

Example 2

Consider the inequality |4x − 5| ≤ 3.

a Solve the inequality for x. Express your solution using interval notation.


Approach

In order to solve this inequality, it will be easier to first remove the absolute value by rewriting the

inequality as a compound inequality. We can then solve as normal.

Solution

Rewrite as a compound inequality

Add 5 to each part

Divide throughout by 4

So the solutions are “all values of x between

notation as the interval

and 2 inclusive”. We can express this using interval

b Represent the solution set on a number line.

Approach

The solution set for this inequality consists of a single interval, so it will only need one region on

the numberline. The endpoints are also included this time, which we represent using filled points.

Solution

−4 −3 −2 −1 0 1 2 3 4

Reflection

We can give a quick check of the answer by thinking about whether this is an “and”-type inequality

or an “or”-type inequality.

In this case, we found the solution to the inequality |4x − 5| ≤ 3 to be the compound inequality

≤ x ≤ 2. A compound inequality is a way of writing an “and”-type inequality, which matches the

fact that the numberline has one region between two endpoints.

Example 3

Write an absolute value inequality to represent the set of “all real numbers x which are at least

5 units away from 12”.

42


Approach

We want to break down the wording of the set into its key parts. These key parts are:

• at least

• away from 12

• 5 units away

Each of these key parts tells us information about the form of the inequality.

Solution

Let’s interpret these key parts:

• “At least” is another way of saying “greater than or equal to”, which tells us the inequality

symbol that we want to use.

• “Away from 12” means that we want to compare the distance between x and 12. We can

represent this as |x − 12|.

• “5 units away from” means that the absolute value will be compared to 5.

Putting this all together, we get “the distance between x and 12 is greater than or equal to 5”,

which we can represent as the absolute value inequality

| x − 12 | ≥ 5


1 Rewrite the following inequalities without using absolute values:

a | x | < 2 b | x | ≥ 5

c | x | > 13 d | x | ≤ k, for a positive value of k

2 Represent the following inequalities on a number line:

a | x | > 6 b | x | ≠ 3 c | x | < 7

d | x | ≤ 9 e | x | ≥ 5 f | x | > −5

3 Rewrite the following as absolute value inequalities:

a −7 ≤ x ≤ 7 b x < −8 or x > 8 c −0.6 ≤ 2x + 3 ≤ 0.6

4 Solve the following inequalities:

a | x − 7| ≥ 2 b | x − 4| ≤ 9

c | 2x + 3 | ≤ 5 d 2 + | x | ≥ 9

e 4 | x | − 5 < 19 f |11 − 2x | > 5

g 3 | x − 7 | − 12 ≥ 0 h 2 |4.5 − x | > x

i | 3x + 4 | ≥ x + 8 j


5 For each of the following inequalities:

i Express the solution using interval notation.

ii Represent the solution on the number line.

a | x | ≤ 4 b | x | > 1 c | x − 5 | ≥ 2 d | 2x | < 10

e 3 | x − 1| ≥ 9 f g |1 + 4x | < 5 h | 5x + 4| + 3 > 2

Let’s practice

6 Write an absolute value inequality that represents each of the following:

a All real numbers x that are less than 8 units away from 0.

b All real numbers x that are more than 8 units away from 0.

c All real numbers x that are at least 2 units away from 8.

d All real numbers x that are at most 5 units away from −2.

7 In a certain company, the measured thickness, m, of a helicopter blade must not differ from

the standard, s, by more than 0.17 millimeters. The manufacturing engineer expresses this

as the inequality | m − s | ≤ 0.17.

Find the range of values that m can take if s is 17.92 millimeters.

8 Cell phone cases have dimension requirements to ensure the phone will fit properly in the

case. The manufacturing engineer has written a specification that the new length, n, of the

case can differ from the previous length, p, by at most 0.04 centimeters.

Find the range of values for the new length of a cell phone case if the previous length was

18.9 centimeters.

9 If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be

heads. A coin is deemed to be unfair if h, the number of outcomes that result in heads,

satisfies

a

b

Solve the inequality.

State the largest and smallest integer values for the number of heads outcomes that

could occur without the coin being labeled as unfair.

10 For each of the following, write two distinct absolute value inequalities that have the

given solution:

a −2 < x < 2 b x ≤ −9 or x ≥ 9

c 1 ≤ x ≤ 7 d x < −4 or x > 0

e

g

f

−5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5

h

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 0 1 2 3 4 5 6 7 8 9 10

44



11 Use the graph of f (x) = | 2x − 4 | and g (x) = 8

to solve the following equation

and inequalities:

a | 2x − 4 | = 8

b | 2x − 4 | ≤ 8

c | 2x − 4 | > 8

14

12

10

8

6

4

2

−8 −6 −4 −2

−2

−4

y

2 4 6 8 10

x

12 Toya claims that −5 < x < −1.4 is the solution to 3 | x + 2 | > 2x + 1. Explain why her answer

is incorrect.

13 Determine the solution(s) to the inequality | 2x − 3 | ≤ 1 − x. Explain your answer.

14 The speedometer in Najah’s car is displaying 35 mph. They know that the speedometer

is accurate to within 10% of the actual speed.

a

b

Write an absolute value inequality to represent the situation.

The road Najah is driving down has a speed limit of 40 mph. Is it possible for Najah to be

driving over the speed limit?

15 Floyd measures the distance between two cities on a map, which has a scale factor of

1 : 50 000. He determines that the cities are approximately 16 inches apart, with a possible

error of measurement of up to half an inch.

a

b

If x represents the actual distance between the two cities, write an absolute value

inequality to represent the situation.

The actual distance between the two cities is 12.9 miles (to one decimal place).

Determine if Floyd’s measurement was accurate.

16 If x represents temperature in Miami, Florida, describe a possible interpretation of

the inequality | x − 84 | ≤ 7.


2

Equations and

Inequalities in

Two Variables

Chapter outline

2.01 Slope-intercept form 48

2.02 Standard form 60

2.03 Point-slope and connecting forms 68

2.04 Parallel and perpendicular lines 78

2.05 Linear inequalities in two variables 84

2.01 Slope-intercept form

Concept summary

The key features of a linear relationship help us to draw its graph, given an equation or table of

values, or to write the equation given a graph.

Graph

A diagram showing the relationship between two things

on a coordinate plane

y

x

Table of values

Numeric information arranged in columns and rows

The slope-intercept form of a line is:

y = mx + b

m The slope of the line

b The y-intercept of the line

A benefit of slope-intercept form is that we can easily identify two key features from the equation.

Rate of change

The ratio of change in one quantity to the corresponding change in another quantity

Slope

The ratio of change in the vertical direction (y-direction) to

change in the horizontal direction (x-direction).

To interpret the slope, it can be helpful to look at the units.

y

x

48


y-intercept

A point where a line or graph intersects the y-axis.

The value of x is 0 at this point, so it often represents

the initial value or flat fee.

y

x

x-intercept

A point where a line or graph intersects the x-axis.

The value of y is 0 at this point.

y

x

Domain constraint

A limitation or restriction of the possible x-values, usually written as an equation, inequality,

or in set-builder notation. The constraint often comes from a context or scenario.

Example: Inequality notation: −2 ≤ x < 3 Set-builder notation: {x ∈ ¡ | − 2 ≤ x < 3}

Worked examples

Example 1

Draw the graph of the line y = −2x + 5 using the slope and y-intercept.

Approach

We will:

1. Identify the slope and y-intercept from the equation

2. Plot the y-intercept

3. Use the rise and run to plot another point

4. Draw the graph of the line

Chapter 2 Equations and Inequalities in Two Variables 49

Solution

The y-intercept is (0, 5).

The slope is m = −2, so we can have:

rise = −2 and run = 1

This means that going down 2 units and right by

1 unit gives another point on the line.

7

6

5

4

3

2

y

y = −2x + 5

1

−4 −3 −2 −1 1 2 3 4

−1

x

Reflection

Sometimes we will see this equation written as y = 5 − 2x, but this does not change the graph.

Example 2

Find the equation of a line that has the same slope as

y = −7x − 9.

and the same y-intercept as

Approach

We will identify the slope of

y = mx + b.

and the y-intercept of y = −7x − 9 and then substitute into

Solution

The slope of

The y-intercept of y = −7x − 9 gives us b = −9.

Substituting in these values, we get:

Reflection

A common error would be to say that 2 is the slope of

because it is the first number

after the equals sign. Remember that the slope is the coefficient of the x-term.

50


Example 3

A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of

water. The table of values shows the linear relationship of the amount of water remaining in the

tub, y, after x minutes.

Time in minutes (x) 0 1 2 3

Water remaining in gallons (y) 30 28 26 24

a Determine the linear equation in slope-intercept form that represents this situation.

Approach

We can pick two points to calculate the rate of change for the slope. Then we can recognize that

the y-intercept is given in the table of values.

Solution

Find the slope using the values (0, 30) and (1, 28):

Notice that the initial value, or y-intercept is given in the table as (0, 30).

The equation that represents this situation is y = −2x + 30.

Reflection

If we had not noticed that the y-intercept was given, then we could have substituted in any pair of

values for x and y, and solved for b.

b Draw the graph of this linear relationship with the domain constraint of {x Î ¡ | 0 ≤ x ≤ 15}.

Solution

From the table, the point at x = 0 is (0, 30).

For the point at x = 15, we need to substitute to find the y value.

y = −2x + 30

y = −2 × 15 + 30

y = 0


The point at x = 15 is (15, 0)

y

30

25

20

15

10

5

x

5

10 15


1 For each of the following linear equations:

i State the value of m, the slope. ii State the value of b, the y-intercept.

a y = 6x + 4 b y = −3x − 8

2 For each of the following graphs:

i Determine the slope of the line. ii Determine the y-intercept of the line.

iii

Determine the equation of the line.

a

5

y

b

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5

52


Let’s practice

3 Consider the following linear equations:

• Line 1: y = 2 + 6x

• Line 2: y = 5x + 6

• Line 3: y = 6x

State the three lines that have the same slope.

• Line 4:

• Line 5: y = 5 − 6x

• Line 6: y = 6x − 2

4 For each of the following lines, state whether the following lines have a y-intercept or not.

If the line does have a y-intercept, state the coordinates of it.

a y = 2 b 5x = 4y c y = −4x + 1 d x = 1

5 Suppose the Line L 1 has the equation

a Simplify L 1 to the form y = mx + b.

b

c

State the slope of the line.

Find the y-value of the y-intercept of the line.

6 Determine whether each line has the same slope as y = 6x + 3.

a y = 6x b y = 3x + 3 c 12x − 2y = 3 d 18x + 3y = 0

7 Find the equation of the lines matching the following descriptions:

a A line that has the same slope as y = 2 − 4x and the same y-intercept as y = −7x − 3

b A line whose slope is and goes through the point (0, 5)

c A line that is horizontal and goes through the point (9, −2)

d A line that is vertical and goes through the point (11, −9)

8 Graph the following lines on a coordinate plane:

a The line that has a y-intercept of −4 and

whose slope is

b The line y = 2x + 4

c The line y = −2x + 4

6

4

y

d

e

The line

The line x = −3

−6 −4 −2

2

2 4 6

x

9 Select the linear equation that could

represent the following graph:

A y = 2x - 3 B y = −2x - 3

−2

−4

C

D

−6


10 Select the linear equation that could represent the following table of values:

x −2 −1 0 1 2 3

y −1 1 3 5 7 9

A y = 2x + 3 B y = −2x − 3 C D

11 A linear function is defined by the equation y = 4x − 3.

a

Select the graph that represents the linear function.

A

5

y

B

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5

C

5

y

D

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5

b

c

Identify the domain of the function.

Identify the range of the function.

54


12 Consider the following line:

a

Complete the table of values.

5

4

y

b

c

x -1 0 1 2

y

State the values of the slope, m and the

y-intercept, b.

Write the linear equation expressing the

relationship between x and y.

−5 −4 −3 −2

3

2

1

−1

−1

−2

−3

1 2 3 4 5

x

−4

−5


a

State the values of the slope, m and the

y-intercept b.

5

4

y

b

Write the linear equation expressing the


3

2

1

x

−5 −4 −3 −2

−1

−1

1 2 3 4 5

−2

−3

−4

−5

14 For the following lines passing through the given two points:

i Find the slope of the line.

ii Find the equation of the line in the form y = mx + b.

a (0, 5) and (2, 9) b (−3, 9) and (4, 2)

c (−6, 8) and (6, 0) d (−7, 3) and (−7, −2)


15 A 12-inch candle decreases in length by 0.4 in/s after it was lit.

a

Select the graph of the linear function that could describe the relationship between the

height of the candle in inches, y, and the number of minutes after it was lit, x.

A

y

B

y

10

10

5

5

x

x

10 20

10 20

C

y

D

y

10

10

5

5

x

x

10 20

10 20

b

c

Identify the slope of the function.

Identify the y-intercept of the function.

16 A tank containing 220 gallons of water is being drained at the rate of 0.1 gallons per minute.

Select the equation of the linear function that could describe the relationship between the

gallons of water, y, remaining in the tank and the time in minutes, x.

A y = 220x + 0.1 B y = 0.1x + 220 C y = −200x + 0.1 D y = −0.1x + 220

17 Consider the following table of values for a linear function:

x -2 -1 0 1 2

y -6 -3 0 3 6

a

Select the graph of the linear function that could represent the table of values.

56


A

4

y

B

4

y

3

3

2

2

1

x

1

x

−4 −3 −2

−1

−1

1 2 3 4

−4 −3 −2

−1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

C

4

y

D

4

y

3

3

2

2

1

x

1

x

−4 −3 −2

−1

−1

1 2 3 4

−4 −3 −2

−1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

b

c

Identify the slope of the function.


18 A race car starts the race with 60 gallons of fuel. From there, it uses fuel at a rate of 1 gallon

per minute.

a


Number of minutes passed (x) 0 5 10 15 20 60

Amount of fuel left in tank (y)

b

c

Write the equation relating the number of minutes passed (x) and the amount of fuel left

in the tank (y).

Graph the function.

19 A cleaner charges an initial fee of $110 plus $20 per hour. Write an equation to represent

the total amount charged by the cleaner, y, as a function of the number of hours worked, x.



20 Neil is running a 30-mile ultramarathon.

a

b

Write an equation to represent the

distance Neil has left to run, y, in terms

of the number of hours since the start, x.

The domain has been restricted to

x Î [0, 6]. Explain why this restriction

has been put in place.

40

35

30

25

20

15

10

Distance remaining

5

1

Time (hours)

2 3 4 5 6 7 8

21 A train leaves Station A and moves along a straight railroad towards Station B, which is

5280 ft ahead. The average speed of the train during the travel is 88 ft/min.

a

A

Select the graph of the linear function that describes the relationship between the

remaining distance of travel in feet, y, and the time of travel in minutes, x.

y

B

y

5000

5000

4000

4000

3000

3000

2000

2000

C

1000

y

10 20 30 40 50

x

D

1000

y

10 20 30 40 50

x

5000

5000

4000

4000

3000

3000

2000

2000

1000

1000

x

x

10 20 30 40 50

10 20 30 40 50

58


b

Identify the domain restriction of the function.

22 The table shows the water level of a well that is being emptied at a constant rate with a

pump where the time is measured in minutes and the water level in feet.

Time (x) 3 7 10

Water level (y) 56.7 48.3 42

a

b

c

d

e

Write an equation in the form y = mx + b to represent the water level (y), as a function of

the minutes passed (x).

Interpret the slope of this linear function.

Interpret the y-intercept of this linear function.

Find the water level after 18 minutes.

State an appropriate domain constraint on this linear function. Justify your answer.

23 A carpenter charges an initial fee of $250 plus $35 per hour for labor.

a

b

c

d

Write an equation to represent the total amount charged, y, by the carpenter as a

function of the number of hours worked, x.

Interpret the slope of this linear function.

Interpret the y-intercept of this linear function.

Peta has budgeted $2500 to have a ramp built at the community center to increase

accessibilty. The carpenter works 8 hour days and estimated that the project would take

one and a half days. He also priced the materials at $1800.

Determine if Peta has budgeted enough for the ramp. Justify your answer.

24 A diver starts at the surface of the water and begins to descend below the surface at a

constant rate. The table shows the depth below the surface of the diver over the first

5 minutes.

Number of minutes passed (x) (min) 0 1 2 3 4

Depth of diver (y) (ft) 0 1.63 3.26 4.89 6.52

a

b

c

d

e

f

State the rate of change with units of ft/min.

Write an equation for the relationship between the number of minutes passed (x) and

the depth below the surface (y) of the diver.

Determine the depth of the diver after 6 minutes.

Find how long the diver takes to reach 11.41 ft beneath the surface.

The diver has two air tanks, each of which last for 45 minutes. One is used for going

down and one is used for returning to the surface. Determine an appropriate domain

restriction for the function from part (b).

Assuming the diver comes back up at the same rate they go down, state the equation

for the linear function which could describe the return to surface including the domain

restriction. Graph the full scenario.


2.02 Standard form

Concept summary

The standard form of a linear relationship is a way of writing the equation with all of the variables

on one side:

Ax + By = C

A is a non-negative integer

B, C are integers

A, B are not both 0

To draw the graph from standard form, we can find and plot the x and y-intercepts or convert to

slope-intercept form.

The standard form is helpful when looking at scenarios that have a mixture of two different items.

When we identify the intercepts in a mixture scenario, it can be interpreted as the amount of that

item when none of the other item is included.

Worked examples

Example 1

Draw the graph of the line 5x − 3y = −15 on the coordinate plane.

Approach

We can find both the x and y-intercept and then graph the line using those two points.

Solution

Find the x-intercept by setting y = 0 and solving:

5x − 3y = −15 State the given equation

5x − 3(0) = −15 Set y = 0

5x = −15 Simplify

x = −3 Divide both sides by 5

Find the y-intercept by setting x = 0 and solving:

5x − 3y = −15 State the given equation

5(0) − 3y = −15 Set x = 0

−3y = −15 Simplify

y = 5 Divide both sides by −3

y

7

6

5

4

3

5x − 3y = −15

2

1

−5 −4 −3 −2 −1 1 2 3

−1

x

60


Reflection

We could have also converted to slope-intercept and graphed using the slope and y-intercept.

We can decide unless the question specifies how to do it. If the x and y-intercept happened to

be same point, (0, 0), then we need to find another point by substituting another x-value into

the equation and solving for y.

Example 2

Darius wants to buy a mix of garlic and chipotle powders for seasoning tacos. Garlic powder costs

$4/lb. Chipotle powder costs $7/lb. Darius spends exactly $14 on spices.

Let x represent the amount of garlic powder Darius buys and let y represent the amount of chipotle

powder he buys. Write an equation to represent this scenario.

Approach

The cost of each spice will be the cost per pound multiplied by the amount purchased.

The total cost will be the sum of the two spice costs and is equal to $14.

Solution

Cost of just the garlic: 4 × x

Cost of just the chipotle: 7 × y

Total cost: 4x + 7y = 14


1 State the standard form of a linear equation.

2 State which of the following linear equations are in standard form:

a 2x − 5y = 0 b x = 10

c

d


Let’s practice


a

b

State the y-intercept.

Find the slope, m, of the line.

4

3

y

c

d

Find the equation of the line in the form

y = mx + b.

Rewrite the equation of the line in

standard form.

4 Consider the following table of values:

2

1

−4 −3 −2 −1

−1

−2

1 2 3 4

x

x 1 2 3 4 5

−3

y −1 −9 −17 −25 −33

−4

a

b

Write the relationship between x and y in

slope-intercept form.

Rewrite the relationship in standard form.

5 For each of the following equations:

i Calculate the x-value of the x-intercept of the line.

ii Calculate y-value of the y-intercept of the line.

iii Draw the graph of the equation of the line on the coordinate plane.

a 5x − 3y = −15 b 2x + y = 10


i Rewrite the equation in slope-intercept form.

ii Draw the graph of the equation of the line on the coordinate plane.

a 4x − 3y = −12 b 2x + y = 10

7 Draw the graph of each of the following linear equations:

a 2x = 6 b 6y = −3 c 3x − 2y = −12 d 5x + 4y = 40

8 Select the linear equation that could represent the following table of values.

x 2 4 6 8 10

y 4 5 6 7 8

A 2x − y = 6 B 2x − y = −6 C x − 2y = 6 D x − 2y = −6

62


9 Select the linear equation that could represent

the following graph.

A 5x + y = 20 B 5x − y = 20

C x + 5y = 20 D x − 5y = 20

5

y

−5

5

x

−5


x 0 3 6 9 12

y −5 −4.25 −3.5 −2.75 −2

a

Select the graph that could represent the table of values.

A

y

B

y

5

10

5

x

x

−5

5

−10

−5 5 10

−5

−5

−10

C

y

D

y

10

10

5

5

x

x

−10

−5 5 10

−10

−5 5 10

−5

−5

−10

−10

b Identify the slope of the line. c Identify the y-intercept of the line.


11 A linear function is defined by the equation 2x − 5y = −15.

a

Select the graph that represents the equation.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5

C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

b Identify the domain of the function. c Identify the range of the function.

12 On a particular day, the air temperature on a mountain decreases by 5.5°F for every 1000 ft

increase in elavation. The temparature at the base of the mountain is 55°F. Select the linear

equation that could describe the relationship between the height x in ft above the base of

the mountain and the temperature y in °F.

A 1000x − 5.5y = 5500 B 1000x + 5.5y = 5500

C 5.5x − 1000y = 302.5 D 5.5x + 2y = 302.5

64


13 A white oak tree stands 20 ft tall in 2009. It has been determined that the tree grows by

a rate of 1.25 ft per year.

a Select the graph that could represent the relationship between the height y in feet and

the number of years x since 2009.

A

y

B

y

25

25

20

20

15

15

10

10

5

5

x

x

1

2 3 4 5

1

2 3 4 5

C

y

D

y

25

25

20

20

15

15

10

10

1 2 3 4 5

1 2 3 4 5

b Identify the slope of the line. c Identify the y-intercept of the line.

14 Marvin went to the grocery store to buy x lb of granulated sugar for $0.60/lb and y lb of

powdered sugar for $0.90/lb. The total cost of all the sugar is $12.

a Select the graph of the linear function that relates the number of pounds of granulated

sugar x and the number of pounds of powdered sugar y.

A

21

5

y

x

B

21

5

y

x

18

18

15

15

12

12

9

9

6

6

3

x

3

x

3

6 9 12 15 18 21

3

6 9 12 15 18 21


C

21

y

D

21

y

18

18

15

15

12

12

9

9

6

6

3

x

3

x

3

6 9 12 15 18 21

3

6 9 12 15 18 21

b

Identify the domain restriction of the function.

15 Salvina needs to buy a mix of red and green apples. She is going to spend $15 in total.

Green apples cost $1.50 lb and red apples cost $3 lb.

a

b

c

d

Write a linear equation in standard form

to represent the scenario. Use the

variables x for green and y for red.

Calculate and interpret the x-intercept of

the linear equation from part (a).

Determine the number of green

apples she could buy if she bought

2 red apples.

State the domain of the linear function.

16 The line kx + 3y = 15 passes through the point (13, −8).

a Find the value of k.

b

c

Solve for the x-value of the x-intercept of the line.

Solve for the y-value of the y-intercept of the line.


17 Write an expression for the slope, m, of the line represented by the equation Ax + By = C.

State any restrictions. Explain how this expression might be helpful.

18 Describe a scenario where using the standard form of a linear equation is more useful than

the slope-intercept form.

19 Compare and contrast the standard form and slope-intercept form of a linear equation.

Include how they are similar, how they are different, and the benefits of each form.

66


20 Effie is a entomologist and is currently studying mosquitos and spiders. She knows that

mosquitos have six legs and spiders have eight legs. In her lab, she has a mix of mosquitos

and spiders. Between all the bugs, there is a total of 240 legs.

a

b

c

d

Write an equation to represent this scenario.

Draw a graph of the linear equation.

State and describe the domain of the equation.

Explain whether or not every point on the line represents a possible solution.

21 Nakisha is selling the baby outfits her daughter has outgrown and is buying second-hand

toddler outfits. She is able to sell each baby outfit for $3 and she buys each toddler outfit for

$5. In the end, she wants to make a profit of exactly $30.

Profit = revenue − cost

Revenue is the money earned

Cost is the money spent

a

b

c

d

Write an equation to represent the scenario, using x for the number of baby outfits sold,

y for the number of toddler outfits bought, and the profit of exactly $30.

Calculate and interpret the x-intercept.

Nakisha has 25 baby outfits available to sell and needs 15 toddler outfits.

Determine if she is able to make her desired profit of $30.

Using the additional information from part (c), state and interpret the domain restriction.


2.03 Point-slope and

connecting forms

Concept summary

When we are given the coordinates of a point on the line and the slope of that line, then pointslope

form can be used to state the equation of the line.

Coordinates can be given as an ordered pair, in a table of values, read from a graph, or described

in a scenario. The slope can be stated as a value, calculated from two points, read from a graph, or

given as a rate of change in a scenario.

Point-slope form of a linear relationship:

y − y 1 = m(x − x 1 )

m The slope of the line

x 1 The x-coordinate of the given point

y 1 The y-coordinate of the given point

Coordinate

A number used to locate a point on a number line. One of the numbers in an ordered pair,

or triple, that locates a point on a coordinate plane or in coordinate space, respectively.

Ordered pair

A point on a graph written as (x, y). Also called coordinates, or a coordinate pair.

We can also find the equation in point-slope form when given the coordinates of two points on

the line, by first finding the slope of the line. We now have three forms of expressing the same

equation, and each provides us with useful information about the line formed. Point-slope form is

useful when we know, or want to know:

• The slope of the line

• A point on the line

Slope-intercept form is useful when we know, or want to know:

• The slope of the line

• The y-intercept of the line

Standard form is useful when we know, or want to know:

• Both intercepts of the line

68


Worked examples

Example 1

For each of the following equations, determine if they are in point-slope form, standard form,

or slope-intercept form. If they are not in standard form, convert them to standard form.

a 2x + 8y = 10

Solution

This is in standard form as all of the variables are on one side and the constant is on the other side.

b y = 4x − 10

Solution

This is in slope-intercept form.

y = 4x − 10

−4x + y = −10

4x − y = 10

4x − y = 10 is in standard form.

State the given equation


Divide both sides by −1

c

Solution

This is in point-slope form.

State the given equation

5y − 15 = −2(x + 7) Multiply both sides by 5

5y − 15 = −2x − 14

2x + 5y = 1

2x + 5y = 1 is in standard form.

Reflection

Distribute the multiplication

Add 2x and 15 to both sides

There are more steps to convert from point-slope form to standard form, than from point-slope

form to slope-intercept form.


d x = 8

Solution

Yes, this is in standard form.

Reflection

This is a vertical line.

Example 2

A line passes through the two points (−3, 7) and (2, −3). Write the equation of the line in pointslope

form.

Approach

We will first find the slope of the line using the two points. Then we will pick one of the points to

substitute into the point-slope equation.

Solution

Find the slope of the line:

Formula for slope of a line

Substitute in the points

Simplify the subtraction

= −2 Simplify the fraction

Find the equation of the line in point-slope form:

Reflection

y − y 1 = m(x − x 1 )

y − 7 = −2(x − (−3))

y − 7 = −2(x + 3)

Formula for point-slope form

Substitute known values into the formula

Simplify

We can confirm we have correctly written this in point-slope form: y − y 1 = m (x − x 1 ), as we can

see the coordinates (−3, 7) and a slope of −2 are represented correctly. This is easier to see in the

previous line: y − 7 = −2(x − (−3))

70


Example 3

A carpenter charges for a day’s work using the given equation, where y is the cost and x is the

number of hours worked:

y − 125 = 50(x − 2)

a Draw the graph of the linear equation from point-slope form.

Approach

We need two points to plot a line. We can read from the equation that one point on the line is

(2, 125). To get another point, we can use the slope from the given point or substitute in an x-value

in the domain and solve for y.

Solution

The given point is (2, 125) Since the slope is 50, we can go up 50 units and right 1 unit from our

given point to plot another point on the line.

225

y

200

175

150

125

100

75

50

25

1 2 3 4

x

b The carpenter works a maximum of 10 hours per day. State the domain constraints for this

scenario.

Solution

Domain: 0 £ x £ 10



1 State the point-slope form of a linear equation.

2 For each of the following equations, state the form they are written in:

a y = −4x + 5 b y − 1 = 5(x − 4) c 5x − 3y = 8

3 Write the equation of the following lines in point-slope form:

a A line passing through the point (7, 1) has a slope of 4.

b A line passing through the point (4, 0) has a slope of −5.

c A line passing through the point (1, 6) has a slope of 0.

4 For each of the following lines:

i Find the slope, m, of the line.

ii Write the equation of the line in point-slope form.

a A line passes through the two points (7, 6) and (9, 12).

b A line passes through the two points (4, −8) and (2, 0).

5 For each of the following table of values:

i Find the slope, m, of the line represented in the table.

ii Write the equation of the line in point-slope form.

a

x 1 2 3 4

b

x 1 2 3 4

y −3 2 7 12

y 9 7 5 3

Let’s practice

6 Rewrite the following linear equations into slope-intercept form:

a 2x + 3y = 12 b y − 2 = 3(x − 7)

7 Rewrite the following linear equations into standard form:

a b y + 7 = −3(x + 2)

c d y − y 1 = m(x − x 1 )

8 Select the linear equation that could represent the following table of values.

x -3 -2 -1 0

y 2 -2

A B C D

72


9 Select the linear equation that could

represent the following graph.

A y + 3 = 2(x + 3)

B y − 7 = 2(x − 2)

C y − 7 = 2(x − 5)

D y − 4 = 2(x − 5)

8

6

4

2

y

x

−8 −6 −4 −2

−2

2 4 6 8

−4

−6

−8

10 State the equation of the given linear

function in point-slope form.

5

4

3

2

y

1

x

−5 −4 −3 −2 −1

−1

1 2 3

−2

−3

11 Sketch the graph of the lines on the coordinate plane given the following equations:

a y + 8 = 3(x + 2) b y + 1 = −2(x + 1)

c

d


x 0 5 10 15 20

y 7 4 1 −2 −5


a

Select the graph that could represent the table of values.

A

y

B

y

10

10

5

5

x

x

−10

−5 5 10

−10

−5 5 10

−5

−5

−10

−10

C

y

D

y

10

10

5

5

x

x

−10

−5 5 10

−10

−5 5 10

−5

−5

−10

−10

b

c

Identify the slope of the line.

Identify the y-intercept of the line.

13 A linear function is defined by the equation

a

Select the graph that could represent the equation.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5

74


C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

b

c


Identify the range of the function.

14 The graph of a linear function passes thorugh the point (1, 6) and has a slope of

a

Select the graph that could represent the function.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5

C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

b

c




15 A race car uses fuel at the rate of 0.9 gallons per minute. After running for 12 minutes,

the car has 48 gallons of fuel left in the tank. Select the linear equation that could describe

the relationship between the number of minutes x the car is running and the number of

gallons of fuel y left in the tank.

A y + 12 = 0.9(x + 48) B y + 48 = 0.9(x + 12)

C y − 12 = 0.9(x − 48) D y − 48 = 0.9(x − 12)

16 A plumber charges a fixed amount for a call out fee plus $30 per hour of work. He charged a

total of $80 for 2 hours of work. He cannot work for more than 8 hours in any given day.

a

A

Select the graph of the linear function that relates the number of hours of work x and

the total charge y for the day.

y

B

y

80

80

60

60

40

40

20

20

x

x

C

y

1

2 3 4

D

y

1

2 3 4

80

80

60

60

40

40

20

20

x

x

b

1 2 3 4


1

2 3 4

17 Sally receives an order to make key chains for an upcoming festival. She is told to make

at least 50 and at most 100 key chains, and will be paid $8 for each key chain she makes.

Sally makes 80 key chains and is paid $665 altogether.

a

b

c

Let x represent the number of key chains Sally makes and y represent the amount she

is paid. Write an equation in point-slope form relating x and y.

Convert the equation from part (a) to slope-interept form.

Determine the flat fee that Sally was paid in addition to the $8 per key chain that

she received.

76


d

State the domain of this linear relationship

18 Gary is flying in a plane which is moving directly away from his home at a constant speed.

At 2 hours into the flight, he checked that he was 1042 miles from his home. He checks

again at 7 hours into the flight to find that he is now 3542 miles from his home.

a

Find the speed of the plane, in miles per hour.

b Write an equation in point-slope form relating the number of hours flying, x,

to the distance Gary is from his home, y, in miles.

c

If the destination of Gary’s flight is 5542 miles from his home, find the duration of

the flight.


19 Create a scenario where you would choose to use each of the following forms and explain

why you would use that form over the others:

a Point-slope form, (y − y 1 ) = m(x − x 1 )

b

c

Slope-intercept form, y = mx + b

Standard form, Ax + By = C

20 Rewrite the point-slope form of a linear equation in slope-intercept form. Draw connections

between the two forms.

21 A cup is placed under a faucet which is leaking at a rate of 2 oz per hour. After 4 hours,

the cup contains 15 oz of water.

a

b

c

d

Write an equation in point-slope form relating the volume in the cup, y, to the amount of

time that has passed in hours, x.

Now, rewrite the equation into slope-intercept form.

State and interpret the y-intercept of the linear equation.

Suppose that the cup has a maximum capacity of 18 oz. Determine how long it can be

under the faucet before it starts to over flow. Now, state the domain and range.

22 Paige is planning an event and the

brochure for the caterer at the venue

shows the following graph for the meal

costs.

a

b

c

State and interpret the domain of the

linear function.

Paige concludes that the unit cost per

person is $16.25. Determine if this is

correct. If so, explain how to calculate

this value. If not, explain the error and

state the correct value.

State the equation of the linear

function in point-slope form.

1900 Cost ($)

1800

1700 (100, 1625)

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500 (10, 500)

400

300

200

100

People

10 20 30 40 50 60 70 80 90 100 110


2.04 Parallel and

perpendicular lines

Concept summary

Knowing that two lines are parallel or perpendicular to each other gives us information about the

relationship between their slopes and graphs.

Parallel lines

Two or more lines that have the same slope and never intersect.

Distinct vertical lines are parallel to each other.

y

x

Perpendicular lines

Two lines that intersect at right angles. Their slopes are opposite

reciprocals. Horizontal and vertical lines are also perpendicular to

each other.

y

x

Opposite reciprocals

Two numbers whose product is −1.

Example:

To determine if two lines are parallel or perpendicular, we will need to find the slopes of the two

lines. We often do this by rearranging to y = mx + b and comparing the values of m.

If m 1 = m 2 , then the two lines are parallel.

If m 1 × m 2 = −1, then the two lines are perpendicular.

If the two lines are parallel, then m 1 = m 2 .

If the two lines are perpendicular, then m 1 × m 2 = −1.

78


Worked examples

Example 1

For each of the following pairs of lines, determine whether they are parallel, perpendicular, or neither.

a y = −3x + 7 and y = 3x − 4

Approach

We need to identify the slopes of both lines to see if they are the same, opposite reciprocals,

or neither.

Solution

For y = −3x + 7, the slope is m = −3.

For y = 3x − 4, the slope is m = 3.

The slopes are not the same. The slopes are not opposite reciprocals.

The lines y = −3x + 7 and y = 3x − 4 are neither parallel nor perpendicular.

b 3x − 5y = 15 and 6x − 10y = 60

Approach

We need to identify the slopes of both lines to see if they are the same, opposite reciprocals, or

neither. First, we want to rearrange both lines into the form y = mx + b to identify their slopes.

Solution

For 3x − 5y = 15,

3x − 5y = 15

−5y = −3x + 15

Given equation



Simplify

The slope for

For 6x − 10y = 60,

6x − 10y = 60

−10y = −6x + 60

Given equation



Simplify


The slope for

The slopes are the same, so lines 3x − 5y = 15 and 6x − 10y = 60 are parallel.

Reflection

Notice that in standard form, Ax + By = C, the values of A and B for 3x − 5y = 15 and

6x − 10y = 60 were in the same ratio, while C was different.

c y = −2x + 3 and x − 2y = 10

Approach

We need to identify the slopes of both lines to see if they are the same, opposite reciprocals,

or neither. We will need to rearrange x − 2y = 10 to slope-intercept form.

Solution

For y = −2x + 3, the slope is m = −2.

For x − 2y = 10,

x − 2y = 10

−2y = −x + 10

Given equation



Simplify

The slope of

The slopes are opposite reciprocals, so lines y = −2x + 3 and x − 2y = 10 are perpendicular.

Example 2

Find the equation of the line, L 1 that is parallel to the line

(0, 4). Give your answer in slope-intercept form.

and goes through the point

Approach

The slope of parallel lines are the same, so we can to identify the slope from the line

The point given happens to be the y-intercept. We can substitute the slope in for m and the

y-intecept in for b in y = mx + b.

80


Solution

The slope of

The y-intercept is b = 4.

The equation of the line

Reflection

If we had been given a point that was not the y-intercept, then we would have needed to substitute

the point in for x and y and solved for b. We could also have used point-slope form and rearranged

to slope-intercept form.


1 Determine whether the following lines are parallel, perpendicular, or neither:

a

10

y

b

10

y

8

8

6

6

4

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−4

−6

−6

−8

−8

−10

−10

c

10

y

8

6

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−6

−8

−10

2 State the relationship between the slopes of:

a Parallel lines b Perpendicular lines


3 Find the slope, m, of a line that is:

a Parallel to a line which has a slope of 4

b Parallel to

c Perpendicular to a line which has a slope of −9

d Perpendicular to y = −5x + 6

e Perpendicular to y = 10 − x

4 Determine if the following lines are parallel to or not:

a b 4x − 6y = 37 c 3x + 2y = −13 d

5 Determine if the following lines are perpendicular to y = 4x − 3 or not:

a b 4x + 16y = 96 c x + 4y = −3 d y = −4x − 3

Let’s practice

6 For each of the following pair of lines, determine whether they are parallel, perpendicular, or

neither.

a y = 3x − 7 and y = 3x + 4 b x − y = 6 and x − y = −6.

c d 2x + 3y = 5 and −3x − 2y = 12

7 Find the equation the line that is:

a Parallel to the line and passes through the point (0, −2)

b Parallel to x = 1 and passes through (−4, 6)

c Parallel to the x-axis and passes through (−8, 5)

d Parallel to the y-axis and passes through (−7, 5)

8 Find the equation the line that is:

a Perpendicular to x = 4 and passes through (−7, 1)

b Perpendicular to y = 2x − 1 and passes through the point (2, 3)

c Perpendicular to the x-axis and passes through the point (10, 3)

d Perpendicular to the y-axis and passes through (1, −7)

9 Consider a line that passes through points A(4, 3) and B(8, −10).

a

b


Find the equation of a line that passes through (−1, 2) and is parallel to the line passing

through A(4, 3) and B(8, −10). Express the equation in slope-intercept form.

82


10 Consider a line that passes through A(2, −4) and B(3, 9).

a

b


Find the equation of a line that has a y-intercept of 5 and is perpendicular to the line that

goes through A(2, −4) and B(3, 9). Express the equation in slope-intercept form.

11 Consider line L 1 with equation 3x − 4y − 4 = 0.

a

Write L 1 in slope-intercept form.

b Find the slope of a line, L 2 , that is perpendicular to L 1 .

c Find the equation of L 2 in standard form, given that it passes through the point A(−6, 4).

12 Consider line L 1 with equation

Select the equation of L 2 that is parallel to L 1 and passes through the point the point (−6, 3).

A B C D

13 Consider line L 1 with equation 2x − 3y = 21.

Select the equation of L 2 that is parallel to L 1 and passes through the point the point (0, −6).

A 2x − 3y = −12 B 3x + 2y = −18 C 2x − 3y = 18 D 3x + 2y = −12

14 Consider line L 1 with equation

Select the equation of L 2 that is perpendicular to L 1 and passes through the point the point

(5, −7).

A y = −2x + 3 B y = −2x − 9 C D

15 Consider line L 1 with equation 32x + 24y = 21. Select the equation of L 2 that is

perpendicular to L 1 and passes through the point the point (−9, 11).

A B C D


16 Consider the lines 2x − 7y = 2 and mx − 35y = −7.

Given that the two lines are parallel, determine the value of m.

17 Explain why two lines with the standard forms ax + by = c 1 and bx − ay = c 2

are perpendicular.

18 Ammon and Minh work part-time at the same pizzeria, both earning $360 per week. Both

Ammon and Minh are saving up, so they don’t spend any of the money they earn. Ammon

currently has a bank balance of $212 while Minh has $275. If we sketch linear graphs for Ammon

and Minh’s bank balances over time, can we expect to get two parallel lines? Justify your answer.

19 Suppose that the lines px − 4y = 7 and qx + 5y = −14 are perpendicular.

Write an equation relating the values of p and q.


2.05 Linear inequalities

in two variables

Concept summary

If a linear inequality involves two variables, we can represent it as a region on a coordinate plane

rather than an interval on a number line.

Linear inequality in two variables

An inequality whose solution is a set of ordered pairs represented by a region of

the coordinate plane on one side of a boundary line.

Example: 2y > 4 − 3x

Boundary line

A line which divides the coordinate plane into two regions.

A boundary line of a linear inequality is solid if it is included

in the solution set, and dashed if it is not.

y

x

Depending on the inequality sign, the boundary line will be solid or dashed, and region shaded will

be above or below the boundary line.

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

y > x

y ≥ x

84


4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

y < x

y ≤ x

Worked examples

Example 1

Write the inequality that describes the region shaded on the given coordinate plane.

4

y

3

2

1

x

−4 −3 −2 −1

−1

1 2 3 4

−2

−3

−4

Approach

First, we want to determine the equation of the boundary line. Then we need to determine what

inequality sign to use.


Solution

To find the equation of the boundary line in

the form y = mx + b:

• The line crosses the y-axis at (0, −3)

• The rise is 3

• The run is 1

So, b = −3 and m = 3.

Now we have y = 3x − 3 as the boundary line.

−4 −3 −2 −1

4

3

2

1

−1

y

1 2 3 4

x

−2

−3

(0, −3)

−4

To determine the inequality:

• The boundary line is solid, so it is ≤ or ≥.

4

y

• Any point in the shaded region would be above

the boundary line.

This means that for every x-value, the y-value for any

point in the region is greater than the y-value of the

point on the line.

So our linear inequality is y ≥ 3x − 3.

3

2

1

−4 −3 −2 −1

−1

−2

1 2 3 4

x

−3

−4

Reflection

It is a good idea to check our answer by substituting in a point in the region to make sure it satisfies

the inequality. For example, the origin, (0, 0), is in the region, so should satisfy the inequality.

y ≥ 3x − 3

0 ≥ 3(0) − 3

0 ≥ −3

It does satisfy the inequality, so we have selected the correct inequality sign.

86


Example 2

A pick-up truck has a maximum weight capacity of 3000 pounds. Each box of oranges weighs

8 pounds and each box of grapefruits weighs 12 pounds. Let x represents the number of boxes of

oranges in the truck. Let y represents the number of boxes of grapefruit in the truck.

a Write an inequality to represent the number of boxes of oranges and grapefruit that can be in

the truck.

Solution

The weight of all the orange boxes is the product of the weight of one box, 8, and the number

of boxes, x.

Weight of orange boxes: 8x

The weight of all the grapefruit boxes is the product of the weight of one box, 12, and the number

of boxes, y.

Weight of grapefruit boxes: 12y

The total weight is the sum of the weights of orange boxes and grapefruit boxes.

Total weight: 8x + 12y

The truck can carry at most 3000 pounds, so we get our inequality: 8x + 12y ≤ 3000

b Create a graph of the region containing the points corresponding to all the different numbers

of orange and grapefruit boxes that can be loaded into the truck.

Approach

We will graph the region representing 8x + 12y ≤ 3000.

We need to identify our boundary line, then graph it. Then, we need to decide which side of

the boundary line to shade.

Solution

The boundary line is 8x + 12y = 3000. This is a line in standard form, so we can graph it by finding

the intercepts. Find the x-intercept by setting y = 0 and solving:

8x + 12y = 3000

8x − 12(0) = 3000 Setting y = 0

8x = 3000

Stating the given equation

Simplifying

x = 375 Dividing both sides by 8

Find the y-intercept by setting x = 0 and solving:

8x + 12y = 3000

8(0) + 12y = 3000 Setting x = 0

12y = 3000


Simplifying

y = 250 Dividing both sides by 12


The boundary line will be solid because we have ≤ as the sign.

We will shade below the line as the point (0, 0) satisfies the inequality.

300

275

250

225

200

175

150

125

100

75

50

25

y

8x + 12y ≤ 3000

x

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400

Reflection

The actual possible solutions are the positive integer coordinates since we are working with whole

fruits and have both x ≥ 0 and y ≥ 0. This region shows where all those possible solutions can be.


1 State the equation of the boundary line for each inequality.

a y ≥ 4x − 3 b c y ≤ −x + 1 d x > 12

2 For each of the following inequalities, state whether they are represented using a solid or

dashed line on a graph:

a y ≤ 4x − 4 b y < 3x − 5 c y ≥ 2x − 3 d y > 5

3 For each of the following shaded regions, determine whether the following points lie in the

shaded region:

i (2, 4) ii (−3, 2) iii (−1, 6) iv (3, −3)

88


a

6

5

4

3

2

1

−3 −2 −1 −1

−2

−3

−4

−5

−6

y

1 2 3 4 5 6 7

x

b

6

5

4

3

2

1

−7 −6 −5 −4 −3 −2 −1

−1

1 2 3

−2

−3

−4

−5

−6

y

x

4 For each of the following inequalities

i Determine if the origin (0, 0) satisfies the inequality.

ii Graph the region that satisfies the inequality.

a x > 3 b y ≤ −2 c x ≥ −1

Let’s practice


i Determine whether the point (2, 3) satisifies the inequality or not.


a y ≤ 3x + 5 b c y ≥ 2x + 4 d y < 3x + 2


i State the coordinates of the x- and y-intercepts of the boundary line.


a 3x + 2y < 12 b 5x + 2y ≥ 10

7 Select the graph that represents the inequality 2x − 7y > 14.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5


C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

8 Select the graph that represents the inequality y ≤ −3x + 4.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5

C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

90


9 Select the graph that represents the inequality y > −x − 3.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5

C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

10 Select the graph that represents the inequality y ≥ 3x + 1.

A

y

B

y

5

5

x

x

−5

5

−5

5

−5

−5


C

y

D

y

5

5

x

x

−5

5

−5

5

−5

−5

11 Select the inequality that describes

the following region:

A

8

6

y

B

C

D

4

2

−8 −6 −4 −2

−2

−4

2 4 6 8

x

−6

−8



A

8

6

y

B

C

D

4

2

−8 −6 −4 −2

−2

−4

2 4 6 8

x

−6

−8

92




A

8

6

y

4

B

C

2

−8 −6 −4 −2

−2

2 4 6 8

x

D

−4

−6

−8



A

8

6

y

B

C

D

4

2

−8 −6 −4 −2

−2

−4

2 4 6 8

x

−6

−8

15 Write the inequality that describes the following shaded regions:

a

5

y

b

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5


c

5

4

3

2

1

−5 −4 −3 −2 −1 −1

−2

−3

−4

−5

y

1 2 3 4 5

x

d

6

5

4

3

2

1

−4 −3 −2 −1

−1

−2

−3

−4

y

1 2 3 4 5 6

x

16 Applicants for a particular job are asked to sit a numeracy test and verbal reasoning

test. Successful applicants must obtain a minimum combined score of 43 for both tests.

Write an inequality relating the applicant’s score on the numeracy test, x, and the verbal

reasoning test, y.

17 Throughout university, Luigi works as a mentor, getting paid $10 per hour, and as a barista

getting paid $13 per hour. The number of hours he works in each job can vary from week

to week, and he needs to be able to at least cover his weekly expenses of $260. Write an

inequality relating the number of hours he works as a mentor, x, and barista, y.

18 Lisa is being careful with her spending so she can later purchase a car. She has allocated no

more than $450 each month for both travel expenses and eating out. On average, each time

she eats out costs $18, and each travel journey costs $5. Write an inequality relating the

number of times she eats out, x, and the number of travel journeys each month, y and then

solve for y.

19 In the new basketball season, Mario is looking to beat his personal best of scoring

102 points in total for the whole season (this does not include points from fouls).

Write an inequality relating the number of ‘two pointers’, x, and the number of ‘three pointers’,

y, he scores throughout the new season and then solve for y.


20 Oprah is thinking about how to use her last arcade tokens. A game of Table Tennis costs

2 tokens and a game of Frog Days costs 5 tokens. She has a total of 30 tokens left.

a

b

c

d

Write an inequality relating the number of times Oprah plays Table Tennis, x and the

number of times she plays Frog Days, y.

Graph the region containing the points corresponding to all the different ways Oprah

could spend her remaining tokens across the two games.

Interpret the value of the x-intercept of the boundary line in the given context.

Interpret the value of the y-intercept of the boundary line in the given context.

94


21 Fiona has set aside $13.20 in her shopping budget for fruit this week. Currently, cantal are

on sale for $1.65 each, while watermelon are on sale for $2.20 each.

a

b

c

Write an inequality relating the number of cantaloupe that Fiona buys, x, and the number

of watermelon that Fiona buys, y.

Graph the inequality on the number plane, and shade the solution set.

Determine whether Fiona can buy the following combinations of fruits. Justify your

answer.

i 4 cantaloupes and 2 watermelons ii 4.9 cantaloupes and 0.1 watermelons

iii 2 cantaloupes and 7 watermelons iv 6 cantaloupes and 5 watermelons

22 A book seller makes a profit of $9 on every book sold online and $5 on every book sold

in store. The company wants to make a profit of at least $270 a day selling books through

online and in-store sales.

a

b

c

Write an inequality relating the number of books sold online, x, and the number of books

sold in store, y on one day.

Graph the region that satifies the inequality.

Given the context, is the ordered pair (−5, 100) a valid solution to the inequality? Justify

your answer.

23 Sean is thinking about how to use his remaining spending money for snacks. A pack of dried

fruit costs $6 and a bag of mixed nuts costs $4. He has $48 remaining.

a

b

c

Write an inequality relating the number of packs of dried fruit, x and the number of bags

of mixed nuts, y that Sean buys.

Graph the region containing the points corresponding to all the different ways Sean

could spend his remaining spending money on these snacks.

Supposing that Sean does not need to spend all of it, how many different ways can Sean

spend his remaining money on some combination of dried fruit and mixed nuts?


3

Systems of

Linear Equations

and Inequalities

Chapter outline

3.01 Solving systems of equations by graphing 98

3.02 Solving systems of equations by substitution 106

3.03 Solving systems of equations by elimination 111

3.04 Systems of linear inequalities 116

3.01 Solving systems of

equations by graphing

Concept summary

A system of equations is a set of equations which have the same variables.

A solution to a system of equations is any set of values of all variables in that system which is a

solution to each equation in the system.

A solution can also be thought of graphically as the point(s) of intersection of the graphs of the

equations (the points in common to all graphs):

When two lines are parallel and distinct, they have

no points of intersection. The corresponding system

of equations has no solutions.

y

x

When two lines are identical, they intersect at

every point. The corresponding system of equations

has infinitely many solutions.

y

x

98


When two lines are not parallel, they have exactly

one point of intersection. The corresponding system

of equations has one solution.

y

x

A solution to a system of equations in a given context is said to be viable if the solution makes

sense in the context, and non-viable if it does not make sense within the context, even if it would

otherwise be algebraically valid.

Worked examples

Example 1

Consider the system of two equations shown in the graph:

10

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

y

2 4 6 8 10

x

Chapter 3 Systems of Linear Equations and Inequalities 99

a How many solutions does this system of equations have?

Approach

The solution(s) to a system of equations can be represented graphically as their point(s)

of intersection.

Solution

This system has one point of intersection and therefore one solution.

Reflection

This system consists of two linear equations. Is it possible for a pair of linear equations to have

more than one solution? Is it possible for a system of two linear equations to have no solutions?

b Determine the solution to the system of equations as an ordered pair (x, y).

Solution

The point of intersection occurs at (−4, −2).

Example 2

Tyson is saving money in order to purchase a new smart-phone for $800 when the latest model

is released. He currently has $350 saved up, and is able to put away $100 each month.

a Write a system of equations to represent the situation.

Approach

To write a system of equations, we will need to define some variables. Let’s choose y to represent

an amount of money (in dollars), and x to represent the number of months that have passed.

Solution

Using these variables, the amount of money Tyson has saved over time can be represented by

y = 350 + 100x. The price of the smart-phone can be represented by y = 800.

100


b Sketch the two lines representing these equations on the coordinate plane.

Approach

All of the values involved in the question are multiples of $50, so we can use this for the scale of

the y-axis. Also, both x and y only make sense for positive values in this context, so we only need

to think about the first quadrant.

Solution

y

900

800

700

600

500

400

300

200

100

−1 1 2 3

4 5 6 7 8

x

c If the new phone is to be released in 5 months time, determine if Tyson will be able to afford

it on release.

Solution

The point of intersection on the graph occurs at (4.5, 800), meaning that in 4.5 months Tyson will

have saved $800. Therefore Tyson will have saved enough money before the phone is released.

Reflection

While the model equation of Tyson’s savings is linear, in reality he probably puts money away once

per month or once per week depending on how often he gets paid.

So although the point of intersection is at x = 4.5 months, Tyson might not actually reach $800 in

savings until the end of the 5th month.



1 Select the solution to the system of equations

on the given graph.

A (−1, −7)

4

y

B (−6, −7)

C (−7, −6)

D (1, −6)

E

No solution

−8

−6

2

−4 −2 2 4

−2

x

−4

−6

−8

2 Select the solution to the system of equations

on the given graph.

A (−3, 0)

B (−3, 5)

C (3, 5)

D

E

Infinitely many solutions

No solution

8

6

4

2

−8 −6 −4 −2 2 4 6 8

−2

−4

−6

−8

y

x

3 Solve each system of equations

graphed below:

a

4

2

−4 −2

−2

−4

−6

−8

−10

−12

−14

−16

−18

y

2 4 6 8 10 12

x

b

6 78 5

2 34 1

x

−5 −4 −3 −2 −1 −1 1 2 3 4 5

−2

−3

−4

−5

−6

−7

−8

y

102


c

y

d

y

6 7 9

8

1 2 3 4 5 6 7 8 9

2 34 5

1

−3 −2 −1 −1

−2

−3

−4

−5

−6

−7

−8

−9

x

−9−8 −7−6−5−4−3−2 −1 −1

−2

−3

−4

−5

−6

−7

−8

−9

6 7 9

8

1 2 3 4

2 34 5

1

x

4 For each of the following system of equations:

i Sketch the two lines representing the equations on the coordinate plane.

ii Solve the system of equations graphically.

a b c d

5 Consider the system of equations:

a

b

Sketch the two lines representing these equations on the coordinate plane.

How many solutions does this system of equations have?


Select the solution to this system of equations.

A (0, 0) B (−4, 0) C (−4, 8)

D No solution E Infinitely many solutions



A (0, −5) B (0, −8) C (1, −5)

D (−8, −5) E No solution


Let’s practice

8 Sarah is determining the dimensions of a rectangular frame. The length, l, of the frame is

5 cm more than the width, w. The perimeter of the frame is equal to 50 cm.

Sarah came up with the following equations to solve for the length and width of the frame:

a

Select the correct dimensions of the rectangular frame.

A l = 10 cm, w = 15 cm B l = 15 cm, w = 10 cm

C l = 15 cm, w = 20 cm D l = 20 cm, w = 15 cm

b

Is the solution viable in terms of the context?

9 The sum of two mystery numbers is 4. The difference of the two numbers is −2.

a

b

c

Write a system of equations to represent the situation.


Use the graph to determine the two mystery numbers.

10 Rochelle and Mohamad are sister and brother. Rochelle’s age is 11 more than 4 times the

age of Mohamad. The sum of their ages is 21.

a

b

c



How old are Rochelle and Mohamad?

11 Lauren and Nicole are working on an assignment together. They have broken down the work

into 9 parts of the same size. Lauren works at 2 times the speed of Nicole but has 9 pieces

of work to do for another subject.

a

b

c



How many parts of the assignment did each student do?

12 Katrina has to read a book for class and only has 8 days before the test. So far, Katrina has

read 90 pages. She is planning on reading 15 pages each day between now and the test.

The book is 215 pages long.

a

b

c



Does Katrina finish her book before the test?

104



13 Write a scenario to represent the system of equations and its solution. Explain what the

solution to the system means in terms of the scenario.

−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

y

2 4 6 8 10 12 14 16 18 20 22 24

x

14 Two equations, y 1 and y 2 represent the growth of two different house plants over time. Use

the graph of y 1 and y 2 to support the claim that the two plants will never reach the same

height on the same day.

y 2

y 1

25

20

15

10

−40 −30 −20 −10 10 20 30

−5

5

−10

−15

−20

−25

y

x

15 Describe a situation where it would be unfeasible to solve a system of equations

by graphing.



equations by substitution

Concept summary

When at least one equation in a system of equations can be solved quickly for one variable,

the system can be solved efficiently by using substitution.

Substitution Method

A method of solving a system of equations by replacing a variable in one equation with an

equivalent expression from another equation

If after solving a system of equations the result is always true, independent of the variables,

then the system has infinitely many solutions. If the result is always false, independent of

the variables, then the system has no solutions.

Worked examples

Example 1

Solve the following system of equations using the substitution method:

Approach

We first want to number our equations to make it easier to work with.

1 y = x + 11

2 y = 3x + 19

Since both equations already have y isolated, we can start by substituting equation 1 into

equation 2 to eliminate y from the equation. We can then solve for x, and substitute this value

back into one of the equations to solve for y.

106


Solution

x + 11 = 3x + 19 Substitute equation 1 into equation 2

11 = 2x + 19 Subtract x from both sides

−8 = 2x


−4 = x Divide both sides by 2

y = −4 + 11 Substitute back into equation 1

y = 7

Evaluate the addition

So the solution to the system of equations is x = −4, y = 7.

Example 2

A mother is currently 7 times as old as her son. In 3 years time, she will be 5 times as old as him.

a Write a system of equations for this scenario, where y represents the mother’s current age

and x represents the current age of her son.

Solution

1 y = 7x

2 y + 3 = 5(x + 3)

b Solve the system of equations for their ages.

Approach

Since 1 is already in the form y =

, we can substitute 1 into 2 without rearrangement.

Solution

7x + 3 = 5(x + 3) Substitute equation 1 into equation 2

7x + 3 = 5x + 15

Distribute the multiplication

2x + 3 = 15


2x = 12



y = 7(6) Substitute back into equation 1

y = 42

Evaluate the multiplication

So the mother’s age is 42 and her son’s age is 6.


c Does the solution make sense in terms of the context? Explain your answer.

Solution

Yes. 6 years earlier, when her son was born, the mother would have been 36 years old.

It makes sense that the mother is older than the son, and both values are valid ages.




A (−1, −5) B (1, 5) C (−5, −1) D (5, 1)



A (−8, −2) B (−2, 2) C (−2, −8) D (2, −2)

3 Solve the following systems of equations by substitution:

a b c

d e f

4 The total cost of 5 rulers and 3 books is $15.30. If the cost of a ruler is a and a book is

$2.70 more expensive, find a.

5 A rectangular garden bed has a perimeter of 13.8 m. The length of the garden bed is 2.5 m

longer than the width.

a Find the width of the garden bed. b Find the length of the garden bed.

6 Judy and Jorge both walk from their houses to the bus stop every morning. Judy walks

1.1 mi further, and together they walk 3.1 mi. Find the distance that Jorge walks.

108


Let’s practice

7 At a bookstore, Stella purchased a novel and a magazine for a total cost of $59.

The price of the novel, n, is $10 less than twice the price of the magazine, m.

This situation can be represented by the following system of equations:

a

Select the correct price of the novel and the magazine.

A m = $23, n = $36 B m = $26, n = $33

C m = $33, n = $26 D m = $36, n = $23

b


8 Luke bought some fresh produce from the farmers’ market. He picked up 4 oranges and

5 plums. The total cost of Luke’s fruit was $13.33. Valentina went to the same shop and

bought 1 orange and 1 plum. The total cost of Valentina’s fruit was $2.89.

a

b

Write a system of equations where f represents the price of an orange and g represents

the price of a plum.

Solve for the prices of oranges and plums at the farmers’ market.

9 A man is five times as old as his son. Four years ago the man was thirteen times as old as

his son.

a

Write a system of equations where x represents the age of the man and y represents

the age of his son.

b Solve for x and y.

c

Does the solution make sense in terms of the context? Explain your answer.

10 The number of new jobs created in Miami varies greatly each year. The number of jobs

created in 2013 was 440 000 less than double the number of jobs created in 2004.

This is equivalent to an increase of 10 000 jobs created from 2004 to 2013.

a

Write a system of equations where x represents the number of jobs in 2004 and

y represents the number of jobs in 2013.


c

Is the solution viable in terms of the context? Explain your answer.

11 Valentina has $2000 to invest, and wants to split it up between two accounts: Account A

earns 8% annual interest, while Account B earns 9% annual interest. Her target is to earn

$177 total interest from the two accounts in one year.

a

Write a system of equations x represents the amount invested in Account A and

y represents the amount invested in Account B.


c

Would you make the same investment as Valentina? Explain your answer.



12 The function f (x) = 0.47x + 8.9 represents the US annual bottled water consumption

(in billions of gallons) and the function g(x) = −0.17x + 14.2 represents the US annual soda

consumption (in billions of gallons). For both functions, x is the number of years since 2009.

a

b

Determine the year in which the bottled water and soda consumption in the U.S is

the same.

Is the solution viable in terms of the context? Explain your answer.

13 Create a scenario for each system of equations where the solution is viable in terms of

the scenario. Explain your answer.

a b c

110



equations by elimination

Concept summary

If at least one pair of variables in a system of equations can be quickly made to have the same

coefficient, or already have the same coefficient, the system can be solved efficiently by using

elimination.

Elimination Method

A method of solving a system of equations by adding or subtracting the equations until only

one variable remains

Worked examples

Example 1

Solve the following system of equations by using the elimination method:

Approach

We first want to number our equations to make it easier to work with.

1 9x + y = 62

2 5x + y = 38

Since the coefficents of y are the same with the same sign, we can start by subtracting 2 from

1 to eliminate y from the equation. We can then solve for x, and substitute this value back into

one of the equations to solve for y.

Solution

(9x + y) − (5x + y) = 62 − 38 Subtract equation 2 from equation 1

4x = 24

Combine like terms



5(6) + y = 38 Substitute back into equation 2

30 + y = 38 Evaluate the multiplication

y = 8


So the solution to the system of equations is x = 6, y = 8.

Reflection

We could also have solved this by subtracting 1 from 2 . We could also have substituted x = 6

back into equation 1 .

(5x + y) − (9x + y) = 38 − 62 Subtract equation 1 from equation 2

−4x = −24

x = 6

Combine like terms


9(6) + y = 62 Substitute back into equation 1

54 + y = 62 Evaluate the multiplication

y = 8

Subtract from both sides

It doesn’t matter which way we choose, but one method will usually be preferable. In this case we

didn’t have to deal with negative numbers when using the first approach.

Example 2

When comparing test results, Verna noticed that the sum of her Chemistry and English test scores

was 128, and that their difference was 16. She scored higher on her Chemistry test.

a Write a system of equations for this scenario, where x represents Verna’s Chemistry test score

and y represents her English test score.

Approach

Since we know that Verna’s Chemistry test score is higher than her English test score, we should

express the difference of the two scores as x − y to get a positive result.

Solution

1 x + y = 128

2 x − y = 16

112


b Solve the system of equations to find her test scores.

Approach

Since the equations have the same coefficient of y with opposite sign, we can add the two

equations to eliminate y and solve for x first.

Solution

(x + y) + (x − y) = 128 + 16 Add equation 1 and equation 2 together

2x = 144

Combine like terms


72 + y = 128 Substitute back into equation 1

y = 56


So Verna scored 72 on her Chemistry test and 56 on her English test.

Reflection

In this case, the two equations also had the same coefficient of x with the same sign. So we could

have instead approached this by subtracting the two equations and solving for y first.

c Does the solution make sense in terms of the context? Explain your answer.

Solution

Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain.


1 Consider the following system of equations:

a Describe how to use multiplication to eliminate the variable y.

b Now, combine the two equations using the method you explained.


a What value can we multiply each equation by in order to eliminate the variable y?





A (1, 2) B (−3, 4) C (2, 1) D (4, −3)



A (−3, 1) B (5, 9) C (1, −3) D (9, 5)

5 Solve each system of equations using the elimination method:

a b c d

e f g h

Let’s practice

6 Solve each system of equations using the elimination method:

a b c d

7 The train in an amusement park has 15 cabs that can hold a total of 72 people. Some cabs

hold 4 people while some hold 6 people.

Let x be the number of four-passenger cabs and y be the number of six-passenger cabs.


a

Select the correct number of four-passenger cabs and six-passenger cabs.

A x = −2, y = 17 B x = 3, y = 12 C x = 12, y = 3 D x = 17, y = −2

b


114


8 When comparing her test, results Judy noticed that the sum of her Geography test score

and Math test score was 137, and that their difference was 29. Judy scored higher on her

Geography test than her math test.

a

b

c

Write a system of equations where x represents Judy’s Geography score and y

represents her Math score.

Solve for Judy’s Geography score using the elimination method.

Now, solve for Judy’s Math score.

9 12 pens and 5 rulers cost $70 while 3 pens and 25 rulers cost $65.

a

b

c

Write a system of equations where x represents the price of a pen and y represents the

cost of a ruler.

Solve for the price of a pen using the elimination method.

Now, solve for the price of a ruler.

10 Fred spent $55.25 to purchase 9 flowers. He bought rhododendrons which cost $6.45 each

and chrysanthemums which cost $5.75 each.

a

b

If R is the number of rhododendrons and C is the number of chrysanthemums that Fred

bought, contruct two equations describing the total number of flowers bought and the

total amount spent in dollars.

Solve for the number of rhododendrons and chrysanthemums that Fred purchased.



Explain how to rewrite the system of equations so that it has integer coefficients.

12 Ricardo solved the following system of equations used to model wildlife populations in a

wildlife sanctuary, where r represents the number of rhinos and h represents the number of

hippopotami.

a

b

Explain why the solution to the system of equations is not viable.

Explain how you might interpret the solution to the system of equations to make sense in

terms of the context.


3.04 Systems of linear

inequalities

Concept summary

A system of inequalities is a set of inequalities which have the same variables.

The solution to a system of inequalities is the set containing any ordered pair that makes all of the

inequalities in the system true.

A solution can also be represented graphically as the region of the plane of the plane that satisfies

all inequalities in the system.

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

The solution to a system of inequalities in a given context is viable if the solution makes sense in

the context, and is non-viable if it does not make sense.

Worked examples

Example 1

Consider the following system of inequalities:

116


Sketch a graph of the solution set to the system of inequalities.

Solution

To sketch the system of inequalities we can first construct the boundary lines for each inequality,

namely y = 3 and y = 4x + 5. When given a strict inequality we will draw a dashed line.

When given a nonstrict inequality we will draw a solid line.

9

8

7

6

5

4

3

2

1

−4 −3 −2 −1 −1

y

1 2 3 4

x

−2

−3

−4

−5

−6

To determine which side of each inequality will be shaded, we can choose some test points that

satisfy each inequality. The test points will indicate which side of the inequality will be shaded.

For y ≤ 3 we will plot (−2, 2) and (4, 0). For y < 4x + 5 we will plot (−2, 5) and (−3, −3).

9

8

7

6

5

4

3

2

1

−4 −3 −2 −1 −1

y

1 2 3 4

x

−2

−3

−4

−5

−6


The region that will be shaded is the region which

satisfies both inequalities.

Using the test points we can see that the shading

will occur below the boundary line for y ≤ 3 and to

the left of y < 4x + 5.

9

8

7

6

5

4

3

2

1

−4 −3 −2 −1 −1

y

1 2 3 4

x

−2

−3

−4

−5

−6

Reflection

Since each inequality in the system was already written in terms of y, it would have been possible

to determine the direction of the shading without first plotting the test points.

With some systems of inequalities written in terms of x or in general form, however, it can be less

intuitive to know which direction to shade.

Example 2

Applicants for a particular university are asked to sit a quantitative reasoning test and verbal

reasoning test. Successful applicants must obtain a minimum score of 14 on a quantitative

reasoning test and a minimum combined score of 29 for both tests.

a Write a system of inequalities for this scenario, where x represents the quantitative reasoning

test score and y represents the verbal reasoning test score.

Approach

Since we know that the minimum accepted score for quantitative reasoning is 14, we can

represent this with an inequality showing 14 as the lowest possible solution. A minimum combined

score of 29 means the total of the two scores must sum to 29 or more.

Solution

118


b. Sketch a graph of the system of inequalities.

Solution

30

y

25

20

15

10

5

5 10 15 20 25 30

x

c. Does the solution (15, 22. ) make sense in terms of the context? Explain your answer.

Solution

No. This would mean that the score for the quantitative reasoning test was 15 and the verbal

reasoning test 22. . By viewing the graph we can see that this point technically satisfies both

inequalities, but a test score is typically a positive integer value or a simple fraction such as

or

and not a non-terminating decimal.



1 Select the graph of the solution set of:

A

10

y

B

10

y

8

8

6

6

4

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−4

−6

−6

−8

−8

−10

−10

C

10

y

D

10

y

8

8

6

6

4

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−4

−6

−6

−8

−8

−10

−10


A

10

y

B

10

y

8

8

6

6

4

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−4

−6

−6

−8

−8

−10

−10

120


C

10

y

D

10

y

8

8

6

6

4

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−4

−6

−6

−8

−8

−10

−10


A

10

y

B

10

y

8

8

6

6

4

4

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−10 −8 −6 −4 −2 −2

2

2 4 6 8 10

x

−4

−4

−6

−6

−8

−8

−10

−10

C

30

y

D

30

y

24

24

18

18

12

12

−10 −8 −6 −4 −2 −6

6

2 4 6 8 10

x

−10 −8 −6 −4 −2 −6

6

2 4 6 8 10

x

−12

−12

−18

−18

−24

−24

−30

−30



A

10

8

6

4

2

−10 −8 −6 −4 −2 −2

y

2 4 6 8 10

x

B

10

8

6

4

2

−10 −8 −6 −4 −2 −2

y

2 4 6 8 10

x

−4

−6

−8

−10

−4

−6

−8

−10

C

10

8

6

4

2

−10 −8 −6 −4 −2 −2

y

2 4 6 8 10

x

D

10

8

6

4

2

−10 −8 −6 −4 −2 −2

y

2 4 6 8 10

x

−4

−6

−8

−10

−4

−6

−8

−10

5 Graph each of the following system of inequalities on the coordinate plane:

a b c d

6 Write the system of inequalities that is represented on each graph:

a

y

12

10

8

6

4

2

−10 −8 −6 −4 −2 −2

2 4 6 8 10

−4

−6

−8

b

x

5

4

3

2

1

−5 −4 −3 −2 −1

−1

−2

−3

−4

−5

y

1 2 3 4 5

x

122


Let’s practice

7 Consider the graph of

a

b

If the system was changed to

graph the new system of inequalities on

a coordinate plane.

Is (−6, −8) a solution to the system in

part (a)?

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

−12

y

2 4 6 8 10

x

8 Consider the graph of

a

b

If the system was changed to

graph the new system of inequalities on

the coordinate plane.

Is (0, 6) a solution to the system in part (a)?

9 Consider the following graphs:

Graph 1 Graph 2

y

12

10

8

6

4

−6 −4 −2

−2

−4

−6

−8

−10

2 4 6 8 10 12

x

12

10

8

6

4

−6 −4 −2

−2

−4

−6

−8

−10

y

12

10

8

6

2

−12 −10 −8 −6 −4 −2

−2

4

−4

−6

−8

y

2 4 6 8

2 4 6 8 10 12

x

x


Graph 1 shows the solution set to the system:

a

b

If Graph 2 shows the solution set when one of the inequalities in the system is changed,

state the new system of inequalities.

Is (3, 0) a solution to either system? Explain how you know.

10 Sheila sells popcorn to raise money during summer break. The popcorn comes in two

flavors, classic butter which costs $2 each and cheese which costs $3 each. Sheila needs to

sell at least $50 worth of popcorn with at least 8 of the cheese popcorn.

Let b be the number of butter popcorn and c be the number of cheddar popcorn.


Select all the viable solutions.

A b = 6, c = 13 B b = 10, c = 10 C b = 11, c = 9

D b = 13, c = 8 E b = 18, c = 6

11 Yreka Bakery makes two types of cookies: plain and iced. They have enough oven space

to bake 15 dozen cookies each day. Each dozen iced cookies requires 0.8 pounds of icing.

Yreka Bakery can make no more than 32 pounds of icing per day.

If x is the number of dozens of iced cookies and y is the number of dozens of plain cookies,

write a system of inequalities to represent the scenario.

12 Throughout university, Jimmy works as a barista, getting paid $10 per hour, and as a mentor

getting paid $13 per hour. The number of hours he works in each job can vary from week to

week, but he never works more than 27 hours in total each week, and he needs to be able

to at least cover his weekly expenses of $260.

If b represents the number of hours worked as a barista and m represents the number of

hours worked as a mentor, write a system of inequalities to represent the scenario.


13 David’s Pizza makes two types of pizzas: Vegan and Meat Lover’s. Based on recent sales

they know they will sell at least twice as many Meat Lover’s Pizzas as they do Vegan Pizzas.

David’s Pizza can use 42 pounds of vegan cheese each day, and each Vegan Pizza requires

0.6 pounds of vegan cheese.

a

b

c

If v is the number of Vegan Pizzas and m is the number of Meat Lover’s Pizzas, write a

system of inequalities to represent the scenario.

Is (28, 70) a viable solution in terms of the scenario? Explain.

Is (25.25, 71) a viable solution in terms of the scenario? Explain.

124


14 In a team crossfit competition, each team is to be made up of no more than 11 people. For a

particular challenge, each team member must complete the course and the team’s total time

in completing the course must be under 26 minutes. Women take on average 2.6 minutes

and men take on average 2.5 minutes to complete the course.

a

b

c

If w is the number of women and m be the number of men on the team, write a system

of inequalities to represent the scenario.

Graph the system of inequalities.

Give an example of one viable and one non-viable solution to the number of men and

women on the team. Explain your answer for both.

15 Throughout university, Tom works as a mentor, getting paid $15 per hour, and as a fitness

instructor getting paid $18 per hour. The number of hours he works in each job can vary

from week to week, but he never works more than 29 hours in total each week, and he

needs to be able to at least cover his weekly expenses of $270.

a

b

c

d

If x represents the number of hours Tom works as a mentor and y represents the number

of hours he works as a fitness instructor, construct a system of inequalities to represent

the scenario.

Graph the system of inequalities.

If Tom works 6 hours as a mentor in one week, what is the minimum number of hours he

can work as a fitness instructor so that he can cover his expenses?

Is it possible for him to work the same number of hours in both jobs and still be able to

cover his expenses?


4 Functions

and Linear

relationships

Chapter outline

4.01 Evaluating functions 128

4.02 Domain and range 132

4.03 Average rate of change 140

4.04 Characteristics of functions from a graph 146

4.05 Linear relationships 153

4.06 Transforming linear functions 162

4.07 Characteristics of linear functions 172

4.08 Comparing linear and nonlinear functions 183

4.09 Linear absolute value functions 189

4.10 Transforming absolute value functions 194

4.01 Evaluating functions

Concept summary

A mathematical relation is a mapping from a set of input values, called the domain, to a set of

output values, called the range. A relation can also be described as a set of input-output pairs.

Input

The independent variable of a relation; usually the x-value

Output

The dependent variable of a relation; usually the y-value

Relations in general have no further restrictions than mapping domain elements to range elements.

By adding the restriction that each input value maps to exactly one output value, we define

a particularly useful type of relation, called a function.

Function

A relation for which each element of the domain corresponds to exactly one element of

the range

Functions are usually written using a particular notation called function notation: for a function f

when x is a member of the domain, the symbol f ( x ) denotes the corresponding member of

the range.

To evaluate a function at a point is to calculate the output value at a particular input value.

Worked examples

Example 1

Consider the equation

where x is the independent variable.

x − 3y = 15

128


a Rewrite the equation using function notation.

Approach

Since x is the independent variable, we want to rearrange the equation to isolate y, and then

replace y with function notation. We can choose a symbol to represent the function, such as f.

Solution

Rearranging the equation:

−3y = −x + 15



We can now rewrite the equation using function notation as

b Evaluate the function when x = 9.

Solution

Substituting x = 9 we have


1 Consider the function y = 2x − 3.

a State the independent variable. b State the dependent variable.

2 For a function f, what does f ( 3 ) represent?

3 Suppose that ( 7, −6 ) is an ordered pair that satisfies the function g. Write this situation using

function notation.

4 Suppose that the value of the function f is 22 when x = 7. Write this situation using

function notation.

5 Write each of the following equations using function notation:

a y = 2x b y = −x + 7 c y = x 2 + x + 1 d y = 12

Chapter 4 Functions and Linear relationships 129

Let’s practice

6 Assume that y is the dependent variable. Write each of the following equations using

function notation:

a 3x + 4y = 0 b y − 7 = 3( x + 2 ) c 4x + 2y = 18 d xy = 1

7 For each of the following functions, evaluate the function outputs when:

i x = 2 ii x = −3 iii x = a + 1

a f ( x ) = 4x + 1 b g( x ) = ( x + 1)( x + 2)

c h( x ) = x 2 − 3x − 2 d

8 For each of the following functions, identify the effect of the output of the function as:

i the domain increases. ii the domain decreases.

a b f ( x ) = −5

c

x 0 1 2 3 4

d

x −2 −1 0 1 2

y 7 5 3 1 −1

y −2 1 2 1 −2

9 Consider the functions f ( x ) = 7x + 2 and g( x ) = 15 − 2x. Find the value of the following:

a f ( 1) + g( 2 ) b f ( −2 ) − g( 7 ) c f ( 0 ) ⋅ g( 0 ) d

10 Consider the function f ( x ) = x 2 + 2x + k.

a Simplify the expression for f ( 2 ). b If f ( 2 ) = 15, find the value of k.

11 Han is painting a mural which requires gallons of paint per square foot. Han also needs

3 additional gallons of paint to go over the outline of the mural once they are finished.

a

b

c

State the dependent and independent variables in this scenario.

Express the relationship between the independent and dependent variables of the

scenario using function notation, letting x represent the square footage of the mural.

Find the number of gallons of paint that Han will need if the mural has a size of

24 square feet.

130



12 The model P ( t ) = 3500 × 2 t describes the population of fish in a lake after t years,

where t = 0 is the initial time when the population of fish was first recorded.

Find and interpret the value of P ( 2 ).

13 Sharon works at a pizza shop and makes $15 per hour. Her wages can be represented by

the function W ( t ) = 15t, where t represents the number of hours that she works.

a

b

c

Explain what the expression W ( 19 ) represents.

Explain what the equation W ( n ) = 165 represents.

Find her wages after working 26 hours.

14 Mindy can model her tomato plant’s height in inches over time using the function h( t ) = 2 t ,

where t is the number of weeks after Mindy planted it.

a

b

Explain what the equation h( 4 ) = 16 represents.

Find the height of Mindy’s tomato plant after 6 weeks.

15 Consider the function f ( x ) = ax + b. If f ( 4 ) = 7 and f ( −1) = −3, find the values of a and b.

16 Describe a scenario that can be represented by the function


4.02 Domain and range

Concept summary

Two defining parts of any function are its domain and range.

The set of all possible input values ( x-values ) for

a function or relation is called the domain.

In the example shown, the domain is the interval of

values −3 < x ≤ 1. Notice that −3 is not included in

the domain, which is indicated by the unfilled point.

−3

Domain

−2

−1

2

1

−1

y

1 2

x

−2

−3

−4

−5

The set of all possible output values ( y-values ) for a function or relation is called the range.

In the example shown, the range is the interval of

values −4 ≤ y ≤ 0. Notice that both endpoints are

included in the range, since the function reaches

a height of y = 0 at the origin.

2

1

y

x

−3

−2

−1

−1

1 2

−2

Range

−3

−4

−5

132


A domain which is made up of disconnected values is said to be a discrete domain.

A function with a discrete domain.

It is only defined for distinct x-values.

y

x

A domain made up of a single connected interval of values is said to be a continuous domain.

A function with a continuous domain.

It is defined for every x-value in an interval.

y

x

It is possible for the domain of a function to be neither discrete nor continuous.

The domain and range of a function are commonly expressed using inequality notation or

set-builder notation.

Note that if two functions have different domains, then they must be different functions,

even if they take the same values on the shared parts of their domains.


Worked examples

Example 1

Consider the function shown in the graph.

8

y

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

2 4 6 8

x

a State whether the function has a discrete or continuous domain.

Solution

The function is defined at every value of x across an interval, so it has a continuous domain.

b Determine the domain of the function using set-builder notation.

Solution

We can see that the function is defined for every x-value between −6 and 8, including −6 but not

including 8.

So the domain of the function can be written as

Domain: {x | − 6 ≤ x < 8}

134


c Determine the range of the function using set-builder notation.

Solution

We can see that the function reaches every y-value between −6 and 4, including 4 but not

including −6.

So the range of the function can be written as

Range: {y | − 6 < y ≤ 4}


1 State the definition for the:

a Domain b Range

2 State when the domain of a function is:

a Continuous b Discrete


i

ii

a

State the domain of the relation using inequality notation.

State the range of the relation using inequality notation.

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x

b

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x

c

8

y

d

8

y

6

6

4

4

2

x

2

x

−8 −6 −4 −2

−2

2 4 6 8

−8 −6 −4 −2

−2

2 4 6 8

−4

−4

−6

−6

−8

−8



i State the domain of the relation using set builder notation.

ii State the range of the relation using set builder notation.

a

8

y

b

8

y

6

6

4

4

2

x

2

x

−8 −6 −4 −2

−2

2 4 6 8

−8 −6 −4 −2

−2

2 4 6 8

−4

−4

−6

−6

−8

−8

c

8

y

d

8

y

6

6

4

4

2

x

2

x

−8 −6 −4 −2

−2

2 4 6 8

−8 −6 −4 −2

−2

2 4 6 8

−4

−4

−6

−6

−8

−8

5 For each of the following graphs, identify whether the domain of the function is discrete

or continuous:

a

−4 −3 −2

4

3

2

1

−1

−1

−2

−3

−4

y

1 2 3 4

x

b

−4 −3 −2

4

3

2

1

−1

−1

−2

−3

−4

y

1 2 3 4

x

136


Let’s practice

6 State the domain and range of the collection

of points shown on the graph.

8

6

4

y

2

−8 −6 −4 −2

−2

−4

−6

−8

2 4 6 8

x

7 Determine for each graph whether or not it has a domain consisting of all the real numbers.

If it does not, state the domain.

a

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x

b

14

12

10

8

6

4

2

−8 −6 −4 −2

−2

y

2 4 6 8

x

8 Determine for each graph whether or not it has a range consisting of all the real numbers. If it

does not, state the range.

a

10

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

y

2 4 6 8

x

b

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x


9 For each of the following:

i Identify an independent variable and a dependent variable.

ii State the domain and range for the scenario.

a

b

c

Uma earns between $200 and $450 selling between 35 and 50 burritos, inclusive.

Jermaine starts reading his book from page 112. Over the course of 17 days,

he reads up to and including page 237.

Brigid Kosgei completes a 26.2 mile marathon in 134 minutes.

10 State the domain and range for each of the following graphs:

a

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

−10

y

x

2 4 6 8 10

b

−4

8

6

4

2

−2

−2

−4

−6

−8

y

2 4 6 8 10

x


11 The graph of a function is shown.

a

State the range of the function.

125

y

b

A ball is thrown from an apartment

window in a high-rise building. The height

of the ball above ground over time can

be modelled by the function shown in

the graph, where the ball is thrown at

x = 0. State the range of the function for

this context.

100

75

50

25

x

−6

−4

−2

2 4 6 8 10

−25

138


12 A comet is travelling through the solar system. It passes by the Earth before curving around

the Sun and heading back out into deep space. The distance of the comet from Earth, y, is

tracked by satellite telescopes and expressed as a function of time, x, since the comet was

first spotted.

a

b

State the domain of this function.

The satellites detect that, at its closest point, the comet was a distance of 0.14 AU

( astronomical units ) from Earth. State the range of the function.

13 A function describing the strictly increasing relation between temperature and a person’s

average resting heart rate has a domain of 50 ≤ T ≤ 105 ( Fahrenheit ) and a range of

40 ≤ f ( T ) ≤ 90 ( beats per minute ), based on experimental results.

a Determine the average resting heart rate at a temperature of 50 °F .

b

c

State the temperature that is expected if a person has an average resting heart rate of

90 beats per minute.

Explain whether or not the function output f ( 10 ) be a reliable estimate for the

corresponding real life scenario.


4.03 Average rate

of change

Concept summary

The average rate of change of a function over an interval is the change in value of the dependent

variable per unit change in the independent variable.

The average rate of change of a function can be calculated by dividing the change in function

values between the start and the end of the interval by the length of the interval.

For the function shown, the function values increase

from 3 to 7 over the interval 1 ≤ x ≤ 5.

This means that over the interval 1 ≤ x ≤ 5,

the function has an average rate of change of

We can see this graphically as well: the dashed lines

show an increase of 1 unit in the y-values per unit

increase in the x-values over this interval.

8

7

6

5

4

3

2

y

Worked examples

1

1 2 3

4 5 6 7 8

x

Example 1

A flock of birds migrate to a new island.

The population of birds on that island over the

next six years is shown on the graph.

60

50

40

y

30

20

10

1 2 3

4 5 6

x

140


Determine the average rate of change of the bird population over the six year period.

Approach

To find the average rate of change, we want to divide the difference between the initial and final

populations by the length of the time period.

Solution

Looking at the graph we can see that the initial population is 15 birds, and after 6 years the

population is 60 birds.

So we can calculate the average rate of change as

Therefore, the average rate of change is 7.5 birds per year.

Example 2

Consider the function f ( x ) = 2x 2 − 1.

Calculate the average rate of change over the interval −4 ≤ x ≤ 0.

Approach

The average rate of change can be thought of as:

To determine the change in f (x), we want to identify the values of f (x) when x = −4 and when x = 0.

Solution

We know that

So we have that

f (x) = 2x 2 − 1

f (−4) = 2(−4) 2 − 1

= 2 ⋅ 16 − 1

= 31

and

f (0) = 2(0) 2 − 1

= −1


Going back to the idea that the average rate of change can be thought of as

we have:


1 State the definition for the average rate of change over an interval.

2 Consider the interval from the point ( x 1 , y 1 ) to ( x 2 , y 2 ).

a State the change in the dependent variable over the interval.

b State the change in the independent variable over the interval.

c State the average rate of change over this interval.


i Find the value of y when x = 2.

ii Find the value of y when x = 7.

iii Find the average rate of change of the equation over the interval 2 ≤ x ≤ 7.

a y = 2x + 3 b y = 4x − 10 c y = −x + 8 d y = −7x + 2

4 Stanton went for a run. The following

graph shows the distance in meters

covered by Stanton at x minutes. Is the

distance’s average rate of change

between 20 and 30 minutes positive,

negative, zero or undefined?

1400

1200

1000

800

Distance

600

400

200

5

10 15 20 25 30 35 40 45

Time

142


5 Tori is controlling a drone for a video shoot.

The following table shows the height ( in feet ) of

the drone x minutes after starting its descent.

Is the height’s average rate of change between

0 and 6 minutes positive, negative, zero or

undefined?

Time in minutes Height in feet

0 140

2 135

4 132

6 126

8 120

6 A clock factory’s weekly profit, in dollars, can be modeled by the equation

y = −1875 + 150x − x 2 where x is the selling price of a clock in dollars. Is the profit’s average

rate of change between the $40 selling price and the $110 selling price positive, negative,

zero or undefined?

7 The graph shows Yoichi’s distance from

home over a 10-minute interval.

Is the distance’s average rate of change

between 3 and 8 minutes positive, negative,

zero or undefined?

6

5

4

Distance from home

8 The population of river otters in a particular

area can be modelled by the equation

y = 15 + 5 x , where x is the number of years

from now.

3

2

a Find the value of y when x = 0.

b Find the value of y when x = 2.

c

Determine the population’s average rate

of change between x = 0 and x = 2.

1

1

Time

2 3 4 5 6 7 8 9 10

9 The side profile of a bridge

is overlapped with the coordinate axes

such that an ordered pair ( x, y ) represents a

point on the arch of the bridge that is

x meters horizontally from one end of the

bridge and y meters vertically above the

road. The bridge meets the road at

coordinates ( 0, 0 ) and ( 100, 0 ). Pedestrians

are able to climb the full arch of the bridge.

Find the average steepness of the climb

between x = 10 and x = 60.

54

48

42

36

30

24

18

12

y

(10, 18)

(60, 48)

6

x

20

40 60 80 100


Let’s practice

10 Consider the input and output pairs of a function shown in the table.

x 2 7 8 12

y 5 15 16 36

a Find the average rate of change between x = 2 and x = 7.

b Find the average rate of change between x = 7 and x = 8.

c Find the average rate of change between x = 8 and x = 12.

d Determine whether or not the given points display a constant rate of change.

11 The graph shows the price, in dollars,

of a new digital toy for different levels

of supply.

a

b

Find the average rate of change.

Interpret the average rate of change

in the given context.

120

100

80

Price

60

40

20

Supply (thousands of units)

5

10 15 20

12 The graph shows the total number of

tickets, in thousands, purchased for a

premiering movie, with pre-ordered tickets

counted as being purchased on the

0th day.

a

b




25

20

15

10

Tickets (thousands)

5

Days

2

4 6

8 10 12

144


13 The graph shows the number

of earthquakes that a particular country

experiences in each year.

a

b




10

8

6

Number of earthquakes

4

2

Year

1

2 3

4 5

14 The path of water projected from a fountain can be modelled by the equation

y = −20x 2 + 80x, where x is the horizontal distance from the nozzle and y is the height.

Find the average rate of change of the water’s height from when it is projected to when it is

two meters horizontally from the nozzle.

15 The value of a particular painting can be modeled by the equation y = 100 ⋅ 2 x , where x is

the number of years from when it was painted. If the painting was made in 2009, find the

average rate of change of its value between 2010 and 2015.


16 Sarah and William are finding an expression for the average rate of change of a function

f ( x ) between x = a and x = b.

a

b

Sarah states that, to find the average rate of change, we find the total change in the

function values and divide it by the difference between the x-values. Use Sarah’s method

to find an expression for the average rate of change.

William claims that finding the average rate of change between two points is the same as

finding the slope of the straight line that passes through those two points. Is he correct?

17 Explain why the average rate of change of a linear equation y = mx + b over any interval is

always equal to the slope m.


4.04 Characteristics of

functions from a graph

Concept summary

The important characteristics, or key features, of a function or relation include its

• domain

• range

• x-intercepts

• y-intercepts

• maximum value ( the highest output value )

• minimum value ( the lowest output value )

• rate of change over specific intervals

Key features of a function are useful in helping to sketch the function, as well as to interpret

information about the function in a given context. Note that not every function will have each type

of key feature. The rate of change of a function over a specific interval can be broadly categorized

by one of the following three discriptions:

A function is increasing over an interval if, as the input values become higher, the output values

also become higher.

A function is decreasing over an interval if, as the input values become higher, the output values

become lower.

A function is constant over an interval if, as the input values become higher, the output values

remain the same.

4

3

y

Constant

2

1

−4 −3 −2 −1

−1

Increasing −2

−3

−4

1 2 3 4

Decreasing

x

146


Worked examples

Example 1

Consider the function shown in the following graph:

y

14

12

10

8

6

4

2

−2 −1 1 2

−2

−4

−6

3

4 5 6 7 8

x

a Identify whether the function has a maximum or minimum value, and state this value.

Solution

This function has a minimum value of −4.

b State the range of the function.

Approach

In part (a) we identified that the function has a minimum value of −4. So we know that the function

can’t take values smaller than −4.

Solution

Looking at the function, we can see that it stretches up towards infinity on both sides of

the minimum point. So the function can take any value greater than or equal to −4. That is,

the range of the function is

Range: {y | y ≥ −4}


c State the x-intercept( s ) of the function.

Approach

The x-intercept(s) of a function are the points where the function crosses the x-axis. In this case,

by looking at the graph we can see that there are two x-intercepts.

Solution

The x-intercepts of this function are the points (1, 0) and (5, 0).

d Determine the largest interval over which the function is increasing.

Approach

In part (a) we identified that the function has a minimum value. Looking at the graph,

we can see that the function values are decreasing on the left side of the minimum and increasing

on the right side.

Solution

The minimum point occurs at x = 3. The function values are increasing for all x-values to the right of

this minimum. That is to say, the function is increasing for x > 3.


1 State the definition for the following function characteristics:

a Maximum b Minimum c x-intercept d y-intercept

2 State the definition for when a function over an interval is:

a Increasing b Decreasing c Constant

Let’s practice

3 For the following graphs of the function y = f ( x ):

i Identify whether the function has a maximum or minimum and state its value.

ii State the range of the function.

iii State the interval over which the function is increasing.

iv State the interval over which the function is decreasing.

148


a

10

y

b

4

y

8

6

4

2

−4 −2

−2

x

2 4 6 8 10 12

2

−6 −4 −2

−2

2 4 6 8 10

x

−4

−6

−8

−4

−10

−6

−12

c

16

y

d

8

y

12

6

8

4

4

x

2

x

−10 −8

−6 −4 −2 2 4

−4

−8

−6

−4 −2 2 4 6

−2

−8

−4

−12

−6

4 For each of the following graphs, identify the interval( s ) of the domain where the function is:

i Increasing ii Decreasing iii Constant

a

4

y

b

8

y

3

6

2

4

1

x

2

x

−5 −4 −3

−2 −1

−1

1 2 3 4 5

−8 −6 −4 −2

−2

2 4 6 8

−2

−4

−3

−6

−4

−8


c

7

y

d

4

y

6

3

5

2

4

3

2

−5 −4 −3

1

−2 −1

−1

1 2 3 4 5

x

−4 −3 −2

1

−1

−1

1 2 3 4

x

−2

−3

−4

5 For the following functions:

i State the coordinates of the x-intercept. ii State the coordinates of the y-intercept.

a

5

4

3

2

1

−5 −4 −3 −2 −1 −1

y

1 2 3 4 5

x

b

10

8

6

4

2

−5 −4 −3 −2 −1 −2

y

1 2 3 4 5

x

−2

−3

−4

−5

−4

−6

−8

−10

6 Consider the functions shown on the following graphs:

i State the coordinates of the x-intercepts.

ii State the coordinates of the y-intercept.

iii As x → ∞, describe the end behavior of the corresponding y-values.

iv As x → −∞, describe the end behavior of the corresponding y-values.

a

12

y

b

10

y

8

8

4

−6 −4 −2

−4

−8

−12

2 4 6 8 10

x

−4 −3 −2

6

4

2

−1

−2

1 2 3 4

x

−16

−4

−20

−6

150


7 For each graph, state the following:

i Domain ii Range iii x- and y-intercepts iv Rate of change

a

2

y

b

7

y

1

−2 −1 1 2

−1

3 4 5 6

x

6

5

4

−2

3

−3

2

−4

−5

−6

1

−1 1 2 3

−1

4 5 6 7

x

c

5

y

d

4

y

4

3

3

2

−4 −3 −2

2

1

−1

−1

1 2 3 4

x

−3 −2

1

−1

−1

−2

1 2 3 4 5

x

−2

−3

−3

−4


8 The graph shows the height ( in meters )

of a soccer ball against the time ( in

seconds ) that has passed after it has been

kicked.

12

10

Height

a

b

c

Find the coordinates of the y-intercept.

Interpret the y-value of the y-intercept in

this context.

Find the coordinates of the x-intercept.

8

6

d

Interpret the x-value of x-intercept in

this context.

4

e

Determine the average rate of change

of the height of the soccer ball.

2

Time

1

2 3 4


9 Two construction workers are competing

to see who can lay the most bricks in one

hour. The graph shows the number of

bricks layed and the time, in minutes.

320

280

Bricks layed

a

b

Interpret the rate of change of

the given lines.

Determine who can lay more bricks

in 60 minutes.

240

200

160

Charlie

Neville

c

Explain why the y-intercept of both

lines is 0.

120

80

40

Minutes

10

20 30 40

50 60 70 80

10 Alicia has concluded that the rate of change

is greater in f ( x ) than g( x ) because it is higher

on the given graph. Explain and correct

her error.

5

4

3

y

2

1

x

−5 −4 −3 −2

f(x)

−1

−1

−2

1 2 3 4 5

g(x)

−3

−4

−5

152


4.05 Linear relationships

Concept summary

A function that has a constant rate of change is called a linear function. Linear functions can be

written in the form:

f ( x ) = mx + b

m the slope of the line

b the y-value of the y-intercept

The graph of a linear function is a straight line.

In general, any relationship which has a constant rate of change is a linear relationship.

A relationship which does not have a constant rate of change is called a nonlinear relationship.

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

A linear relationship

A nonlinear relationship

Worked examples

Example 1

Emanuel is selling raffle tickets to raise money for charity. The table below shows the cumulative

number of tickets he has sold each hour for the first three hours:

Time ( hours ) 1 2 3

Total ticket sales 14 28 42


a State whether Emanuel’s ticket sales represent a linear or nonlinear function.

Approach

A linear function will have a constant rate of change. We can compare the values in the table and

see how much the total ticket sales are increasing by each hour.

Solution

Emanuel sells 14 tickets in the first hour. He then sells 28 − 14 = 14 tickets in the second hour,

and 42 − 28 = 14 tickets in the third hour.

So the rate of change is constant and therefore the ticket sales represent a linear function.

b Determine the rule which relates Emanuel’s ticket sales and time.

Solution

From part (a) we know that Emanuel is able to sell 14 additional raffle tickets each hour.

c If Emanuel’s ticket sales continue in this way, determine the total number of tickets he will have

sold after 6 hours.

Solution

From part (b) we know that Emanuel is selling 14 tickets per hour. So after 6 hours, if the pattern

stays the same, he will have sold 14 ⋅ 6 = 84 raffle tickets.

Example 2

Tiles were stacked in a pattern as shown:

Stack 1 Stack 2 Stack 3 Stack 4

A table of values representing the relationship between the height of the stack and the number of

tiles was partially completed.

Height of Stack 1 2 3 4 5 10 100

Number of tiles 1 3

154


a Identify the pattern for a relationship represented between the height of a stack and

the number of tiles.

Solution

The number of tiles from one stack to the next increases by 2. The number of tiles can also be

determined by taking the height of the stack, multiplying by 2, then subtracting 1.

b Complete the table of values representing the relationship between the height of the stack and

the number of tiles.

Solution

We can use the pattern identified in part (a) to help us fill in the table.

Height of Stack 1 2 3 4 5 10 100

Number of tiles 1 3 5 7 9 19 199

c Identify if the relationship between the stack height and number of tiles is linear or not.

Solution

There is a constant rate of change in the table of values and the pattern described a constant

increase, so the relationship is linear.


1 State whether each of the following graphs represents a linear or nonlinear relationship:

a

−4 −3 −2

4

3

2

1

−1

−1

−2

−3

−4

y

1 2 3 4

x

b

−4 −3 −2

4

3

2

1

−1

−1

−2

−3

−4

y

1 2 3 4

x


c

4

y

d

4

y

3

3

2

2

1

x

1

x

−4 −3 −2

−1

−1

1 2 3 4

−4 −3 −2

−1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

e

4

y

f

4

y

3

3

2

2

1

x

1

x

−4 −3 −2

−1

−1

1 2 3 4

−4 −3 −2

−1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

2 Consider the given table of values and

its graph:

Time

Revenue

1 −4

2 3

3 10

4 17

5 24

24

20

16

12

8

4

−4

Revenue

Time Period

1 2 3 4 5

a

b

Determine if the revenue is changing at a constant rate with respect to the time period.

State if the relationship between time and revenue is linear.

156


3 The number of goldfish in a pond over a six-month period is recorded in the table and

plotted in a graph:

Number of fish

Number of

months

Number of

fish

200

180

a

b

1 6

2 12

3 24

4 48

5 96

6 192

Determine if the the number of fish

is changing at a constant rate with

respect to the number of months.

State if the relationship between the number of months and the number of fish is linear.

4 The growth of a potted plant over a week is recorded in the table below, with measurements

being taken at the end of each day.

160

140

120

100

80

60

40

20

1

2 3 4 5 6

Months

Day 1 2 3 4 5 6 7

Height ( inches ) 1 3 5 7 9 11 13

Determine if the growth of the potted plant can be represented by a linear function.

5 The following table shows the decay rate of 1000 grams of radioactive element D:

Day 0 1 2 3 4

Mass of D ( grams ) 1000 500 250 125 62.5

Determine if the decay can be modeled by a linear function.

Let’s practice

6 Consider the pattern.

12, 19, 26, 33, …

a Describe the pattern. b Determine if this pattern is linear or not.

c Determine the next value in the pattern.


7 Consider the pattern shown with the given figures:

Step 1 Step 2 Step 3 Step 4

a

b

Describe the pattern.

Determine the number of squares there will be in the next step in the pattern.

8 Consider the pattern.

Step 1 Step 2 Step 3 Step 4

a

b

Describe the pattern.

Determine the number of circles there will be in the next step in the pattern.

9 The total volume of water that has dripped from a tap is measured each minute and

displayed in the table.

Time ( minutes ) 1 2 3 4 5 6 7 8

Volume ( mL ) 5 10 15 20 25

a

b

c

State the rule that relates the time and volume.

Determine if the rate of change of the volume is constant with respect to time.

Copy and complete the table.

10 The total distance traveled by a cyclist is measured each hour and displayed in the table.

Time ( hours ) 1 2 3 4 5 6 7 8

Distance ( miles ) 9 18 27 36 45

a

b

c

State the rule that relates the time and distance.


Determine the distance the cyclist would be able to travel in 15 hours.

158


11 Scarlett records the number of push ups she does each day and records them in the table.

Time ( days ) 1 2 3 4 5 6 7 8

Push ups 18 23 28 33 38

a

b

c

State the rule that relates the time and push ups.


Determine how many push ups Scarlett will be doing on the 20th day.

12 Eileen saves some money each week and deposits it into her savings account.

The weekly balance of this account at the end of each week is displayed in the table.

Time ( weeks ) 2 3 4 5 6 10 20 50

Balance ( dollars ) 425 450 475 500 525

a

b

c

State the rule that relates the time and balance.


Determine the balance be at the start of the first week.

13 Complete each of the following tables so that they could represent a linear function:

a Input 1 2 3 4 5

Output 8 12 20 24

b Input 3 4 5 6 7

Output −8 −20 −24

c Input 5 10 15 20 25

Output −13 −73

14 The cumulative rainfall over a series

of 4 days is represented on the graph.

a

b

Describe the relationship between time

and volume.

Determine if points on the graph lie on

a straight line.

16

14

12

10

8

6

4

Volume (mL)

2

1

Time (days)

2 3 4 5 6


15 The total number of matchbox cars

owned by Sally over a series of years is

represented on the graph.

a

b


and the total number of matchbox cars.

Determine if the points on the graph lie

on a straight line.

28

26

24

22

20

18

16

14

12

10

8

6

4

2

Number of matchbox cars

Time (years)

1

2 3 4 5 6

16 John is handing out flyers on the street to

passersby. The total number of flyers John

has handed out is recorded each hour and

displayed on the graph.

a

Describe the relationship between

time and the total number of flyers.

180

160

140

120

Total number of flyers

b

c

Determine if points on the graph lie on

a straight line.

If the pattern continues over the next

few hours, determine what the total

number of flyers will be after six hours.

100

80

60

40

20

Time (hours)

1

2 3 4 5 6

160



17 Determine whether the following statements are always, sometimes, or never true.

Explain your answer.

a

A linear function has a constant rate of change.

b A linear function has an end behavior where y → ∞ when x → ∞.

c

A linear function appears as a curve on a graph.

18 State whether the function that would be used to model each scenario is linear or nonlinear.

a

b

c

Water drops from a leaking tap are falling into a bucket at a constant rate. It has to model

the volume of water in the bucket as time passes.

A snowball rolling down a mountain doubles its volume every five seconds. It has to

model the volume of the snowball as time passes.

A raindrop falling from the sky speeds up by 9.8 m/s every second as it is pulled down

by gravity. It has to model the distance travelled by the raindrop as time passes.

19 Explain why a linear function can always be written in the form y = mx + b.


4.06 Transforming linear

functions

Concept summary

A transformation of a function is a change in the position, size, or shape of its graph. There are

many ways functions can be transformed:

Vertical compression

A transformation that scales all of the y-values of a function by a constant factor towards

the x-axis

Vertical stretch

A transformation that scales all of the y-values of a function by a constant factor away from

the x-axis

A vertical compression or stretch can be represented algebraically by

g( x ) = af ( x )

where 0 < a < 1 corresponds to a compression and a > 1 corresponds to a stretch.

Reflection

A transformation that produces the mirror image of a figure across a line.

A reflection across the x-axis can be represented algebraically by

g( x ) = −f ( x )

A reflection across the y-axis can be represented algebraically by

Translation

g( x ) = f ( −x )

A transformation in which every point in a figure is moved in the same direction and by the

same distance.

Translations can be categorized as horizontal ( moving left or right, along the x-axis ) or vertical

( moving up or down, along the y-axis ), or a combination of the two.

Vertical translations can be represented algebraically by

g( x ) = f ( x ) + k

162


where k > 0 translates upwards and k < 0 translates downwards.

Similarly, horizontal translations can be represented by

g( x ) = f ( x − k )

where k > 0 translates to the right and k < 0 translates to the left.

Functions that can be obtained by performing one or more of these transformations on each other

can be collected into groups or families of functions. The function in any family with the simplest

form is known as the parent function, and we frequently consider transformations as coming from

the parent function.

The parent function of the linear function family is the function y = x. Some examples of

transformations are shown below. In each example, the parent function is shown as a dashed line:

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

Vertical compression with scale factor of 0.5:

Reflection across y-axis:

g( x ) = 0.5f ( x ) g( x ) = f ( −x )

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

Vertical translation of 4 units upwards: Horizontal translation of 3 units to the right:

g( x ) = f ( x ) + 4 g( x ) = f ( x − 3 )


Worked examples

Example 1

A graph of the function

a Describe the transformation given by

g( x ) = −2f ( x )

is shown below.

10

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

y

2 4 6 8 10

x

Solution

The transformation g(x) is a combination of a vertical stretch of the graph of f (x) by a factor of 2,

and a reflection across the x-axis.

b Draw a graph of g( x ) on the same plane as the graph of f ( x ).

Approach

To perform a vertical stretch, we want to multiply the y-coordinate of each point by 2 without

changing the x-coordinate, so that it is twice as far away from the x-axis. To perform a reflection

across the x-axis, we want to change the sign of the y-coordinate of each point, so that it is on the

opposite side of the x-axis.

Solution

10

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

y

f(x)

x

2 4 6 8 10

g(x)

164


Example 2

The linear functions f ( x ) and g( x ) are represented on the given graph.

3

y

2

g(x) 1

−4 −3 −2 −1

−1

f(x)

−2

−3

−4

−5

1 2 3 4

x

a Describe the type of transformation( s ) that transforms f ( x ) to g( x ).

Approach

We can look at the slope and intercepts to help describe which transformation has taken place.

Solution

f (x) is increasing, while g(x) is decreasing so there has been a reflection over the x-axis.

g(x) is steeper than f (x), so there has also been a dilation.

b Write an equation for g( x ) in terms of f ( x ).

Approach

The equation will be of the form g(x) = k ⋅ f (x), where k is a real number.

Solution

The slope of f (x) is and the slope of g(x) is −1, so it has be reflected and is twice as steep.

The transformation is g(x) = −2 ⋅ f (x).


c Create a table of values for f ( x ) and g( x ) on the same coordinate plane to confirm your answer

to parts ( a ) and ( b ).

Approach

We can use about five corresponding points to check that g(x) = −2 ⋅ f (x). Since f (x) has slope of

we can use even x-values to avoid using fractions or decimals.

Solution

Getting this data from the graph:

3

y

2

g(x) 1

−4 −3 −2 −1

−1

f(x)

−2

−3

−4

−5

1 2 3 4

x

x −4 −2 0 2 4

f (x) −1 0 1 2 3

g(x) 2 0 −2 −4 −6

This table does confirm that g(x) = −2 ⋅ f (x).


1 Assuming the value of k is positive, state whether the linear function f ( x ) has been translated

up, down, left, or right to produce the graph of g( x ).

a g( x ) = f ( x ) + k b g( x ) = f ( x ) − k c g( x ) = f ( x ) + 3k

2 For the what values of k will the transformation from f ( x ) to g( x ) = kf ( x ) involve a reflection

across the x-axis?

166


3 For each of the following, state whether f ( x ) has been translated up, down, left, or right:

a g( x ) = f ( x ) + 4 b g( x ) = f ( x ) − 2 c d

4 For each of the following, describe the type transformation from f ( x ) to g( x ):

a g( x ) = 6 f ( x ) b c d g( x ) = − f ( x )

5 For each of the following tables:

i Write an equation for g( x ) in terms of f ( x ) in the form g( x ) = f ( x ) + k or g( x ) = kf ( x )

depending on the transformation.

ii Describe the type of transformation from f ( x ) to g( x ).

a f (x) 2 5 8 11 14

g(x) 4 7 10 13 16

c f (x) −2 −1 0 1 2

g(x) −4 −3 −2 −1 0

b f (x) 1 2 3 4 5

g(x) 4 8 12 16 20

d f (x) 4 6 8 10 12

g(x) −2 −3 −4 −5 −6


i Describe the transformation applied to f ( x ) to produce g( x ).

ii Write an equation for g( x ) in terms of f ( x ) in the form g( x ) = f ( x ) + k or g( x ) = kf ( x )


a

y

b

y

4

4

−4 −3 −2

g(x)

f(x)

3

2

1

−1

−1

−2

−3

−4

1 2 3 4

x

−4 −3 −2

f(x)

3

2

1

−1

−1

−2

−3

g(x)

−4

1 2 3 4

x

c

4

y

d

f(x)

4

y

3

3

2

1

g(x)

f(x)

x

g(x)

2

1

x

−4 −3 −2

−1

−1

1 2 3 4

−4 −3 −2

−1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4


Let’s practice

7 For each of the following pairs of functions:

i Copy and complete the table.

ii Graph f ( x ) and g( x ) on the same coordinate plane.

a x 0 1 2

f (x) 4 6 8

g(x) = f (x) + 3

b x 0 1 2

f (x) 3 5 7

g(x) = f (x) − 7

c x 0 1 2

f (x) −2 0 2

g(x) = −3 f (x)

d x 0 1 2

f (x) 18 24 30

8 For each graph of f ( x ), graph g( x ) using the given transformation.

a g( x ) = f ( x ) + 5 b g( x ) = 2 f ( x )

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2

−1

−1

−2

1 2 3 4

−4 −3 −2

f(x)

−1

−1

−2

1 2 3 4

−3

f(x)

−4

−3

−4

c g( x ) = −f ( x ) d

4

y

4

y

f(x)

3

2

3

2

1

x

1

x

−4 −3 −2

−1

−1

−2

1 2 3 4

−4 −3 −2

−1

−1

−2

f(x)

1 2 3 4

−3

−3

−4

−4

168


e g( x ) = f ( x ) − 2 f g( x ) = f ( x ) + 2

f(x)

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2

−1

−1

1 2 3 4

−4 −3 −2

−1

−1

1 2 3 4

−2

−3

f(x) −2

−3

−4

−4

9 For each transformation, describe how the graph of g( x ) is related to the graph of f ( x ).

a g( x ) = f ( x ) − 8 b c g( x ) = −3f ( x ) d

10 Consider the graph of f ( x ) and the table of values for g( x ).




a

4

3

2

y

x −1 0 1 2 3

g(x) 6 4 2 0 −2

1

x

−4 −3 −2 −1

−1

1 2 3 4

−2

−3

f(x)

−4


b

4

3

y

x −2 −1 0 1 2

g(x) −4 −3 −2 −1 0

2

1

x

−4 −3 −2 −1

−1

1 2 3 4

f(x)

−2

−3

−4

c

f(x)

4

3

2

y

x −4 −2 0 2 4

g(x) 12 8 4 0 −4

1

x

−4 −3 −2 −1

−1

1 2 3 4

−2

−3

−4

170



11 Stavros has a bank account with $100.

Every week, Stavros deposits an

additional $50 into the account. A graph

of Stavros’ bank balance reveals that the

growth of his account is linear. Suppose

that Stavros was instead depositing an

additional $100 every week. Explain

whether or not this could be represented

by a vertical stretch of the current graph.

500

400

300

200

Balance

100

Weeks

1

2 3 4 5 6 7

12 Consider the transformation g( x ) = kf ( x ). Under what conditions will the graphs of f ( x ) and

g( x ) have:

a No points of intersection b Infinitely many points of intersection

13 Oreste draws the graph of g( x ) = −f ( x ),

as shown.

a

b

Describe the error Oreste made.

Describe the type of graphs that f ( x ) could

be for Oreste’s error to still give him the

correct answer.

g(x)

4

3

2

1

y

x

−4 −3 −2 −1

−1

1 2 3 4

−2

−3

f(x)

−4



linear functions

Concept summary

The characteristics, or key features of a function include its:

• domain and range

• x- and y-intercepts

• maximum or minimum value( s )

• rate of change over specific intervals

Key features of a function are useful in helping to sketch the function, as well as to interpret

information about the function in a given context. In the case of linear functions, they will either be

constant everywhere ( a horizontal line ), increasing everywhere, or decreasing everywhere, due to

their constant rate of change. This also means that they will never have a maximum or minimum

turning point, and will have at most one x-intercept ( except for the horizontal line y = 0 ).

4

y

4

y

3

3

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4

An increasing linear function. It has an

A decreasing linear function. It has an

x-intercept at ( −2, 0 ) and a y-intercept at ( 0, 2 ). x-intercept at ( 2, 0 ) and a y-intercept at ( 0, 1).

172


Worked examples

Example 1

Consider the function f ( x ) = 4x − 8.

a Find the value of f ( 0 ).

Approach

To find the value of a function at a particular x-value, we simply substitute that value into the

function.

Solution

We have

f (0) = 4(0) − 8

= −8

Reflection

The y-intercept of a function is the point on the function where x = 0. So we know that this function

has a y-intercept at (0, −8).

b Find the value of x which gives a function value of 0.

Approach

This means that we want to solve the equation f (x) = 0 using the expression we have for f (x).

Solution

We have

f (x) = 0

4x − 8 = 0

4x = 8

x = 2

Reflection

The x-intercepts of a function are the points on the function where y = 0. So we know that this

function has a single x-intercept at (2, 0).


c Sketch a graph of the function and label each intercept.

Approach

We found the two intercepts in parts (a) and (b). Since two points determine a line, we can connect

the intercepts to create the graph of the function.

Solution

10

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

y

(2, 0)

x

2 4 6 8 10

(0, −8)

Example 2

Whitney is traveling across the city by taking an Uber ride. The cost of the ride, in dollars,

is given by

C( x ) = 2.4x + 2.8

where x is the distance traveled in miles. The minimum charge for a ride is $10.

a State the range of the function.

Solution

Since x is the distance traveled in miles, we know that x > 0, which corresponds to C (x) > 2.8.

We are also told that the minimum charge is $10.

Putting these together, we have that the range is

Range: {y | y ≥ 10}

174


b Determine the rate of change of the function, and state what it represents in context.

Approach

The rate of change of a linear function in the form y = mx + b is m, the constant being multiplied by

the variable.

Solution

In this case, the function is C (x) = 2.4x + 2.8, so the rate of change is 2.4.

Since the output of C (x) is a value in dollars, and x is a value in miles, the rate of change is 2.4

dollars per mile, and it represents the additional cost of the trip per each extra mile traveled.

c If Whitney is travelling a distance of 4 miles, determine the cost of her trip.

Approach

We want to find the value of C (4), and compare it to the minimum cost. The larger of the two values

will be the cost of Whitney’s trip.

Solution

We have

C(4) = 2.4(4) + 2.8

= 9.6 + 2.8

= 12.4

Since this is larger than the minimum amount, the cost of her trip will be $12.40.

Example 3

Consider functions f and g shown in the graph.

Compare the following features of each function:

4

3

y

g(x)

−4 −3 −2 −1

2

1

−1

−2

1 2 3 4

x

−3

f(x) −4


a Domain

Approach

The domain of a function is the set of all x-values that correspond to a point on the graph.

Solution

As these are linear functions, with no domain restrictions from context, all real values of

x correspond to a point on each line.

So the domain of both functions is all real values of x.

b Range

Approach

The range of a function is the set of all y-values that correspond to a point on the graph.

Solution

We can see that all values of y correspond to a point on the graph of f, so its range will be all real

values of y.

On the other hand, g is a constant function and so it only takes one y-value - in this case, 2.

So the range of g is just y = 2.

c Intercepts

Approach

The x-intercepts of a function are the points where it intersects the x-axis. Similarly, the y-intercept

of a function is the point where it intersects the y-axis.

As these are linear functions, they will have at most one of each type of intercept.

Solution

We can see that function f crosses the x-axis at

the point (2, 0), and crosses the y-axis at the point

(0, −2).

g(x)

4

3

2

1

y

x

−4 −3 −2 −1

−1

1 2 3 4

−2

−3

f(x) −4

176


Function g crosses the y-axis at the point (0, 2). Since it is a constant function, it is parallel to the

x-axis and so it does not have an x-intercept.

4

y

g(x)

−4 −3 −2 −1

3

2

1

−1

−2

1 2 3 4

x

−3

f(x) −4

d Slope

Approach

The slope of a line is the ratio of its change in the vertical direction to the change in the horizontal

direction.

That is, the slope is a measure of how far up (or down) the y-values change for each unit change in

the x-values.

Solution

For function f we can see that at any point on the graph, if we move 1 unit to the right

(in the x-direction), we also move 1 unit up (in the y-direction). So the slope of f is 1.

Function g is a constant function, which has no change in its y-values. That is, at any point on the

graph, moving 1 unit to the right corresponds to 0 units of change in the vertical direction.

So the slope of g is 0.


1 For each function, complete the following:

i Find the coordinates of the intercepts. ii Determine the slope of the function.

iii Determine the domain of the function. iv Determine the range of the function.

a f ( x ) = 35x − 15 b f ( x ) = −3x + 7


2 For each of the following functions, evaluate the function at two values for x and find the

slope of the function:

a f ( x ) = 6x + 3 b f ( x ) = −2x + 4

c

d

3 Using the following graphs, write the equation in function notation:

a

8

6

4

2

−8 −6 −4 −2

−2

f(x)

−4

−6

−8

y

2 4 6 8

x

b

g(x)

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x

Let’s practice

4 Consider the following table of values of a function.

x −2 −1 0 1 2

f ( x ) −5 −1 3 7 11

a

Select the graph that could represent the function:

A

5

4

3

2

1

−5 −4 −3 −2 −1 −1

y

1 2 3 4 5

x

B

5

4

3

2

1

−5 −4 −3 −2 −1 −1

y

1 2 3 4 5

x

−2

−3

−4

−5

−2

−3

−4

−5

178


C

5

y

D

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5

b

c

Find the coordinates of the y-intercept.

Find the slope of the function.

5 Consider the function

a

Select the graph of the function:

A

8

y

B

8

y

6

6

4

4

2

x

2

x

−8 −6 −4 −2

−2

2 4 6 8

−8 −6 −4 −2

−2

2 4 6 8

−4

−4

−6

−6

−8

−8

C

8

y

D

8

y

6

6

4

4

2

x

2

x

−8 −6 −4 −2

−2

2 4 6 8

−8 −6 −4 −2

−2

2 4 6 8

−4

−4

−6

−6

−8

−8

b

c

Find the domain of the function.

Find the range of the function.


6 Consider the function whose graph passes through points ( 0, −5 ) and ( −1, 1).

a


A

5

y

B

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5

C

5

y

D

5

y

4

4

3

3

2

2

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−5 −4 −3 −2 −1 −1

1

1 2 3 4 5

x

−2

−2

−3

−3

−4

−4

−5

−5

b

c

Find the slope of the function.

Find the coordinates of the x-intercept.

7 For each of the following equations, assuming that y is a function of x:

i Rewrite the equation using the function notation f ( x ).

ii

iii

iv

Find the value of f ( −2 ). Write answer using function notation.

Find the value of f ( 3 ). Write answer using function notation.

Graph the function.

a y = −3x + 4 b y − 2 = 3( x + 4 ) c 3x + 5y = 14 d 7x − 5y = −4

8 For each of the following functions:

i Find the value of f ( 0 ).

ii

iii

Find the value of x which makes f ( x ) equal to zero.

Graph the function, labelling each intercept.

a f ( x ) = −3x + 12 b f ( x ) = 5x − 15 c f ( x ) = −x − 4 d f ( x ) = 5( x − 2 )

180


9 For each of the following functions, use the given domain to find the range of the function:

a f ( x ) = 3x + 2 has a domain of x > 7

b f ( x ) = −x − 13 has a domain of x ≤ 2

c f ( x ) = 2x − 5 has a domain of −5 ≤ x ≤ 2

d f ( x ) = −5x + 6 has a domain of 3 ≤ x < 6

10 Two functions f and g are shown in the graph.

Compare the following features of each

function:

a

b

c

d

Domain

Range

Intercepts

Slope

g

−4 −3 −2 −1

4

3

2

1

−1

y

1 2 3 4

x

−2

f

−3

−4

11 Two functions are described by the equations f ( t ) = 3t − 6 and g( t ) = 3t + 9. Compare the

following features of each function:

a Domain b Range c Intercepts d Slope

12 Each time Serena finishes reading a book, her mother adds a dollar to Serena’s reading jar.

At the start of the holidays, Serena had $31 in her reading jar. During the holidays, Serena

reads at a rate of five books a week.

a

b

Write a function which represents the number of dollars in Serena’s reading jar in terms

of how many holiday days have passed.

If Serena’s holidays last for four weeks, determine how many dollars will be in her

reading jar by the end of the holidays.

13 The height of a candle in inches can be represented as a linear function of time in minutes

since the candle was lit. At 2 minutes after being lit, the candle had a height of 11 inches.

At 4 minutes after being lit, the candle had a height of 8 inches.

a

b

c

Determine the rate of change of the candle’s height per minute.

Find the height of the candle before it was lit.

Determine how long the candle can be lit for before it completely melts.


14 During a fundraiser marathon, Conner sprains his ankle and needs to walk for the rest of the

race. Connor’s distance in miles remaining in the race can be represented as a function of

how many hours have passed since his injury: f ( t ) = 18 − 2t.

a

b

c

d

Describe what the variable t represents in the function.

Find the remaining distance to the finish line after Connor has been walking for 3 hours.

Determine how many hours of walking it will take for Connor to reach the finish line.

Determine the constraints on the domain and range in terms of the context.

15 Tyrell and Veronica live 500 meters away from school, and they often race each other home.

As Tyrell is younger, Veronica gives him a 20 second head start so that they can arrive home

at approximately the same time. Tyrell takes 32 seconds to jog 100 meters, on average.

Let T ( x ) represent Tyrell’s distance from home x seconds after they started a race, and V ( x )

represent Veronica’s distance from home at the same time.

a

b

c

Determine the range of each function in terms of the context.

Determine the domain of each function in terms of the context.

Compare the x-intercept of each function, and describe what they represent in terms of

the context.


16 Use numbers 1 through 5 at most once to write a linear equation and to identify a point on

the function. The linear function has the following key features:

• Rate of change is greater than −1

• y-intercept is greater than 2

Write an equation of a line in the form f ( x ) = mx + b and a point in ( x, y ) form.

17 Give an example of a scenario that could be modeled by the function f ( x ) = 25x − 12.

18 Describe how to find the x- and y-intercepts of a linear equation when it is presented as a

function f of x.

182


4.08 Comparing linear

and nonlinear functions

Concept summary

We can use key features to compare linear and non-linear functions. Some additional key features

that we might look at are:

End behavior

Describes the trend of a function or graph at its left and right ends; specifically the y-value that

each end obtains or approaches

Positive interval

A connected region of the domain in which all function values lie above the x-axis

Negative interval

A connected region of the domain in which all function values lie below the x-axis

Worked examples

Example 1

Consider the two functions shown in the graphs below.

y

4

3

2

4

3

2

y

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

−4

−4


a State the intercepts of each function.

Solution

Both functions have a y-intercept at (0, −1).

Also, both functions have an x-intercept at (1, 0).

The second function has an additional x-intercept at (−1, 0).

b Compare the end behavior of the two functions.

Solution

On the right side, both functions take larger and larger positive values as x gets further from zero.

That is, as x → ∞, y → ∞ for both functions.

On the left side, the first function takes larger and larger negative values as x gets further from

zero. That is, as x → −∞, y → −∞ for the first function.

On the other hand, the second function takes larger and larger positive values as x gets further

from zero on the left side. That is, as x → −∞, y → ∞ for the second function.

Reflection

Note that although both functions tend towards infinity to the right, the way they do so is different.

The first function increases at a constant rate, while the second function increases at an

increasing rate.

c State the interval( s ) over which each function is positive or negative.

Solution

The first function is positive for x > 1 and negative for x < 1.

The second function is positive for both x > 1 and x < −1, and is negative for −1 < x < 1.

d Determine whether each function is linear or non-linear.

Approach

A linear function has a constant rate of change. It also has no turning points, with at most one

x-intercept.

Solution

The first function is linear, while the second function is non-linear.

184



1 Consider the functions shown below:

4

y

4

y

3

3

2

1

x

2

g(x)

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

f(x)

−2

−2

−3

−3

a

b

c

−4

Identify the domain of each function

Identify the range of each function

State whether each function is linear or nonlinear

−4


4

y

4

y

3

g(x)

3

f(x)

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

−2

−3

−3

a

b

c

−4

Identify the increasing interval( s ) for each function

Identify the decreasing interval( s ) for each function


−4



4

y

4

y

3

3

f(x)

2

2

1

x

1

x

−4 −3 −2 −1

−1

1 2 3 4

−4 −3 −2 −1

−1

1 2 3 4

−2

g(x)

−2

−3

−3

a

b

c

−4

Identify the intercepts of each function

Describe the end behavior of each function


−4

4 A few values for two functions are shown in the following table and plotted on the graph:

x f (x) g(x)

1 −4 −6

2 −2 −3

3 0 0

4 2 3

6

5

4

3

2

1

y

f(x)

x

5 4 0

6 6 −3

−1 1 2 3

−1

−2

−3

−4

−5

−6

4 5 6 7 8 9

g(x)

Assume the domain of each function is all real numbers.

a Compare the intervals where the function is positive and negative for each function

b State whether each function is linear or nonlinear

186


Let’s practice

5 A few values for two functions are shown in the following table:

x f ( x ) g( x )

−2 −8 5

−1 −7 2

0 −5 −1

1 −1 −4

2 7 −7

Assume the domain of each function is all real numbers.

a Draw a graph of each function

b Compare the intervals where the function is increasing and decreasing for each function

c State whether each function is linear or nonlinear

6 The perimeters and areas for a sequence of similar triangles are shown in the following table:

Triangle Perimeter Area

1 18 30

2 27 67.5

3 36 120

4 45 187.5

a

b

Draw a graph of the perimeter and area functions


7 An amount of $1000 is invested into two

different accounts for a fixed period of

time, with account C earning compound

interest and account S earning simple

interest. The balance of each account

over time is shown in the following

graph:

2500

2000

y

S(t)

a

b

State the domain of each function

Describe how each function

increases over its domain

1500

C(t)

c

State whether each function is linear

or nonlinear

1000

1

2 3 4 5 6 7 8 9 10

t


8 Consider the function shown in the

graph:

4

y

Bonita claims that this graph is a linear

function, since it is a horizontal line at each

point in its domain. Explain why this is

incorrect.

−4 −3 −2 −1

3

2

1

−1

1 2 3 4

x

−2

−3

−4


9 Three functions f, g, and h each pass through the origin and have the following end behaviors:

• The values of f tend towards ∞ as x gets far from zero on both sides.

• The values of g tend towards ∞ as x → −∞, while the values tend towards −∞ as x → ∞.

• The values of h tend towards 3 as x gets far from zero on both sides.

Determine which of these could describe a linear function. Use graphs to support your answer.

10 Two species of parrot finch, Red-faced

and Blue-faced, are introduced to a new

location at the same time. The population

of the Red-faced species after t months

can be modeled by the function

R ( t ) = 5t + 100. The population of the

Blue-faced species is modeled by the

function shown in the graph.

a

b

c

Determine which species has the

largest population after the first year

has passed. Explain your answer.

Determine if the two species’ will

ever have the same population at the

same time after the initial introduction.


If both species continue to increase

in population in the same way, determine which species will reach a population of 1000

first. Explain your reasoning.

11 Determine whether each of the following statements are always, sometimes, or never true:

a The valus of a linear function tend towards ∞ as x → ∞.

b

c

Graphs of nonlinear functions are curves.

260

240

220

200

180

160

140

120

100

Linear functions cannot have more than one positive interval.

80

Population

4

B(t)

t (months)

8 12 16 20 24

188


4.09 Linear absolute

value functions

Concept summary

The absolute value of a number is its distance from zero on a number line. An absolute value

is indicated by vertical lines on either side. For example, the absolute value of −3 is 3, which is

written as |−3| = 3.

An absolute value function is a function that contains a variable expression inside absolute value

bars; a function of the form f ( x ) = a |x − h| + k

x −2 −1 0 1 2 3 4

f ( x ) 3 2 1 0 1 2 3

For example, consider the absolute value function

f ( x ) = |x − 1|. We can complete a table of values for

the function.

4

3

2

1

y

x

Since the absolute value of an expression is nonnegative,

the graph of this absolute value function

does not go below the x-axis, as the entire

expression is inside the absolute value. If the

function has a negative value for k, such as

y = |x − 1| − 2, it can go below the x-axis, but will

still have a minimum value, in this case y = −2.

−4 −3 −2 −1

−1

−2

1 2 3 4

If the function has a negative value of a, the graph

of the function will open downwards, and instead

have a maximum value.

−3

−4

Worked examples

Example 1

Consider the function f ( x ) = |3x − 6|.

a Complete the table of values for this function.

x 0 1 2 3 4 5

f ( x )


Solution

x 0 1 2 3 4 5

f (x) 6 3 0 3 6 9

b Sketch a graph of the fuction.

Solution

8

y

7

6

5

4

3

2

1

x

−2 −1 1 2

−1

3

4 5 6 7 8

−2


1 Determine whether the following graphs show an absolute value function:

a

−2

2

1

−1 −1

y b

1 2

x

−2

2

1

−1 −1

y c

1 2

x

−2

2

1

−1 −1

y d

1 2

x

−2

2

1

−1 −1

y

1 2

x

−2

−2

−2

−2

190


2 Consider the table of values of a function.

x −3 −2 −1 0 1 2

y 2 1 0 1 2 3


A

−4

2

−2

−2

−4

y B

x

2

4

2

−2

−2

y C

x

2 4

2

−2

−2

−4

y D

x

2 4

4

2

−4 −2

−2

y

2

x

3 Consider the function y = |x|.

a State the values of x that can be substituted into the function.

b State the values of y that the function could take.

4 Consider the function y = |5x|.

Select the graph that represents the function:

A

−4 −2

12

10

8

6

4

2

y B

2 4

x

4

2

−4 −2

−2

y C 2

2 4

x

−4 −2

−2

−4

−6

−8

−10

−12

y x D 12

10

2 4

8

6

4

2

−4 −2 −2

−4

y

2 4

x

5 Consider the graph of y = f ( x ) as shown.

a For what values of x is f ( x ) positive?

6

y

b For what values of x is f ( x ) negative?

c For what values of x is |f ( x )| = f ( x ).

d For what values of x is |f ( x )| = −f ( x ).

4

2

x

e Graph y = |f ( x )|. −6 −4 −2

−2

2 4 6 8

−4

−6

−8


Let’s practice

6 An absolute value function has its vertex at ( −1, 3 ), and goes through the point A ( 2, 0 ).


A

3

2 1

−6 −4 −2−1

2

−2

−3

−4

y B

x

3

2 1

−6−4−2−1

2 4

−2

−3

−4

y C 3

x

2 1

−4−2−1

−2

−3

2 4 6

−4

y D 3

x

y

2

1

−4−2−1

−2

−3

2 4 6

−4

x

7 Consider the function y = |5 − x|.

a Complete the given table.

b Graph the function.




9 Consider the function y = 2 |x + 3| − 4.









x 1 2 3 4 5 6 7

y

x 2 3 4 5 6 7 8

y

x −5 −4 −3 −2 −1 0

y

x −4 −3 −2 −1 0 1

y

x −15 −10 −5 0 5 10

y

12 Michael is training for the land speed record and takes his new car for a test drive by driving

straight down a closed highway and back. His distance y in miles from the end of the

highway x minutes after he takes off is given by the function y = |5x − 40|.

a


192


A

40

y

B

40

y

35

35

30

30

25

25

20

20

15

15

10

10

5

x

5

x

2 4 6 8 10 12 14 16

2 4 6 8 10 12 14 16

C

40

y

D

40

y

35

35

30

30

25

25

20

20

15

15

10

10

5

x

5

x

2 4 6 8 10 12 14 16

2 4 6 8 10 12 14 16

b

c

d

How far does he drive in total?

How long does Michael take to reach the end of the highway?

The next day Michael goes for another drive down the same route and his distance from

the end of highway is given by y = |4x − 40|. Is Michael driving faster or slower than the

previous day?


13 Compare the function y = 3x − 9 and the function f ( x ) = |3x − 9|.

14 Describe a method of sketching an absolute value function y = |f ( x )|, given a graph of

y = f ( x ).

15 Explain why the function y = |x| doesn’t go below the x-axis.


4.10 Transforming

absolute value functions

Concept summary

Absolute value functions can be transformed in the same ways as linear functions. Recall the types

of transformations that we have looked at:

Translation

A transformation in which every point in the graph of the function is shifted the same distance

in the same direction.

Reflection

A transformation that flips a function across a line of reflection, producing a mirror image of

the original function.

Vertical compression

A transformation that pushes all of the y-values of a function towards the x-axis.

Vertical stretch

A transformation that pulls all of the y-values of a function away from the x-axis.

The parent function of the absolute value function family is the function y = |x|. Other linear

absolute value functions can be obtained by transformations of this parent function.

194


Worked examples

Example 1

A graph of a function f ( x ) is shown below.

10

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

y

2 4 6 8 10

x

a Sketch a graph of the function g( x ) = −f ( x ).

Approach

The function g(x) is formed by changing the sign of the function values of f (x). So the x-intercept

will stay the same, and every point above the x-axis will change to a point the same distance below

the x-axis.

Solution

10

8

6

4

2

−10 −8 −6 −4 −2

−2

−4

−6

−8

−10

y

2 4 6 8 10

x


b Describe the transformation from f ( x ) to g( x ).

Solution

The transformation from f (x) to g(x) is a reflection across the x-axis.

Example 2

Consider the following table of values for two absolute value functions f ( x ) and g( x ):

x 0 1 2 3 4 5 6

f ( x ) 6 3 0 3 6 9 12

g( x ) 12 9 6 3 0 3 6

a Describe the transformation from f ( x ) to g( x ).

Solution

Looking at the values in the table for f (x), we can see that it has a vertex at (2, 0) and has a rate of

change of 3 on either side.

Comparing this to the value for g(x) we can see that it has the same rate of change, but has its

vertex at (4, 0) instead.

So we can get from f (x) to g(x) by translating 2 units to the right.

Reflection

Since absolute value functions are symmetric about a vertical axis, it is also possible to describe

this transformation as a reflection across the line x = 3.

b Write a function that describes the relationship between f ( x ) and g( x ).

Solution

We can write “a translation by 2 units to the right” as g(x) = f (x − 2).

Reflection

Since it is also possible to describe this transformation as a reflection across the line x = 3, we can

also express this transformation as g(x) = f (6 − x).

196



1 Assuming the value of k is positive, state whether the graph of f ( x ) has been translated up,

down, left, or right to produce the graph of g( x ).

a g( x ) = f ( x ) + k b g( x ) = f ( x + k )

c g( x ) = f ( x − k ) d g( x ) = f ( x ) − k





a

10

f(x)

g(x) 8

6

4

2

−10 −8 −6 −4 −2

−2

y

2 4 6

x

b

10

8

6

f(x)

4

2

−10 −8 −6 −4 −2 2

−4

g(x)

−6

−8

−10

y

2 4 6

x

c

g(x)

10

f(x)

8

6

y

d

g(x)

10

8

f(x)

6

y

4

4

2

x

2

x

−8 −6 −4 −2

−2

2 4 6 8

−8 −6 −4 −2

−2

2 4 6 8 10


Let’s practice


i How was f ( x ) transformed to produce the graph of g( x )?

ii State the equation of g( x ) in terms of f ( x ).

a

y

b

y

f(x)

10

g(x)

8

8

f(x)

6

g(x)

6

4

4

2

−8 −6 −4 −2

−2

2 4 6 8

x

2

−8 −6 −4 −2

−2

−4

x

2 4 6 8 10

c

f(x)

10

y

d

g(x)

8

y

8

g(x)

6

4

2

−8 −6 −4 −2

−2

2 4 6 8

x

f(x) 6

4

2

−6 −4 −2

−2

x

2 4 6 8 10

4 Consider the graph of the function y = f ( x ):

a Graph y = g( x ) where g( x ) = f ( −x ).

b Describe the transformation from f ( x )

to g( x ).

10

8

6

4

y

−10 −8 −6 −4 −2

2

−2

2 4

x

198



a Graph y = g( x ) where g( x ) = f ( x ) + 3.


to g( x ).

10

8

6

4

y

−10 −8 −6 −4 −2

2

−2

2 4 6

x


a Graph y = g( x ) where g( x ) = f ( x ) − 5.


to g( x ).

10

8

6

4

y

2

−6 −4 −2 2 4 6 8

−2

−4

10 12

x


a Graph y = g( x ) where g( x ) = f ( x + 3 ).


to g( x ).

10

8

6

y

4

2

−4 −2 2 4 6 8

−2

10

x



a Graph y = g( x ) where g( x ) = f ( x − 5 ).


to g( x ).

10

8

6

4

y

2

−6 −4 −2 2 4 6 8

−2

10

x


a Graph y = g( x ) where g( x ) = 4f ( x ).

b

Describe the transformation from f ( x ) to

g( x ).

8

6

y

4

2

−10 −8 −6 −4 −2 2 4

−2

6

x


y

a

b

Graph y = g( x ) where


g( x ):

8

6

4

2

−6 −4 −2 2 4

−2

6

x

200



a Graph y = g( x ) where g( x ) = −2f ( x ).

b


g( x ).

8

6

y

4

2

−8 −6 −4 −2 2 4

−2

6

x


a Graph y = g( x ) where g( x ) = f ( 2x ).

b


g( x ).

10

8

6

4

y

−8 −6 −4 −2 2 4 6

2

−2

8

x

13 For each of the given f ( x ) and g( x ):

i Complete the table.

ii Describe the transformation from f ( x ) to g( x ).

a f ( x ) = |x − 2| b f ( x ) = |2x|

g( x ) = 3f ( x ) g( x ) = f ( −2x )

c f ( x ) = |2x − 1| d f ( x ) = |3x + 4|

g( x ) = f ( x ) + 5 g( x ) = f ( x ) − 3

e f ( x ) = |3x − 4| f f ( x ) = |2x − 6|

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

x f (x) g(x)

−3

−2

−1

0

1

2

3

4

5


14 Consider the following tables:

i Describe the transformation from f ( x ) to g( x ).

ii Write a function that describes the relationship between f ( x ) and g( x ).

a

x f (x) g(x)

b

x f (x) g(x)

−1 3 12

−4 4 5

0 2 8

−3 2 3

1 1 4

−2 0 1

2 0 0

−1 2 3

3 1 4

0 4 5

4 2 8

1 6 7

5 3 12

2 8 9

c

x f (x) g(x)

d

x f (x) g(x)

−3 2 −4

−4 4 1

−2 1 −2

−3 2 −1

−1 0 0

−2 0 −3

0 1 −2

−1 2 −1

1 2 −4

0 4 1

e

x f (x) g(x)

f

x f (x) g(x)

−2 8 0

−4 4 20

−1 6 2

−3 2 16

0 4 4

−2 0 12

1 2 6

−1 2 8

2 0 8

0 4 4

3 2 10

1 6 0

4 4 12

2 8 4

5 6 14

3 10 8

15 Describe how the graph of y = f ( x ) is shifted to get the graph of the following:

a y = f ( x ) + 4 b y = f ( x + 6 )

202



16 Determine whether each of the following statements are true or false. Explain your thinking.

a

b

c

Absolute value functions are always positive.

The vertex of an absolute value function is always a minimum or maximum point.

The vertex of an absolute value can occur on the x-axis.

17 Maddox and Paulina are going to a lake 2 miles from their home to go ice skating.

Their mother told them that if the ice is not thick enough they must turn around and come

straight home. It took them 20 minutes to walk to the lake at a constant rate and when they

arrived Maddox could tell the ice was not thick enough so they immediately turned around

and went home which took them an additional 20 minutes.

a

b

c

Graph Paulina and Maddox’s distance from home over their entire journey

If Paulina and Maddox had walked twice as fast, describe how your graph would change.

Describe the transformations if instead you were graphing Paulina and Maddox’s

distance from the lake.


5 Exponential

Functions

Chapter outline

5.01 Operations with numerical radicals 206

5.02 Rational exponents 213

5.03 Exponential relationships 220

5.04 Characteristics of exponential functions 228

5.05 Exponential growth and decay 237

5.06 Percent growth and decay 249

5.07 Simple interest 259

5.08 Compound interest 264

5.01 Operations with

numerical radicals

Concept summary

Radical expressions have many parts as shown in the following diagram:

Index

The number on a radical symbol that indicates which type of root it represents. For instance,

the index on a cube root is 3. The index on a square root is usually not written, but would be 2.

Radical

A mathematical expression that uses a root, such as a square root

or a cube root

Radicand

The value or expression underneath the radical symbol

Perfect square

A number that is the result of multiplying two of the same integer together

206


Perfect cube

A number that is the result of multiplying three of the same integer together

If a radical expression is written such that there are no factors that can be removed from

the radicand, and with no radicals in the denominator (if the expression is a fraction),

then the expression is said to be in simplified radical form.

For square roots this means there are no remaining factors of the radicand that are perfect

squares, and for cube roots this means there are no remaining factors of the radicand that are

perfect cubes. The same operations that can be applied to rational expressions can also be

applied to radical expressions.

• Multiplication: For radicals with the same index, multiply the coefficients, multiply the

radicands, and write under a single radicand before checking to see if the radicand can be

simplified further

• Division: For radicals with the same index, divide the coefficients, divide the radicands,

and write under a single radicand before checking to see if the radicand can be

simplified further

• Addition and subtraction: Add or subtract like radicals (radicals with the same index and

radicand) by adding the coefficients and keeping the radicand the same, if there are no like

radicals check to see if any of the radicals can be simplified first.

Worked examples

Example 1

Rewrite the expressions in simplified radical form.

a

Approach

To simplify a square root it can be helpful to see if the radicand has any factors that are perfect

squares. In this case, 150 = 25 ⋅ 6 and 25 is a perfect square.

Solution

Knowing this, we can simplify the original expression as follows:

Since 6 does not have any factors that are perfect squares, the original expression is fully

simplified as

Chapter 5 Exponential Functions 207

b

Approach

To simplify a cube root it can be helpful to see if the radicand has any factors that are perfect

squares. In this case, 72 = 8 ⋅ 9 and 8 is a perfect cube.

Solution

Knowing this, we can simplify the original expression as follows:

Since 9 does not have any factors that are perfect cubes, the original expression is fully simplified

as

Example 2

Simplify the radical expressions.

a

Approach

To simplify the product of two square roots we can multiply them together in the usual way.

Then we can look at the result and factor out the square root of any perfect square factors, so that

the solution is in simplified radical form.

Solution

Following this process, we have:

So the fully simplified form is

Combining the radicals

Multiplying the radicands

Rewriting to find a perfect square factor

Taking out the perfect square factor

208


b

Approach

To simplify the quotient of two radicals it can be helpful to see if the radical in the numerator

can be written as the product of two or more radicals, one of which is equal to the radical in

the denominator.

Solution

In this case, 48 = 24 ⋅ 2, and so we can rewrite the original expression as follows:

So we have that

fully simplifies to

Example 3

Simplify the following expressions:

a

Approach

Since both terms in the expression include

we can combine like terms to simplify.

Solution

b

Approach

Since both terms in the expression include

we can combine like terms to simplify.

Solution


c

Approach

Since the two terms in the expression have different radicands, we first want to rewrite each term

in simplified radical form and see if they have any like terms. Since

and

we can then combine like terms.

Solution


1 How do you know if a radical is in simplified radical form?

2 Simplify each of the following radicals:

a b c d

3 Determine whether each expression is in simplified radical form:

a b c d

4 Rewrite each expression in simplified radical form:

a b c d

5 Simplify the following:

a b c

3 3

2 7

d


a b c d


a b c d

Let’s practice


a b c

d e f

210



a b c d

e f g h

10 Rewrite the following expressions in simplified radical form:

a b c d

11 Find the area of the following rectangles:

a

b

12 Solve for the height of a triangle with and whose base measures

13 Aviva and Jillian are traveling from the shore

at Point A to their home at Point C. Jillian

wants to swim home and travels directly from

A to C. Aviva wants to avoid the water and

decides to run home via point B.

a

b

c

How far does Jillian have to swim to

get home?

How far does Aviva have to run to

get home?

How much further did Aviva travel than Jillian?


14 Kurt and Ursula are comparing answers to the question:

“Which of the following is larger: or ?”

• Ursula claims that is the largest

• Kurt claims is the largest

Who is correct? Explain how you know.


15 Describe a method you could use to compare and without a calculator.

16 Uma solved for the height of a triangle whose area is and whose base measures

ft.

Has Uma made an error? If so, identify and correct the error.

17 Give two different pairs of values for k and m which make the following equation true:

18 Sasha is simplifying the expression and says that the answer is

Is she correct? Explain how you know.

212


5.02 Rational exponents

Concept summary

Expressions with rational exponents are expressions where the exponent is a rational number (can

be written as an integer fraction). In general, a rational exponent can be rewritten as a radical (or a

radical as a rational exponent) in the following ways:

The laws of exponents can also be applied to expressions with rational exponents, where m, n, p

and q are integers and a and b are nonzero real numbers

Product of powers

Quotient of powers

Power of a power

Power of a product

Identity exponent

a 1 = a


Zero exponent

a 0 = a

Negative exponent

Worked examples

Example 1

For each expression, convert between exponential and radical forms.

a

Approach

We can use

be simplified.

to first rewrite the expression, then check to see if the radical can

Solution

b

Approach

We can use

to rewrite the expression.

Solution

Example 2

Rewrite the following as an expression with a single exponent:

a

214


Approach

We can use the product of powers,


Solution

Product of powers law

Evaluate the addition

Evaluate the division

b

Approach

We can use the power of a product


Solution

Power of a product law


c

Approach

We can use the power of a power


Solution

Power of a power law


Simplify the fraction


d

Approach

We can use the quotient of powers


Solution

Quotient of powers law

Evaluate the subtraction


Identity exponent

Example 3

Evaluate the following expressions:

a

Approach

We can use the definition of negative exponents,

to evaluate the expression.

Solution

Definition of negative exponents

Definition of rational exponents

Evaluate the radical

216


b

Approach

We can use the quotient of powers,

to simplify the expression.

Solution

Quotient of powers law

Evaluate the subtraction


Definition of zero exponent

Reflection

It is possible to simplify this expression more quickly by recognizing that a number divided by itself

always results in 1.

c

Approach

We can use the definition of zero exponents to simplify the expression, then apply the law of

negative exponents followed by multiplication to fully evaluate the expression.

Solution

Definiton of zero exponent

Definition of negative exponent

Evaluate the exponent





1 Rewrite the following exponents in radical form:

a b c d

2 Rewrite the following radicals in exponential form:

a b c d

3 Evaluate the following:

a b c d

4 Rewrite each expression using a single exponent:

a b c d

e f g h

Let’s practice

5 The volume of a cube, V, with a surface area, A, is given the formula

Find the volume of a cube with a surface area of 54 in 2 .

6 Find the value of x that makes each equation true.

a b c

d e f

7 Simplify the following expressions:

a b c

d e f

218



8 Kaley and Lucia are discussing the laws of exponents.

Kaley thinks that the exponent can only be written as the radical .

Lucia thinks that the exponent can only be written as .

Is either student correct? Explain your reasoning.

9 Brody and Veena are evaluating the radical expression Here are their works:

Brody:

Veena:

Describe the differences in the two students’ work.

10 Show that 3 5 ⋅ 3 6 is equivalent to 3 3 ⋅ 3 8 .

11 Identify and correct the error in the following work:

12 A colony of bacteria doubles each hour for three hours. After three hours, the scientists change

the temperature conditions and the colony triples each hour for three hours. Represent the

bacteria growth rate over the six hours using three different exponential expressions.

13 Identify and correct the error in the following work:


5.03 Exponential

relationships

Concept summary

Exponential relationships include any relations where the outputs change by a constant factor for

consistent changes in x, and form a pattern.

In the table, we can see the change in output is increasing by a factor of 3, and can describe this

pattern as “the number triples each time”.

x 0 1 2 3

y 1 3 9 27

This relationship can be shown on a coordinate plane, with the curve passing through the points

from the table.

y

• This graph shows an exponential relationship.

30

• y approaches a minimum value as x decreases

and approaches ∞ as x increases.

25

20

15

10

−4 −3 −2 −1

5

f(x)

1 2 3 4

x

An exponential relationship can be modeled by a function with a variable in the exponent,

known as an exponential function:

f (x) = ab x

a The initial value

b The growth or decay factor

The initial value is the output value when x = 0, and the growth or decay factor is the

constant factor.

220


Worked examples

Example 1

Consider the following pattern:

Step 1 Step 2 Step 3

a Describe the pattern in words.

Approach

We can see that the first step is made up of 2 squares, the second step is made up of 4 squares,

and the third step 8 squares. If we only considered the first two steps we would not know if this

relationship was linear or exponential, we would just know it has increased by 2. By considering

the increase from step 2 to step 3, we can see that it has increased by 4 squares, so we know

the relationship is not linear.

Solution

The number of squares doubles each step.

Reflection

We could have constructed a table of values showing the step number and the number of squares

in each step to see the pattern in another form.

b Determine the number of squares the next step if the pattern continues.

Approach

Using the pattern we described in part (a), we have to double the number of squares in step 3 to

find the number of squares in the next step.

Solution

2 ⋅ 8 = 16

There are 16 squares in the next step.


Example 2

For the following exponential function:

x 1 2 3 4

f (x) 5 25 125 625

a Identify the growth factor.

Approach

We can find the growth factor by dividing a term by the previous term, that is by evaluating.

We can see that when x = 1, f (x) = 5 and when x = 2, f (x) = 25.

Solution

Reflection

We could have chosen other values and arrived at the same result. For example

b Determine the value of f (5).

Approach

Using the growth factor found in part (a), we know that as x increases by 1, f (x) increases by a

factor of 5. This means f (5) = 5 × f (4).

Solution

f (5) = 5 × 128 = 3125

Example 3

A large puddle of water starts evaporating when the sun shines directly on it. The amount of water

in the puddle over time is shown in the table.

Hours since sun came out 0 1 2 3 4 5

Volume in mL 1024 512 256 64

222


Assuming the relationship is exponential, complete the table and describe the relationship

between time and volume.

Approach

We can find the value of b by dividing the amount of water in the puddle after one hour by the

amount that was present at the start. Using this value for b, we can then find the missing values.

Solution

Using this value for b, we know that the volume after 3 hours will be half of 256 and the time after

5 hours will be half of 64.

Hours since sun

came out

Volume in mL

0 1024

1 512

2 256

3 128

4 64

5 32

Reflection

This exponential relation is an example of one that decreases over time. We can see that it

represents decay instead of growth because in this case b is less than one.


1 Consider the functions shown in the following tables of values:

i Describe the pattern in words. ii Find the next two terms.

a

x 0 1 2 3

b

x 0 1 2 3

y 5 10 20 40

y 5 10 15 20

c

x 0 1 2 3

d

x 0 1 2 3

y 2.5 7.5 22.5 67.5

y 40000 4000 400 40


2 Consider the pattern:

a

b

Describe the pattern in words.

Determine the number of circles the next

step in the pattern contains.



a

b

Describe the pattern in words.

Determine the number of hexagons the

next step in the pattern contains.


4 Consider the function y = 5 x .

a Copy and complete the following table of values:

x 0 1 2 3 4 5

y 1 5 3125

b

Determine if y = 5 x represents a linear or an exponential relationship.

Let’s practice

5 Determine if the following functions represent a linear, exponential or another type

of relationship:

a y = 4 x b y = 4x + 7 c y = 4x 2 + 7 d y = 4 ⋅ 7 x

6 Determine if the following representations of functions are linear, exponential or another type

of relationship:

a

4

2

−5 −4 −3 −2 −1

−2

−4

−6

−8

−10

y

1 2 3 4 5

x

b

−4 −3 −2

−1

80

70

60

50

40

30

20

10

y

1 2 3 4

x

224


c

f (x) 1 2 3 4

d

f (x) 1 2 3 4

x 0.8 1.6 2.4 3.2

x 0.125 0.5 2.0 8.0

7 For the following exponential functions:

i Find the growth or decay factor. ii Determine the value of f (5).

a

x 1 2 3 4

b

x 1 2 3 4

f (x) 0.123 0.246 0.492 0.984

f (x) 10 35 122.5 428.75

c

x 1 2 3 4

d

x 1 2 3 4

f (x) 2 30 450 6750

f (x) 4096 1024 256 64


a

b

Find the growth factor.

Determine the number of triangles the

next step in the pattern contains.


9 For each of the following scenarios, state whether the type of function to be used should be

linear, exponential or neither:

a

b

c

d

Water drops from a leaking tap are falling into a bucket at a constant rate.

A radioactive element is decaying such that every 10 seconds there is half as many

atoms left.

A snowball rolling down a mountain doubles its volume every five seconds.

A raindrop falling from the sky increases in speed by 9.8 m/s every second as it is pulled

down by gravity.

10 The total volume of water that has dripped from

a tap is measured each minute and displayed in

the following table:

a

b

c


and volume.

Complete the values in the table over the next

two minutes.

Find the volume after 8 minutes.

Time (minutes) Volume (mL)

1 3

2 9

3 27

4 81

5

6


11 Hannah saves some money each week and puts

it in her piggy bank. The total amount in her piggy

bank each week is displayed in the

following table:

a

b

c

Describe the relationship beween time

and balance.

Complete the values in the table over the next

two weeks.

Find the balance after 8 weeks.

Time (weeks) Balance (dollars)

1 4

2 8

3 16

4 32

5

6


12 The cumulative rainfall over a series of

4 days is represented on the graph.

a

b

c


and volume.

Determine if the points represent

a linear or exponential function.

Explain your reasoning

Assuming the pattern continues over

the next few days, find the cumulative

volume on day seven.

36

32

28

24

20

16

12

8

Volume (mL)

4

1

2 3 4 5

Time (days)

13 Consider the two functions shown:

a

Find the average rate of change for

f (x) and g (x) from x = 0 to x = 3.

20

18

y

b

c

d

e







Describe what is happening to the

rates of change for both f (x) and g (x).

16

14

12

10

8

6

4

f(x)

g(x)

f

Determine which is an exponential

function and which is a linear function.

Explain your reasoning.

2

1

2 3 4 5 6 7 8

x

226


14 Consider the table of values for

the functions for x ≥ 1:

f (x) = (1.05) x and g (x) = 5x.

Is Ivan correct to conclude that g (x) is

always greater than f (x) for all values

of x ≥ 1? Explain your answer.

x 1 2 3 4 5

f (x) 1.05 1.10 1.16 1.22 1.28

g (x) 5 10 15 20 25

15 Several points have been plotted on

the coordinate plane:

a

Find the three points that form a linear


b For each 1 unit increase in x,

determine how much the linear function

increases by.

c

Find the three points that form an

exponential relationship between

x and y.

d For each 1 unit increase in x,

determine the constant ratio for

the exponential function.

40

35

30

25

20

15

10

5

y

A

B

C

D

E

F

G

x

1

2 3 4 5 6



exponential functions

Concept summary

To draw the graph of an exponential function we can fill out a table of values for the function and

draw the curve through the points found. We can also identify key features from the equation:

f (x) = ab x

a The initial value gives us the the value of the y-intercept

b We can use the constant factor to identify other points on the curve

The constant factor, b, can be found by finding the common ratio.

We can determine the key features of an exponential function from its graph:

• The graph is increasing

• y approaches a minimum value of 0

• The domain is −∞ < x < ∞

• The range is 0 < y

• The y-intercept is at (0, 3)

• The common ratio is 4

• The horizontal asymptote is y = 0

27

24

21

18

15

12

9

6

3

y

f(x)

1 2

x

• The graph is decreasing

• y approaches a minimum value of 0

• The domain is −∞ < x < ∞

• The range is 0 < y

• The y-intercept is at (0, 10)

• The common ratio is

• The horizontal asymptote is y = 0

25

20

15

10

y

5

f(x)

x

−4

−3

−2

−1

1 2 3 4

228


Asymptote

A line that a curve or graph approaches as it heads toward positive

or negative infinity

y

x

Worked examples

Example 1

Draw a graph of y = 2.5(4) x by first finding the common ratio and the y-intercept.

Approach

The function has a common ratio of 4, and

a y-intercept at (0, 2.5). We can find other

points on the curve using a table of values.

x −2 −1 0 1 2 3

y 0.15625 0.625 2.5 10 40 160

Solution

−2

−1

45

40

35

30

25

20

15

10

5

y

1 2 3

x

Reflection

When drawing the graphs of exponential functions we want to be sure the y-intercept is clearly

displayed and that the exponential curve is also visible. Be sure to choose a scale for the y-axis

that will show all important characteristics. In this case, we chose to scale by 5s which allows us to

read both the y-intercept at 2.5, a second point at (1, 10), the horizontal asymptote at y = 0 and the

steep slope that all exponential functions have.


Example 2

Consider the table of values for the function

x −5 −4 −3 −2 −1 0 1 2 3 4 5 10

y 486 162 54 18 6 2

a Describe the behavior of the function as x increases.

Approach

We want to identify if the values of y are increasing or decreasing as x increases.

Solution

As x increases, the function decreases at a slower and slower rate.

Reflection

We can see that the equation has a constant factor that is less than 1. This is why the function

is decreasing.

b Determine the y-intercept of the function.

Approach

The y-intercept occurs when x = 0. We can read these coordinates from the table.

Solution

(0, 2)

Reflection

We can see that the equation has an initial value of 2. This is the value of the y-intercept,

and the result of substituting x = 0 into the equation.

230


c State the domain of the function.

Approach

The domain is the complete set of possible values for x. For exponential functions, the graph

extends indefinitely in both horizontal directions.

Solution

All real x.

Reflection

All exponential equations of the form y = ab x have a domain of all real x.

d State the range of the function.

Approach

The range is the complete set of possible values for y. We can see graph extends indefinitely up

towards the left but it approaches an asymptote at y = 0 towards the right.

Solution

y > 0

Reflection

All exponential equations of the form y = ab x have a range of y > 0 for positive values of a.


1 Consider the exponential functions shown in the following tables of values:

i Determine the common ratio. ii Graph the exponential function.

a

x −1 0 1 2 3

b

x −1 0 1 2 3

y 1 3 9 27 81

y 2 10 50 250

c

x −1 0 1 2 3

d

x −1 0 1 2 3

y

y 4000 400 40 4 0.4


2 State whether the following are increasing or decreasing exponential functions:

a b y = 9 × 3 x c d

3 Graph the exponential function with the following properties:

a An initial value of 1 and a common ratio of 4.

b An initial value of 4 and a common ratio of 2.

c An initial value of 64 and a common ratio of

4 Select the exponential function that

represents the following graph.

A

B

C

4 x

5 x

5(4) x

10

5

y

D

−5(4) x

x

−10

−5

5 10

−5

−10

5 Select the exponential function that

represents the following graph.

10

y

A

y = −5 x

B

C

y = −9 x

5

x

D

−10

−5

5 10

−5

−10

6 Consider the table of values:

x 1 2 3 4 5

y 3

232


Select the exponential function that could represent the table.

A B C D

7 Consider the table of values:

x −4 −3 −2 −1 0 1 2 3 4

y 112 56 28 14 7

Select the exponential function that could represent the table.

A B C D

Let’s practice

8 Consider the graph of the equation y = 4 x :

a

b

State the equation of the horizontal

asymptote.

Explain what the horizontal asymptote

means for an exponential function.

5

4

3

2

y

1

x

−3

−2

−1

1 2 3

−1

9 Do either of the functions have x-intercepts? Explain your answer.

10 Draw the graphs of the functions

Then answer the following questions:

a State whether the following statements are true for all of the functions:

i All of the curves have a maximum value.

ii All of the curves pass through the point (1, 2).

iii All of the curves have the same y-intercept.

iv None of the curves cross the x-axis.


b

c

State the y-intercept of each curve.

Describe what happens to the values of y as x gets increasingly larger.

11 Consider the graph of the functions

y = 3 x and

15

y

a

b

c

d

State the coordinates of the point of

intersection of the two curves.

Describe what happens to the values

of y for each function as x gets

increasingly larger.

Describe the rate of change for

each function.

Describe what other features these

functions have in common.

y = 3 x

−5 −4 −3

−2

12

9

6

3

−1

−3

−6

1

⎛ 1 ⎞

y = ⎜ ⎟

⎝ 3 ⎠

2 3 4 5

x

x

12 Consider the table of values for the function .

x −5 −4 −3 −2 −1 0 1 2 3 4 5 10

y 32 16 8 4 2 1

a

b

c

d

Describe the behavior of the function as x increases.

Determine the y-intercept of the function.

State the domain of the function.

State the range of the function.

13 Consider the function y = 4(2 x ).

a Find the y-value of the y-intercept of the curve.

b Can the function values ever be negative?

c As x approaches infinity, determine the value that y approaches.

d Graph y = 4(2 x ).

e List the domain and range for the function.


a

b

Find the y-value of the y-intercept of the curve.

Complete the table of values for

x −3 −2 −1 0 1 2 3

y

234


c

d

e

Find the horizontal asymptote of the curve.

Graph

List the domain and range of the function.

15 Explain how the answers to part (a) for questions 8 and 9 relate to the functions and the

general form of an exponential function f (x) = a (b) x .

16 Compare the domain and range for f (x) = 4(2) x and from questions 8 and 9.

What do you notice?

17 Consider the given graph of y = 5 x .

a

Describe a transformation of the graph

of y = 5 x that would obtain y = −5 x .

10

8

y

b

Graph y = 5 x and y = −5 x on the same

coordinate plane.

6

4

y = 5 x

c

Compare the domain and range of

y = 5 x and y = −5 x .

−3

−2

−1

2

−2

1

2 3

x

−4

−6

−8

−10


18 Consider the graphs of the two exponential

functions R and S:

18

y

a

b

One of the graphs is of y = 4 x and

the other graph is of y = 6 x .

Identify which is the graph of y = 6 x .


For x < 0, is the graph of y = 6 x above

or below the graph of y = 4 x ?


16

14

12

10

8

6

S

4

R

2

x

−3

−2

−1

1

2 3



a

State whether the following functions are equivalent to

i ii y = 2 -x iii y = −2 x iv y = −2 -x

b Describe a trasformation that would obtain the graph of from the graph of

y = 2 x .

c Graph the functions y = 2 x and on the same coordinate plane.

20 Consider the equation y = −10 x .

a

Jenny thinks she has found a set of solutions for the equation as shown in the table:

x −2 −1 0 1 2 3

y −1 −10 −100 −1000

She notices that all the y values are negative and concludes that for any value of x, y must

always be negative. Is she correct? Explain your answer.

b Graph y = −10 x .

c Find the values of x for which y = 0.

21 Consider the original graph y = 3 x . The function values of the graph are multiplied by 2 to

form a new graph.

a

For each point on the original graph, find the point on the new graph.

Point on original graph (0, 1) (1, 3) (2, 9)

Point on new graph (−1, ) (0, ) (1, ) (2, )

b State the equation of the new graph.

c Graph the functions y = 3 x and y = 2 × 3 x on the same coordinate plane.

d For negative x-values, is the graph of y = 2 × 3 x above or below y = 3 x ?

e For positive x-values, is the graph of y = 2 × 3 x above or below 3 x ?

236


5.05 Exponential growth

and decay

Concept summary

Exponential functions can be classified as exponential growth or exponential decay based on the

value of the constant factor.

Exponential growth

f (x) = ab x

a The initial value of the exponential function

b The constant factor of the exponential function

The process of increasing in size by a constant percent rate of change.

Sometimes called percent growth. This occurs when b > 1

y

8

7

1 2 3 4

−4−3−2−1

6

5

4

3

2

1

x

Exponential decay

The process of reducing in size by a constant percent rate of change.

Sometimes called percent decay. This occurs when 0 < b < 1

y

8

7

1 2 3 4

−4−3−2−1

6

5

4

3

2

1

x

Growth factor

The constant factor of an exponential growth function

Decay factor

The constant factor of an exponential decay function


Worked examples

Example 1

Consider the exponential function:

a Classify the function as either exponential growth or exponential decay.

Approach

To classify an exponential function we want to identify the constant factor, b, and determine if b > 1

or 0 < b < 1.

Solution

In this function b = 4. Since 4 > 1, we would classify this function as exponential growth.

b Identify the initial value.

Approach

In the general form of an exponential function, y = ab x , the initial value is represented by the

variable a, which is the factor, or coefficient, that does not have a variable exponent.

Solution

The initial value is

c Identify the growth or decay factor.

Approach

In the general form of an exponential function, y = ab x , the growth or decay factor is represented

by the variable b, which is the factor with a variable exponent.

Solution

The growth factor is 4.

238


Example 2

Write an equation that models the function shown in the table.

x 0 1 2 3

y 3 9 27 81

Approach

To create the equation we need to identify the intial value and growth or decay factor for the

function modeled in the table.

For functions in the form y = a(b) x , the y-intercept represents the initial value and the growth or

decay factor can be found using the ratio of two successive outputs.

Solution

The initial value is 3 and the growth factor is so the equation for this function is y = 3(3) x .

Example 3

Write an equation that models the function shown in the graph.

32

y

−4 −3 −2 −1

28

24

20

16

12

8

4

1 2 3 4

x


Approach

To create the equation we need to identify the intial value and growth or decay factor for the

function modeled in the graph.

For functions in the form y = a(b) x , the y-intercept represents the initial value and the growth or

decay factor can be found using the ratio of two successive outputs.

Solution

The initial value is 16. To find the growth factor we will take two points, (0, 16) and (1, 4) and create

a ratio of their outputs:

The equation for this function is

Reflection

When determining the growth or decay factor be sure to use the first output as the denominator

and the second output as the numerator.

Example 4

A sample contains 300 grams of carbon-11, which has a half-life of 20 minutes.

a Write a function, A, to represent the amount of the sample remaining after n minutes.

Approach

We will start with the general form of an exponential function, f (x) = a(b) x and input all known values

to find the function.

Solution

The initial value of the function is 300 grams and half-life means that the function has a

decay factor of so we can start with the function Since the decay factor is

happening every 20 minutes we need to adjust the exponent of the function to be

Reflection

By using

in the exponent we can see that the function will halve its value once in 20 minutes

since n = 20 gives us twice in 40 minutes since n = 40 gives us etc.

240


b Evaluate the function for n = 30 and interpret the meaning in context.

Approach

In this context n represents the time, in minutes, and the output represents the amount of the

sample remaining. We will evaluate the function and apply these units to interpret the meaning of

the solution.

Solution

remaining after 30 minutes had passed.

which tells us that there were approximately 106.1 grams of carbon-11


1 Classify each of the following situations as exponential growth, decay, or neither.

a A store owner checked the sales report for the previous month and found that each

week the sales were of the previous week’s sales.

b

c

d

In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are

found to triple each hour.

1 gram of sugar is poured into a cup of water and immediately begins to dissolve.

Each second the amount of sugar remaining is

previous second.

of the amount present in the

Search engines give every web page on the internet a score (called a Page Rank)

which is a rough measure of popularity/importance. One such search engine uses a

logarithmic scale so that the Page Rank is given by: R = log 11 x, where x is the number of

views in the last week.

2 For each of the following exponential functions, where x represents time:

i Classify as either exponential growth or exponential decay.

ii State the initial amount.

iii State the growth or decay factor.

a y = 0.68(6) x b c d


3 Consider the points given in the graph.

a

Copy and complete the table of values.

20

18

y

b

c

x 1 2 3 4

y

State the growth factor.

Determine the equation relating

x and y.

16

14

12

10

8

6

4

4 Consider the given table of values.

a

b

State the initial amount.

Classify as either exponential

growth or exponential decay.

c State the growth or decay factor. d Write an equation relating x and y.

5 Consider the table of values.

a

Number of days passed (x) 1 2 3 4 5

Population of shrimp (y) 5 25 125 625 3125

Classify the population of shrimp over time as either exponential growth

or exponential decay.

b Find the equation linking population, y, and time, x, in the form y = a x .

c

Graph the population over time.

6 Some bacteria double every hour which means 1 bacterium can be 2 x bacteria after x hours.

This can be expressed as f (x) = 2 x . Select the graph that represents the function.

A

y

2

B

1

2 3 4 5

x 0 1 2 3 4

y 60000 12000 2400 480 96

y

x

10

10

5

5

x

x

−10

−5

5 10

−10

−5

5 10

−5

−5

−10

−10

242


C

y

D

y

10

10

5

5

x

x

−10

−5

5 10

−10

−5

5 10

−5

−5

−10

−10


x 0 1 2 3 4 5 6

y 9 3 1


A

y

B

y

10

10

5

5

x

x

−10

−5

5 10

−10

−5

5 10

−5

−5

−10

−10

C

y

D

y

10

10

5

5

x

x

−10

−5

5 10

−10

−5

5 10

−5

−5

−10

−10


8 Select the graph that represents

A

y

B

y

10

10

5

5

x

x

−10

−5

5 10

−10

−5

5 10

−5

−5

−10

−10

C

y

D

y

10

10

5

5

x

x

−10

−5

5 10

−10

−5

5 10

−5

−5

−10

−10

9 The population of a certain city can be expressed by the function, P (t) = 10000(2) t where t is

the number of years since 1995. Select the graph that represents the function.

A

60000

P(t)

B

60000

P(t)

50000

50000

40000

40000

30000

30000

20000

20000

10000

t

10000

t

1

2 3 4 5

1

2 3 4 5

244


C

60000

P(t)

D

60000

P(t)

50000

50000

40000

40000

30000

30000

20000

20000

10000

t

10000

t

1

2 3 4 5

1

2 3 4 5

Let’s practice

10 The number of layers, y, resulting from

a rectangular piece of paper being folded in

half x times, is shown in the graph.

a

Write the equation linking y and x in

the form y = a x .

10

9

8

7

y

b

c

d

Interpret the meaning of the y-intercept

in this context.

If a rectangular piece of paper is folded

10 times, find the resulting number

of layers.

If a rectangular piece of paper of

thickness 0.02 mm is folded 11 times,

find the total resulting thickness.

6

5

4

3

2

1

1

2 3 4 5

x

11 For each of the following graphs of a

population, P, over time, find the equation of the curve in the form P (x) = Ab x :

a

250

200

P

b

450

400

350

P

150

300

100

50

t

250

200

150

t

1 2 3 4

1 2 3 4


12 A futsal ball has less bounce than a full size soccer ball, and is expected to bounce back off

the ground to 25% of the height from which it falls, and after every subsequent bounce.

a

b

Write a function, y, to represent the height of the nth bounce, if it was dropped from an

initial height of 5 meters.

Find the height of the fourth bounce in centimeters. Round your answer to two

decimal places.

13 In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found

to double each hour.

a


Number of hours passed (x) 0 1 2 3 4

Number of bacteria (y) 1

b Find the equation linking the number of bacteria, y, and the number of hours passed, x.

c

d

e

At this rate, how many bacteria will be present in the petri dish after 15 hours?

Graph the number of bacteria over time.

Interpret the meaning of the y-intercept in this context.

14 One person in a city is infected with a virus. During the first day, before they start to show

symptoms and decide to stay home sick, they infect five more people with the virus.

a

Given each new person infects an average of five new people on their first day of being

sick, copy and complete the table showing the number of newly infected people for

that day.

Day 0 1 2 3 4

Number of people infected 1

b If this trend continues, write an expression for the number of people infected on day n.

c

d

There are 345 800 people in the city in total. Given the trend continues, after how many

days will the entire city have been infected?

If everyone wears a mask and socially distances they can restrict the spread so that each

new person only infects an average of two new people on their first day, determine the

number of people that would be infected on day 10

15 During a sudden outbreak, scientists must decide between two anti-bacterial treatments

that are currently being trialled to try to control the outbreak. In the laboratory, they apply

Treatment A and Treatment B to two samples of the bacteria, each containing 300 microbes.

They keep track of the number of microbes in each sample. The table shows the results:

Number of hours (t) 0 2 4 6

Number of microbes using Treatment A 300 310 320 330

Number of microbes using Treatment B 300 900 2700 8100

246


a

b

c

Identify which treatment causes the number of microbes to increase linearly,

and describe the rate of change.

Determine which treatment will better control the number of microbes.

Justify your choice.

Scientists approximate that they’ll have a more effective treatment in 10 hours.

By that time, what will be the difference between the number of microbes in the

two samples?


16 Winston and Rasiah want to organise a lunch time dominos tournament. They want a

knockout tournament, where the winner of each round progresses to the next round until

there are only two players left. The diagram shows the draw for the final, semi-final and

quarter final rounds.

a

Complete the table of values for the total number of players, p, that the competition can

accommodate given a number of rounds, r.

b

Number of rounds (r) 1 2 3 4

Number of players (p)

Winston and Rasiah want to make sure that each round of the tournament has every spot

filled. Find the values of p for which a tournament can be formed.

c Determine the equation relating r and p.

d

Over two weeks they can fit in 8 rounds of play. Determine how many players can they

accept into the tournament.


17 An online business that sells phones online has been growing very quickly over the last year.

They have asked you to model their growth over the next three quarters, and further into

the future.

a

Looking at the quarterly data, you start by assuming the number of customers changed

in the same way each quarter over the first year.

Complete the table under this assumption to predict the number of phones sold over the

next three quarters.

Quarter (N) 1 2 3 4 5 6 7

Online Customers (C) 480 2880 17 280 103 680

b

c

d

Find an equation for the number of sales (C) in terms of the number of quarters that have

passed (N).

Predict the number of sales they will have 2.75 years after they started the business.

Considering your answer to part (c), determine if this is a reasonable goal for the company.

18 A teaspoon of sugar, weighing 4 grams, is poured into a cup of hot water and immediately

begins to dissolve. Each second the amount of sugar remaining is

the previous second.

a

Complete the table of values:

of the amount present in

Seconds passed (t) 0 1 2 3 4 5

Undissolved sugar in grams (y)

b Write an equation linking undissolved sugar, y, and time, t.

c

d

e

f

g

Find the difference in undissolved sugar between the first and second seconds.

Find the difference in undissolved sugar between the second and third seconds.

Describe the change in the amount of undissolved sugar over time.

According to this model, will all the sugar eventually dissolve? Explain your answer.

The water is heated more so that when the experiment is performed again,

the sugar dissolves twice as quickly. Determine if more or less sugar will dissolve

between the second and third seconds. Explain your reasoning.

19 Carbon-14 is a form of carbon that is found in all living plants and animals. When the plant

or animal dies, carbon-14 is no longer taken in, and as such, it is useful in estimating the age

of fossils. It has a half-life of 5730 years, which means it takes approximately 5730 years for

half of any amount of carbon-14 to decay.

a

b

c

Write a function, A, to represent the amount of carbon-14 remaining in a sample

containing 15 grams of carbon-14, after n years.

Find the amount of isotope left after 20 000 years. Round your answer to two

decimal places.

Carbon-11 is far more reactive and has a half-life of 20 minutes. Determine how much

carbon-11 would remain from a sample of 15 grams after 3 hours. Round your answer to

two decimal places.

248


5.06 Percent growth

and decay

Concept summary

Exponential functions, both growth and decay, can be thought of in terms of their percent change:

f (x) = a(1 ± r) x

a The initial value of the exponential function

r The growth or decay rate of the exponential function, usually expressed as a decimal value

For functions in the form f (x) = a(1 + r) x , r is a growth rate, while for functions in the form

f (x) = a(1 − r) x , r is a decay rate.

Growth rate

The fixed percent by which an exponential function increases

Decay rate

The fixed percent by which an exponential function decreases

Worked examples

Example 1

Classify the exponential function f (x) = 5(1 − 0.03) x as either exponential growth or exponential

decay and identify both the initial value and the rate of growth or decay.

Approach

We know that an exponential function represents growth if the function is in the form f (x) = a(1 + r) x

and decay if it is in the form f (x) = a(1 − r) x .

Solution

This function represents exponential decay because it is in the form f (x) = a(1 − r) x .

The initial value is 5 and the decay rate is 0.03 or 3%.

Reflection

Growth and decay rates are represented as percentages.


Example 2

Consider the exponential function modeled by the graph.

18

y

−4 −3 −2 −1

16

14

12

10

8

6

4

2

1 2 3 4

x

a Identify the initial value and growth rate.

Approach

The initial value is represented by the y-intercept. To find the growth rate, first find the constant

factor by evaluating the ratio of two consecutive outputs, then find |constant factor − 1|.

Solution

The initial value of the function is 4. The constant factor is

and the growth rate is

|1.5 − 1| = 0.5 = 50%.

b Write the equation of the function in the form y = a(1 ± r) x .

Solution

y = 4(1 + 0.5) x

250


Example 3

Justin purchased a piece of sports memorabilia for $2900, and it is expected to increase in value

by 9% per year.

a Write a function, y, to represent the value of the piece of sports memorabilia after v years.

Approach

Since the memorabilia is predicted to increase in value we will use the growth rate form of

the function, f (x) = a(1 + r) x .

Solution

The initial value of the function is $2900 and the growth rate is 9% so the function is

y = 2900(1 + 0.09) v

Reflection

Remember to convert the rate as a percentage to a decimal by dividing it by 100.

b Evaluate the function for v = 8 and interpret the meaning in context.

Approach

In this context v represents the time, in years, and the output, y represents the value of Justin’s

sports memorabilia. We will evaluate the function and apply these units to interpret the meaning of

the solution.

Solution

A = 2900(1 + 0.09) 8 ≈ 5778.43 which tells us that the memorabilia will be worth approximately

$5778.43 after 8 years have passed.


1 Classify each of the following situations as exponential growth, decay, or neither.

a

b

Mr. Alvarado bought $3800 worth of stocks in 2019. The value of the stocks has been

decreasing by 1% each year.

A piece of land was purchased for $80 000. The value of the land has slowly been

decreasing by 2% annually.


c

d

Claire owns a chain of bakeries that consisted of 100 stores in 2000. There are 7% more

stores in the chain each year.

Michael has $1300 in his checking account. He gets paid $600 every 2 weeks.

2 For each of the following exponential functions, where x represents time:

i Classify as either exponential growth or exponential decay.

ii State the initial amount.

iii State the growth or decay rate.

a f (x) = 0.723(1 − 0.05) x b f (x) = 5.9(1 + 0.25) x

c f (x) = 10 000(1 + 0.13) x d f (x) = 2500(1 − 0.123) x

3 Consider the points shown on an

exponential growth function.

a

b

c

Copy and complete the table

of values.

x 0 1 2 3

y

State the initial value.

State the growth rate.

d Write the function relating x and y.

4 The local rat population is changing

according to the exponential function

f (t) = 840(1 − 0.04) t where t is the number

of years that have passed.

a

b

950

900

850

800

750

700

650

600

550

500

450

Determine the size of the initial population of rats.

Determine the rate at which the population is decreasing.

y

1 2 3 4

x

5 Consider the function where t represents time.

a

b

c

d

Find the initial value of the function.

Express the function in the form f (t) = 2(1 − r) t , where r is a decimal.

Does the function represent growth or decay of an amount over time?

State the rate of change per time period as a percentage.


x 0 1 2 3 4

y 200 188 176.72 166.12 156.15

252



A y

B

y

200

200

150

150

100

100

50

50

x

x

1 2 3 4

1 2 3 4

C

y

D

y

200

200

150

150

100

100

50

50

x

x

1 2 3 4

1 2 3 4


x 0 1 2 3 4

y 1500 1470 1440.60 41411.79 1383.55


A y

B

y

2500

2500

2000

2000

1500

1500

1000

1000

500

x

500

x

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9


C

y

D

y

2500

2500

2000

2000

1500

1500

1000

1000

500

x

500

x

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

8 The number of members at the local country club is expected to increase by 19% per year

from the current number of 160. The manager of the club is set to receive a bonus when the

number of members rises above 727. This can be expressed as y = 160(1.19) x .

Select the graph that represents the function.

A

950

900

850

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

50

y

1 2 3 4 5 6 7 8 9

x

B

950

900

850

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

50

y

1 2 3 4 5 6 7 8 9

x

C

950

900

850

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

50

y

1 2 3 4 5 6 7 8 9

x

D

950

900

850

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

50

y

1 2 3 4 5 6 7 8 9

x

254


9 Candice purchased a rare vase worth $2000 and it is expected to increase in value by 7%

per year. This can be expressed as y = 2000(1 + 0.07) x . Select the graph that represents

the function.

A

4500

4000

3500

3000

2500

2000

1500

1000

500

y

1 2 3 4 5 6 7 8 9 10

x

B

4500

4000

3500

3000

2500

2000

1500

1000

500

y

1 2 3 4 5 6

x

C

4500

y

D

4500

y

4000

4000

3500

3500

3000

3000

2500

2500

2000

2000

1500

1500

1000

1000

500

x

500

x

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

Let’s practice

10 For each of the following graphs of a population, P over time, write the exponential growth

or decay function that models it:

a

100

90

80

70

60

50

40

30

20

10

P

1 2 3 4

t

b

360

320

280

240

200

160

120

80

40

P

1 2 3 4

t


c

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

P

t

d

320

300

280

260

240

220

200

180

160

140

120

100

P

t

1 2 3 4 5 6 7 8 9

1 2 3 4

11 Marlow purchased a piece of sports memorabilia for $1300, and it is expected to increase in

value by 8% per year.

a

b

Write a function, y, to represent the value of the piece of sports memorabilia after

v years.

Find the value of the piece of sports memorabilia after 10 years.

12 Due to favorable market conditions, a company’s annual costs are expected to decrease

by 5% per year for the next few years. Their annual costs for the current financial year

are $322 000.

a

b

Write a function, y, to represent the company’s annual costs in m years.

Find the company’s annual costs in 5 years.

13 Tina currently follows 174 people on Snapface. The number of people she follows is growing

at a rate of 5% per month. Find how many people she will be following in 6 months.

14 The population of Juanita’s home town is 27 400 and has a growth rate of 6% each year.

Find the estimated population after 7 years.

15 The population of some native bees is declining at a rate of 8% per year. If there are 10 000

bees in a hive now, find the number of bees in 5 years time.

16 A new car purchased for $38 200 depreciates at a rate r each year. The value of the car for

the first two years is shown in the table below:

Years passed (n) 0 1 2

Value of car ($A) 38 200 37 818 37 439.82

a Find the value of r.

b

c

d

Write the rule for A, the value of the car n years after it is purchased.

Find the value of the car after 6 years.

A new motorbike purchased for the same amount depreciates according to the model

V = 38 200(1 − 0.03) n . Determine which vehicle depreciates more rapidly.

256


17 A sample contains 60 grams of iodine-131, an isotope of iodine, which has a half-life of

8 days.

a

b

Write a function, A, to represent the amount of iodine-131 remaining in the sample

after n days.

Find the amount of the isotope left after 72 days. Round your answer to two

decimal places.

18 When a heated substance such as water starts to cool, its temperature T at time t minutes

after being left to cool is given by an exponential function.

a

b

The function can be written in the form T = a (1 − r) t . Explain what the variables a AND

r mean within context.

Substance P starts off at a temperature of 145°C and is left to cool in a room whose

temperature is a constant 0°C. By using the table below, find the equation for the

temperature of the substance.

c

d

Minutes passed (t) 0 1 2 3 4

Temperature (T) 145 87 52.2 31.32 18.792

Find the temperature after 10 minutes, correct to two decimal places.

Will the temperature of the substance ever reach 0°C?

e The temperature of another substance Q which also starts off at a temperature of 145°C

is modelled by T = 145(1 − 0.1) t . Which substance will cool more rapidly?


19 The population of a town is expected to double in the next 20 years. Determine the rate, r,

the town will it grow each year, assuming growth is exponential. Give your answer as a

percentage correct to two decimal places. Explain your reasoning.

20 A flashmob suddenly appears at the foodcourt of the local mall and starts dancing.

They originally start with 16 performers, and by the end of every minute the number of

performers increases by 50%. The song ends after 4 minutes.

a

b

c

d

Write a function, f (t), to model the number of performers over time.

Determine the domain and range of the function.

Explain what the range means within context.

Determine if this function could continue to model this situation if the song continued.



21 Juanita and Pedro are taking part in a study, and have to eat the same food and in equal

quantity. They were chosen such that Juanita incorporates lots of sugar into their diet while

Pedro consumes limited amounts of sugar in their diet.

After eating, their blood sugar level peaked, and was then continually measured,

with the following functions modelling their blood sugar level at time t hours:

• Juanita: f (t) = 179(1 − 0.7) t

• Pedro: g (t) = 132(1 − 0.4) t

a

State the highest blood sugar level measured for:

i Juanita ii Pedro

b

Find the rate, as a percentage, that the blood sugar level decreased each hour for:

i Juanita ii Pedro

c

Compare the reactions to sugar between Juanita and Pedro.

22 In a memory study, subjects are asked to memorize some content and then asked to recall

what they can remember each day after. The given graphs represent the percentage of

content remembered by two participants Maximilian (P) and Lucy (Q) after t days.

100%

Recall percentage

80%

60%

40%

20%

2

P

Q

4 6 8 10 12 14 16 18 20

Days

a

b

Each day, Maximilian finds that he has forgotten 15% of what he could recount the day

before. Form a model for the percentage P that Maximilian can still recall after t days.

Determine which of the following could be the model for the percentage that Lucy can

still recall after t days.

A Q = (1 − 0.15) t B Q = (1 − 0.17) t C Q = (1 − 0.13) t D Q = (1 − 0.19) t

c

d

Using your answer from part (c), determine the percentage of the previous day’s content

did Lucy forgets.

According to the models, determine if either of them will completely forget the content.


258


5.07 Simple interest

Concept summary

Simple interest is a method for computing interest where the interest is computed from the original

principal alone, no matter how much money has accrued so far.

A = P (1 + rt)

A future value, or final amount

P principal, or present value

t number of years

r rate of interest per year

If we want to determine the total amount of interest, we can use the formula:

I = Prt

I Interest



t number of years

Because P and r are constant and t, the time, is variable, this is a linear function. Therefore,

the growth of a sum of money by simple interest is linear growth. We can see this in the fact that

simple interest causes an account balance to grow by a constant amount per interval of time.

Worked examples

Example 1

Taylor’s investment of $5000 earns interest at a simple interest rate of 2.5% per year for 5 years.

a Use the simple interest formula to find the final balance of their investment.

Approach

To apply the simple interest formula we need to identify which variables we have given

information for. In this example we know that the principal is P = 5000, the rate is r = 0.025 and the

time is t = 5 years.

Solution

Substituting all given values into the formula gives us:

A = P (1 + rt)

A = 5000(1 + 0.025(5))

Simple interest formula

Substitute in values for the variables


A = 5000(1.125) Simplify

A = 5625

Taylor will have $5625 after 5 years.

Reflection

Evaluate the product

When solving using the simple interest formulas, the rate must be in decimal form. To find the

decimal form of a percentage rate, divide the percentage value by 100.

In this example, we found the rate was 0.025 because

b Taylor wanted to have $6000 at the end of their investment term. Find the initial amount

they would need to invest at the same rate for the same length of time to have an ending value

of $6000.

Approach

In this example we know the ending balance is A = 6000, the rate is still r = 0.025, the time is still

t = 5, and the unknown value is the principal P.

Solution


A = P (1 + rt)


6000 = P (1 + 0.025(5)) Substitute in values for the variables

6000 = 1.125P Simplify

5333.33 = P Divide

Taylor would need to invest $533.33 to have an ending balance of $6000 after investing their

money for 5 years at a simple interest rate of 2.5%.

Example 2

Rei borrows $1200 at a simple interest rate of 8.2%, to help buy a new car. She doesn’t want to

pay more than $300 in interest over the duration of the loan. Calculate the maximum number of

years Rei can take to repay the loan while keeping her total interest at $300 or less.

Approach

We can use either formula to help solve this, but because we know the total maximum interest Rei

wants to pay, we will use the formula I = Prt. In this example we know the maximum total interest is

I = 300, the rate is 0.082 and the unknown value is the number of years t.

Since we don’t want the interest to exceed $300, we will setup an inequality.

260


Solution


I ≥ Prt


300 ≥ 1200(0.082) t Substitute in values for the variables

300 ≥ 98.4t Simplify

3.048 ≥ t Divide

Rei would need to repay the loan within approximately 3 years in order to pay no more than

$300 in interest.


1 Calculate the ending balance of the following investments, assuming simple interest is

constantly accruing:

a

b

c

d

e

$1030 at 8% per year for 2 years

$1050 at a semiannual rate of 1.1% for 9 years

$7000 at 1.8% per quarter for 9 years

$8510 at a rate of 7% per year for 25 months

$6730 at 7% per year for 33 days (Note: 1 year = 365 days)

2 Calculate the simple interest charged on the following loans, assuming interest is constantly

accruing:

a

b

c

d

$9300 at 2% per year for 6 years

$7580 at a rate of 6% per year for 13 months

$9040 at 7% per year for 41 weeks (Note: 1 year = 52 weeks)

$7180 at 6% per year for 193 days (Note: 1 year = 365 days)

Let’s practice

3 Amelia made loan repayments totalling to $4320 on a loan of $4000 with simple interest

gained over 4 years.

a

b

Calculate the total interest charged on the loan.

Calculate the annual interest rate.

4 Dave invested $848 at 8% per year simple interest. He wants to know the number of years

it will take the investment to grow to $1458.56.

a

Calculate the total amount in interest that will be earned on the investment.

b Calculate the number of years it will take the investment to grow to $1458.56.


5 $3973.00 is to be invested to earn $148.99 at a simple interest rate of 3% per year.

Find the number of years it will take to earn the interest. Round your answer to two

decimal places.

6 $85 620 is to be invested to earn $8990.10 at a simple interest rate of 7% per year.

Find the number of years it will take to earn the interest.

7 Dave invests in government bonds with a simple interest rate of 3% per year.

Calculate the size of Dave’s initial investment if he earned $517.20 interest after 10 years.

8 Find the principal investment that will earn:

a

b

c

d

$865 simple interest, at 3% annually over 6 years

$390 simple interest, at 2.5% annually over 4 years

$1200 simple interest, at 1% per month over 2 years

$3260 simple interest, at 2.5% per week over 6 months

9 Beth takes out a car loan of $3000 at a simple interest rate of 8% per year.

She plans to repay the loan over 4 years through regular monthly payments.

a

b

Calculate the total interest that Beth will be charged over the duration of the loan.

Calculate the value of each loan repayment.

10 Bianca takes out a loan of $800 to pay for an online course. Simple interest is added to the

loan value at 9% per year, charged monthly. Calculate the interest she pays in total if she

repays the loan in 9 months.

11 James’s investment of $3000 earned simple interest of 2% per year for the first 10 years and

6% per year for the next 7 years. Calculate the total amount of interest earned.

12 For each of the following investments find the annual interest rate, r, as a percentage

rounded to one decimal place.

a

b

c

$2220 invested over 8 years earns $728.16 in interest

$7500 invested over 10 quarters earns $1781.25 in interest

$6600 invested over 20 months earns $528.00 in interest

13 Asha was calculating the principal that would earn $520 simple interest over 6 months at a

rate of 4% annually, but her answer of approximately $2167 didn’t make sense.

Find and explain the error in Asha’s work:

I = Prt

520 = P (0.04)(6) Substitute the known values into the equation

520 = 0.24P Multiply

2166.67 = P Divide both sides by 0.24

262



14 Scott and Kirsten are looking to invest $15 600 and $26 700 respectively for 9 years.

Principal

Rate per year

Between $7400 and $17 400 3%

Between $17 400 and $27 400 4%

Greater than $27400 5%

a

b

Use the interest rates in the table to compare their investments.

Scott suggests that instead they should combine their investments to earn more in

interest. Determine the additional interest they would earn by combining their principals.

15 If $2196 is invested at 2% simple interest annually for 8 years, what simple interest rate, r,

would earn the same amount of interest in only 5 years?

16 Jolanda has $5000 to invest at a 1% simple interest rate and Idan has $5 000 000 to invest

at the same rate.

a

b

Calculate the difference in their earnings after 5 years.

Determine the interest rate Jolanda would need to earn the same interest as Idan after

5 years.

17 Explain the similarities between simple interest and linear growth.


5.08 Compound interest

Concept summary

Compound interest is a method for computing interest where the interest is computed from the

original principal combined with all interest accrued so far.

A future value, or final amount



n number of times interest is compounded per year

t number of years

Because compound interest has the variable t in the exponent while P, r, and n are constant,

it is considered an exponential function. We can see this in how a sum of money earning

compound interest grows by a constant percent rate per unit of time.

We can think of compound interest as a repeated application of simple interest.

Worked examples

Example 1

Frasier’s investment of $200 earns interest at a rate of 4% per year, compounded annually for

5 years.

a Use the compound interest formula to find the final balance of his investment.

Approach

To apply the compound interest formula we need to identify which variables we have given

information for. In this example we know that the principal is P = 200, the rate is r = 0.04 and the

time is t = 5. If the interest is compounded annually, then we also know n = 1 since annually means

once per year.

Solution

Frasier will have $243.33 after 5 years.

264


b Frasier wanted to have $300 at the end of his investment term. Find the initial amount would he

need to invest at the same rate to have an ending value of $300.

Approach

In this example we know the ending balance is A = 300, the rate is still r = 0.04, the time is still

t = 5 and the interest is still compounded annually so n = 1.

Solution


Compound interest formula


Simplify

Divide

Frasier would need to invest $246.58 to have an ending balance of $300 after investing his money

for 5 years at a rate of 4% compounded annually.

Example 2

A $1610 investment earns interest at 4.5% per year compounded quarterly over 10 years.

Use the compound interest formula to calculate the value of this investment to the nearest cent.

Solution

We are given that the present value is P = 1610 and the annual rate is r = 0.045. The investment

is compounded quarterly which means four times per year so n = 4 for t = 10 years. Substituting

these values into the compound interest formula gives us:

Compound interest formula


Simplify

If we evaluate the expression for A, rounding to two decimal places, we find that the value of this

investment is $2518.65


Example 3

Decide if each expression represents simple interest or compound interest and explain your decision.

a A = 500(1 + 0.02) 6

Approach

Simple interest will take the form, A = P (1 + rt) and compound interest will take the form

Solution

Compound interest, since the expression is exponential.

Reflection

This is compound interest where P = 500, r = 0.02, n = 1 and t = 6.

b A = 500(1 + 0.05(3))

Approach

Simple interest will take the form, A = P (1 + rt) and compound interest will take the form

Solution

Simple interest, since the expression is linear.

Reflection

This is simple interest where P = 500, r = 0.05 and t = 3.


1 State whether the following statements are true or false about compound interest:

a Interest is earned on the principal.

b The interest in any time period is calculated using only the original principal.

c Interest is earned on any accumulated interest.

d The amount of interest earned in any time period changes from one period to the next.

266


2 $4400 is invested for three years at a rate of 5% per year, compounded annually.

a Copy and complete the table to determine the final value of the investment.

Balance + interest Total balance Interest earned

First year − $4400 $220

Second year $4400 + $220 $4620 $231

Third year $4620 + $ $ $

Fourth year $4851 + $ $ −

b

Calculate the total interest earned over the three years.

3 Roberto’s investment of $70 000 earns interest at 5% per year, compounded annually

over 3 years.

a

b

Calculate the value of the investment after 3 years.

Calculate the amount of interest earned.

4 Hannah borrows $5000 at a rate of 4.2% per year, compounded annually.

a

b

After 4 years, Hannah repays the loan all at once. How much money does she pay back

in total?

How much interest was generated on the loan over the four years?

Let’s practice

5 $1000 is invested at 2% per year, compounded annually. The table tracks the growth of the

principal over three years.

Time Period

n (years)

Value at beginning

of time period

Value at end

of time period

Interest earned

in time period

1 $1000 $ $

2 $ $1040.40 $

3 $1040.40 $1061.21 $

a

b


Calculate the total interest earned over the three years.

6 Calculate the value of the following investments:

a A $7230 investment earns interest at 3% per year, compounded quarterly over 7 years.

b A $5520 investment earns interest at 0.8% per year, compounded monthly over 2 years.

c A $1980 investment earns interest at 0.45% per year, compounded weekly over 2 years.

d A $3000 investment earns interest at 4.2% per year, compounded daily over 14 years.


7 Calculate the amount of interest earned by the following investments:

a

b

c

d

Beth’s investment of $4800 earns interest at 3.3% per year, compounded monthly over

13 years.

James’s investment of $4210 earns interest at 0.78% per year, compounded weekly

over 16 years.

Davinda’s investment of $5390 earns interest at 2.1% per year, compounded daily over

11 years.

Una’s investment of $80 000 earns interest at 0.5% per year, compounded semiannually

over 10 years.

8 When Yuri got his first job, he invested $10 000 in an account with a 1.3% annual

growth rate.

a

b

Write a function, A (t), that represents the value of this investment t years after Yuri

started his job.

Calculate how much more the investment will be worth when Yuri has been working for

12 years than when he had been working for 5 years.

9 Jerome has just won $50 000. When he retires in 15 years, he wants to have $94 000 in

his fund which earns 6% interest annually. Determine how much of his winnings he needs to

invest now to achieve this.

10 Victoria has been promised an inheritance of $70 000 in 5 years time. Determine the

maximum she can borrow now at a rate of 7% compounded annually, and still be able to pay

off the loan with her inheritance.


11 Grace wants to invest $1600 at 5% per year for 7 years. She has three investment options,

compounded quarterly, compounded monthly, or compounded weekly.

a

Calculate the value of the investment if it is compounded:

i quarterly ii monthly iii weekly

b

c

Find the difference between the final investment values of the three options.

Describe the effect that changing the compound frequency has on the interest earned.

12 AMS Bank of offers two investment opportunities to investors:

• Option 1: 6.86% simple interest for 10 years.

• Option 2: 6.76% interest for 6 years, compounding semiannually.

a

b

Determine which investment earns the most in interest. Use calculations and reasoning

to explain your answer.

Determine which of the two investment opportunities has the larger effective annual

interest rate. Use calculations and reasoning to explain your answer.

268


13 Frank is working out the compound interest accumulated on his loan. He writes down the

following working:

a

b

c

d

e

How much did he borrow in dollars?

What is the annual interest rate as a percentage?

Is the interest being compounded weekly, monthly, quarterly or annually?

For how many years is he accumulating interest?

How much interest does he pay?

14 Maria has $5000 to invest for 6 years and would like to know which investment plan to enter

into out of the following three:

• Plan 1: invest at 4.70% per year interest, compounded monthly

• Plan 2: invest at 5.77% per year interest, compounded quarterly

• Plan 3: invest at 5.40% per year interest, compounded annually

Determine which is the best option for Maria. Use calculations and reasoning to explain

your answer.

15 Explain the similarities between compound interest and exponential growth.

16 Explain how we can use the concept of simple interest to calculate compound interest


6

Polynomials and

Factoring

Chapter outline

6.01 Adding and subtracting polynomials 272

6.02 Multiplying polynomials 276

6.03 Multiplying special products 280

6.04 Dividing polynomials by a monomial 284

6.05 Factoring GCF 287

6.06 Factoring by grouping 291

6.07 Factoring trinomials where a = 1 295

6.08 Factoring trinomials where a > 1 300

6.09 Factoring special products 305

6.01 Adding and

subtracting polynomials

Concept summary

In the same way that we can add and subtract algebraic expressions, we can also add and

subtract polynomials.

When adding or subtracting polynomials, we add or subtract the whole expressions and simplify by

combining any like terms.

• To find the sum/difference of two polynomials using a horizontal algorithm:

1. Group each polynomial inside a pair of parentheses and add/subtract one from the other.

2. If we are subtracting one polynomial from the other, we then distribute the subtraction and

remove any parentheses.

−(ax 2 + bx + c) = −ax 2 − bx − c

3. Combine any like terms. We can use the commutative and associative property of addition

(for both positive and negative terms) to group the like terms beforehand.

• To find the sum or difference of two polynomials using a vertical algorithm:

1. Align the like terms in a vertical algorithm with the appropriate operation.

2. Apply the operation to each column of the algorithm. For polynomials, there is no carry

over or borrowing between columns.

The sum or difference of any two polynomials will always be a polynomial.

Worked examples

Example 1

Simplify the expression:

(3x 2 − 5x + 1) − (x 2 + 7x − 10)

Solution

(3x 2 − 5x + 1) − (x 2 + 7x − 10) = 3x 2 − 5x + 1 − x 2 − 7x + 10 Distribute the subtraction

= (3x 2 − x 2 ) + (−5x − 7x) + (1 + 10) Group the like terms together

= 2x 2 − 12x + 11 Simplify

272


Example 2

Consider the polynomials x 3 − 6x + 2 and x 2 + 9x + 7.

a Find the sum of the two polynomials.

Approach

Since we want to find the sum of the two polynomials, we want to add them together.

Solution

Reflection

Sum = (x 3 − 6x + 2) + (x 2 + 9x + 7) Add the polynomials together

= x 3 − 6x + 2 + x 2 + 9x + 7 Remove the parentheses

= x 3 + x 2 + 3x + 9 Combine the like terms

If we want to use the vertical algorithm method, we need to make sure we correctly align the

like terms.

b Explain why the sum of two polynomials is also a polynomial.

Solution

By definition, a polynomial is the sum or difference of terms which have variables raised to

non-negative integer powers and which have coefficients that may be real or complex.

Adding one polynomial to the other is the same as adding more terms to one polynomial.

This doesn’t change the fact that it is a polynomial, so the sum of two polynomials will always

be a polynomial.

Reflection

We can use the same explanation for why the difference of two polynomials is also a polynomial,

and we can extend this explanation to include the sum or difference of any number of polynomials.

Chapter 6 Polynomials and Factoring 273


1 Simplify:

a (5x + 1) + (2x − 3) b (x + 1) − (2x − 1)

c (−6x 3 − 2) − (4x 3 − 3x 2 ) d (−2x 2 − x − 5) + (6x 2 − 9x − 8)

e

f

g (2x 2 − 3x − 9) − (−6x 2 + 6x − 2) h (9x 3 − 3x + 2) − (−6x 3 − 5x 2 − 5x)

i (2x 2 + 1) + (x 2 + 3x) + (4x − 5) j

2 A rectangle with the given dimensions is to have a

right triangle cut out from one corner.

Write and simplify a polynomial sum or difference to

model each of the following:

2x

a An expression for the length represented by y.

b

c

An expression for the perimeter of the rectangle

before the triangle has been removed.

If the area of the rectangle is 5x 2 + 16x + 3 and the

area of the triangle is x 2 + 3x, determine the area

leftover once the triangle has been removed.

y

x + 3

5x + 1

Let’s practice

3 Explain how two trinomials can be added together to produce a binomial.

4 A polynomial of degree m and a polynomial of degree n, with m ≥ n, are added together.

What is the highest possible degree of the result?

5 Fill in the blanks to make each equation true.

a (x 3 − x + 7) + ( x 3 + x + 3) = 10 b ( x 2 − 5x − 5) − ( x + 3) = 4x 2 − 2x − 2

c ( x ) + (2x 3 − 3x + 1) = 2x 3 + x + 1 d (4x + 1) − ( x + ) = 0

6 A piece of paper with dimensions 8.5 inches

by 11 inches will have a square with length

x cut out from each corner to form a box.

Write and simplify a polynomial sum that can

be used to represent the perimeter of the

shape formed once the corners have

been removed.

274



7 Determine whether each statement is always, sometimes, or never true.

Explain your reasoning with examples.

a

b

c

d

Two polynomials added together will result in a polynomial

A linear function is a polynomial

A polynomial added to a non-polynomial will result in a polynomial.

A non-polynomial added to a non-polynomial will result in a polynomial.

8 Compare and constrast adding two-digit integers and adding binomials.


6.02 Multiplying

polynomials

Concept summary

To multiply two polynomials together, we make use of the distributive property:

Distributive property

a(b + c) = ab + ac

If one of the polynomials is a monomial, and the other is fully simplified, then distributing the

multiplication will be the only step.

If neither polynomial is a monomial, then we apply the distributive property twice. For multiplication

of two binomials, this works as follows:

(w + x)(y + z) = (w + x) y + (w + x) z Distribute once

= y(w + x) + z(w + x) Reorder variables

= yw + yx + zw + zx Distribute twice

In general, we can multiply any two polynomials using this process. Here is an example of a

trinomial multiplied by a binomial:

(x 2 + 3x + 1)(x − 2) = x(x 2 + 3x + 1) − 2(x 2 + 3x + 1) Distribute once

= x 3 + 3x 2 + x − 2x 2 − 6x − 2 Distribute twice

= x 3 + x 2 − 5x − 2 Combine like terms

Notice that we can summarize the process of distributing twice as “multiply each term in the first

polynomial by each term in the second polynomial, and add the results”. We then simplify by

combining like terms, if possible.

276


Worked examples

Example 1

Multiply 3x(2x 2 − 5x + 4).

Solution

We can use the distributive property to get the product of the monomial 3x and the trinomial

2x 2 − 5x + 4.

3x(2x 2 − 5x + 4) = 6x 3 − 15x 2 + 12x

Since there are no more like terms and the expression is already in standard form, the final answer

is 6x 3 − 15x 2 + 12x.

Example 2

Multiply (7y + 2)(4y − 5).

Solution

We can use the distributive property to get the product of the two binomials 7y + 2 and 4y − 5.

(7y + 2)(4y − 5) = 4y(7y + 2) − 5(7y + 2) Distribute (7y + 2)

= 28y 2 + 8y − 35y − 10 Distribute 4y and −5

= 28y 2 − 27y − 10 Combine like terms

Since the expression is already in standard form, the final answer is 28y 2 − 27y − 10.


1 Distribute:

a (3y − 2)(−5y 2 ) b 7wy(y + w)

c 7y(y 2 + 8) d y(y − 4) + 10

e 10u 2 (7u 4 + 8u 3 ) f 8a(a − 4b − 5c)

g h y(y − 9)

i −m(m + 1) j


2 Simplify each product:

a (x − 2)(x − 3) b

c (7y − 6)(6y + 6) d 9(y + 8)(y + 2)

e (v + 5)(5v 2 − 3v − 5) f

g (5c 2 − 2)(3 − 2c + 2c 2 ) h 9y(y + 6) + 8y − 2

i j x(x + 6) + 4(5x + 3)

k l 3y + 6y 3 − 7y(y 2 − 5)

m 3n 2 (n − 2) − 2n(n 2 + 8) n 4x(5x(x − 3) + 2x)

Let’s practice

3 Find the missing values that make each equation true.

a (x + )(x − 7) = x 2 − 3x +

b (2x − 1)( x − 5) = 2x 2 + x + 5

c ( x + 3)( x + 6) = 12x 2 + 30x +

4 Sonya wants to frame a 5 × 7 picture frame. If the frame is meant to have a constant width of

x all around, write an expression for

a

b

c

The new dimensions of the picture with its frame.

The area of the picture with its frame.

The perimeter of the picture with its frame.

5 A flat rate large box from USPS has dimensions of 12 × 12 × 5.5 inches. If Jerome decides to

put a layer of insulation in his box x inches thick, write a simplified polynomial expression that

models

a

b

The dimensions of the open volume left in the box.

The volume he has left in his box to fill.

278


6 Anette decides to use the area model to complete a polynomial multiplication question,

(10x + 2)(10x + 3). To warm-up, she decides to try a simple integer example: x = 1:

10

2

10x

2

10

100

20

10x

100x

20x

3

30

6

3

30x

6

a

100 + 20 + 30 + 6 = 156 100x + 20x + 30x + 6 = 150x + 6

What is the error in Anette’s polynomial product? Correct her work and solution.

b How would you set up the area model to find the product (10x + 3)(x 2 + 10x + 2)?

Find the product using any method.

7 If Mackenzie creates a polynomial with degree 4 and Edgardo creates a polynomial with

degree 3, what do we know about the product of their polynomials?


8 A rectangular garden has a length that is one foot less than twice the width.

a

b

c

Write and simplify an expression for the area of the rectangle.

A 2 foot border is to be placed all around the garden. Write and simplify an expression

for the area of the border.

The landscaper designing the garden has 100 square feet of pavers to use for the

border. What is the largest garden the landscaper can border before running out

of pavers? Assume the landscaper will only use positive-integer dimensions.

9 Consider the following problem: “Find two binomials whose product results in a polynomial

with 5 terms”.

a

b

Yao was given the problem, but they think it’s impossible. Do you agree or disagree?

Explain.

Create two different products of two polynomials that results in an answer with exactly

5 terms.


6.03 Multiplying special

products

Concept summary

Special products are special cases of products of polynomials. With special products,

we can multiply two polynomials without using the distributive property.

For binomials, we have the following special binomial products:

Square of a binomial (Sum)

Square of a binomial (Difference)

(a + b) 2 = a 2 + 2ab + b 2 (a − b) 2 = a 2 − 2ab + b 2

Product of a sum and difference

(a + b)(a − b) = a 2 − b 2

This is the identity of the difference of two squares.

Note: (a + b) 2 a 2 + b 2 and (a − b) 2 a 2 − b 2

Worked examples

Example 1

Multiply and simplify: (x − 4) 2

Approach

We check first whether (x − 4) 2 is a special binomial product and identify its form.

Solution

Since (x − 4) 2 is a square of a binomial in the form (a − b) 2 , we use the formula and simplify

the expression:

(a − b) 2 = a 2 − 2ab + b 2 Formula for the square of a binomial

(x − 4) 2 = x 2 − 2(x)(4) + 4 2 Substitute the given values

= x 2 − 8x + 16 Simplify

280


Example 2

Multiply and simplify: (x + 4)(x − 4)

Approach

We check first whether (x + 4)(x − 4) is a special binomial product and identify its form.

Solution

Since (x + 4)(x − 4) is a product of a sum and difference, we use the formula and simplify

the expression:

(a + b)(a − b) = a 2 − b 2 Formula for the product of a sum and difference

(x + 4)(x − 4) = x 2 − 4 2 Substitute the given values

= x 2 − 16 Simplify

Example 3

Multiply and simplify: (2x + 5)(2x − 5)

Approach

We check first whether (2x + 5)(2x − 5) is a special binomial product and identify its form.

Solution

Since (2x + 5)(2x − 5) is a product of a sum and difference, we use the formula and simplify the

expression:

(a + b)(a − b) = a 2 − b 2 Formula for the product of a sum and difference

(2x + 5)(2x − 5) = (2x) 2 − 5 2 Substitute the given values

= 4x 2 − 25 Simplify

Example 4

Multiply and simplify: 3(2x + 5y) 2

Approach

We check first whether 3(2x + 5y) 2 involves a special binomial product and identify its form.


Solution

Since (2x + 5y) 2 is a square of a binomial in the form (a + b) 2 , we use the formula and simplify the

expression:

(a + b) 2 = a 2 + 2ab + b 2 Formula for the square of a binomial

3(2x + 5y) 2 = 3 [(2x) 2 + 2(2x)(5y) + (5y) 2 ] Substitute the given values and multiply by 3

= 3 [4x 2 + 20xy + 25y 2 ] Distribute the multiplication by 3

= 12x 2 + 60xy + 75y 2 Simplify


1 Complete the expansion of the following perfect squares:

a (x − 3) 2 = x 2 − x + b (x + ) 2 = x 2 + x + 36

2 Multiply and simplify:

a (x 2 + 3) 2 b 3x(7x − 5y) 2

c (4 − 5m) 2 d

e (m − 5) 2 f

g (3x − 8)(3x + 8) h

i (10 − m)(m + 10) j (4x − 3y)(4x + 3y)

k l (−7y − 8)(−7y + 8)

m (11 − 9p)(11 + 9p) n (9x + 2) 2

o p 6(8x − 9y)(8x + 9y)

q 5x(2x − 9y)(2x + 9y) r a 2 + (11 − a)(11 + a) − 10

282


Let’s practice

3 Prove the following results about special products:

a (a + b) 2 = a 2 + 2ab + b 2 b (a − b) 2 = a 2 − 2ab + b 2

c (a + b)(a − b) = a 2 − b 2

4 Show how you can use one of the special products (a ± b) 2 to find each perfect square

without a calculator.

a 21 2 b 19 2 c 33 2 d 47 2

5 Find an expression, in expanded form, for the product of three consecutive integers, where

the middle integer is m.

6 A square with side lengths measuring x − 1 centimetres has each side enlarged by a factor

of 2. Find an expression, in expanded form, for the area of the new square.

7 Show how the difference of squares pattern can be used to make the problem 105 2 − 104 2

easier to evaluate without the use of a calculator.


8 Explain how the area of a square given a binomial side length will always be a

special product.

9 Polynomial multiplication can be used to find the total area of a rectangle that has a border,

where the length and width each represent a factor in the product.

a

b

Write an algebraic model that can be used to find the total area of a square with an

unknown side length and a border with an unknown border width.

Use your model to find the total area of a square with a side length of 2m + 1 and a

border width of 3

10 Determine whether the conjectures are true or false. Create algebraic expressions to justify

your response.

a

b

The product of any two consecutive even numbers is one more than a perfect square.

The product of any two consecutive odd numbers is one less than a perfect square.


6.04 Dividing

polynomials by

a monomial

Concept summary

When dividing a polynomial by a monomial, we divide each term of the polynomial by the

monomial then simplify each individual fraction using the rules of exponents.

Dividing a polynomial by a monomial

Note: Final answers are usually written without any negative exponents.

Worked examples

Example 1

Simplify the following:

Solution

Divide each term by x

Simplify

Since there are no negative exponents and the expression is already in standard form, the final

answer is 3x 4 + 4x.

Reflection

We can check the answer by multiplying it with the monomial in the denominator. The product

should be the numerator in the original expression.

x(3x 4 + 4x) = 3x 5 + 4x 2

Check

284


Example 2

Simplify the following:

Solution

Divide each term by 3y

Simplify

Since there are no negative exponents and the expression is already in standard form, the final

answer is 2y 2 − 5y + 8.

Reflection

We can check the answer by multiplying it with the monomial in the denominator. The product

should be the numerator in the original expression.

3y(2y 2 −5y +8) = 6y 3 − 15y 2 + 24y

Check


1 Simplify the following expressions:

a b c d

e f g h

i (30j 2 ) ÷ (35j) j

k (b 3 − 4b 2 + 2b) ÷ (4b) l (9w 3 v 2 + 45w 2 v 2 + 18wv) ÷ (9wv)

Let’s practice

2 Fill in the blanks to make a true algebraic statement.

a

b

c


3 Consider the following statement: x 9 ÷ x 3 = x 3 for all nonzero real numbers x

a What is x 9 ÷ x 3 actually equal to?

b Identify a nonzero. Real number for which the original statement is true.

4 The rectangle shown has an area of

4x 4 − 8x square units. Find a polynomial

expression for its length.

?

4x

5 The triangle shown has an area of 13n 3 + 11n 2 +

29n. Find a polynomial expression for its height.

6 Consider the problem

Identify and correct the error in each of the

following student’s work.

Lawrence:

Marika:

n


7 Determine whether each statement regarding the division of polynomials by monomials is

true or false. Justify your conclusion.

a

b

c

d

When dividing a polynomial by a monomial, it is possible to reduce the number of terms.

The degree of the polynomial will always decrease after dividing by a monomial.

The result of dividing a polynomial by a monomial will have a constant term if the degree

of the monomial matches the degree of any term in the polynomial.

The result of dividing a polynomial by a monomial will be another polynomial.

8 Explain the difference between the two expressions:

5x 2 + 10x + 5/5

9 Create an example and describe the result when a polynomial is divided by a monomial that

a Has a coefficient larger than the coefficients of the terms in the polynomial.

b Has a variable with a larger degree than the terms in the polynomial.

286


6.05 Factoring GCF

Concept summary

When multiplying polynomials, we apply the distributive property:

a(b + c) = ab + ac

We can also apply this in reverse, which is known as factoring an expression:

xy + xz = x(y + z)

There can be many steps when factoring a polynomial expression. To begin with, we first want to

identify the greatest common factor, or GCF, of the terms in the expression.

Greatest common factor (GCF)

The largest whole number or algebraic expression that evenly divides the given expression.

The GCF of two or more terms includes the largest numeric factor of the coefficients of each term,

and the lowest power of any variable that appears in every term.

(If a variable does not appear in a term, it can be thought of as if it had an exponent of 0.)

Once an expression has been factored, we can verify the factored form by multiplying.

The product should be the original expression.

Worked examples

Example 1

Identify the greatest common factor between 12x 5 y 3 z 6 and 18w 2 x 7 yz 4

Approach

We find the largest whole number that evenly divides the coefficients of 12x 5 y 3 z 7 and 18w 2 x 4 yz 3

and find the expression with the lowest power of each of the variables.

Solution

The largest whole number that evenly divides the coefficients 12 and 18 is 6. The expression

with the lowest power of each of the variables is x 5 yz 4 .

Putting this together, the GCF of this expression is 6x 5 yz 4 .


Example 2

Factor the expression 8xy 4 z 3 − 16x 3 y 2 z + 4xy 3 z 5

Approach

We will first find the GCF of 8xy 4 z 3 , −16x 3 y 2 z, and 4xy 3 z 5 . We can then rewrite each expression as

a product of the GCF and any remaining factors, and then factor out the GCF.

Solution

The GCF of 8xy 4 z 3 , −16x 3 y 2 z and 4xy 3 z 5 is 4xy 2 z. So we have:

Reflection

8xy 4 z 3 − 16x 3 y 2 z + 4xy 3 z 5 = 4xy 2 z (2y 2 z 2 ) − 4xy 2 z (4x 2 ) + 4xy 2 z(yz 4 )

= 4xy 2 z(2y 2 z 2 − 4x 2 + yz 4 )

We can check the answer by distributing the multiplication:

4xy 2 z (2y 2 z 2 − 4x 2 + yz 4 ) = 4xy 2 z (2y 2 z 2 ) − 4xy 2 z (4x 2 ) + 4xy 2 z (yz 4 )

= 8xy 4 z 3 − 16x 3 y 2 z + 4xy 3 z 5

Notice that, if no mistakes have been made, these are the same steps just in reverse.

Example 3

Factor the expression 3x(x − 4) + 7(x − 4)

Approach

This time, the expressions are themselves already factored. We can use this to help identify

the GCF.

Solution

In particular, notice that both terms 3x(x − 4) and 7(x − 4) have a factor of (x − 4).

Furthermore the remaining parts of each expression, 3x and 7, have no factors in common.

So the GCF is (x − 4), which we can use to factor the expression:

3x(x − 4) + 7(x − 4) = (x − 4)(3x + 7)

288



1 Identify the greatest common factor between the following sets of terms:

a 8a and 9a b 9ax and 8ba

c 3mxy and 13mxb d 2x 2 n and 7 xn 2

e 12b and 15b 2 x f 6b 2 y and 11b 2 a

g 11n 2 mb and 13xn 2 y h 15mxya and 20xnmb

i 20mb 7 n 3 y 2 and 15a 3 x 2 bm 6 j 45n, 55n 2 y and 40mn 3

k 3a 2 mn, 2ya 4 x and 5xma 3 l 3a 2 bx, 4na 4 m and 12mba 3

2 Factor the following expressions:

a 6v + 30 b −2s − 10

c −12s + 10 d y 2 + 4y

e 2u 2 − 8u f 4t + 2t 2

g 42x − x 2 h 9x 2 y 2 z − 18xyz

i pqr + p 2 q 2 r + p 3 q 3 r j 9r 2 t 2 v 3 + 6r 3 tv − 12rt 3 v 2

k 9(3x + 4) + 4(3x + 4) l 8t(t − 3) + 9(t − 3)

3 Determine which of the following represent a factored form of the expression −12x + 20x 2 :

a −4 × 3x + 5 b −2(6x − 10) c 4x(5 − 3x) d x(−12 + 20x)

Let’s practice

4 Create an expression with at least three factorizations. Then, write out each factorization

including the fully factored form.

5 Identify the common factor of any two terms.

6 Xander was asked to factor the expression 35x 2 y + 10xy 2 − 5xy. Identify his error.

Xander:

The greatest common factor is 5xy so the factored form of the expression is 5xy(7x + 2y).

7 A farmer wants to create a set of adjacent fields which all have the same width. He plans

to create the smallest field in the shape of a square, with the largest field 9 times the size

of the smallest field, and the middle field to have a length that is 5 units more than the

smallest field.

a

b

Write expressions for the area of each field.

Use factoring to determine the dimensions of the entire field.



8 Explain how dividing monomials relates to factoring the greatest common factor.

9 Explain why the greatest common factor of the variables in any expression has the least

possible exponent of any of the terms.

10 You are to design a photo collage made of two large square photos with a side length of

x and four smaller rectangular photos that have a height of x and a width of 4 inches.

a

b

Find an algebraic expression for the area of the rectangle formed if the photos are all

placed in a single row. Draw an example of what this arrangement would look like.

Fully factor your answer from part (a) and then draw a photo arrangement that would

match these dimensions.

290


6.06 Factoring by

grouping

Concept summary

Polynomials with four terms can sometimes be factored using a method called factoring by

grouping. This method involves grouping the terms into two pairs and taking a common factor out

of each pair, then taking a common binomial factor out of the two resulting terms.

ax + ay + bx + by = a(x + y) + b(x + y)

= (x + y)(a + b)

Sometimes the terms will need to be rearranged before common factors can be taken out of each

pair. It is also possible to factor some polynomials with more than four terms using this method - for

instance, a polynomial with six terms might be able to split into three pairs which each leave

behind a common binomial.

Note that not every four-term polynomial is factorable in this way. For instance, the polynomial

3x + 6y + x + 5z

doesn’t have enough common factors between any possible pairing of terms.

Worked examples

Example 1

Factor the expression 10xy + 4x + 15y + 6.

Approach

We arrange the terms first, grouping those with common factors. We factor out the GCF on each

pair and the common binomial factor afterwards.

Solution

10xy + 4x + 15y + 6 = (10xy + 4x) + (15y + 6)

Split based on common factors

= 2x(5y + 2) + 3(5y + 2) Factor out each GCF (2x and 3)

= (5y + 2)(2x + 3) Factor out the common binomial factor

Since (5y + 2)(2x + 3) cannot be factored further, it is the final answer.


Reflection

Alternatively, we can group 10xy and 15y and 4x and 6 together and get the same answer.

10xy + 4x + 15y + 6 = (10xy +15y) + (4x + 6)


= 5y(2x + 3) + 2(2x + 3) Factor out each GCF (5y and 2)

= (2x + 3)(5y + 2) Factor out the common binomial factor

We can check the answer by multiplying the factored form (5y + 2)(2x + 3).

(5y + 2)(2x + 3) = 2x(5y + 2) + 3(5y + 2) Distribute (5y + 2)

= 10xy + 4x + 15y + 6 Check

Example 2

Factor the expression 4a 2 + 5ab − 8a − 10b.

Approach

We arrange the terms first, grouping those with common factor. We factor out the GCF on each pair

and the common binomial factor afterwards.

Solution

4a 2 + 5ab − 8a − 10b = (4a 2 + 5ab) + (−8a − 10b)


= a(4a + 5b) − 2(4a + 5b) Factor out the GCF (a and − 2)

= (4a + 5b)(a − 2) Factor out the common binomial factor

Since (4a + 5b)(a − 2) cannot be factored further, it is the final answer.

Reflection

Alternatively, we can group 4a 2 and −8a and 5ab and −10b together and get the same answer.

4a 2 + 5ab − 8a − 10b = (4a 2 − 8a) + (5ab − 10b)


= 4a(a − 2) + 5b(a − 2) Factor out the GCF (4a and 5b)

= (a − 2)(4a + 5b) Factor out the common binomial factor

We can check the answer by multiplying the factored form (4a + 5b)(a − 2).

(4a + 5b)(a − 2) = a (4a + 5b) − 2(4a + 5b) Distribute (4a + 5b)

= 4a 2 + 5ab − 8a − 10b Check

292


Example 3

Factor the expression 3x 3 + 3x 2 − 6x − 6.

Approach

We arrange the terms first, grouping those with common factor. We factor out the GCF on each pair

and the common binomial factor afterwards.

Solution

3x 3 + 3x 2 − 6x − 6 = (3x 3 + 3x 2 ) + (−6x − 6)


= 3x 2 (x + 1) − 6(x + 1) Factor out the GCF (3x 2 and −6)

= (x + 1)(3x 2 − 6) Factor out the common binomial factor

= 3(x + 1)(x 2 − 2) Factor out 3 in 3x 2 − 6

Since 3(x + 1)(x 2 − 2) cannot be factored further, it is the final answer.

Reflection

• Alternatively, we can also factor out 3 at the start.

• We can check the answer by multiplying the factored form 3(x + 1)(x 2 − 2).

3(x +1)(x 2 − 2) = 3(x 2 (x + 1) −2(x + 1)) Distribute (x + 1)

= 3(x 3 + x 2 − 2x − 2) Multiply

= 3x 3 + 3x 2 − 6x − 6 Check



a 8y(y − 4) + 10(4 − y) b 8x(2y + 3w) − z(2y + 3w)

c 8z(5x 2 + 4y) − (5x 2 + 4y) d (y + 4)(y + 7) + x(y + 7)

e 2f ( g + h) + ( g + h) 2 f 5mp + 6 + 2p + 15m

g 2x + xz − 40y − 20yz h 50 + 5y + 10x + xy

i 12xy + wx + 12yz + wz j 17x 3 + 5x 2 + 17x + 5

k x 2 + 5x + 8x + 40 l x 2 − 5x + 10x − 50

m a 3 + 5a 2 + a + 5 n 3y 3 + 6y 2 − 15y − 30


Let’s practice


a 12xy + z + 1 + 12xyz b 7x + 54yz + 63xy + 6z

3 Identify and explain the error:

5xy − 5xz + 7z − 7y = 5x( y − z) + 7( z − y)

= (5x + 7)( y − z)

4 For each polynomial:

i Find three pairs of values that make the polynomial factorable.

ii Determine what the pairs from part ( i) have in common.

a x 3 − 3x 2 + x +

b x 3 + x 2 + x +

5 Fill in the blanks to make x 3 + 5x 2 − 5x + factorable.


6 Without repeating any values, fill in each blank with a single digit number that makes the

polynomial factorable. Then, factor the polynomial.

a x 3 + x 2 + x +

b x 3 + x 2 + x +

294


6.07 Factoring trinomials

where a = 1

Concept summary

To factor a quadratic trinomial in the form ax 2 + bx + c where a = 1, we aim to find two numbers p

and q whose product is c and whose sum is b.

Factoring quadratic trinomials where a = 1

x 2 + bx + c = ( x + p)( x + q)

It is usually easiest to find these numbers by first looking at the factors of the constant term c. Steps

in factoring a quadratic trinomial where a = 1:

1. Identify the factors of the constant term, c.

2. Find p and q as a pair of factors of c which also have a sum of b.

3. Write the quadratic expression in the form ( x + p)( x + q).

4. Check whether the answer will not factor further and verify the factored form by multiplication.

Note: Some trinomials are not factorable with integer values of p and q. Such polynomials are

sometimes called prime polynomials.

Worked examples

Example 1

Consider the trinomial x 2 + 5x + 6.

a Find two integer values that have a sum of b and a product of c.

Approach

We find the factors of the constant term, 6, and choose the pair of factors that add up to 5.

Solution

The factors of 6 include 1, 2, 3 and 6. Among these factors, 2 and 3 are the ones that add up to 5.


b Factor the quadratic trinomial.

Approach

We use the values of p and q found in (a) to write the trinomial in the form (x + p)(x + q).

Solution

Since we already identified that 2 and 3 are the values of p and q respectively, we can now write

the trinomial in the form (x + p)(x + q).

Substituting the values, we get (x + 2)(x + 3) as the final answer.

Reflection

We can check the answer by multiplying the factored form (x + 2)(x + 3).

(x + 2)(x + 3) = (x)(x + 2) + (3)(x +2) Distribute the multiplication of x + 2

= x 2 + 2x + 3x + 6 Distribute the multiplication of x and 3

= x 2 + 5x + 6 Combine like terms

Example 2

Factor x 2 + 10x − 24.

Approach

First, we find the factors of the constant term −24 then choose the pair of factors, p and q, that add

up to 10. After finding p and q, we factor the trinomial by writing it in the form (x + p)(x + q).

Solution

• Since the constant term is a negative number, we know that one factor will be positive and

the other will be negative. The factors of −24 include ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24.

Among these factors, 12 and −2, are the pairs that add up to 10. Therefore 12 and −2 are the

values of p and q respectively.

Note: While 6 and 4 have a sum of 10, they don’t have a product of −24 as they are both

positive numbers.

• Since we already identified that 12 and −2 are the values of p and q respectively, we can now

write the trinomial in the form (x + p)(x + q). Substituting the values, we get (x + 12)(x − 2) as the

final answer.

296


Reflection

We can check the answer by multiplying the factored form (x + 12)(x − 2).

(x + 12)(x − 2) = (x)(x + 12) + (−2)(x + 12) Distribute the multiplication of x + 12

= x 2 + 12x − 2x − 24 Distribute the multiplication of x and −2

= x 2 + 10x − 24 Combine like terms


1 For each of the following quadratic equations:

i List all factors of the constant term. ii Factor the quadratic expression.

a x 2 + 11x + 18 b x 2 + 16x + 64 c x 2 + 17x + 72

2 For each trinomial in the form ax 2 + bx + c:

i Find two integer values that have a sum of b and a product of c.

ii

Write the quadratic in the form ( x + p)( x + q).

a x 2 + 2x − 24 b x 2 − 4x − 77

c x 2 − 17x + 66 d x 2 + 10x + 24

3 Factor the following quadratic expressions:

a x 2 − 8x + 15 b x 2 + 11x + 24

c x 2 + x − 90 d x 2 + 22x + 120

e x 2 − 34x − 72 f x 2 + x − 56

g 3x 2 − 27x − 30 h 6ab 2 − 36 ab − 162a

i 35m 4 + 140m 3 + 105m 2 j 2x 2 + 28x + 96

4 Determine whether each of the following trinomials can be written in the form (x + p)(x + q),

where p and q are positive integers:

a x 2 + 12x + 15 b x 2 − 17x + 70

c x 2 + x − 42 d x 2 + 19x + 90

e x 2 + 9x + 5 f x 2 + 7x + 12


Let’s practice

5 A square carpet with side length x is modified by shortening one side by 6 inches and

lengthening the other by 7 inches.

a

Find the area of the carpet in the form (x + p)(x + q).

b Find the area of the modified carpet in the form x 2 + bx + c.

c Describe the relationship between the values 7, −6, 1 and −42.

6 A square table of side length x has one of

its dimensions decreased by 4. This can be expressed

visually by the area model shown.

Determine if the following expressions are equivalent

to the area x(x − 4):

a Area I + Area II b x 2 - Area II

c Area I d x 2 - Area I

x

4

x

7 A square rug originally had a side length of

x inches. One of its dimensions is extended by

3 inches. We can model the area of the rug as a

collection of rectangles, as shown in the diagram.

The square on the left of the diagram has a side

length of x inches and the short side of each

rectangle is 1 inch.

a

b

c

Write a factored expression that represents the area of the rug.

Find the areas of each section from the rug.

Find the total area of the rug in terms of x. Give your answer in the standard form

ax 2 + bx + c.

8 The area of a quilt can be expressed as x 2 + 6x + 4x + 24.

a

Label the area model so that it represents the area of the quilt.

298


b

Express the area of the quilt in factored form (x + p)(x + q).

9 Suppose we want to write the trinomial x 2 + bx + c in the form (x + p)(x + q). Read each

condition and explain what must be true regarding the signs of p and q.

a

b

c

c is negative.

c and b are both positive.

c is positive and b is negative.


10 Find two digits (1 to 9) to fill in the blanks and create a trinomial in the form ax 2 + bx + c that

a Maximizes the value of b. (x + )(x + )

b Minimizes the value of c. (x + )(x + )

c Minimizes the value of b. (x + )(x − )

11 A square field with side length x will have its length

extended by a and its width extended by b.

x

a

a

b

What values of a and b would maximize the

length of the entire field?

What values of a and b would minimize the

difference in area of the bottom left and top

right field?

x

x 2

b

36


6.08 Factoring trinomials

where a > 1

Concept summary

To factor a quadratic trinomial in the form ax 2 + bx + c where a > 1, we first look for any common

factors that can be taken out of the whole expression. If this leaves behind a polynomial with a

leading coefficient of 1, then we can proceed as normal.

If, even after taking out any common factors, the leading coefficient is greater than 1, we then want

to look for two integers whose sum is b and whose product is ac. We can use then these integers,

r and s below, to rewrite the trinomial with four terms and use the method of factoring by grouping.

Factoring quadratic trinomials where a > 1

ax 2 + bx + c = ax 2 + rx + sx + c

Steps in factoring a quadratic trinomial where a > 1:

1. Factor out any GCF.

(If a is negative, we can also take out a factor of −1 before continuing.)

2. Find two numbers, r and s, that multiply to ac and add to b.

3. Rewrite the trinomial with four terms, in the form ax 2 + rx + sx + c.

4. Factor by grouping.

5. Check whether the answer will not factor further and verify the factored form by multiplication.

Remember to include any common factors that are taken out at the start, so that each step results

in an equivalent expression.

Worked examples

Example 1

Factor 2x 2 + 11x + 5.

Approach

Since there are no common factors for all three terms, we proceed with finding the value of

two integers that multiply to ac = 2 ⋅ 5 = 10 and add up to b = 11. After finding these integers,

we use them to rewrite the middle term 11x as a sum of two terms, and then factor the trinomial

by grouping.

300


Solution

The factors of 10 are 1, 2, 5, and 10. Among these factors, 10 and 1 are the pair that add up to 11.

We can use this to rewrite the trinomial and factor by grouping as follows:

2x 2 + 11x + 5 = 2x 2 + x + 10x + 5

= x (2x + 1) + 5(2x + 1) Factor each pair

Rewrite polynomial with four terms

= (2x + 1)(x + 5) Take out common factor of (2x + 1)

There are no more factors to be taken out, so the fully factored form of the polynomial is (2x + 1)

(x + 5).

Reflection

We can check the answer by multiplying the factored form (2x + 1)(x + 5).

(2x + 1)(x + 5) = x(2x + 1) + 5(2x + 1) Distribute the multiplication by (2x + 1)

= 2x 2 + x + 10x + 5 Distribute the remaining multiplication

= 2x 2 + 11x + 5 Combine like terms

Also note that we could have also rewritten the polynomial as 2x 2 + 10x + x + 5. This would have

resulted in a different middle step in factoring by grouping, but the same end result.

Example 2

Factor 5x 2 − 18x + 9.

Approach

Since there are no common factors for all three terms, we proceed with finding the value of two

integers that multiply to ac = 5 ⋅ 9 = 45 and add up to b = −18. After finding these integers, we use

them to rewrite the middle term −18x as a sum of two terms, and then factor the trinomial by grouping.

Solution

The factors of 45 are ±1, ±3, ±5, ±9, ±15, and ±45. Note that since the middle term of the trinomial

is negative, we need to consider negative factors as well as positive ones. Of these factors, −15

and −3 are the pair that add up to −18.

We can use this to rewrite the trinomial and factor by grouping as follows:

5x 2 − 18x + 9 = 5x 2 − 15x − 3x + 9

Rewrite polynomial with four terms

= 5x(x − 3) − 3(x − 3) Factor each pair to leave behind a common binomial

= (x − 3)(5x − 3) Take out common factor of (x − 3)

There are no more factors to be taken out, so the fully factored form of the polynomial is (x − 3)

(5x − 3).


Reflection

We can check the answer by multiplying the factored form (5x − 3)(x − 3).

(5x − 3)(x − 3) = x (5x − 3) − 3(5x − 3) Distribute the multiplication by (5x − 3)

= 5x 2 − 3x − 15x + 9 Distribute the remaining multiplication

= 5x 2 − 18x + 9 Combine like terms

Example 3

Factor 15x 2 − 27x − 6.

Approach

This polynomial has a common factor between all three terms. We can start by taking out this

factor, then proceeding to look for integers that we can use to rewrite the polynomial and use

factoring by grouping.

First, we find any common factors for all three terms and factor it out. Then, from the simpler

trinomial, we find the values of r and s, that multiply to ac and add up to b. After finding r and s,

we write the trinomial in the form ax 2 + rx + sx + c and factor it by grouping.

Solution

The greatest common factor of the three terms is 3. Taking out this factor, we have

15x 2 − 27x − 6 = 3(5x 2 − 9x − 2)

From the trinomial 5x 2 − 9x − 2, we want to find two numbers that have a product of

ac = (5) (−2) = −10 and a sum of b = −9. The factors of −10 include ±1, ±2, ±5, and ±10.

Of these factors, −10 and 1 are the pair that add up to −9.

We can use this to rewrite the expression and factor by grouping as follows:

15x 2 − 27x − 6 = 3(5x 2 − 9x − 2)

Previous work

= 3(5x 2 + x − 10x − 2) Rewrite polynomial with four terms

= 3(x(5x + 1) − 2(5x + 1)) Factor each pair to leave behind a

common binomial

= 3(5x + 1)(x − 2) Take out common factor of (5x + 1)

There are no more factors to be taken out, so the fully factored form of the polynomial is 3(5x + 1)

(x − 2).

302


Reflection

We can check the answer by multiplying the factored form 3(5x + 1)(x − 2).

3(5x + 1)(x −2) = 3(x(5x + 1) −2(5x + 1)) Distribute the multiplication by (5x + 1)

= 3(5 x 2 + x − 10x − 2) Distribute the multiplication by x and −2

= 15 x 2 − 27x − 6 Combine like terms and distribute the

multiplication by 3


1 Consider each quadratic expression. The factored form of the expression is (Ax + C)(Bx + D).

Determine the following:

i

ii

iii

iv

the product of A and B

the signs for A and B: opposite or the same

the product of C and D

the signs for C and D: opposite or the same

a 15x 2 + 14x − 16 b −45x 2 + 29x − 4

2 Rewrite the following quadratic expression in factored form (as a product of two

linear factors):

a 4x 2 − 32x + 15 b 5x 2 − 47x + 18 c 2x 2 − 19x + 45 d 10x 2 − 77x − 24

e 24x 2 + 22x − 35 f 6x 2 − 19x + 15 g −6x 2 + 13x − 5 h −35x 2 + 97x − 66

Let’s practice

3 Brandon claims that the polynomial 3x 2 + 15x − 42 will follow the factoring form of trinomials

with a leading coefficient a ≠ 1.

Explain Brandon’s error, then fully factor the expression.

4 Fully factor each expression.

a 60x 2 − 70x − 100 b 2x 4 − 25x 3 + 42x 2

c 9x 2 + 12x + 3 d 27 − 123x − 60x 2

5 Using the digits 1 to 9 with no repeats, fill in the blanks to create a factorable trinomial and

provide the factored form: 4x 2 + x + .


6 Consider the following polynomials:

a

b

2x 2 + x + 1, 3x 2 − 2x − 4, 6x 2 + 10x − 5, 12x 2 − 12x + 1

What do the polynomials have in common?

How do you determine if a polynomial where a ≠ 1 is not factorable?

7 A photographer wants to put a border around her photo.

a

b

c

Why might the photographer use x units to represent the width of the border?

Use factoring to find expressions for the dimensions of the photograph with its border

if the total area is 4x 2 + 24x + 35 square units.

What is the area of the photograph? Explain how you reached your solution.


8 Describe the similarities and differences between factoring a polynomial in the form

ax 2 + bx + c where a = 1 or where a 1.

9 Consider the factorable polynomial 12x 2 + bx − 6, where b is a positive integer.

a

b

Find the largest possible value for b and list the factors.

Find the smallest possible value for b and list the factors.

304


6.09 Factoring special

products

Concept summary

When factoring there are a few special products that, if we learn to recognize, can help us factor

polynomials more quickly. Two of these special products are perfect square trinomials and

difference of two squares.

Perfect square trinomials

A trinomial that is made by multiplying a binomial by itself

a 2 + 2ab + b 2 = (a + b) 2 or a 2 − 2ab + b 2 = (a − b) 2

Difference of two squares

Two perfect square expressions being subtracted from each other

a 2 − b 2 = (a + b) (a − b)

Note: The sum of squares, a 2 + b 2 , is always prime (non-factorable).

Note: To factor some polynomials completely, we may need to combine different factoring methods

(finding common factors, grouping, and using special products).

Worked examples

Example 1

Factor x 2 + 6x + 9.

Approach

We check first whether x 2 + 6x + 9 is a special product and identify its type.


Solution

Since x 2 + 6x + 9 is a perfect square trinomial, we use the formula in factoring:

a 2 + 2ab + b 2 = (a + b) 2

Formula of a perfect square trinomial

x 2 + 6x + 9 = x 2 + 2(3x) + 3 2 Substitute the given values

= (x + 3) 2 Final answer

Reflection

We can check the answer by multiplying the factored form (x + 3) 2 .

(x + 3) 2 = x 2 + 6x + 9 Square of a binomial

Example 2

Fully factor 9x 2 − 24x + 16.

Approach

We check first whether 9x 2 − 24x + 16 is a special product and identify its type.

Solution

Since 9x 2 − 24x + 16 is a perfect square trinomial, we use the formula in factoring:

a 2 − 2ab + b 2 = (a − b) 2

Formula of a perfect square trinomial

9x 2 − 24x + 16 = (3x) 2 −2(4)(3x) + 4 2 Substitute the given values

= (3x − 4) 2 Final answer

Reflection

We can check the answer by multiplying the factored form (3x − 4) 2 .

(3x − 4) 2 = 9x 2 − 24x + 16 Square of a binomial

Example 3

Factor x 2 − 25.

Approach

We check first whether x 2 − 25 is a special product and identify its type.

306


Solution

Since x 2 − 25 is a difference of two squares, we use the formula in factoring:

a 2 − b 2 = (a + b)(a − b)

x 2 − 25 = x 2 − 5 2

Formula of difference of two squares

Substitute the given values

= (x + 5)(x − 5) Final answer

Reflection

We can check the answer by multiplying the factored form (x + 5)(x − 5).

(x + 5)(x − 5) = x 2 − 25 Product of a sum and difference

Example 4

Factor 3x 2 − 27.

Approach

We find a common factor first then check whether the polynomial is a special product.

Solution

The common factor for the two terms is 3. We factor out 3 and get 3(x 2 − 9).

Since x 2 − 9 is a difference of two squares, we use the formula in factoring:

a 2 − b 2 = (a + b)(a − b)

x 2 − 9 = x 2 − 3 2

Formula of difference of two squares

Substitute the given values

= (x + 3)(x − 3) Factors of x 2 − 9

Adding the common factor 3, the final answer is 3(x + 3)(x − 3).

Reflection

We can check the answer by multiplying the factored form 3(x + 3)(x −3).

3(x + 3)(x − 3) = 3(x 2 − 9) Product of a sum and difference

= 3x 2 − 27 Distribute 3



1 Fully factor the quadratic expressions:

a x 2 + 10x + 25 b x 2 + 14x + 49

c x 2 − 16x + 64 d 81 + 18x + x 2

e 36 − 12x + x 2 f 4x 2 + 40x + 100

g 3x 2 − 24xy + 48y 2 h 81x 2 + 36x + 4

i 49x 2 − 28x + 4 j −3x 2 + 12x − 12

k n 2 − 36 l v 2 − 49

m 4 − u 2 n 121m 2 − 64

o 4 − 49y 2 p 3t 2 − 12

q 5x 2 − 320 r 121x 2 − 49y 2

Let’s practice

2 A square has an area of x 2 + 12x + 36. Determine the length of the sides of this square.

3 A cube has a surface area of 6x 2 + 36x + 54. Find an expression for the length of one side of

the cube

4 Fully factor each expression:

a 27x 2 − 48 b x 3 + 10x 2 + 25x c x 4 − 16 d 2x 4 − 1250

5 The expression 16x 2 − 24x + is a perfect square trinomial. Determine the missing value.

6 A quadratic polynomial is of the form ax 2 + bx + c, where a is non-zero, and has a special

factored form.

a If b = 0, which form could this be? What must be true about the sign of c?

b If b < 0, which form could this be? What must be true about the sign of c?

7 A shape is formed from a square with side

lengths of 3x has a smaller square,

with side lengths of 4, cut out from the center.

a

b

Write a simplified expression for the area of

this shape.

A rectangle is created to have the same

area found in part (a). Determine the

dimensions for the rectangle.

4

3x ?

?

308



8 Using the digits 1 to 9 with no repeats, fill in the blanks to create a perfect square trinomial:

x 2 + x +

9 Rewrite the expression (a 2 − b 2 ) (c 2 − d 2 ) as a difference of two squares.

10 Sheldon and Gabriella each factored 16x 2 + 48x + 36. Whose work is incorrect? Identify and

explain the error.

Sheldon:

Gabriella:

16x 2 + 48x + 36 = 16x 2 + 24x + 24x + 36 16x 2 + 48x + 36 = (4x + 6)(4x + 6)

= 8x(2x + 3) + 12(2x + 3) = 2(2x + 3)(2x + 3)

= (8x + 12)(2x + 3)

= 4(2x + 3)(2x + 3)

11 Maribel is crafting a large blanket to give to

her parents for their anniversary. Her plan is

to combine a patchwork of family photographs

with the flags from her parents’ home

countries. She hasn’t determined the exact

size of the blanket yet, and instead has

expressions for the possible dimensions of

each piece of fabric.

a

Write an expression for the total area of

the blanket. Show at least three possible

factorizations for the area.

(2x + 3)

(x + 3)

(x + 6)

(x)

(x + 3)

b

Find a value for x that creates a blanket

with reasonable dimensions.

Be sure to include units.

(x + 11)

(2x − 5)


7 Quadratic

Functions

Chapter outline

7.01 Quadratic relationships 312

7.02 Key features of quadratic graphs 321

7.03 Quadratic functions in factored form 334

7.04 Quadratic functions in vertex form 344

7.05 Quadratic functions in standard form 355

7.06 Modeling with quadratic functions 366

7.07 Linear, quadratic, and exponential models 374

7.08 Honors: Combining linear and quadratic functions 384

7.01 Quadratic

relationships

Concept summary

A quadratic relationship is any relationship where the change in output values increases or

decreases by a non-zero constant value for each consistent change in x. We can identify a

quadratic relationship or function from its equation, a table of values, and by the shape of its graph.

Standard form of a quadratic equation

A form where the quadratic function is expressed as separate polynomial terms, written in

descending order of exponents as f ( x ) = ax 2 + bx + c.

To determine the value by which a quadratic relationship increases or decreases we look at the

first difference, which is the difference between consecutive y-values, and then identify the

difference between consecutive first differences, known as the second difference. If the second

difference is a non-zero constant, we have a quadratic relationship.

Consider a table of values for y = x 2 .

x -3 -2 -1 0 1 2 3

f ( x ) 9 4 1 0 1 4 9

We can see the first differences are: -5, -3, -1, +1, +3, +5.

Notice that these values are increasing by 2 each time. This means the second differences have

a constant value of 2.

We can draw the graph of y = x 2 y

and see the

10

general shape of all quadratic relationships.

9

The curve that is formed is known as a parabola.

8

• This graph shows a quadratic relationship.

• The graph is symmetrical about a vertical line.

In this case x = 0.

• The input, x, can be any value ( positive or

negative ), and the graph continues endlessly to

the left and right.

• The output, y, has a minimum or maximum value.

In this case a minimum value of y = 0.

−4

−3

−2

7

6

5

4

3

2

1

f(x)

−1 1 2 3 4

−1

x

312


Example 1

Functions f ( x ), g( x ) and h( x ) are shown below using different representations.

4

y

3

2

f(x)

1

x

−4 −3 −2 −1 1 2 3 4

−1

−2

−3

−4

x -3 -2 -1 0 1 2 3

g( x ) 4 1.5 -1 -3.5 -6 -8.5 -11

h( x ) = 2 x

a Identify if f ( x ) is linear, quadratic, or exponential.

Approach

We want to examine the graph and determine how the y-values are changing for a consistent

change in x.

In a linear relationship, the y-value changes at a constant rate.

In a quadratic relationship, the increase/decrease in the output increases/decreases by a constant

value for each consistent change in x. The graph of a quadratic relationship is symmetrical and

y has a minimum or maximum value.

In an exponential relationship, the y-value changes by a constant percent.

Solution

The graph has a maximum value of y = 4 and is symmetrical, so therefore the graph is quadratic.

Chapter 7 Quadratic Functions 313

b Identify if g( x ) is linear, quadratic, or exponential.

Approach

We want to determine if the table of values has constant first differences ( linear ) or non-zero

constant second differences ( quadratic ), or if there is a constant percent rate of change

( exponential ).

Solution

The y-values are decreasing by a constant value of 2.5 for each consecutive increase in x by 1.

Therefore, g( x ) is linear.

c Identify if h( x ) is linear, quadratic, or exponential.

Approach

A linear equation contains a term of degree 1, but no higher degree.

A quadratic equation contains a term of degree 2, but no higher degree.

An exponential equation has the variable in the exponent.

Solution

The function h( x ) has a variable in the exponent, so the function is exponential.

Example 2

Complete the table for the following quadratic function:

x -3 -2 -1 0 1 2 3 4 5

f ( x ) 0 -3 -4 -3 5

Approach

We first want to determine if the minimum or maximum value of the quadratic function is shown in

the table. If it is, we can use the property of symmetry to help complete any missing values. To find

any remaining values, we want to find the first and second differences and use these to complete

the table.

314


Solution

In this case since f ( -1) and f ( 1) are equal, the minimum value occurs at the x-value directly

between -1 and 1. The minimum value occurs at ( 0, -4 ).

Since the quadratic is symmetric about this point, we can use this to find other points. We are given

the point ( -2, 0 ) so using symmetry we know ( 2, 0 ) must also be on the function. In other words,

when we move 2 units to the left of x = 0, the output is 0 so when we move 2 units to the right of

x = 0 the output must also be 0. For the same reason we know when x = -3, the output is 5.

x -3 -2 -1 0 1 2 3 4 5

f ( x ) 5 0 -3 -4 -3 0 5

To find the last two values we want to find the first and second differences. Comparing the output

values we have found so far, we can see the first differences are: -5, -3, -1, +3, +5, …. This means

the second difference is +2.

From this information, we can find the remaining values. When x = 4, the output will be 5 + 2 = 7

more than when x = 3, and the value for x = 5 will be 7 + 2 = 9 more than this value.

x -3 -2 -1 0 1 2 3 4 5

f ( x ) 5 0 -3 -4 -3 0 5 12 21

Reflection

The minimum or maximum will not necessarily be shown in the table, but it is helpful if it is shown

due to the symmetry of quadratic relationships.

We could have also noticed that this table follows a similar pattern to the parent quadratic function,

f ( x ) = x 2 , but is translated down 4 units. If we noticed this, we could use the rule to complete the

missing values.



1 Determine if the following are linear, exponential, or quadratic:

a

y

b

y

5

5

x

x

−5 5

−5 5

−5

−5

c

y

d

y

5

5

x

x

−5 5

−5 5

−5

−5

e 2x - 3y - 2 = 0 f y = 2x 2 - 3x - 2

g y = 0.5 x - 1 h y = 1 - 3x - 5x 2

i

x f ( x )

j

x g( x )

-3 27

-2 12

-1 3

0 0

-3

-2

-1

1

27

1

9

1

3

0 1

1 3

1 3

316


k

x h( x )

l

x k( x )

-3 -9

-2 8

-2 -6

-1 2

-1 -3

0 0

0 0

1 2

1 3

2 8

Let’s practice

2 Consider the following patterns:

i

ii

iii

iv

a

Describe how you see the pattern growing.

Determine how many squares or hexagons will be in the next figure, assuming the

pattern continues.

Determine how many squares or hexagons will be in the 8th figure.

Determine if the rule for the number of squares or hexagons in any case would be a

linear, exponential, or quadratic expression.

b


3 Draw figures for the first four cases, knowing the rule for the pattern is represented by

f ( n ) = n 2 + 3. Assume for the first case that n = 1.

4 Consider the pattern shown:

a

b

c

d


Determine how many squares will be in the next figure, assuming the pattern continues.

Determine how many squares will be in the 11th figure.

Determine if the rule for the number of squares in any case would be a linear,

exponential or quadratic expression.

5 A pinecone falls from the top of a tree that is 18 ft tall. The function h( t ) = -16t 2 + 18

represents the height of the pinecone above ground in feet after t seconds.

a


t 0 0.25 0.5 0.75 1 1.25

h( t ) = -16t 2 + 18 9

b

Approximate the time that the pine cone hits the ground. Justify your reasoning.

6 Copy and complete the tables for the following quadratic functions:

a

x -3 -2 -1 0 1 2 3 4 5

f ( x ) 5 2 1 2 10

b

x 1 3 4 5 6 7 8 10

g( x ) 4 1 0 1

c

x -2 -1 0 1 2 3 4 5 6

g( x ) 2 3 2 -1

318



7 Consider the pattern shown:

a

b


Determine a rule to find the number of squares in any case, assuming the

pattern continues.

8 The perimeter for a rectangular community garden is 60 ft.

a Copy and complete the table for all missing measurements.

Width ( ft ) Length ( ft ) Area ( sq ft )

6 24

12

15

18 216

22

24

b

Determine the length and width that would produce the largest possible area.


c For any width w and any length l, determine an expression to find the area in terms of w.

9 Consider the table shown:

x -2 -1 0 1 2

f ( x ) 16 4 0 4 16

a

b

c

Describe a rule to find the nth term.

Determine if the rule to find the nth term would be a linear, exponential, or quadratic

expression.

Determine a rule to find the nth term.


10 Compare and contrast the following equations, tables, and graphs for the three functions:

a( x ) = 3x + 1

y

x a( x )

-3 -8

-2 -5

-1 -2

0 1

1 4

2 7

3 10

5

−5 5

−5

x

b( x ) = 2 x

x b( x )

-3 0.125

-2 0.25

-1 0.5

0 1

1 2

2 4

3 8

y

5

−5 5

−5

x

c( x ) = x 2 - 5

y

x c( x )

-3 4

-2 -1

-1 -4

0 -5

1 -4

2 -1

3 4

5

−5 5

−5

x

320


7.02 Key features of

quadratic graphs

Concept summary

A quadratic function is a polynomial function of degree 2. A quadratic function can be written in

the form f ( x ) = ax 2 + bx + c where a, b, and c are real numbers.

A parabola is the graph of a quadratic function.

From the graph of a quadratic function, we can identify key features including domain and range,

x and y-intercepts, increasing and decreasing intervals, positive and negative intervals, and end

behavior. The parabola also has the following two features that help us identify it, and that we can

use when drawing the graph:

Axis of symmetry

A line that divides a figure into two parts, such that the reflection of

either part across the line maps precisely onto the other part.

For a parabola, the axis of symmetry is a vertical line passing through

the vertex.

y

x

Vertex

The point where the parabola crosses the axis of symmetry.

The vertex is either a maximum or minimum on the parabola.

y

x


We can determine the key features of a quadratic

function from its graph:

This is a graph of the parent quadratic function:

f ( x ) = x 2

• Axis of symmetry: x = 0

• Vertex: ( 0, 0 )

• y-intercept at ( 0, 0 )

• x-intercept at ( 0, 0 )

• Domain: {x|-∞ < x < ∞}

• Range: {y|y ≥ 0}

• As x → ∞, f ( x ) → ∞

• As x → -∞, f ( x ) → ∞

y

10

9

8

7

6

5

4

3

2 f (x)

1

−4 −3 −2 −1 1 2 3 4

−1

x

Worked examples

Example 1

Graph the quadratic function: f ( x ) = x 2 - 2x + 1

Approach

We can create a table of values that satisfy f ( x ) and use it to help graph the function. It can be

useful to choose values for x that are positive and negative, as well as x = 0:

x -2 -1 0 1 2 3 4

f ( x )

To complete the table, evaluating the function for each x-value.

Here is how we can obtain f ( -2 ):

f ( -2 ) = ( -2 ) 2 - 2( -2 ) + 1

f ( -2 ) = 4 + 4 + 1

f ( -2 ) = 9

Repeat this process for each x-value in the table.

x -2 -1 0 1 2 3 4

f ( x ) 9 4 1 0 1 4 9

Now we can use these points to graph the quadratic function f ( x )

322


Solution

9

8

7

6

5

y

4

f (x)

3

2

1

x

−2

−1

1 2 3 4

Reflection

Having the vertex in your table is useful, since it tells you where the parabola will change direction.

Sometimes the table values you select will not include the vertex of the function, depending on the

quadratic function being graphed. If you plot your initial table values and find you are unsure where

the parabola changes direction, you can add additional values to your table until you can identify

where f ( x ) changes direction.

Example 2

Consider the graph of the quadratic function g( x ):

9

8

7

6

5

4

3

2

1

y

g(x)

x

−5 −4 −3 −2 −1

−1

−2

−3

1 2 3


a Determine the x and y-intercepts.

Approach

To find the x-intercepts, locate the places where the parabola crosses the x-axis.

To find the y-intercepts, locate the place where the parabola crosses the y-axis.

Solution

We can identify the intercepts on the graph:

From the graph we can see the there are two

x-intercepts, at ( -4, 0 ) and ( 2, 0 ), and there is one

y-intercept at ( 0, 8 ).

9

8

7

6

5

4

3

2

1

y

g(x)

x

−5 −4 −3 −2 −1

−1

−2

−3

1 2 3

b Determine the domain and range.

Approach

To find the domain of g( x ), we want to find all possible x-values for which g( x ) could be graphed.

To find the range, we want to find all possible values of g( x ). The vertex of a parabola affects the

range of the function, as it will be the maximum or minimum value of g( x ).

Solution

We can see that for a parabola, there are no restrictions on which x-values can be graphed as each

side of the parabola continues infinitely in either x-direction.

This parabola opens down, so the y-value of the vertex is the maximum value of the function.

The parabola continues infinitely in the negative y-direction.

Domain: -∞, x < ∞

Range: -∞, < y ≤ 9

324


c Identify each interval where the function is either increasing or decreasing.

Approach

g( x ) is increasing where the output values become larger as its input values become larger.

g( x ) is decreasing where the output values become smaller as its input values become larger.

Parabolas change from increasing to decreasing or decreasing to increasing about the x-value of

the vertex.

Solution

We can see from the graph of g( x ) that the parabola has positive slope up until x = -1 and negative

slope after x = -1.

The function is neither increasing or decreasing at the vertex.

Increasing: x < -1

Decreasing: x > -1

d Identify each interval where the function is either positive or negative.

Approach

In order to find the intervals for where the function is positive, we want to find the set of x-values

where the parabola lies above the x-axis.

Similarly, to find the intervals where the function is negative, we want to find the set of x-values

where the parabola lies below the x-axis.

The x-intercepts are where the function crosses the x-axis, in other words where the function is

equal to 0. These values will help us determine where the function is positive and negative.

Solution

We can see that the parabola lies above the x-axis

between x = -4 and x = 2. We can also see that the

parabola lies below the x to the left of x = -4 and to the

right of x = 2.

Positive: -4 < x < 2

Negative: x < -4, x > 2

9

8

7

6

5

4

3

2

1

y

g(x)

x

−5 −4 −3 −2 −1

−1

−2

−3

1 2 3


e State the end behavior of the function.

Approach

In order to find the end behavior, we want to find the y-value that the function approaches as the

x-values approach positive and negative infinity. We can see on the graph that as the x-values on

the parabola get larger in either direction, the parabola continues downwards, or towards -∞.

Solution

x → ∞, y → -∞

x → -∞, y → -∞

Example 3

The graph shows the height of a softball above ground ( in ft. ) x seconds after it was thrown in

the air.

14

y

12

10

8

6

4

2

0.5 1 1.5 2 2.5 3 3.5

x

a Find the y-intercept and describe what it means in context.

Approach

We want to find the place where the parabola crosses the y-axis.

Once we find the y-intercept, we want to connect this to the context of the softball. Since the y-axis

represents the height of the softball in feet above ground, we can use it to identify the height of the

softball at 0 seconds.

326


Solution

We can identify the y-intercept on the graph:

The y-intercept is ( 0, 6 )

The y-intercept tells us that the softball was

thrown from a height of 6 feet above

the ground.

14

12

10

8

y

6

4

2

0.5 1 1.5 2 2.5 3 3.5

x

b Find the value of the x-intercept and describe what it means in context.

Approach

We want to find the place where the parabola crosses the x-axis.

Once we find the x-intercept, we want to connect this to the context of the softball. Since the x-axis

represents the time in seconds after being thrown, we can use it to identify at how many seconds

does the softball hit the ground.

Solution

We can identify the x-intercept on the graph:

The x-intercept is ( 3,0 )

The x-intercept tells us that the softball hits the

ground 3 seconds after it was thrown in the air.

14

12

10

y

8

6

4

2

0.5 1 1.5 2 2.5 3 3.5

x


c Find the value of the vertex and describe what it means in context.

Approach

In order to find the vertex, we want to find the maximum point of the parabola.

Once we find the vertex, we want to connect this to the context of the softball. We know that the

x-value of the vertex represents time in seconds after the softball is thrown and the y-value of the

vertex represents the height of the softball above ground in feet.

Solution

We can identify the vertex on the graph:

The vertex is ( 1.25, 12 )

After 1.25 seconds, the softball reaches a

maximum height of 12 feet above ground.

14

12

10

y

8

6

4

2

0.5 1 1.5 2 2.5 3 3.5

x


1 Consider the graph of the function y = f ( x ).

a

b

c

d

e

f

g

h

Find the coordinates of the x and

y-intercepts.

Determine the domain of the function.

Determine the range of the function.

Determine the intervals of the domain

where the function is decreasing.


where the function is increasing.


for where the function is negative.


for where the function is positive.

State the end behavior of the function.

−8

−6

−4

10

8

6

4

2

−2

−2

−4

−6

−8

−10

y

2 4 6 8

x

328


2 For each table, complete the following:

i Graph the quadratic function shown in the following tables.

ii Find the coordinates of the vertex.

iii Determine whether the vertex is a maximum or minimum point.

iv Determine the axis of symmetry.

a

x 0 1 2 3 4 5 6

y -7 -2 1 2 1 -2 -7

b

x -7 -6 -5 -4 -3 -2 -1

y 11 6 3 2 3 6 11

Let’s practice

3 Consider the function g( x ) = -( x + 5 )( x + 1).

a Copy and complete the table.

x -6 -5 -4 -3 -2 -1 0

g( x )

b

c

d

e

f

Determine the equation of the axis of symmetry.

Determine the intervals of the domain where the function is increasing and decreasing.

Determine the intervals of the domain where the function is positive and negative.

Determine the end behavior of the function.

Graph the function.

4 Consider the function h( x ) = x 2 - 4x + 4.

a Copy and complete the table:

x -1 0 1 2 3 4 5

h( x )

b

c

d

e

f

g

Find the coordinates of the x and y-intercepts.

Find the coordinates of the vertex.

Determine the domain and range.

Determine the equation of the axis of symmetry.

Determine the end behavior of the function.

Graph the function.


5 Zahra jumps off a diving platform and the path of her dive is modeled by the function

f ( x ) = −x 2 + 2x + 8, where f ( x ) is her height in meters above the pool, and x is the horizontal

distance in meters from the edge of the diving platform.

a

b

c

d

e

f

A

Find the height of the diving platform.

Find the maximum height of Zahra’s dive.

Determine the interval of the domain where the diver is ascending.

Determine the interval of the domain where the diver is descending.



8

6

4

2

−6 −4 −2

−2

−4

−6

−8

y

2 4 6

x

B

9

8

7

6

5

4

3

2

1

y

1 2 3 4 5

x

C

9

8

7

6

5

4

3

2

1

y

1 2 3 4 5

x

D

8

6

4

2

−6 −4 −2

−2

−4

−6

−8

y

2 4 6

x

6 A frisbee is thrown upward and away from the top of a cliff. The height, y meters, of the

frisbee at time, x seconds, is given by the equation y = −20( x − 6 )( x + 2 ).

a

b

c

d

e

At what the height is the frisbee thrown?

Find the maximum height the frisbee reached.

Determine the interval of the domain where the frisbee is ascending.

Determine the interval of the domain where the frisbee is descending.


330


f


A

y

B

y

360

360

300

300

240

240

180

180

120

120

60

x

60

x

1 2 3 4 5 6

1 2 3 4 5 6

C

320

y

D

320

y

240

240

160

160

80

x

80

x

−6 −4 −2

−80

2 4 6

−6

−4

−2 2 4 6

−80

−160

−160

−240

−240

−320

−320

7 The graph shows the height of a soccer ball above ground, in feet, after it is kicked in terms

of x seconds.

y

a Find the y-intercept.

b

c

d

e

f

Describe what the y-intercept means in

context.

Find the x-intercept.

Describe what the x-intercept means in

context.

Find the coordinates of the vertex.

Describe what the vertex means

in context.

10

8

6

4

2

1 2 3 4 5

x


8 A clothing company is designing a new jacket and wants to determine how to maximize their

profit once the jacket is ready to be sold. The graph represents the total profit, P, the shop

will make at each price point, x, the jacket could sell for.

700

600

500

y

400

y = P(x)

300

200

100

5 10 15 20 25 30 35 40 45 50 55 60

x

a

b

c

d

Find the value of the vertex and describe what the vertex means in context.

Determine the domain that results in a profit for the clothing company.

Determine the corresponding range of profit.

The manager believes that selling a jacket at a higher price will always result in a larger

profit. Explain how increasing the price of the jacket affects the profit.

9 Graph quadratic functions with the following key features:

a • Increasing when x > 2

• Decreasing when x < 2

• Range: y ≥ -3

c

• End behavior:

x → ∞, y → −∞

x → −∞, y → −∞

• Positive when -2 < x < 3

• Negative when x < -2 or x > 3

• y-intercept: ( 0, 6 )

b • x-intercepts: ( -3, 0 ) and ( 3, 0 )


• Range: y ≥ -9


y

16

10 Graham, Habib, and Joel throw or kick 14

footballs around the same time. The vertical 12

height of Graham’s football is shown in the

10

8

graph. The function G( t ) represents the 6

vertical distance of the football above the 4

ground, in feet, and t represents time, in

2

seconds. 1

y = G(t)

2 3 4 5 6 7 8

t

332


The vertical height of Habib’s football is shown in the table. H( t ) represents the vertical

distance of the football above the ground, in feet, and t represents time, in seconds.

t 0.172 2 3 4 5 5.828

H( t ) 0 14 16 14 8 0

The height of Joel’s football can also be modeled with a quadratic function that has the

following key features. Let J( t ) represent the vertical distance of the football above the

ground, in feet, and t represent time, in seconds.

• y-intercept: ( 0, 5 )

• Vertex at ( 1, 5.5 )

• t-intercept: ( 4.317, 0 )

Use the above information to complete the following:

a

b

c

d

e

Graph the three quadratic functions on the same coordinate plane.

Determine whose football reached the ground the quickest after being kicked or thrown.


Determine whose football reached the greatest height. Explain your answer.

Describe what G( 0 ) and H( 0 ) mean in context.

Habib claims that his football reaches the maximum height the quickest. Determine

whether or not Habib is correct. Explain your answer.

11 Elise is the owner of a restaurant. Elise wants to install new wooden floors in several rooms.

The rooms in the restaurant are square. The wood costs $6.25 per square foot. The cost of

the flooring in terms of its side length is shown by the quadratic function C( x ) = 6.25x 2 .

a

b

Determine how much Elise should expect to spend on flooring if the room has side

lengths of 12 ft.

Determine how much the price would change if the side lengths decreased by 3 ft.

c Graph the quadratic model from part ( a ).

d

Describe what changes and what stays the same about the graph of the quadratic model

if the cost per square foot decreases.

12 Rafael is trying to explain the design for a painting to his

friend, over the phone. Describe how Rafael could use key

features of quadratic functions to share his idea.


7.03 Quadratic functions

in factored form

Concept summary

One way to represent quadratic functions is using the factored form. This form allows us to identify

the x-intercepts, direction of opening, and scale factor of the quadratic function.

y = a( x - x 1 )( x - x 2 )

x 1 , x 2 x-values of the x-intercepts

a The scale factor, so tells us about the shape of the graph

y

y

4

4

3

3

2

2

1

x

1

x

−4

−3

−2

−1 1 2 3 4

−1

−4

−3

−2

−1 1 2 3 4

−1

−2

−2

−3

−3

−4

−4

If a > 0, then the quadratic function opens

upwards and has a minimum value.

If a < 0, then the quadratic function opens

downwards and has a maximum value.

To draw the graph of a quadratic function, we generally want to find three different points on the

graph, such as the x- and y-intercepts. As the graph has a line of symmetry passing through the

vertex, we know the vertex lies half way between the two x-intercepts. We can also determine the

direction in which the graph opens by identifying if the scale factor, a, is positive or negative.

334


Worked examples

Example 1

A quadratic function in factored form has the equation:

y = 2( x - 4 )( x + 6 )

a State the coordinates of the x-intercepts.

Approach

In the factored form y = a( x - x 1 )( x - x 2 ) the values of x 1 and x 2 are the x-values of the x-intercepts.

The y-value of the x-intercepts is y = 0.

Solution

The x-intercepts are ( 4, 0 ) and ( -6, 0 ).

Reflection

Notice that x + 6 is the same as x - ( -6 ).

b Determine the coordinates of the y-intercept.

Approach

The y-value of the y-intercept is the result when x = 0. We can substitute x = 0 into the factored

form to find this value.

Solution

To find the y-value of the y-intercept:

y = 2( x - 4 )( x + 6 )

Given quadratic function

y = 2( 0 - 4 )( 0 + 6 ) Substitute in x = 0

y = 2( -4 )( 6 )

y = -48

The y-intercept is ( 0, -48 ).

Simplify the factors

Simplify


c Determine the coordinates of the vertex.

Approach

The vertex lies on the axis of symmetry, so the x-coordinate of the vertex will be the average of

4 and -6. We can then substitute this value into the funtion to find the y-coordinate.

Solution

To find the x-coordinate:

The average of 4 and -6 is half way between them. We can calculate that 4 + (−6)

2

x-coordinate of the vertex and the axis of symmetry is x = -1.

To find the y-coordinate:

y = 2( x - 4 )( x + 6 ) Given quadratic function

y = 2( -1 - 4 )(-1 + 6 ) Substitute in x = -1

y = 2( -5)( 5 )

Simplify the factors

y = -50

Simplify

The vertex is (-1, -50).

= −1, so the

d Draw the graph of the function.

Approach

The scale factor is 2 which is positive, so the graph will open up. We can draw the graph through

any three points that we know are on it.

Solution

20

10

y

x

Reflection

Any three points is enough to draw the graph,

but knowing where the vertex is can make

it easier since the vertex is on the axis of

symmetry.

−8

−6

−4

−2 2 4 6 8

−10

−20

−30

−40

−50

−60

336


Example 2

Consider the graph of a quadratic function:

4

3

2

y

1

x

−4

−3

−2

−1 1 2 3 4

−1

−2

−3

−4

a Identify the coordinates of the x- and y-intercepts of the function.

Solution

The x-intercepts are ( -2, 0 ) and ( 3, 0 ).

The y-intercept is ( 2, 0 ).

b Find the equation of the quadratic function in factored form.

Solution

Since the x-values of the x-intercepts are -2 and 3, we know that the factored form will be:

y = a( x + 2 )( x -3 )

for some value of a. We can find a by substituting in the coordinates of the y-intercept into the function.

To find a:

y = a( x + 2 )( x -3 )

Factored form

2 = a( 0 + 2 )( 0 - 3 ) Substitute in the y-intercept

2 = a( 2 )( -3 ) Simplify the factors

2 = -6a Simplify

1

− = a

Divide both sides by -6

3

The equation of the quadratic function in factored form:

1

y = −

3

(x + 2)(x − 3)



1 State the factored form for a quadratic equation and describe its features.

2 Describe the axis of symmetry of a quadratic function.

3 For each of the following quadratic equations, state the x-values of the x-intercepts:

a y = ( x - 1)( x - 4 ) b y = ( x - 3 )( x + 7 )

c y = x( x - 4 ) d y = 2( x - 1)( x + 5 )

4 For each of the following quadratic functions, determine the y-value of the y-intercept:

a y = 3( x - 1)( x - 8 ) b y = x( x + 9 )

5 Determine the axis of symmetry of this quadratic function.

y

8

6

4

2

x

−6

−4

−2

2 4 6 8 10

−2

−4

−6

−8

6 Consider the following graph of a function.

Select the equation that represents

the function:

A y = x( x + 5 )

B y = x( 5 − x )

C y = −x( x − 5 )

D y = x( x − 5 )

y

1

−2 −1 1

−1

−2

−3

−4

2 3 4 5 6

x

−5

−6

−7

338




the function:

A h = ( 2 − t )( t − 4 )

B h = −( 2 − t )( t − 4 )

C h = ( t + 2 )( t − 4 )

D h = −( t − 2 )( t + 4 )

9

8

7

6

5

4

3

2

1

h

t

−6 −5 −4 −3 −2

−1

−1

1 2 3 4

−2


x 0 2 4 6 8

y 12 0 −4 0 12


A

−10 −8 −6 −4

y

16

14

12

10

8

6

4

2

−2

−2

2 4

−4

x

B

−10 −8 −6 −4

y

4

2

−2

−2

2 4

−4

−6

−8

−10

−12

−14

−16

x

C

y

16

14

12

10

8

6

4

−4

2

−2

−2

−4

2

4 6 8 10

x

D

y

4

2

−4 −2

−2

2

−4

−6

−8

−10

−12

−14

−16

4 6 8 10

x




A

y

6

5

4

3

2

1

−3 −2 −1 1

−1

−2

−3

−4

2

x

3 4 5 6 7

B

−7 −6 −5 −4 −3 −2

y

4

3

2

1

x

−1 1 2 3

−1

−2

−3

−4

−5

−6

C

−7 −6 −5 −4 −3 −2

y

6

5

4

3

2

1

x

−1 1 2 3

−1

−2

−3

−4

D

y

4

3

2

1

−3 −2 −1 1

−1

−2

−3

−4

−5

−6

2

3 4 5 6 7

x

10 A quadratic function has x-intercepts of ( 5, 0 ) and ( −3, 0 ), a y-intercept of ( 0, 60 ), and its

vertex is a maximum.


A

−4 −3 −2

30

20

10

−40

−50

−60

−70

y

−1 1

−10

−20

−30

2 3 4 5 6

x

B

70

60

50

40

30

20

10

y

−4 −3 −2 −1 1

−10

−20

−30

2 3 4 5 6

x

340


C

70

60

50

40

30

20

10

−6 −5 −4 −3 −2 −1

−10

−20

−30

y

1 2 3 4

x

D

30

20

10

−6 −5 −4 −3 −2 −1 −10

−20

−30

−40

−50

−60

−70

y

1 2 3 4

x

Let’s practice

11 For each quadratic function:

i State the coordinates of the x-intercepts.

ii Determine the axis of symmetry.

iii Determine the coordinates of the vertex.

iv Graph the function.

a y = ( x - 2 )( x + 4 ) b y = -2( x - 1)( x - 5 )

12 For each graph:

i State the x-values of the x-intercepts.

ii State the y-value of the y-intercept.

iii Write the equation of the quadratic function in factored form.

a

10

y

b

8

y

8

6

6

4

−2

4

2

−1

−2

1 2 3 4 6 6

x

−8

−6

−4

2

−2 2 4 6 8

−2

−4

x

−4

−6

−6

−8

13 A quadratic function has x-intercepts of ( 5, 0 ) and ( -3, 0 ), and a y-intercept of ( 0, 60 ).

Write the equation of the function in factored form.


14 A quadratic function passes through the points ( -2, 0 ), ( 9, 0 ) and ( 0, -6 ).


15 The x-values of the x-intercepts of a quadratic function are x = -4 and x = -7.

The graph of the function passes through the point ( -3, 12 ).


16 Satellite dishes follow the shape of a parabola to optimally receive signals. Winston models

the cross section of a satellite dish with the points ( −2, 0 ) and ( 2, 0 ) being the edges of the

satellite and the x-axis representing the top opening of the satellite dish. The satellite dish is

-foot deep.

a

A

Select the graph that represents the problem:

y

B

y

2

2

1

1

x

x

−2

−1

1 2

−2

−1

1 2

−1

−1

−2

−2

C

y

D

y

2

2

1

1

x

x

−2

−1

1 2

−2

−1

1 2

−1

−1

−2

−2

b

Find the vertex of the satellite dish.

342


17 Xia notices that the Sunshine State Arch is

in the shape of a quadratic function. They

know that the arch has a height of 110 ft and

the feet of the arch are 100 ft apart. Xia

chooses to let the x-intercepts of the arch be

the origin and ( 100, 0 ).

a

b

c

Determine the coordinates of the vertex of the

arch.

Write the equation of the quadratic function

for Xia’s representation of the Sunshine State

Arch in factored form.

Find the domain and range of the Sunshine

State Arch.


18 Ami throws a javelin forwards in a parabolic arc from the ground. Using photos that her friend

Maryellen is taking from the stands, she determines that 10 yards horizontally from where

she threw the javelin, it reaches a maximum height 5 yards above the ground. Ami models

her throw with the origin of a cartesian plane being the point 20 yards behind her.

a

b

For Ami’s model, state the coordinates of the x-intercepts.

Write the equation for the quadratic function modelling the path of Ami’s throw in

factored form.

19 Wilson tosses an eraser into the air and counts how long it takes for the eraser to return

to his hand. He estimates that it takes 2 seconds and that he is tossing the eraser 6 ft into

the air.

Wilson models the height of the eraser above his hand in feet as a function h of time t in

secondes, letting t = 0 be when he tosses the eraser and h = 0 be the height of his hand.

a

b

c

d

Explain how to find the intercepts of the function and state them.

Assuming that the path of the eraser is symmetric going up and coming down, state the

coordinates of the vertex of the function.

Graph the function.


20 A quadratic function has the factored form equation y = k( x - x 0 )( x -12 ).

If the function has a vertex at the point ( 8, 15 ), determine the values of x 0 and k.

Justify your answers.



in vertex form

Concept summary

One way to represent quadratic functions is using vertex form. This form allows us to identify the

coordinates of the vertex of the parabola, as well as the direction of opening and scale factor that

compresses or stretches the graph of the function.

f ( x ) = a( x - h ) 2 + k

( h, k ) The coordinates of the vertex.

a The scale factor, so tells us about the shape of the graph.

If we know the coordinates of the vertex, we actually only need to know one other point on the

graph, such as the y-intercept, to be able to draw the graph of quadratic function. As the parabola

is symmetric across the line of symmetry, which the vertex lies on, we can use the properties of

symmetry to find a third point on the graph.

y

As an example, consider the graph of

y = ( x - 2 ) 2 - 3

• Vertex at ( 2, -3 )

• Axis of symmetry x = 2

• y-intercept at ( 0, 1) has a corresponding point

at ( 4, 1)

−4

−3

−2

4

3

2

1

(0, 1) (4, 1)

−1 1 2 3 4

−1

−2

−3

−4

x

Worked examples

Example 1

The table of values below represents a quadratic function. Write the function p( x ) in vertex form.

x -4 -3 -2 -1 0 1 2

p( x ) -5 0 3 4 3 0 -5

344


Approach

We can use the fact that a quadratic function has symmetry about its vertex to identify the location

of the vertex from the table.

Solution

Looking at the values of p( x ), we can see that it has a maximum value of 4 and falls off

symmetrically on either side. So we know that the vertex is the point ( -1, 4 ), and therefore the

quadratic is of the form

p( x ) = a( x + 1) 2 + 4

We can now find the value of a by substituting any other pair of values from the table, such as

( 0, 3 ). Doing so, we get

which we can solve to get a = -1.

3 = a( 0 + 1) 2 + 4

So the quadratic function shown in the table of values is

p( x ) = -( x + 1) 2 + 4

Example 2

The quadratic function f ( x ) = 2x 2 has been

transformed to produce a new quadratic function

g( x ), as shown in the graph:

8

6

y

g (x)

4

2

f (x)

−8

−6

−4 −2 2

−2

x

a Describe the transformation from f ( x ) to g( x ).

Solution

The function g( x ) has the same shape and size as f ( x ), but has been shifted to the left. Comparing

the vertices of the two parabolas, we can see that this is a translation of 6 units to the left.


b Write the equation of the function g( x ) in vertex form.

Approach

Remember that vertex form for a quadratic is g( x ) = a( x - h ) 2 + k, where the vertex is at the point

( h, k ).

Solution

f ( x ) = 2x 2 has been translated 6 units to the left to produce g( x ), and we can see that its vertex is at

( -6, 0 ). So g( x ) has can be written as g( x ) = 2( x + 6 ) 2 .

Example 3

Consider the quadratic function A( x ) = 5 - ( x - 3 ) 2 .

a Determine the coordinates of the vertex.

Approach

A quadratic function in the form A( x ) = a( x - h ) 2 + k has a vertex at ( h, k ).

Solution

In this case the quadratic is A( x ) = -( x - 3 ) 2 + 5, and so the vertex is at ( 3, 5 ).

b Determine the equation of the axis of symmetry.

Approach

The axis of symmetry of a parabola is the vertical line x = h, which passes through its vertex.

Solution

We found that the vertex is the point ( 3, 5 ), and therefore the axis of symmetry is the line x = 3.

c Determine the y-intercept.

Approach

The y-intercept of a function is the point on the function at which x = 0.

Solution

Substituting x = 0 into the function, we have

y = A( 0 ) = 5 - ( 0 - 3 ) 2

Evaluating this, we find that y = -4 and so the y-intercept is the point ( 0, -4).

346


d Draw the graph of the parabola.

Approach

We have found the vertex, axis of symmetry, and y-intercept of this parabola. We can use these

features to help draw the graph.

Solution

7

6

5

4

3

2

1

−4 −3 −2 −1

−1

−2

−3

−4

−5

y

(3, 5)

x

1 2 3 4 5 6 7 8

(0, −4)


1 Write an equation for each quadratic function in vertex form.

a

−8 −7 −6 −5 −4 −3 −2

y

b

16

14

12

10

8

6

4

2

x

−1 1 2

−2

2

−2 −1

−2

−4

−6

−8

−10

−12

−14

y

1 2 3 4 5 6 7 8

x

c

x 0 1 2 3 4 5 6

h( x ) -6 -1 2 3 2 −1 −6


d

x -6 -5 -4 -3 -2 -1 0

j( x ) 11 6 3 2 3 6 11

2 Write the equation of the transformed graph in vertex form.

a

b

c

y = x 2 is horizontally translated 10 units to the right and vertically translated 2 units up

y = x 2 is horizontally translated 9 units to the left and is vertically stretched by a factor of

9 units

y = x 2 is reflected across the x-axis and vertically translated 8 units down

3 For each of the following, state whether the transformation from the parent function f ( x ) = x 2

to the function g( x ) is a translation up, down, left, or right:

a g( x ) = ( x − 8 ) 2 b g( x ) = x 2 − 5

c g( x ) = ( x + 9 ) 2 d g( x ) = x 2 + 0.75



the function:

A y = −( x + 5 ) 2 + 25

B y = ( x − 5 ) 2 + 25

C y = ( x − 5 ) 2 − 25

D y = −( x + 5 ) 2 − 25

−12 −10 −8

−6

−4

y

5

−2 2

−5

−10

x

−15

−20

−25



the function:

A p = −10( x − 1) 2 − 4

B p = −10( x + 1) 2 − 4

C p = −10( x − 1) 2 + 4

D p = −10( x + 1) 2 + 4

−1

2

−2

−4

−6

−8

p

1

2 3

x

−10

−12

−14

−16

348



x 1 2 3 4 5

y −6 0 2 0 −6


a

3

2

1

y

x

b

3

2

1

y

x

−1 1 2

−1

−2

−3

−4

−5

−6

3 4 5 6

−6 −5 −4 −3

−2 −1

−1

1

−2

−3

−4

−5

−6

c

3

2

1

y

x

d

3

2

1

y

x

−6 −5 −4 −3

−2 −1

−1

1

−2

−3

−4

−5

−6

−1 1 2

−1

−2

−3

−4

−5

−6

3 4 5 6



a

y

x

−1

−1

1 2 3 4 5 6 7 8 9

−2

−3

−4

−5

−6

−7

−8

−9

−10

b

y x

−9 −8 −7 −6 −5 −4 −3 −2 −1

−1

1

−2

−3

−4

−5

−6

−7

−8

−9

−10


c

10

9

8

7

6

5

4

3

2

1

y

x

d

10

9

8

7

6

5

4

3

2

1

y

x

−9 −8 −7 −6 −5 −4 −3 −2 −1 1

−1 1 2 3 4 5 6 7 8 9

Let’s practice

8 For each of the following quadratic functions:

i

State whether the transformation from f ( x ) to g( x ) is a horizontal translation,

vertical translation, vertical stretch, or vertical compression by k units.

ii State the value of k.

a

y

10

g(x)

8

6

4

2 f(x)

x

−10 −8 −6 −4 −2

−2

2 4 6 8 10

−4

−6

−8

−10

b

y

10

8

6

4

g(x) 2 f(x)

x

−10 −8 −6 −4 −2

−2

2 4 6 8 10

−4

−6

−8

−10

c

y

10

8

6

f(x)

4

2 g(x) x

−10 −8 −6 −4 −2

−2

2 4 6 8 10

−4

−6

−8

−10

350


Let’s practice


i Describe in words the transformation of f ( x ) to g( x ).

ii Write an equation for g( x ) in vertex form.

a

x f ( x ) x g( x )

b

x f ( x ) x g( x )

-3 9 -3 5

-3 9 -1 9

-2 4 -2 0

-2 4 0 4

-1 1 -1 -3

-1 1 1 1

0 0 0 -4

0 0 2 0

1 1 1 -3

1 1 3 1

2 4 2 0

2 4 4 4

3 9 3 5

3 9 5 9

c

x f ( x ) x g( x )

-3 9 -3 18

-2 4 -2 8

-1 1 -1 2

0 0 0 0

1 1 1 2

2 4 2 8

3 9 3 18


i Determine the coordinates of the vertex.

ii Determine the zeros of the function, if there are any.

iii Determine the y-intercept.

iv Draw a graph of the quadratic function.

a m( x ) = ( x - 3 ) 2 b n( x ) = ( x + 4 ) 2 - 1

c p( x ) = -( x - 1) 2 - 7 d r( x ) = 3( x + 5 ) 2


11 Rodney observed that the water stream of a fountain is in the shape of the parabola. This

water stream lands on an underwater spotlight. He models the path of the water stream with

the maximum height of 8 feet, represented by the vertex ( 4, 8 ) and the underwater spotlight,

represented by the point ( 8, 0 ).

a

A

Select the graph that represents the problem:

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x

B

9

8

7

6

5

4

3

2

1

y

1 2 3 4 5 6 7 8 9

x

C

9

8

7

6

5

4

3

2

1

y

x

2 4 6 8 10 12 14 16 18

D

8

6

4

2

−16 −12 −8 −4

−2

−4

−6

−8

y

4 8 12 16

x

b


12 For each quadratic equation:

i Determine the coordinates of the vertex.

ii Determine the equation of the axis of symmetry.

iii Determine the intervals of the domain where the function is increasing and decreasing.

iv Determine the end behavior.

v Determine the domain and range.

vi Draw a graph of the quadratic function.

1

a y = − (x − 4) 2 + 1

2

b

352


13 Ashtyn throws a rock into Crescent Lake. The height of the rock above ground is a quadratic

function of time. The rock is thrown from 4.4 ft above ground. After 1.5 seconds, the rock

reaches its maximum height of 24 ft. Write the quadratic equation in vertex form.

14 A roller coaster has a part that is

modeled by a quadratic function.

Hau loves roller coasters and wants

to be able to build a small model

with his 3D Printer. Assume that the

roller coaster passes through points

( 0, 0 ) and ( 57, 0 ) and reaches a

maximum height of 90 ft. Help Hau

build a model of the roller coaster by

writing an equation for the parabola

in vertex form.

15 Karima and Riley are playing soccer. Karima just kicked a soccer ball and Riley is playing

goalie 2 ft in front of the goal post. The following diagram shows the situation on a coordinate

plane. The soccer ball is kicked at ( 1, 0.6 ) and the goal post is 13 ft away from Karima’s back

foot. The maximum height of the soccer ball is 5.5 ft and this occurs when it is 8 ft away

from Karima.

10

Height (ft)

5

(8, 5.5)

(1, 0.6)

0 5

Distance (ft)

10 15

a

b

Write a quadratic function that models the height of the soccer ball in feet in terms of the

horizontal distance from Karima’s starting position.

Riley jumps to try to block the soccer ball and will block soccer balls that are between

4 ft and 8 ft high. Will Riley be able to block Karima’s kick? Explain your answer.



16 Determine how many quadratic equations could share a common vertex.

17 Consider the following diagram:

a

b

Describe a situation that could be modeled by the

following three quadratic functions. For each part, make

sure to identify the vertex and another possible point on

the function.

Write equations for the quadratic functions from your

above description.

18 Consider the family of quadratic equations of the form y = ( x - ) 2 - where the boxes

contain the integers 2, 3, and 5.

a

b

Write a quadratic equation with the lowest possible minimum value.

Describe the similarities and differences between members of the family.

19 Compare and contrast the following functions:

f ( x ) = -3a( x - h ) 2 + k and g( x ) = a( x - h ) 2 - k

354



in standard form

Concept summary

The standard form of a quadratic equation allows us to quickly identify the y-intercept and

whether the parabola opens up or down.

The standard form of a quadratic equation is:

y = ax 2 + bx + c

a The scale factor, so tells us about the shape of the graph

b Helps us to find the axis of symmetry and vertex

c The y-value of the y-intercept

The axis of symmetry is the line:

x

b

= −

2a

As the vertex lies on the axis of symmetry, we know the x-coordinate of the vertex. We can substitute

the x-coordinate of the vertex into the original equation to get the y-coordinate of the vertex.

Worked examples

Example 1

For the quadratic function y = 3x 2 - 6x + 8:

a Identify the axis of symmetry.

Approach

We will use the formula x

Solution

x

b

= − , so we need to identify the values of a and b from the equation.

2a

a = 3, b = -6

b

= − Equation for axis of symmetry

2a

x = − −6

2(3)

x = − −6

6

x = 1

The axis of symmetry is x = 1.

Substitute in the values

Simplify the denominator

Simplify fully


b State the coordinates of the vertex.

Approach

Once we have the x-coordinate of the vertex from the axis of symmetry, we can substitute it into

y = 3x 2 - 6x + 8 and evaluate to get y.

From part ( a ), we know that the axis of symmetry is x = 1 so the x-coordinate of the vertex is x = 1.

Solution

y = 3x 2 - 6x + 8


y = 3( 1) 2 - 6( 1) + 8 Substituting x = 1

y = 3 - 6 + 8

y = 5

The vertex is ( 1, 5 ).

Simplify

Simplify fully

c State the coordinates of the y-intercept.

Approach

Since the y-intercept occurs when x = 0, we can substitute x = 0 into the equation and evaluate y.

When we are given an equation in standard form, the y-value of the y-intercept will be y = c.

Solution

In this case, the value of c in the equation is 8.

So we have that the coordinates of the y-intercept are ( 0, 8 ).

d Draw a graph of the corresponding parabola.

Approach

We have all the key features we need to create a graph. For more accuracy, we can use the axis

of symmetry and y-intercept to find another point. This point will be a reflection of the y-intercept

across the axis of symmetry.

We know that the parabola will open upwards because a > 0.

356


Solution

Axis of symmetry: x = 1

Vertex: ( 1, 5 )

y-intercept: ( 0, 8 )

Another point: ( 2, 8 )

Reflection

From the graph we can identify that the vertex

form of the equation would be:

y = 3( x - 1) 2 + 5

10

8

6

4

2

y

−1 1 2 3

x

Example 2

For the quadratic function y = -x 2 + 7x - 10:

a State the coordinates of the y-intercept.

Approach

Since the y-intercept occurs when x = 0, we can substitute x = 0 into the equation and determine

the value of y.

When we are given an equation in standard form, the y-value of the y-intercept will be y = c.

Solution

In this case, the value of c in the equation is -10.

So we have that the coordinates of the y-intercept are ( 0, -10 ).

b State the coordinates of the x-intercept( s ).

Approach

Since the x-intercept( s ) occur when y = 0, we can substitute y = 0 into the equation and solve for

the values of x.

Since the equation is given in standard form, this will involve factoring the quadratic.


Solution

Setting y = 0, we have the equation -x 2 + 7x - 10 = 0. We can now solve this for x as follows:

-x 2 + 7x -10 = 0 Substituting y = 0

x 2 - 7x + 10 = 0 Multiply both sides by -1

( x - 2 )( x - 5 ) = 0 Factoring the quadratic

The two values of x which satisfy this equation are x = 2 and x = 5. This tells us that the coordinates

of the two x-intercepts are ( 2, 0 ) and ( 5, 0 ).

c Draw a graph of the corresponding parabola.

Approach

We now know the coordinates of the x- and y-intercepts of the function. We can use these to plot the

parabola. For more accuracy, we can also note that the axis of symmetry will occur exactly halfway

between the x-intercepts - so in this case, the axis of symmetry will be the line

We also know that the parabola will open downwards because a < 0.

Solution

y-intercept: ( 0, -10 )

x-intercepts: ( 2, 0 ) and ( 5, 0 )

−4

−2

2

y

2 4 6 8

x

−2

−4

−6

−8

−10

Example 3

To start a play in a game of Kapucha Toli, a ball is thrown in the air. The given equation models the

height of the ball above the ground, where y represents the height in feet and t represents time in

seconds.

y = -5t 2 + 10t + 4

358


a Determine the height that the ball was thrown from.

Approach

When the ball is thrown, t = 0. This statement is asking for the value of the y-intercept.

Solution

The y-intercept is y = 4.

This means that the ball was thrown from a height of 4 feet above the ground.

b Calculate when the ball reached its maximum height.

Approach

The maximum occurs at the vertex. We are asked “when”, so we are looking for the time ( t-value of

the vertex ).

Remember that the equation to find the axis of symmetry is

In this case, x = t, a = -5, and b = 10

Solution

Finding the t-coordinate of the vertex:

b

t = −

2a

Equation for axis of symmetry

t = − 10

2(−5)

Substitution

t = − 10 −10

t = 1

Simplify

Simplify fully

The axis of symmetry is t = 1.

The ball reached the maximum height after 1 second.

c Calculate the maximum height reached by the ball.

Approach

This is asking for the y-coordinate of the vertex. From part ( b ), we know that the axis of symmetry is

t = 1, so we want to use this to find the y-coordinate of the vertex.

Solution

y = -5t 2 + 10t + 4


y = -5( 1) 2 + 10( 1) + 4 Substitute t = 1

y = -5 + 10 + 4

y = 9

Simplify

Simplify fully

The maximum height reached by the ball is 9 feet.




i Determine the axis of symmetry. ii State the coordinates of the vertex.

iii State the coordinates of the y-intercept.

a y = x 2 - 4x + 8 b y = -x 2 - 4x - 9



the function:

A y = x 2 − 6x + 4

2

1

y

x

B y = x 2 + 6x − 4

−7 −6 −5 −4

−3 −2 −1 1

−1

C y = x 2 − 6x − 4

−2

D y = x 2 + 6x + 4

−3

−4

−5

−6



the function:

A y = −x 2 + 10x − 25

y

1

−1 1 2

−1

3 4 5 6 7 8

x

B y = −x 2 − 10x − 25

−2

C y = x 2 + 10x − 25

−3

D y = x 2 − 10x − 25

−4

−5

−6

−7

360



x −6 −5 −4 −3 −2 −1

y −8 −9 −8 −5 0 7


A

9

6

3

y

x

B

9

6

3

y

x

−12 −10 −8

−6 −4 −2

−3

2

−6

−9

−12

−15

−18

−2 2

−3

−6

−9

−12

−15

−18

4 6 8 10 12

C

18

15

12

9

6

3

y

x

D

18

15

12

9

6

3

y

x

−12 −10 −8

−6 −4 −2

−3

2

−6

−9

−2 2

−3

−6

−9

4 6 8 10 12



A

y

x

B

y

x

−2 2

−3

4 6 8 10 12

−12 −10 −8

−6 −4 −2 2

−3

−6

−6

−9

−9

−12

−12

−15

−15

−18

−18

−21

−21

−24

−24


C

16

y

D

16

y

14

14

12

12

10

10

8

8

6

6

4

4

2

x

2

x

−2 2

4 6 8 10

−10

−8

−6 −4 −2 2

Let’s practice


i Determine the axis of symmetry. ii State the coordinates of the vertex.

iii State the coordinates of the y-intercept. iv Draw a graph of the corresponding

parabola.

a y = x 2 - 2x + 5 b y = 2x 2 + 24x + 75

c y = -x 2 + 6x - 8 d y = 4x 2 - 64

e y = 2x 2 + 2x + 9 f y = −

1 3

x 2 + x

4 2


i

State the coordinates of the y-intercept.

ii Determine the coordinates of the x-intercept( s ).

iii

Draw a graph of the corresponding parabola.

a y = x 2 - 3x - 10 b y = x 2 - 9

c y = 4 - 3x - x 2 d y = 2x 2 + 12x + 18

e y = 4x 2 + 8x - 5 f y = -6x 2 + 10x + 4

8 Darnell throws a bag of cookies to his friend, Ike. The height of the bag is modeled by the

quadratic function

h( t ) = −16t 2 + 32t + 3.25

where t is the time, in seconds, the cookies have been in the air and h is the distance of the

bag above the ground in feet.

a Select the graph of h( t ):

362


A

−2

−1

20

18

16

14

12

10

8

6

4

2

−2

−4

y

1 2 3

x

B

20

18

16

14

12

10

8

6

4

2

y

1 2 3 4

x

C

20

18

16

14

12

10

8

6

4

2

y

1 2 3 4

x

D

−2

−1

20

18

16

14

12

10

8

6

4

2

−2

−4

y

1 2 3

x

b

c

d

e

Determine the height that the cookies are thrown from.

Determine when the cookies reach the maximum height in the air.

Determine the maximum height of the cookies during Darnell’s throw.

If Ike caught the cookies after 2 seconds, determine the height of the catch.

9 Carliss is starting a summer car wash business. Her profit function relates the total profit to

the rate she charges for each car wash. The rate and profit are in dollars.

a

P( x ) = -x 2 + 50x - 95

Carliss wants to make at least $525 this summer. Determine if she could make enough

money based on her quadratic profit model. Explain your reasoning.

b Draw a graph of P( x ).

10 The table shows points on a quadratic function.

x -3 -2 -1 0 2

A( x ) -5 -8 -9 -8 0

Another quadratic function has the equation B( x ) = x 2 - 2x - 8.

Determine what the quadratic functions have in common and what is different.


11 Orland and Delfino are comparing the vertex of the following graph and equation of

quadratic functions:

g( x ) = -0.5x 2 y

+ x + 7.5

5

Identify and explain their error( s ):

Orland’s work:

1 x −1

= Finding axis of symmetry for g( x )

2(0.5)

2 x = -1

3 g( -1) = -0.5(-1)2 + (-1) + 7.5 Finding y-value of vertex

4 g( -1) = -0.5 - 1 + 7.5

5 g( -1) = 6

Vertex of g( x ) is ( -1, 6 ) and vertex of f ( x ) is ( 1, -4 )

Delfino’s work:

1 x −1

= Finding axis of symmetry for g( x )

−0.5

2 x = 2

3 g( 2 ) = -0.5( 2 )2 + ( 2 ) + 7.5 Finding y-value of vertex

4 g( 2 ) = -2 + 2 + 7.5

5 g( 2 ) = 7.5

Vertex of g( x ) is ( 2, 7.5 ) and vertex of f ( x ) is ( 1, -5 )

f (x)

4

3

2

−6 −5 −4 −3 −2 −1

−1

1

−2

−3

−4

−5

1 2 3 4 5 6 7 8

x

364



12 Kinsey wants to understand how vertex form and standard form of a quadratic function are

related. Consider the vertex form of the function.

y = a( x - h ) 2 + k

a Expand a( x - h ) 2 + k and write the equivalent equation to y = a( x - h ) 2 + k.

b State the value of b in the equation from part ( a ).

c State the value of c in the equation from part ( a ).

d

Explain how the equations are related.

13 Fill in the boxes to create a quadratic equation with the lowest possible minimum value. Use

the digits 1 to 5 at most once.

y = x 2 + x +

14 A Happy Birthday banner is modeled by the quadratic function h( x ) = 0.2x 2 - x + 1.25 where

x is the distance from the left side of the banner and h, is the height above the ground.

Both x and h are in feet. Currently, the lowest part of the banner touches the ground.

Immanuel is trying to create a virtual card and needs to know how to affect the look of the

banner by changing parts of the quadratic function.

a

b

Determine a domain and range for the quadratic function that models the initial Birthday

banner. Explain your reasoning.

Describe to Immanuel how changes to the quadratic function affect the shape of the

Birthday banner.


7.06 Modeling with

quadratic functions

Concept summary

The different quadratic forms are useful for modeling different quadratic scenarios based on what

information is given.

y

The factored form of a quadratic function is useful

when we have two points that can be represented as

the x-intercepts.

The x-intercepts often represent the points when a

parabolic arc meets the ground.

For example, the feet of an arch can be thought of as

x-intercepts when modeling the arch as a quadratic

function.

8

7

6

5

4

3

2

−4 −3 −2 −1

1

1 2 3 4

x

The vertex form of a quadratic function is useful when we know the maximum or minimum point

of a quadratic situation.

y

For the vertex, the x-coordinate is where or when

the maximum occurs and the y-coordinate is the

minimum or maximum value.

For example, the point where a basketball shot

reaches its peak can be thought of as the vertex, with

the player being set as the origin.

8

7

6

5

4

3

2

1

1 2 3 4

5 6 7 8

x

The standard form of a quadratic function is more useful in algebraic scenarios when we know

that we are combining a constant, some variable, and the square of that variable together, or when

we are comparing different types of functions.

366


Worked examples

Example 1

The sum of two whole numbers is 24. Let one of the numbers be x.

a Let y represent the product of the numbers. Form an equation for y in terms of x.

Approach

If we let one of the integers be represented by x, then the other integer must be 24 - x.

Solution

The equation modeling the product of the two integers is:

y = x( 24 - x )

b Find the largest possible value for the product.

Approach

Notice that our model is a parabola that opens down, the largest product will be the maximum

value of y and will occur at the vertex.

Solution

To find the vertex, we first need to find the axis of symmetry. We can rewrite the model equation in

factored form by taking a factor of -1 from the term ( 24 - x ). This gives us:

y = -x( x - 24 )

This tells us that the x-intercepts of the model are ( 0, 0 ) and ( 24, 0 ).

This means the axis of symmetry is the vertical line x = 12, so the maximum value for y is when

x = 12. Substituting this into the model equation:

y = -x( x - 24 )

Model equation

y = -12( 12 - 24 ) Substitute in x = 12

y = 144

Simplify

So the largest possible value for the product is 144.

Example 2

Neva’s yo-yo has a string length of 2 ft. When using the yo-yo, it takes 1 second for the yo-yo to go

as low as possible. Assume that yo-yo’s position with respect to time is a quadratic relation.

Write an equation modeling the path of the yo-yo.


Approach

If we let time and height be our two variables ( t, h ), we can set t = 0 to be when Neva releases the

yo-yo and h = 0 to be at the height of Neva’s yo-yoing hand in feet. When the yo-yo reaches its

lowest point, we can say that it is at the vertex ( 1, -2 ). Since we have vertex coordinates for our

model, we want to use vertex form.

Solution

Substituting the coordinates of the vertex into the vertex form of a quadratic function gives us:

for some value of a.

y = a( x - 1) 2 - 2

We know also that the yo-yo starts at the origin, so we can substitute the coordinates of the origin

into the model equation to find a.

y = a( x - 1) 2 - 2

Model equation

0 = a( 0 - 1) 2 - 2 Substitute in the origin

0 = a - 2 Simplify

2 = a Add 2 to both sides

We can model the path of the yo-yo with the quadratic equation:

y = 2( x - 1) 2 - 2

Example 3

The size of a snowball rolling down a hill can be modelled by the equation:

S = 1 2 t2 + 2t + 7

where S is the diameter of the snowball in inches and t is the time in seconds after the snowball

starts rolling.

a Find the value of S when t = 4 and interpret the result in the given context.

Solution

To find S when t = 4:

S = 1 2 t2 + 2t + 7

Model equation

S = 1 2 ⋅ 42 + 2 ⋅ 4 + 7 Substitute in t = 4

S = 23

Simplify

This means that the diameter of the snowball will be 23 inches after the snowball has rolled down

the hill for 4 seconds.

368


b Identify the initial diameter of the snowball.

Approach

The snowball will be at its initial diameter when t = 0.

Solution

S = 1 2 t2 + 2t + 7

Model equation

S = 1 2 ⋅ 02 + 2 ⋅ 0 + 7 Substitute in t = 0

S = 7

Simplify

The initial diameter of the snowball was 7 inches.

Reflection

In standard form, the constant term is the y-value of the y-intercept.


1 Given f ( x ) = 4x 2 + 5x - 1, evaluate the following:

a f ( -1)

b f ( 10 )

2 Write a quadratic function that has the following features, graph, or table of values:

a • Vertex at ( 5, 8 )

• Point at ( 2, 35 )

b

−9

−8

−7

−6

−5

−4

−3

−2

18

16

14

12

10

8

6

4

2

−1

−2

y

1

x


c

x f ( x )

d

x g( x )

−3

−3

8 -20

−2 0.5

−1 3

0 4.5

1 5

2 4.5

3 3

9 -5

10 0

11 -5

12 -20

13 -45

14 -80


Select the equation that represents the

function:

A

q = −5x 2 + 5x

B q = −5x 2 + 5x + 1

C

q = −5x 2 − 5x

−2

−1

2

−2

p

1 2 3

x

D q = −5x 2 − 5x + 1

−4

−6

−8

−10


Select the equation that represents the

function:

A p = 4x 2 − 8x + 4

B p = 4x 2 − 8x − 4

C p = 4x 2 + 8x + 4

−2

−1

2

−2

p

1 2 3

x

D p = 4x 2 + 8x − 4

−4

−6

−8

−10

5 A quadratic function has x-intercepts of and ( 2,

0 ), and ( −4, 0 ) passes through ( −2, 8 ). Select the equation for the function:

A f ( x ) = −x 2 − 2x + 8 B f ( x ) = −x 2 + 2x + 8

C f ( x ) = x 2 + 2x − 8 D f ( x ) = x 2 − 2x − 8

370


6 A ball is thrown upwards from a height of 4 ft with an initial velocity of 44 ft/s. The height of

the ball at time t is given by the equation h( t ) = -16t 2 + 44t + 4.

a

b

Determine how long it takes for the ball to reach its maximum height.

Find the height of the ball at its highest point, correct to two decimal places.

Let’s practice

7 When a fire hose is turned on, the height h, of the stream of water, in feet, is described by

h( x ) = −0.25( x − 16 ) 2 + 64, where x is the horizontal distance from the hose in feet. Let the

hose be positioned at ( 0, 0 ).

a

A

C


64

56

48

40

32

24

16

8

64

56

48

40

32

24

16

8

h(x)

4 8 12 16 20 24 28 32 36

h(x)

4 8 12 16 20 24 28 32 36

x

x

B

D

64

56

48

40

32

24

16

8

64

56

48

40

32

24

16

8

h(x)

4 8 12 16 20 24 28 32 36

h(x)

4 8 12 16 20 24 28 32 36

x

x

b

c

d

e

Determine the horizontal distance from the hose when the water reaches its maximum

height above ground.

Determine the maximum height reached by the hose.

Determine the height of the water above the ground when it is a horizontal distance of

6 ft from where the hose is.

Determine the horizontal distance from the hose when the the water hits the ground.



8 The sum of two integers is 80.

a

b

Let y be the product of the integers. Write an equation to find the product for y in terms

of x.

Determine the integer values that have the largest product. Explain your reasoning.

9 The height h, in feet, reached by a ball thrown in the air after t seconds is modeled by a

quadratic function. Points on the function are shown in the table:

t 1 2 3 4 5 6 7 8 9 10

h( t ) 11 20 27 32 35 36 35 32 27 20

a

b

c

Write the equation of the quadratic function.

Determine the height of the ball after 7.5 seconds have elapsed.

Find the maximum height reached by the ball.

10 Charlene throws a baseball upwards and towards the sports field from the top of the

chemistry building which is 120 ft tall. The baseball reaches a maximum height of 160 ft

above ground level at 2 seconds. After 4 seconds the baseball is 120 ft above ground level.

a

b

Write a quadratic function that models the height of the baseball above ground level, in

feet, after t seconds after being thrown.

Find the y-intercept and explain it in context.

11 Norman throws a stick vertically upwards. After t seconds, its height h feet above the ground

is given by the formula h = -5t 2 + 25t + 3.

a

b

Determine whether or not the height of the stick would reach 36 ft. Explain your

reasoning.

Find the initial height that Norman threw the stick from.

12 A company is building a pool for a hotel. Liana is

deciding how much free space to use for the pool.

She wants the pool to be 4 times as long as it is

wide.

a

b

If Liana wants a 16 ft wide pool, determine

how many square feet Liana will need

available in the hotel’s yard to build the pool

and deck. Explain your answer.

Determine the difference in required available square feet if the surrounding deck is 2 ft

wide instead of 5 ft. Assume the pool is still 16 ft wide.

c Write an equation to find the area of the pool and deck for any deck width, w.

372



13 To minimize the chance of large avalanches, small explosives are fired onto mountain sides.

One such explosive is fired at a height of 24 ft and follows a parabolic flight path. It reaches a

maximum height of 122 ft, 180 ft horizontally from where it is fired. Let x and y represent the

horizontal and vertical distances of the explosive from where it is fired. Assume the position

where it is fired is ( 0, 0 ).

The explosive hits a slope that is 361 ft horizontally from the point where it was fired.

Determine if the explosive lands above, at the same height or below the point where it was

fired from. Explain your reasoning.

14 A rectangular paved labyrinth is to be built in the Florida Botanical Gardens.

The stonemasons have 336 ft of rare marble for the perimeter, but they want to maximize the

area of the labyrinth.

Let x be the width of the labyrinth.

a

b

Describe a plan to find the greatest possible area of the labyrinth.

Find the greatest possible area of the labyrinth.

15 In a game of tennis, the ball is mistimed and hit high up into

the air. Initially ( at t = 0 ), the ball is struck 3.5 ft above the

ground and hits the ground 7 seconds later. It reaches its

greatest height 3 seconds after being hit.

y

a

Determine the coordinates of the points labelled A, B

and C in the diagram.

A

b

Find the greatest height reached by the ball. Explain

your reasoning.

B

C

t


7.07 Linear, quadratic,

and exponential models

Concept summary

Linear, quadratic and exponential functions differ in how the output of the function changes with

regards to the input x.

We can use three things to determine the change in output: the first difference, the second

difference, and the common ratio.

The first difference is the difference between two consecutive outputs.

If the first difference is constant, then the function is linear.

x f ( x ) first difference

-2 -4

-1 -2 -2 - ( -4 ) = 2

0 0 0 - ( -2 ) = 2

1 2 2 - 0 = 2

2 4 4 - 2 = 2

3 6 6 - 4 = 2

The second difference is the difference between two consecutive first differences.

If the second difference is constant and the first difference is not, then the function is quadratic.

x g( x ) first difference

-2 -4

-1 -2 2

second

difference

0 2 4 4 - 2 = 2

1 8 6 6 - 4 = 2

2 16 8 8 - 6 = 2

3 26 10 10 - 8 = 2

374


If neither the first nor second difference is constant, then we look at the common ratio. If the

common ratio is constant then the function is exponential.

x f ( x ) common ratio

-2

-1

1

4

1

2

0 1

1

2

1

1

÷ = 2

4

1

÷ = 2

2

1 2 2 ÷ 1 = 2

2 4 4 ÷ 2 = 2

3 8 8 ÷ 4 = 2

We can also identify a function by the shape of its graph:

• f ( x ) is a quadratic function.

• g( x ) is a linear function

• h( x ) is an exponential function

4

3

2

1

−4 −3 −2 −1

−1

−2

y

f (x)

h (x)

x

1 2 3 4

g (x)

−3

−4

Worked examples

Example 1

Identify if the function f ( x ) is linear, quadratic or exponential.

x -2 -1 0 1 2 3

f ( x ) 13 1 -3 1 13 33


Approach

Construct a table to find the common ratio and first and second difference.

• If the common ratio is constant for all x, then f ( x ) is exponential.

• If the first difference is constant for all x, then f ( x ) is linear.

• If the second difference is constant for all x, and the first difference is not, then f ( x ) is quadratic.

Solution

We get the following table of values:

x f ( x ) Common ratio First difference Second difference

−2 13

−1 1

1

13

0 −3 −3 −4 8

−12

1 1 − 1 3

4 8

2 13 13 12 8

3 33

13

33

20 8

The common ratio and the first difference are not constant, so the function is not exponential or linear.

The second difference is constant, so the function is quadratic.

Reflection

If the common ratio and the first and second differences are not constant, then it cannot be linear,

quadratic or exponential.

Example 2

Find the average rate of change of the

function f ( x ) between x = 2 and x = 6.

8

7

6

5

4

3

2

f (x)

−1

1

1 2 3 4 5 6 7 8

x

376


Approach

Find the values of f ( 2 ) and f ( 6 ) using the graph.

Divide the difference between the y-values by the change in x. In this case getting the expression

Evalute the expression to find the average rate of change.

Solution

Average rate of change =

= 5 − 1

6 − 2

= 4 4

Substitute the values in for f ( 6 ) and f ( 2 )

Evaluate the numerator and denominator

Reflection

= 1 Cancel the common factors in the fraction

The average rate of change is the slope of the straight line between the two points.

Example 3

Would a linear, quadratic or exponential function best model the number of pieces a cake is cut

into, if each piece of cake is cut in half every minute?

Assume you start with a whole cake.

Approach

Construct a table with the number of pieces of cake.

Construct a first differences, second differences and the common ratios table of consecutive terms

to identify whether there are any which are constant.

• If the common ratio is constant for all x, then f ( x ) is exponential.

• If the first difference is constant for all x, then f ( x ) is linear.

• If the second difference is constant for all x, and the first difference is not, then f ( x ) is quadratic.

Determine whether the situation is best described be a linear, quadratic or exponential function.


Solution

We get the following table of values:

Minutes passed Pieces of cake Common ratio First difference Second difference

0 1

1 2 2 1

2 4 2 2 1

3 8 2 4 2

4 16 2 8 4

5 32 2 16 8

The common ratio is constant for all values, therefore the best function to model this scenario is an

exponential function.

Example 4

Compare the key features of quadratic and exponential functions.

Approach

Think about how you would describe each feature for each function type. It can help to sketch a

few examples of each function type to visualize these features.

Solution

The table below is an example of how we could compare the key features of these two types of

functions.

Quadratic

Exponential

number of x-intercept( s ) 0, 1 or 2 0 or 1

y-intercept Yes Yes

Min/Max point Yes No

Domain All reals All reals

Range Limited by min/max Limited by asymptote

Asymptote No Yes

End behavior Same ( both ∞ or -∞ ) Different ( Asymptote and ∞ or -∞ )

Leading term ax 2 ab x

378


Reflection

The example solution above is not the only way to fill the table. You can be more specific, provide

examples, or sketch small diagrams to help you understand each feature.


1 The graphs of f ( x ), g( x ) and h( x ) are

shown below. Use the graphs to

complete the following:

a

b

c

d

Identify the graphs as exponential,

linear, or quadratic.

Find the average rate of change from

x = 1 to x = 2 of f ( x ).


x = 1 to x = 2 of g( x ).


x = 1 to x = 2 of h( x ).

18

16

14

12

10

8

6

4

y

h (x)

g (x)

f (x)

2

1 2 3 4 5

x

2 For each pair of table and its graph, determine whether a linear or quadratic function could

represent it:

a

x −3 −2 −1 0 1

b

x −2 −1 0 1 2

y 3 6 7 6 3

y 0 1 2

−5 −4 −3 −2

7

6

5

4

3

2

1

−1

−1

−2

y

1 2 3

x

−4 −3 −2

4

3

2

1

−1

−1

−2

−3

−4

y

1 2 3 4

x


3 Using the tables below, complete the following:

i Calculate the first and second differences in the table.

ii Identify each function as linear, exponential or quadratic.

a

x f ( x )

b

x g( x )

c

x h( x )

d

-1 -2

-1 4

-1 0.5

0 0

0 1

0 1

1 2

1 0

1 2

2 4

2 1

2 4

3 6

3 4

3 8

4 8

4 9

4 16

x j( x )

-1 -1

0 0

1 -1

2 -4

3 -9

4 -16

Let’s practice

4 For each function, complete the following:

i Find the average rate of change from x = 0 to x = 1.

ii Find the average rate of change from x = 1 to x = 2.

iii Find the average rate of change from x = 2 to x = 3.

iv

Classify the function as linear, quadratic, or exponential.

a y = 3 x b y = 3x c y = 3x 2

5 The side profile of a bridge is shown below such that ( x, y ) represents a point on the arch

of the bridge that is x ft horizontally from one end of the bridge and y ft vertically above the

road. The bridge meets the road at coordinates ( 0, 0 ) and ( 100, 0 ). Pedestrians are able to

climb the full arch of the bridge.

a

b

c

d

Determine whether a linear or

quadratic function best models the

scenario.

Find the domain and range and

explain in context.

Determine the intervals of the

domain where the bridge is

increasing and decreasing.

Find the average rate of change

between x = 10 and x = 60 and

explain in context.

52

48

44

40

36

32

28

24

20

16

12

8

4

y

10

20 30 40 50 60 70 80 90 100 110

x

380


6 The daily net profit of an upmarket restaurant can be modeled by the following equation,

where x is the number of customers:

a

b

y = -18x 2 + 504x

Determine whether the model is a linear or quadratic function.

Determine the average rate of change between x = 0 and x = 13 and explain in context.

7 During a sudden bacterial outbreak, scientists must decide between two anti-bacterial treatments

that are currently being trialed to try to control the outbreak. In the laboratory, they apply

Treatment A and Treatment B to two samples of the bacteria, each containing 200 microbes.

They keep track of the number of microbes in each sample. The table shows the results.

Number of hours ( t ) 0 3 6 9

Number of microbes using Treatment A 200 215 230 245

Number of microbes using Treatment B 200 600 1800 5400

a

b

c

Determine whether the treatments cause the number of microbes to increase at a linear,

exponential or quadratic rate.

Determine which treatment will better control the number of microbes.


Find the average rate of change between t = 3 and t = 9 for each treatment and explain

in context.

8 The table shown below represents the revenue, in thousands of dollars, over time, of two

new shoe companies: Foot Swag, F( x ) and Sweet Kicks, K( x ), where x is the time in weeks.

x F( x ) K( x )

0 1 2.12

1 2 3.52

2 4 5.48

3 8 8

4 16 11.08

5 32 14.72

a

b

c

Each company secured funding from investors to get up and running. Determine the

initial revenue for each function.

Determine when the revenue of Foot Swag and Sweet Kicks will be equal.

Assuming both companies stay in business for a very long time, which company

would you expect to reach a revenue of $1 million first? Foot Swag or Sweet Kicks?




9 Nadia is considering two different career options.

She could accept a job with Kord Enterprise that has offered her an income of $125000 for

the first year, and a projected income increase of $34000 each year after that.

Alternatively, she can start her own business, where she estimates her initial annual income

to be $40000 for the first year, increasing by 40% each year after that.

a

Copy and complete the following table of values:

n Income in year n with Kord Enterprise Income in year n with her own business

1 125 000 40 000

2

3

4

b

c

d

e

f

Determine whether or not Nadia’s income for each situation could be modeled by

a linear, quadratic, or exponential model.

Find the average rate of change for each model between x = 1 and x = 4 and explain in

context.

In the 5th year, determine which option will result in a higher annual income for Nadia.


In the 10th year, determine which option will result in a higher annual income for Nadia.


Determine the domain that will result in a higher annual income at her own company

versus working at Kord Enterprise.

10 Consider the functions shown in the tables below. Draw the graphs of the functions and

compare and contrast their key features ( domain, range, intercepts, intervals where function

is increasing/decreasing, intervals where function is positive/negative, end behavior and

asymptotes ).

x 0 1 2 3 4 5

a( x ) 0 1 4 9 16 25

x 0 1 2 3 4 5

b( x ) 1 3 9 27 81 243

x 0 1 2 3 4 5

c( x ) 0 6 12 18 24 30

382


11 Describe how to determine whether a linear, quadratic or exponential function best

models data.

12 Determine whether the following functions could be linear, quadratic, or exponential based

on the given information. Explain your reasoning.

a

b

c

The average rate of change of a function is less from x = 1 to x = 3 than from x = 6 to

x = 8.

The second differences are zero.

The absolute value of the average rate of change of a function is more from x = -10 to

x = -8 than from x = -3 to x = -1 and less from x = 1 to x = 3 than from x = 8 to x = 10.


Honors:

7.08 Combining linear

and quadratic functions

Concept summary

We can combine functions using the four arithmetic operations ( add, subtract, mulitply, and divide ).

• Add functions: ( f + g )( x ) = f ( x ) + g( x )

• Subtract functions: ( f − g )( x ) = f ( x ) − g( x )

The domain of the resulting function when finding the sum or difference of two functions is the

intersection of the initial two functions, in this case f and g.

• Multiply functions: ( f ⋅ g )( x ) = f ( x ) ⋅ g( x )

• Divide functions:

The domain of the product or quotient of two functions is the set of all real numbers for which the

initial two functions and the resulting new function are defined, in this case f and g and ( f ⋅ g )( x ) or

( x ) respectively.

When finding the domain of the quotient of two functions remember that the denominator cannot

equal 0.

Worked examples

Example 1

Given that f ( x ) = x 2 + 3 and g( x ) = 7x − 2, perform each function operation and find the domain.

a ( f + g )( x )

Approach

We want to add the two functions and combine like terms. So we use the formula:

( f + g )( x ) = f ( x ) + g( x )

We also want to find the domain of f ( x ) and g( x ) to help us determine the domain of ( f + g )( x ).

The domain of the resulting function will be the intersection of the initial two functions, in this case

f and g since we are finding the sum or difference of f and g.

384


Solution

First we find the function ( f + g )( x ):

( f + g )( x ) = f ( x ) + g( x ) Formula

= x 2 + 3 + 7x − 2 Substitution

= x 2 + 7x + 1 Simplify

Next, we identify the domain of f ( x ) and g( x ):

Domain of f ( x ): −∞ < x < ∞

Domain of g( x ): −∞ < x < ∞

Finally, we find the domain of ( f + g )( x ) by taking the intersection of the domains of f ( x ) and g( x ).

Domain of ( f + g )( x ) : −∞ < x < ∞

Reflection

Notice that the degree of the resulting function is the maximum of the degrees of the two functions.

b ( f − g )( x )

Approach

We want to subtract the one function from the other function and combine like terms. So we use

the formula:

( f − g )( x ) = f ( x ) − g( x )

From part (a) we know the domain of f ( x ), g( x ), and the intersection of their domains ( which is the

domain of ( f − g )( x ) ).

Solution

( f − g )( x ) = f ( x ) − g( x ) Formula

= x 2 + 3 − ( 7x − 2 ) Substitution

= x 2 + 3 − 7x + 2 Distribute the −1

= x 2 − 7x + 5 Simplify

Domain of ( f − g )( x ) : −∞ < x < ∞

c ( f ⋅ g )( x )

Approach

We want to multiply the two functions and combine like terms if possible. So initially we start with:

( f ⋅ g )( x ) = f ( x ) ⋅ g( x )


We also want to find the domain of the resulting function and check that domain of the expression

formed by ( f ⋅ g )( x ).

Solution

( f ⋅ g )( x ) = f ( x ) ⋅ g( x ) Formula

= ( x 2 + 3 )( 7x − 2 ) Substitution

= 7x 2 + 2x 2 + 21x − 6 Distribute the parentheses

Find the domain by taking the intersection of f ( x ), g( x ), and the expression 7x 3 − 2x 2 + 21x − 6.

Domain of the expression 7x 3 − 2x 2 + 21x − 6: −∞ < x < ∞

Domain of ( f ⋅ g )( x ): −∞ < x < ∞

Reflection

Notice that the degree of the resulting function is the sum of the degrees of f and g.

d

Approach

We want to divide one function by another function and combine like terms if possible. So initially

we start with:

We also want to find the domain of the resulting function. We do this by checking the domain of the

expression formed by and excluding points where g( x ) = 0( as we cannot divide by 0 ).

Solution

Formula

Substitution

We need to work out where g( x ) = 0 so we can exclude those values from the domain.

386


Example 2

For the given graph, calculate ( g − f )( 0 )

y

9

8

7

6

5

4

3

g(x)

2

1

−8 −6 −4 −2 2 4 6 8

−1

f(x)

x

Approach

First we need to find g( 0 ) and f ( 0 ) using the graph. In other words, we need to find the output for

g and f when the input is 0 and find the difference between the outputs.

It can help to label the two points on the graph with their coordinates.

y

9

8 (0, 8)

7

6

5

4

3

g(x)

2

1 (0, 1)

−8 −6 −4 −2 2 4 6 8

−1

f(x)

x


Solution

g( 0 ) = 8 and f ( 0 ) = 1

So ( g − f )( 0 ) = g( 0 ) − f ( 0 ) and g( 0 ) − f ( 0 ) = 8 − 1

So ( g − f )( 0 ) = 7

Reflection

We could also evaluate the other arithmetic operations from a graph, such as add, multiply,

and divide g and f.

Example 3

Dayana is organizing a trip to a theme park. The theme park has an admission cost of $60 and a

hotel room costs $280. She is trying to decide how many friends to invite.

a If x people go, find the quotient which determines the cost per person.

Approach

The total cost of everyone would be

Total cost of everyone = Cost for hotel room + ( theme park cost per person ) ⋅ ( number of people )

The cost per person will be

Solution

A function which represents the total cost of everyone would be:

T( x ) = 280 + 60x

A quotient to determine the cost per person would be:

Reflection

We could rewrite this as:

388


b Describe the domain in terms of the context.

Approach

We need to consider what x is representing and any constraints based on the context or

restrictions algebraically.

Solution

Algebraically, since we have x in the denominator, we know that x ≠ 0.

Based on the context, x is representing the number of people, so must be positive integer values.

Also, there will be restrictions on the number of people that can stay in one hotel room.

This will likely vary based on the hotel, but is generally around 4.

A possible domain could be x ∈ {1, 2, 3, 4} if we assume a limit of 4 people per room.


1 Perform each operation:

a ( f + g )( 2 )

b ( g − f )( −2 )

c

9

8

7

6

5

4

3

2

1

−8 −6 −4 −2 2 4 6 8

−1

y

g(x)

f(x)

x

2 Let a( x ) = x 2 − 1, b( x ) = 5x − 1, c( x ) = 2x 2 + 8x + 3 and d( x ) = x 2

Find each value or expression:

a ( a + b )( 2 ) b ( a − b )( x )

c ( a − b )( 5 ) d ( a ⋅ b )( x )

e

f

g ( c ⋅ d )( x ) h


3 • f ( x ) = x − 7

• g( x ) = 8x + 7

Perform each function operation then find the domain.

a ( f + g )( x ) b ( f − g )( x ) c ( f ⋅ g )( x ) d

4 • g( x ) = x 2 − 5

• h( x ) = 3x + 4

Find the domain of the following:

a g( x ) ⋅ h( x ) b h( x ) − g( x ) c d

Let’s practice

5 Perform each function operation and state the domain:

a f ( x ) ⋅ g( x ) b

4

3

2

g(x) 1

−8 −6 −4 −2 −1

−2

−3

−4

−5

−6

−7

−8

−9

−10

−11

y

f(x)

2 4 6 8

x

−4

−2

28

26

24

22

20

18

16

14

12

10

8

6

4

2

y

g(x)

2 4

f(x)

x

6 a Give a pair linear functions m( x ) and m( x ) where the domain of is

−∞ ≤ x < 6 ∪ 6 < x ≤ ∞

b Give a pair quadratic functions t( x ) and v( x ) where the domain of is

−∞ ≤ x < −1 ∪ −1 < x ≤ ∞

7 Sunshine Travel Company use various functions to predict the price of airline tickets at

different times of the year. Based on historical data, the company uses the following functions

to predict the cost of a plane ticket, where x is the number of days until the date of travel.

The cost to fly roundtrip from Miami to Tehran is predicted by

S( x ) = x 2 − 200x + 10227

The cost to fly roundtrip from Miami to Mexico City is predicted by

U( x ) = 18x + 542

390


a

b

Find S( x ) + U( x ), the cost to fly roundtrip to both Tehran and Mexico City.

What is the cost to fly roundtrip to both Tehran and Mexico City if you book 100 days

before your travel date?

8 Umberto repairs phones and charges $45 for each phone repair. The materials and labor for

each phone cost $15. Umberto’s business also has fixed costs of $500 each month. Let x

represent the number of phones repaired in a month. Complete the following:

a

b

c

d

If revenue is defined as the total amount of money earned, write a function r( x ) to

represent Umberto’s monthly revenue from repairing x phones.

Write a function c( x ) to represent Umberto’s monthly costs from repairing x phones.

The profit of a business is the difference between the revenue and the cost. Write a

composition function to represent Umberto’s monthly profit.

How many phones does Umberto needs to repair each month to earn a profit?


9 The Blues is a company that sells popular jeans. The demand of jeans sold by The Blues is

modeled by

d ( x ) = −10x + 2000

where x is the the price of each pair of jeans sold.

a

b

If the revenue for a business can be found by multiplying the price of a product by the

demand for the product, write a function to represent the revenue for The Blues.

How much should The Blues sell their jeans for in order to maximize revenue?

Justify your answer.

10 Let h( x ) be a linear function and k( x ) be a quadratic function.

Determine whether each statement is true or false. If true, explain how you know. If false,

provide a counterexample.

a The domain of h( x ) + k( x ) is the same as the domain of k( x ) + h( x )

b The domain of is the same as the domain of

c The graph of h( x ) ⋅ k( x ) is the same as the graph of k( x ) ⋅ h( x )

d The graph of h( x ) − k( x ) is the same as the graph of k( x ) − h( x )

11 If p( x ) is a linear function and q( x ) is a quadratic, is it possible for p( x ) − q( x ) to be a linear

function? Explain.


8 Solving

Quadratic

Equations

Chapter outline

8.01 Solving quadratic equations using graphs and tables 394

8.02 Solving quadratic equations by factoring 403

8.03 Solving quadratic equations using square roots 409

8.04 Solving quadratic equations by completing the square 414

8.05 The quadratic formula and the discriminant 419

8.06 Solving quadratic equations using appropriate methods 425

8.01 Solving quadratic

equations using graphs

and tables

Concept summary

A quadratic equatIon is a polynomial equation of degree 2. The standard form of a quadratic

equation is written in the in the form ax 2 + bx + c = 0 where a, b, and c are real numbers.

We can solve some quadratic equations by drawing the graph of the corresponding function.

This also allows us to determine the number of real solutions it has.

y

y

x

x

OnE real solution

The solutions to a quadaratic equation are the

x-intercepts of the corresponding function.

They also known as the roots of the equation or

the zeros of the function. A quadratic equation with

no real solutions is said to have non-real solutions.

Two real solutions

y

The zeros of an equation can be also be seen

in a table of values, provided the right values of

x are chosen, and the equation has at least one

real solution.

x

No real solutions

394


Worked examples

Example 1

Complete a table of values for y = 2x 2 − 18 and then determine the solutions to the corresponding

equation 2x 2 − 18 = 0.

Approach

When building a table we want to choose values with suitable range so we don’t have to do too

many calculations. Start by finding the values in the domain −4 ≤ x ≤ 4. If the function value is zero

for any of these x-values, then we have found a solution to the corresponding equation.

Solution

x −4 −3 −2 −1 0 1 2 3 4

y 14 0 −8 −16 −18 −16 −8 0 14

We can see the equation has solutions of x = −3, x = 3, which we can also write as x = ±3.

Reflection

We can see that the table of values has both positive and negative values. Whenever this is the

case for a function of the form f ( x ) = ax 2 + bx + c we know that the equation 0 = ax 2 + bx + c must

have two real solutions, and the corresponding parabola will have two x-intercepts.

Example 2

Consider the function y = ( x − 2 ) 2 − 9.

a Draw a graph of the function.

Approach

The function is given in vertex form so we know the vertex is at (2, −9 ). We can substitute x = 0 to

find the y-intercept at (0, −5 ). We can find other points on the curve by substituting in other values,

and by filling a table of values.

Chapter 8 Solving Quadratic Equations 395

Solution

y

6

4

2

x

−4 −3 −2 −1

−2

1 2 3 4 5 6 7 8 9

−4

−6

−8

−10

Reflection

It is important when drawing graphs to clearly show the key features such as the vertex and

the intercepts.

b Determine the solution to the equation ( x − 2 ) 2 = 9.

Approach

The solutions to the equation can be found at the x-intercepts of the graph we just drew,

as ( x − 2 ) 2 − 9 = 0 is an equivalent equation.

Solution

The solutions are x = −1 and x = 5.

Reflection

In this case, the equation has integer solutions, which makes solving graphically an effective

method. In other cases the answer can be irrational and so solving graphically may only help

determine an approximate solution.

396



1 Using the given graphs, determine whether the following equations have real or

non-real solutions:

a x 2 + 3x − 10 = 0 b −x 2 + 4x + 4 = 0

20

y

10

y

15

10

5

x

5

−4 −2 2

−5

4 6 8

x

−8 −6 −4 −2

−5

−10

2 4 6

−10

−15

−15

−20

c x 2 − 6x + 15 = 0 d −2x 2 + 12x − 18 = 0

y

y

25

10

20

15

10

−4

−2

5

−5

2 4 6 8

x

−2

5

−5

2 4 6 8

x

−10

−15

−20

2 Using the given tables, find the solutions to the following equations:

a x 2 + 7x + 12 = 0

x −6 −5 −4 −3 −2 −1

y 6 2 0 0 2 6

b 3x 2 − 27 = 0

x −4 −3 −2 −1 0 1 2 3 4

y 21 0 −15 −24 −27 −24 −15 0 21


3 Using the given graphs, find the solutions to the following equations:

a x 2 + 2x − 8 = 0 b 2x 2 = 12x − 16

y

20

y = x 2 + 2x − 8

15

10

5

x

−7 −6 −5 −4 −3 −2 −1 1 2 3 4 5

−5

−10

−2

18

16

14

12

10

8

6

4

2

−1

−2

−4

y

y = 2x 2 − 12x + 16

1 2 3 4 5 6 7

x

Let’s practice

4 For each of the following quadratic equations:

i

ii

Draw the graph of the corresponding function.

Use the graph to determine the solution( s ) to the equation.

a 0 = x 2 b 0 = − x 2 + 9 c 3x 2 + 3 = 0

d 3x 2 = 3 e 0 = 2x 2 − 2x + 4 f 0 = −( x − 3 ) 2


i

ii

Copy and complete the table of values for the corresponding function.

Set each function equal to zero and use the table to determine the solution( s ) to

the equation.

a b y = −2x 2

x −2 −1 0 1 2

y

x −1 0 1 2 3

y

c y = −2x 2 − 2 d y = ( x − 2 ) 2 − 4

x −2 −1 0 1 2

y

x 0 1 2 3 4

y

398


6 Consider the table of values for y = −2x 2 + 5:

x −3 −2 −1 0 1 2 3

y −13 −3 3 5 3 −3 13

a

b

c

Using the table, determine how many real solutions 2x 2 = 5 has. Explain how you

can tell.

Draw the graph of the equation.

Use the graph to give an approximation of the solutions.

7 Solve the following equations by first completing a table of values or drawing a graph of the

corresponding function:

a x 2 − 15x + 54 = 0 b −( x + 5 ) 2 + 9 = 0

c x 2 − 15x + 50 = 0 d ( x − 5 ) 2 − 4 = 0

8 A compass is accidentally thrown upward and out of an air balloon at a height of 100 feet.

The height, y, of the compass at time x in seconds is given by the equation:

y = −20x 2 + 80x + 100

a Complete the following table of values for y = −20x 2 + 80x + 100:

x 0 1 2 3 4 5 6

y

b Graph the relationship y = −20x 2 + 80x + 100.

c

Find the time it takes for the compass to hit the ground.

9 Sketch the graph of quadratic equations with the following key features:

a • Increasing when x > 1

b

c

• Solutions of x = −2, x = 4

• End behavior:

x → ∞, y → −∞

x → −∞, y → −∞

• One real solution x = 5

• y-intercept: (0, −25 )

• No real solutions


• Range: y ≤ −9

10 Consider the following key features:

a

• Increasing when x > −1

• Solutions of x = −7, x = 5



A

−35

−30

−25

−20

−15

−10

−5

y

−7−6−5−4−3−2−1 −10

1 2 3 4 5 6 7 8 9

−15

−20

−25

−30

−35

x

B

−35

−30

−25

−20

−15

−10

−5

−9−8−7−6−5−4−3−2

−10

−5 −1 1 2 3 4 5 6 7 8 9

−15

−20

−25

−30

−35

y

x

C

−35

−30

−25

−20

−15

−10

−5

y

x

D

−35

−30

−25

−20

−15

−10

−5

y

x

−7−6−5−4−3−2−1 −10

1 2 3 4 5 6 7 8 9

−15

−20

−25

−30

−35

−9−8−7−6−5−4−3−2

−10

−5 −1 1 2 3 4 5 6 7 8 9

−15

−20

−25

−30

−35

b

Consider the following key features:

• End behavior:

x → ∞, y → −∞

x → −∞, y → −∞

• One real solution x = 6

• y-intercept: (0, −36 )


A

−5

−10

−15

−20

−25

−30

−35

y

1 2 3 4 5 6

7 8 9 10 11 12 13 14

x

B

y

x

−12−11−10−9−8 −7−6−5−4−3−2 −1 1

−5

−10

−15

−20

−25

−30

−35

400


C

−5

y

1 2 3 4 5 6

7 8 9 10 11 12

x

D

−11−10−9

−8 −7 −6 −5 −4 −3 −2 −1

−5

x

−10

−10

−15

−15

−20

−20

−25

−25

−30

−30

−35

−35

c

Consider the following key features:

• No real solutions


• Range: y ≤ −6

A

y

x

B

y

x

−5

1 2 3 4 5 6 7 8 9 1011121314151617

−5

1 2 3 4 5 6 7 8 9 1011121314151617

−10

−10

−15

−15

−20

−20

−25

−25

−30

−30

−35

−35

C

y

x

D

y

x

−5

1 2 3 4 5 6 7 8 9 1011121314151617

−5

1 2 3 4 5 6 7 8 9 1011121314151617

−10

−10

−15

−15

−20

−20

−25

−25

−30

−30

−35

−35



11 A rectangular enclosure is to be built for an animal. Zookeepers have 100 ft of fencing,

but they want to maximise the area of the enclosure.

a

Write a quadratic equation for A, the area of the enclosure in terms of x the width.

b Draw the graph of the equation found in part ( a ).

c

d

e

Use the graph to determine the solutions of the equation.

Explain what the solutions mean in context.

Explain why the vertex is a more useful point for the zookeepers.

12 Jessie uses large sprinklers to water crops. When water is sprayed, it takes the path of a

parabola. The stream of water reaches a maximum height of 16.2 m above the ground,

18 m from the sprinkler.

Let the position of the sprinkler be (0, 0 ). Let x be the horizontal distance from the sprinkler

and y be the height of the water above the ground.

a

b

c

d

Determine whether using standard form

or vertex form of a quadratic function

would be most efficient in writing an

equation. Explain your answer.

The graph of the height of the

water in terms of the horizontal

distance from the sprinkler is shown.

Write the quadratic equation.

Determine the solutions to the

quadratic equation.

Jessie wants to adjust the sprinkler

so that it reaches a maximum height

of 10 m above the ground, 20 m from

the sprinkler. Determine the quadratic

function they should use.

13 a Describe how to find solutions of a quadratic equation from a graph and table

b

c

Explain how you can verify the number of real solutions a quadratic equation has by

looking at a table of values of the corresponding funciton.

Explain how you can verify the number of real solutions a quadratic equation has by

looking at the graph of the corresponding funciton.

14 Consider the quadratic equation x 2 + k = 0. Determine what values of k will ensure the

equation has zero, one and two real solutions. Justify your answer with reference to the

graph of y = x 2 + k.

16

14

12

10

8

6

4

2

y

5

10 15 20 25 30 35

x

402



equations by factoring

Concept summary

The zero product property states that if a product of two or more factors is equal to 0, then at least

one of the factors must be equal to 0. That is, if we know that xy = 0 then at least one of x = 0 or

y = 0 must be true.

We can use this property to solve quadratic equations by first writing the equation in factored form:

a( x − x 1 )( x − x 2 ) = 0

If we can write a quadratic equation in the factored form, then we know that either x − x 1 = 0 or

x − x 2 = 0. This means that the solutions to the quadratic equation are x 1 and x 2 . This approach can

be useful if the equation has rational solutions.

If an equation is in the form ( x − x 1 ) 2 = 0 then it has only one real solution, x = x 1 .

We can use the following process for solving most quadratic equations:

1. Gather together all the terms on one side of the equation so that the other side is equal to 0

2. Factor these terms using any appropriate techniques.

3. Use the zero product property to split the quadratic equation into two linear equations.

4. Solve the linear equations to get the solutions to the original quadratic equation

Worked examples

Example 1

Solve x 2 + 6x − 55 = 0 by factoring:

Approach

First we want to find the factors of the constant term −55 then choose the pair of factors

( let’s call them p and q ) that add up to 6. After finding p and q, we factor the quadratic by writing it

in the form ( x + p )( x + q ) = 0.

The prime factors of −55 are ±1, ±5, ±11 and ±55, and we want a factor pair whose sum is 6.


Solution

The factor pair whose sum is 6 is −5 and 11, so we have p = −5, q = 11. Factoring gives us

which leads to the equation

x 2 + 6x − 55 = ( x − 5 )( x + 11)

( x − 5 )( x + 11) = 0

We can then solve the equation by setting each factor equal to zero, giving us x − 5 = 0 and

x + 11 = 0, which gives the solutions x = 5 and x = −11.

Reflection

It is important to pay attention to the sign of the terms when determining the solution from a

quadratic in factored form. In the example above the term of ( x − 5 ) leads to a solution of x = 5

and ( x + 11) to a solution of x = −11.

It is also important to note that in the two forms ( x + p )( x + q ) and ( x − x 1 )( x − x 2 ), we can see that

x 1 = −p, and x 2 = −q.

Example 2

The graph of a quadratic function has x-intercepts at (3, 0 ) and (−1, 0 ) and passes through the

point (4, 10 ). Write an equation in factored form that models this quadratic.

Approach

To write the equation for this quadratic in factored form we need to first identify the solutions of

the equation. We can then substitute these values for x 1 and x 2 . Next, we will need to find a by

substituting one other point and solving the resulting equation. The roots of the equation are at

(−1, 0 ) and (3, 0 ) so we know the equation has solutions of −1 and 3.

Solution

Since the solutions are −1 and 3, we can write the factored form as:

y = a( x + 1)( x − 3 )

for some value of a. We can find a by substituting in the coordinates of the additional point, (4, 10 ),

into the function. To find a:

y = a( x + 1)( x − 3 )

Factored form

10 = a(4 + 1)(4 − 3 ) Substitute in the known values

10 = a(5 )(1) Simplify the factors

10 = 5a Simplify

2 = a Divide both sides by 5

The equation of the quadratic function in factored form:

y = 2( x + 1)( x − 3 )

404



1 Solve the following equations by using the zero product property:

a x( x − 9 ) = 0 b 2m( m − 8 ) = 0

c c(5c − 12 ) = 0 d ( k − 3 )( k − 5 ) = 0

e ( y − 6 )( y + 11) = 0 f (3x − 9 )(2x − 5 ) = 0

g 5( x + 5 )( x − 5 ) = 0 h (7a − 2 ) 2 = 0

2 Consider the quadratic equation:

x 2 + 2x − 35 = 0

Select the solution of the quadratic equation.

A x = −5, x = 7 B x = 5, x = −7 C x = 5, x = 7 D x = −5, x = −7


x 2 − 4x − 12 = 0

Select the solution of the quadratic equation.

A x = −6, x = −2 B x = −6, x = 2 C x = 6, x = −2 D x = 6, x = 2

Let’s practice

4 Solve the following equations by factoring:

a 6x 2 + 54x = 0 b 4y − 8y 2 = 0

c x 2 − 5x − 14 = 0 d f 2 + 6f − 55 = 0

e h 2 + 19h + 88 = 0 f x 2 − 20x + 100 = 0

5 Solve the following equations by first rearranging, and then factoring:

a x 2 − 14 = 5x b 2y 2 = 9y + 5

c 3x 2 − 14x = −8 d

6 For each quadratic function, complete the following:

i Determine the y-intercept. ii Determine the x-intercept( s ).

iii Determine the coordinates of the vertex. iv Draw a graph of the quadratic function.

a y = ( x + 8 )( x + 4 ) b y = − ( x − 10 ) 2

c y = ( x − 3 )( x + 2 ) d y = (1 − x )( x + 5 )

7 For each graph, complete the following:

i Determine the y-intercept. ii Determine the x-intercept( s ).

iii Write the equation in factored form.


a

y

14

12

10

8

6

4

2

−2 −1

−2

−4

1 2 3

4 5 6 7

x

b

18

16

14

12

10

8

6

4

2

y

−2

−2

2 4 6

−4

8 10

x

c

2

−2

−4

−6

−8

−10

−12

−14

−16

y

2 4 6

8

x

d

8

6

4

2

−8 −6 −4 −2

−2

−4

−6

−8

y

2 4 6 8

x

8 Software engineers are designing a self-serve checkout system for a supermarket.

They notice that the traffic through the store during the day is described by the function

C = −t( t − 12 )

where C is the number of customers and t is the number of hours after the store opens.

To meet the peak demand, the engineers allow for an extra checkout machine

to automatically turn on when the number of customers first reaches 32 people,

and to automatically turn off when it next falls below 32 people.

a

b

c

Find the times t when the number of customers is equal to 32 people.

Determine how many hours it takes the extra checkout machine to turn on after opening.

Determine many hours the extra machine will be on for.

9 Delores needs a sheet of paper x in by

12 in for an origami alligator. The local art supply store

only sells square sheets of paper. The lower portion of

the image below shows the excess area A of paper that

will be left after Delores cuts out the x in by 12 in piece.

The excess area, in square inches, is given by

the equation

x

x

12

A = x( x − 12 )

406


a

b

Determine the lengths for which there will be no excess area

Determine the value of x that will allow Delores to make an origami alligator with the

least amount of excess paper.

10 The area of a rectangle is 60 cm 2 . If the area can be expressed as A = 17x − x 2 , what are the

dimensions of the rectangle?

A 6 cm × 10 cm B 3 cm × 20 cm C 4 cm × 15 cm D 5 cm × 12 cm

11 Consider the following quadratic equations.

a y = ( x − 6 )( x − 24 ) b y = − ( x − 8 )( x + 2 )

c y = −( x + 13 ) 2 d y = x( x + 3 )

i

Determine the y-intercept.

ii Determine the x-intercept( s ).

iii

iv

Determine the coordinates of the vertex.

Select the graph that represents each function.

A

160

140

120

100

80

60

40

20

−20

−40

−60

−80

y

5 10 15

20 25 30

x

B

26

24

22

20

18

16

14

12

10

8

6

4

2

y

−4−3−2 −1 −2 1 2 3 4 5 6 7 8 9

−4

x

C

4

3

2

y

D

−45−40−35−30−25−20−15

−10 −5

y

x

1

x

−50

−4 −3 −2

−1

−1

1 2 3 4

−100

−2

−3

−150

−4



12 The school football field is in the shape of a rectangle and has stadium seating all the way

around the field that is of a uniform width. If the total area of the field and the stadium is

8400 yd 2 , determine the width of the stadium seating.

x

100 yd

x

50 yd

A = 8400 yd 2

13 Write the equation of a quadratic in factored form that has the following key features and

draw a graph of the function:

a

b

Two solutions that have the same absolute value but opposite signs

One solution that is a fraction and the other is a prime number

c One unique solution that is less than 1 but greater than 0

d Solutions of x = 7, x = −2 and has a y-intercept at (0, 14 )

e Solutions of x = −1, x = −10 and has a y-intercept at (0, 30 )

14 Determine how many quadratic equations exist for each of the following key features:

a Solutions of x − 7, x = 10

b One unique solution of x = −4 and y-intercept at (0, −20 )

c Vertex at the origin and passes through (2, 14 )

d

Solutions of x = 15, x = −7 and is symmetric about the y-axis

15 The sum of the series 1 + 2 + 3 + 4 + ... + n is given by Determine a method to

solve for n given any sum, and then find the number of integers required for a sum of 66.

408



equations using

square roots

Concept summary

We can solve quadratic equations of the form a( x − h ) 2 = k, for positive values of a and k,

using square roots. This method is particularly useful if k or is a square number, as we will get

rational solutions.

Remember that when we take the square root of a number k, we get two solutions

Worked examples

Example 1

Solve the following equations by using square roots:

a x 2 = 9

Approach

This is a straight forward equation, the solutions are the square roots of 9.

Solution

x = ±3

b ( x − 2 ) 2 − 100 = 0

Approach

In order to use square roots to solve, the squared expression must be isolated. In this example we

want to isolate the term ( x − 2 ) 2 .

Solution

( x − 2 ) 2 − 100 = 0

( x − 2 ) 2 = 100 Add 100 to both sides

x − 2 = ±10

Square root both sides

This leaves us with two equations x − 2 = 10 and x − 2 = −10. Add 2 to solve both equations and

we find that the solutions are x = −8, x = 12.


c (3x − 8 ) 2 = 7

Approach

We can see that 7 is not a square number so our answer will involve radicals. Since the squared

expression is already isolated, we are ready to solve by taking square roots.

Solution

Square root

Add 8

Divide by 3

The solutions are

Example 2

A square field has perpendicular lines drawn across it dividing it into 36 equal sized smaller

squares. If the total area of the field is 200 square feet, determine the approximate side length

of one of the smaller squares.

Approach

We know that there are 36 smaller squares in total on a larger square grid, so there must be 6 by

6 smaller squares on the grid. If we let the side of a smaller square be x, then the side of the

larger square can be 6x. This gives us the quadratic equation (6x ) 2 = 200. We can then solve this

equation by taking square roots.

Solution


Divide by 6

Simplifying the expression gives us the solutions x ≈ ±2.36. We can exclude the negative solution

as the length of the square must be positive. So the smaller square has a side length of 2.36 feet.

Reflection

In most real life applications, we will exclude the negative solution as it will not fit in the range of

possible solutions for that context.

410



1 Solve the following equations by using square roots:

a x 2 = 25 b x 2 = 81 c x 2 = 10

d x 2 = 11 e x 2 = 98 f x 2 − 25 = 0

2 How many real solutions does the equation x 2 + 64 = 0 have?

3 State the quadratic equation that has the solutions x = ±5.


x 2 = 144

Select the solution to the quadratic equation.

A x = 2, x = 72 B

C x = 12, x = −12 D x = −2, x = −72

5 Consider the quadratic equation: x 2 − 167 = 89


A x = 16, x = −16 B x = −167, x = 89

C x = 256, x = −256 D x = 128, x = −128

6 Harry is using a diving board to dive into a swimming pool. The distance from his head to the

surface of the water can be represented as ( x − 7 )( x + 7 ) = 147. Select the viable solution to

the quadratic equation.

A x = 14 B x = -14 C x = 49 D x = 7

Let’s practice

7 Solve the following equations by finding square roots:

a x 2 − 5 = 31 b c 25y 2 = 36 d 5x 2 − 15 = 0

e ( x − 7 ) 2 = 81 f (4 − x ) 2 = 9 g (4x + 3 ) 2 = 64 h 5( p 2 − 3 ) = 705

8 On the graph of y = x 2 − 4, there are two points where y = 12. State the x-coordinates of

these two points.

9 Solve for x:

a

b

6x in

36 in

18 ft

42 ft

6x in

4x ft


10 On Earth, the equation d = 4.9t 2 is used to find the distance ( in meters ) an object has fallen

through the air after t seconds.

Willow is sky diving and wants to release her paraachute once she has fallen 400 m

Determine the time it will take her to fall 400 m, rounding your answer to the nearest second.

11 The Panorama Tower in Miami, Florida, is 868 ft tall.

We can model the behaviour of objects falling from the arch using Galileo’s formula for falling

objects: d = 16t 2 , where d is distance fallen in feet and t is time in seconds since the object

was dropped.

How long would it take an object dropped from the arch to fall to the ground? Round your

answer to the nearest tenth.

12 The kinetic energy E of a moving object is given by where m is its mass in

kilograms and v is its speed in metres/second. If a vehicle weighing 1600 kilograms has

kinetic energy E = 204800, at what speed is it moving?

13 The Widget and Trinket Emporium has released the forecast of its revenue for the next year.

The revenue R( in dollars ) at any point in time t( in months ) is described by the equation:

a

b

R = −( t − 18 ) 2 + 16

Find the times at which the revenue will be zero.

Select the graph that represents the revenue function.

A

−50

y

5 10 15

20 25 30 35

x

B

−300

−250

y

−100

−200

−150

−150

−200

−100

−250

−300

−50

5 10 15

20 25 30 35

x

C

325

300

275

250

225

200

175

150

125

100

75

50

25

−45−40−35−30−25−20−15 −10 −5

y

x

D

−30 −20 −10

−50

−100

−150

−200

−250

−300

−350

y

x

412



14 Eduardo is trying to solve the equation ( x + 7 ) 2 = 28. He thinks that the equation is

equivalent to

Nicolette, however, thinks he has performed the operations

incorrectly, and that he should have subtracted 7 from both sides of the equation first giving

an equivalent equation of x 2 = 21

Describe any errors Eduardo and/or Nicolette have made, and find the correct solution to

the equation.

15 Consider the quadratic equation ax 2 − c = 0 for. Given the following conditions, state the real

solution( s ). If there are no real solutions state “No real solutions”

a a and c are positive. b a and c are negative.

c a and c have opposite signs. d c is equal to 0.

16 For the following graphs:

i

ii

iii

Write a quadratic you can solve to find where the line intersects the parabola

Solve the equation you found

Determine the points where the lines intersect

a

7

y

b

8

y

6

6

5

4

4

3

2

2

−8 −6 −4 −2

−2

2 4 6 8

x

−4 −3 −2

1

−1

−1

1 2 3 4

x

−4

−6

−8

c

y

d

y

2

x

4

−2

−2

−4

−6

−8

−10

2 4 6

−6

2

−4 −2

2

−2

−4

x



equations by completing

the square

Concept summary

Completing the square is a method we use to rewrite a quadratic expression so that it contains a

perfect square trinomial. A perfect square trinomial takes on the form A 2 + 2AB + B 2 = ( A + B ) 2 .

If we can rewrite an equation by completing the square, then we can solve it using square roots.

For quadratic equations where a = 1, we can write them in perfect square form by following

these steps:

1 x 2 + bx + c = 0

2 x 2 + bx = −c Subtract c from both sides

3 Rewrite the x term

4 Add to both sides

5 Factor the trinomial

If a

1, we can first divide through by a to factor it out.

Worked examples

Example 1

Solve the following quadratic equation by completing the square:

x 2 + 18x + 32 = 0

Approach

To solve an equation by completing the square, start by moving the constant term to the other side

of the equation. We will complete the square by finding half the coefficient of the x term, squaring

it, and adding it to both sides of the equation. Since the coefficient of the x term is 18, we will need

to add

to both sides of our equation. Once we’ve completed the square, we can solve.

414


Solution

x 2 + 18x + 32 = 0

x 2 + 18x = −32

x 2 + 18x + 81 = −32 + 81


Complete the square

( x + 9 ) 2 = 49 Factor the perfect square trinomial and simplify

x + 9 = ±7


This leaves us with two equations x + 9 = 7 and x + 9 = −7. We can solve both equations by

subtracting 9 and so we get the solutions x = −2, x = −16.

Example 2

Solve the following quadratic equation by completing the square:

3x 2 − 12x + 10 = 0

Approach

In this example, the coefficient of x 2 is 3 so we will need to divide this coefficient out before

completing the square. We can then perform steps similar to the previous example. Once 3 has

been divided out, the coefficient of x will be 4. Taking

completes the square.

gives us so this is the value that

Solution

Divide by 3

Subtract

from both sides

Complete the square

Factor and simplify


This leaves us with two equations

We need to add 2 to solve both equations and we find that the solutions are



1 Complete the following expressions so they form a perfect square:

a x 2 + 10x + b x 2 − 2x + c x 2 − x + d x 2 + mx +

e f x 2 − x + 81 g x 2 − x + 16 h x 2 − x + 121

2 Complete the following perfect squares:

a x 2 + 4x + = ( x + ) 2 b x 2 − 5x + = ( x − ) 2

c d ( x + ) 2 = x 2 + 20x +

e

f


x 2 − 2x − 10 = 0


A

C

B

D


x 2 + 8x − 18 = 0


A

B

C

D

5 The sales ( in thousands of dollars ) of a restaurant x years after it first started is presented as:

x 2 + 6x + 2 = 0

Select the value of x that would make the quadratic equation true?

A

B

C

D

Let’s practice

6 For each of the following:

i Rewrite the equations in vertex form by completing the square.

ii Find the x-intercepts.

a y = x 2 + 12x b y = x 2 − 7x + 14

c y = 3x 2 + 9x + 8 d y = 4x 2 − 11x + 7

416


7 Solve the following quadratic equations by completing the square:

a x 2 + 18x + 32 = 0 b x 2 − 6x + 8 = 0

c x 2 − 9x + 8 = 0 d 2x 2 − 12x − 32 = 0

e 4x 2 + 11x + 7 = 0 f x 2 − 2x − 32 = 0

8 Solve each quadratic equation by completing the square. Leave your answers in simplest

radical form.

a x 2 + 24x + 5 = 0 b x 2 + 11x + 5 = 0

c 6x 2 + 48x + 24 = 0 d x 2 − 7x + 8 = 0

e f 5x 2 + 55x + 3 = 0

9 Flora and Yang are designing a square shaped

banner to display at the next football match. Yang wants

the banner to be gold with horizontal black bars. They

have enough gold material to make the area of the

rectangular section 49 ft 2 . Determine the dimensions of

the largest banner they can make.

2 ft

2 ft

10 The bulbs for the headlights of a car are

shaped like a parabola, to direct all light in the

same direction. For a particular model of car,

the bottom of the headlight can be modelled by

the function

(0, 18)

y

f ( x ) = x 2 − 6x + 18

where x and f ( x ) are measured in inches, and

the y-intercept represents the top left point of

the headlight. Determine the width and height

of the headlight by first rewriting the function in

vertex form. Explain your reasoning.

x

11 A rectangular playground is 12 ft shorter than it is wide. If it has a total area of 1250 ft 2 ,

determine the perimeter of the playground. Round your answer to two decimal places.

12 A cube has a surface area of 6x 2 + 36x + 54. What is a length of one of the sides?



13 The revenue y( in millions of dollars ) of a company x years after it first started is modelled by

a

b

y = 12.5x 2 − 64x + 135

Use this equation to predict the number of years it will take for the revenue to reach

$1167 million.

Describe another method you could use to calculate this.

14 Explain the relationship between solving equations by completing the square and solving

using square roots.

15 Explain why it is preferable to use the method of completing the square to find the exact

solutions to some quadratic equations rather than by sketching a graph or factoring.

Give one example of a quadratic function for which this is the case.

418


8.05 The quadratic

formula and the

discriminant

Concept summary

So far the methods we have looked at are useful for quadratic equations that are of a particular

form, but we cannot use these methods to solve all quadratic equations. The quadratic formula

is a method we can use to solve any quadratic equation written in the form ax 2 + bx + c = 0 and

also to quickly determine the number of real solutions it has.

The quadratic formula is:

The expression under the radical is known as the discriminant, and we can use the sign of this

value to determine the number of real solutions.

b 2 − 4ac > 0

The equation has two real solutions.

Example: x 2 − 4x − 8 = 0

b 2 − 4ac = 0

The equation has one real solution.

Example: 9x 2 + 6x + 1 = 0

b 2 − 4ac < 0

The equation has no real solutions.

Example: 5x 2 − 3x + 9 = 0


Worked examples

Example 1

Use the discriminant to determine the number and nature of the solutions of the following

quadratic equations:

a 2x 2 − 8x + 3 = 0

Approach

The discriminant is equal to b 2 − 4ac, and for this equation we have a = 2, b = −8, c = 3.

Solution

The discriminant is (−8) 2 − 4(2 )(3) = 40, which is positive, so the equation has two real solutions.

b −5x 2 + 6x − 2 = 0

Approach

The discriminant is equal to b 2 − 4ac, and for this equation we have a = −5, b = 6, c = −2.

Solution

The discriminant has a value of (6 ) 2 − 4(−5 )(−2 ) = −4, which is negative, so the equation has no

real solutions.

Reflection

As the quadratic formula involves taking the square root of the discriminant, if the discriminant is

negative then the formula will involve taking the square root of a negative number, which will in

turn result in no real solutions.

We can also see that if 4ac > b 2 the discriminant will be negative, and the corresponding equation

will have no real solutions.

c x 2 − 6x + 9 = 0

Approach

The discriminant is equal to b 2 − 4ac, and for this equation we have a = 1, b = −6, c = 9.

Solution

The discriminant has a value of (−6) 2 − 4(1)(9 ) = 0, so the equation has one real solution.

420


Reflection

If the discriminant is zero, the quadratic formula simplifies to be

which is equal to the

x-value of the vertex. This means the vertex lies on the x-axis and the x-coordinate of the vertex is

the only solution to the quadratic equation.

Example 2

A ball is launched from a height of 80 ft with an initial velocity of 107 ft per second. Its height,

h feet, after x seconds is given by

h = −16x 2 + 107x + 80

Determine the number of seconds it will take the ball to reach the ground. Explain your reasoning.

Approach

On the ground the ball will have a height of 0 ft. We want to solve the equation

0 = −16x 2 + 107x + 80, and since the numbers are large the quadratic formula is an appropriate

method to solve. The discriminant is equal to (107 ) 2 − 4(−16 )(80 ) = 16569, so we know there must

be two real solutions.

Solution

Substituting into the quadratic equation we get

The two solutions rounded to two decimal places are therefore x = −0.68, x = 7.37. We can

exclude the negative solution as it is outside the domain, which is x ≥ 0.

The ball will reach the ground after 7.37 seconds.

Reflection

In most real life applications, where x is a unit of measurement, such as seconds or feet, the

domain is restricted to non-negative numbers. For these types of problems we need to exclude any

negative solutions.



1 The standard form of a quadratic equation is ax 2 + bx + c = 0. Find the values of a, b and c in

the following quadratic equations:

a x 2 + 7x = 10 b

c 2x 2 + 9x = 0 d −5( x − 3 ) 2 + 4 = 0

2 For the following equations:

i

ii

Find the value of the discriminant.

State the number and nature of the solutions to the equation.

a x 2 + 6x + 9 = 0 b 4x 2 − 6x + 7 = 0

c 2x 2 − 2x = x − 1 d −2x 2 = 6 − 5x

3 For the following equations:

i

ii

Determine the number of real solutions. Explain your reasoning.

Find the real solution( s ) of the equation.

a x 2 − 8x − 48 = 0 b 4x 2 − 4x + 1 = 0

c x 2 − 4x + 7 = 0 d x 2 + 11x + 28 = 0

e f x 2 + 12x + 36 = 0


x 2 + 2x − 5 = 0


A

C

B

D


x 2 − 15x = 5x − 27


A

C

B

D

6 A child throws a ball upward with in intial upward velocity of 10 feet per scond from a height

of 2 feet. If no one catches the ball, the situation canbe modeled by:

x 2 + 10x + 2 = 0


A

C

B

D

422


Let’s practice

7 Find the values of n for which x 2 − 8nx + 1296 = 0 has one solution.

8 Solve the following equations using the quadratic formula. Round your answers to two

decimal places:

a 13 + x 2 = −7x b −x 2 − 5x = 8

c 5x 2 + 13x + 10 = x 2 − 7x − 15 d −2x 2 + 3x − 2 = 0

e 1.8x 2 + 5.2x − 2.3 = 0 f 3x( x + 4 ) = −3x + 4

9 Yuri is playing baseball and hits a homerun with an initial velocity of 101 feet per second,

from a height of 3 feet. After x seconds, its height ( in feet ) is given by

a

b

h = −16x 2 + 101x + 3

Determine the maximum height the baseball reaches, rounding your answer to the

nearest tenth of a foot.

LaDeana is in the crowd and catches the homerun ball in the stands from a height of

15 feet above the ground. Determine the number of seconds after Yuri hits the ball that

LaDeana catches it. Round your answer to two decimal places.

10 The stopping distance of a car when the brakes are applied can be modelled by

the equation

where s is the stopping distance in ft and u is the initial speed

in mph.

a

b

Determine the speed the car was travelling if its stopping distance was 100 ft,

rounding your answer to the nearest hundredth of a mile.

It takes 399 ft to stop when travelling at the speed limit of 70 mph. If it takes Ray 496 ft

to stop, determine how fast over the speed limit Ray was driving.

11 Consider a right-angled triangle with side

lengths x units, x + p units and x + q units,

ordered from shortest to longest. No two sides

of this triangle have the same length.

a

b

c

d

Complete the statement: p and q have lengths

such that 0 < < .

Write a quadratic equation in standard form that

describes the relationship between the sides of

the triangle in terms of x.

Find the discriminant of this quadratic equation.

Determine the number of real solutions.

e Find the value of x when q = 2p.

Give your answer in terms of p.

x + p

x + q

x


12 For each of the given solutions to a quadratic equation:

i Find the values of a, b and c.

ii Write down the quadratic equation that has these solutions.

a

b


13 Consider the equation in terms of x:

mx 2 − 3x − 5 = 0

a Given that it has two unique solutions, determine the possible values of m.

b

There is one value of m that must be eliminated from the range of solutions found in the

previous part. Determine the solution and explain why it must be eliminated.

14 With reference to the discriminant, explain what determines the nature of the solutions to a

quadratic equation.

15 Determine the range of values of the constant k such that the equation 3x 2 + kx + 12 = 0 has

no real solutions. Justify your answer.

16 Use an algebraic method to show that the graphs of the f ( x ) = 3x 2 − x + 8 and

g( x ) = −x 2 + 2x − 4 do not intersect.

17 A quadratic equation has two real solutions whose difference is k, for some positive value k.

Determine algebraically a simplified expression for the discriminant of this equation.

424



equations using

appropriate methods

Concept summary

We have several methods we can use to solve quadratic equations. To determine which method is

the most suitable we need to look at the form of the quadratic equation. Some are easily solved by

factoring for example, or by taking the square root.

If we are unable to solve the quadratic easily using one of these methods, the quadratic formula

is often the best approach since it can be used to solve any quadratic equation once it’s written in

standard form. If we have access to technology, drawing the graph of the corresponding quadratic

function can help us find exact solutions or approximate a solution if it is not an integer value.

Worked examples

Example 1

For the following quadratic equations, find the solution using an efficient method. Justify which

method you used.

a x 2 − 7x + 12 = 0

Approach

The leading coefficient of x 2 is 1, so we can check if this can be easily factored. The prime

factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, and we want to find two factors that have a product of

12 and sum to −7. As the product is positive but the sum is negative we know both factors must

be negative.

Solution

The two factors that have a product of 12 and a sum of −7 are −3 and −4. We can write the

equation in factored form as ( x − 4 )( x − 3 ) = 0, which gives us two solutions x = 3, x = 4.

Reflection

In general, if the coefficients are small, and especially if a = 1, it is worth checking to see if we can

easily factor the equation to solve.


b x 2 − 6 = 21

Approach

Here we have b = 0, and can easily isolate the x 2 , which means we can solve this by using

square roots.

Solution

x 2 − 6 = 21

x 2 = 27 Add 6

Square root

Reflection

giving us two solutions x ≈ 5.196, x ≈ −5.196

In general, if we can easily rearrange the equation into the form ( x − h ) 2 = k for some positive value

of k then solving using square roots is a suitable method.

Example 2

A rectangular enclosure is to be constructed from 100 meters of wooden fencing. The area of the

enclosure is given by A = 50x − x 2 , where x is the length of one side of the rectangle. If the area is

525 m 2 , determine the side lengths.

Approach

We can set up and solve a quadratic equation, 50x − x 2 = 525. Since the values are large we will

try solving this problem with the quadratic formula. The two solutions will be the side lengths of

the enclosure.

Solution

Rearranging the equation into standard form we get x 2 − 50x + 525 = 0.

We can solve this using the quadratic equation:

1 Quadratic formula

2 Subtitute values for a, b, c

3 Simplify

4 Evaluate the square root

426


This leaves us with two values,

Evaluating each expression for x we

get x = 35 and x = 15.

Reflection

We can confirm our answer is correct by checking the conditions of the problem.

We had 100 meters of fencing and 2(35 + 15 ) = 100. We needed the area to be 525 m 2 and

35 (15 ) = 525 as required.

Since there are two rational solutions, the quadratic equation was also factorable:

x 2 − 50x + 525 = ( x − 35 )( x − 15 ), but these factors are not immediately obvious.



a ( x − 3 ) 2 = 64 b (2 − x ) 2 = 81

c x( x + 7 ) = 0 d (10x − 9 ) 2 = 0

e ( x − 6 )( x + 7 ) = 0 f

g x 2 − 8x + 15 = 0 h x 2 − 4x − 22 = 0

2 The formula for the surface area of a sphere is S = 4pr 2 , where r is the radius.

Determine the radius of a sphere that has a surface area of 804 in 2 , rounding your answer to

two decimal places.


x 2 − 3x − 108 = 0


A x = 9, x = −12 B x = −9, x = 12 C x = −9, x = −12 D x = 9, x = 12


x 2 − 200 = −79


A

C x = 200, x = −79 D x = 11, x = −11

5 A square lot has an area of 289 m 2 . Find the length of one side of the square lot if A = s 2 .

A s = 144.5 m B s = 72.25 m C D s = 17 m

B


6 Tara is practising diving. She springs up off a board 32 feet high, and after t seconds,

her height in feet above the water is described by the function:

a

b

h( t ) = −16t 2 + 16t + 32

Determine how many seconds it will take for Tara to hit the water.

Select the graph that represnts the quadratic function.

A

35

30

25

20

15

10

5

−2 −1 −5

−10

−15

−20

−25

−30

−35

y

B

y

35

30

25

20

15

x

10

1 5

x

−1

−5

−10

−15

−20

1

2

C

−1

45

40

35

30

25

20

15

10

5

y

1

x

D

−1

−5

−10

−15

−20

−25

−30

−35

−40

−45

y

1

x

7 Thirty rabbits are released into the wild. In the absence of predators, the population P is

expected to grow every x years according to the function:

P( x ) = 24x 2 + 6x + 30

a Determine approximately how many years it will take for the population to reach 10 000.

b

Select the graph that represnts the quadrartic function.

428


A

35

30

25

20

15

10

5

−2 −1 −5

−10

−15

−20

−25

−30

−35

y

B

y

35

30

25

20

15

x

10

1 5

x

−1

−5

−10

−15

−20

1

2

C

−1

45

40

35

30

25

20

15

10

5

y

1

x

D

−1

−5

−10

−15

−20

−25

−30

−35

−40

−45

y

1

x

Let’s practice

8 For the following quadratic equations, find the solution using an efficient method. Justify

which method you used.

a x 2 − 10 = 15 b x 2 − 7x + 6 = 0

c 25y 2 = 36 d 4x 2 + 5x + 1 = 0

e x 2 + 24x + 63 = 0 f x 2 − 7x = 0

g 5k 2 − 17k + 13 = 0 h x 2 + 9x + 20 = 0


a ( x − 6 ) 2 − 2 = 0 b 24x 2 = 71x − 35

c x 2 + 27x + 23 = 3x − 40 d 3x 2 − 36x + 33 = 0


10 The following rectangle has a length

of L = 56y + 11 and a width of W = 5y 2 :

a Write the perimeter of the figure in terms of y.

b If the perimeter is equal to 630 find y.

L

W

11 At time t seconds, the distance, s, travelled by an

object moving in a straight line is given by

where u is its starting speed and a is its acceleration.

When u = 16 and a = 8, find how long it would take for the object to travel 128 meters.

12 The base of a triangle is 3 meters more than twice its height. The area of the triangle is

115 square meters. Let x be the height of the triangle.

a Find the height by solving for x. b Find the length of the base.

13 A rectangular swimming pool is 16 m long

and 6 m wide. It is surrounded by a pebble path of

uniform width x m. The area of the path is 104 m 2 .

x

a

Find an expression for the area of the path in

terms of x.

x

6 m

x

b

Write an equation and solve for x, the width of

the path.

16 m

x


14 For each of the following equations determine, without solving them, the most efficient

method for solving them. Explain your thinking.

a x 2 − 3x + 2 = 0 b 8x = x 2

c 16x 2 − 81 = 0 d

15 Executives at the Widget Emporium are discussing whether to merge their company with

the Trinket Bazaar, a large competitor. Market analysis shows that the extra revenue the

company will receive can be modelled by the equation R = 0.25t 2 , and the extra costs by

the equation C = 3.5t. R and C are measured in thousands of dollars and t is measured in

months after the merger.

a Find the times at which the extra revenue R will match the extra cost C.

b

The executives decide that they can only afford to operate at a loss for one year.

Based on this requirement, would you advise that the Widget Emporium merge with the

Trinket Bazaar?

430


16 An interplanetary freight transport company has won a contract to supply the space station

orbiting Mars. They will be shipping stackable containers, each carrying a fuel module and

a water module, that must meet certain dimension restrictions.

The design engineers have produced a sketch for the modules and container, shown below.

The sum of the heights of both modules equal to the height of the container.

6426

cm

x 2

663 cm

x

17 cm

Water Module Fuel Module Container

a

b

Write an equation that equates the height of the container and the sum of the heights of

the modules.

Find the tallest possible height of the container, rounding your answer to two decimal

places. Explain your method.

17 The shaded area in the rectangle has a

uniform width and an area of 11 ft 2 :

a

b

Determine a quadratic equation

to represent this situation.

Find the value of x, rounding your answer

to two decimal places.

x

x + 1

3

4


9 Descriptive

Statistics

Chapter outline

9.01 Populations and sampling 434

9.02 Classifying data 440

9.03 Displaying and interpreting univariate data 445

9.04 Two-way frequency tables 457

9.045 Honors: Two-way relative frequency tables 466

9.05 Associations in bivariate data 478

9.06 Scatter plots and lines of fit 483

9.07 Analyzing lines of fit using residuals 490

9.01 Populations and

sampling

Concept summary

A sample survey is a research method used to collect information from a group of individuals.

This information, called data, is then used to make conclusions about the population.

Population

The entire set of cases or individuals under consideration in a statistical analysis.

Sample

A set of individuals that data is collected on in a statistical experiment; a subset of

the population

When the sample is representative of the population, the statistics describing the sample mean can

be extended to estimate the population mean and the proportions from the sample can be used

to estimate the population percentage or the size of specific groups in the population by using the

population total.

Population mean

The average value, over the entire population, of a certain characteristic being measured

Population total

The total count of all cases or individuals in the population

When analyzing data, we are sometimes given a margin of error which represents a range for how

far the results may differ from the actual value.

434


Worked examples

Example 1

For a statistical survey, the population is deemed to be all people in a city who play in

any organized sporting competition. State whether the following could be a sample from

this population:

a All students in a school in the city who compete in a school sporting competition.

Approach

To determine if a sample is representative of a population, consider whether there are any people

in the sample who do not fit in the described population.

Solution

Yes. The students in the school in the city are in the city and since they are participating in

the school sporting competition they are participating in an organized sporting competition.

b A random 100 people chosen from a park in the city.

Approach

To determine if a sample is representative of a population, consider whether there are any people

in the sample who do not fit in the described population.

Solution

No. Although people in a city park are in the city, there is no guarantee that they also play in an

organized sporting competition so this sample does not fit in the described population.

Example 2

A local council wanted to monitor the number of rabbits in the area. They used the capturerecapture

technique to estimate the population of rabbits. 219 rabbits were caught, tagged,

and released. Later, 42 rabbits were caught at random. 15 of these 42 rabbits had been tagged.

Find the estimated population of the rabbits. Round your answer to the nearest whole number

if necessary.

Chapter 9 Descriptive Statistics 435

Approach

The sample is representative of the population and should be proportional. Identify the sample

values and population values to write a proportion and solve for the missing quantity.

Solution

In this case, 15 of the 42 rabbits caught in the sample were tagged and can be used for one side

of the proportional equation. Let x represent the unknown rabbit population.

The estimated population is 613 rabbits.

Reflection

Reciprocate both fractions

Multiply by 219

Be sure to round to a reasonable value. 613.2 rabbits does not make sense so we round this value

down to 613.

Example 3

A recent survey of 250 registered adults found that only 20% of them planned to re-elect their

city’s mayor with a margin of error of 3%. If there are 5970 adults in the town, determine how many

votes the current mayor can expect to receive.

Approach

The margin of error provides a range of possible values. To start the problem we need to add and

subtract the margin of error from the sample statistic. Then, we can use the proportion percentages

with the population total to estimate the total votes.

Solution

Using the margin of error we know that 20% ± 3% of the sample, plan to vote for the current mayor.

This gives a range between 17% and 23%. We need to convert the percentages to decimals and

multiply them to the population total to find the range of votes the mayor can expect.

0.17(5970 ) = 1014.9 and 0.23(5970 ) = 1373.1

The mayor can expect to receive between 1014 and 1373 votes.

436



1 Identify the population and sample in each scenario.

a

b

c

The mayor of Anytown wants to determine the opinion of his residents on the new mall

planned to be built. He surveys 500 residents at random.

A bookstore owner wants to know what genres of books are most popular amongst her

customers. She decides to pull up the receipts for 20 purchases out of the 650 sales

made over the weekend.

Antony wants to know what teams the students at his school support. He finds

that 30 students support the Tampa Bay Buccaneers, and 55 students support

the Miami Dolphins.

2 For a statistical survey, the population is deemed to be all the students that attend the local

high school. State whether the following are samples of that population:

a

b

c

d

e

All of the seniors.

The first 30 students to arrive at school that day.

All of the students that were in the library at lunch time.

All of the teachers.

Some of the teachers.

3 For a statistical survey, the population is deemed to be people aged under 18 who live in

Florida. State whether the following are samples of that population:

a

b

c

d

e

40 professors chosen from University of Florida.

Every person who lives in Miami.

All elementary students who are visiting DisneyWorld in a single day.

Students enrolled in the on-campus preschools of high schools in Orlando.

100 children chosen from a Pensacola little league team.

4 A sample of 50 people is drawn from a population. In this sample, the youngest person

is 18 years old and the oldest person is 64 years old. State whether the following are

populations that the sample could have been chosen from:

a

b

c

d

Students from elementary schools.

People stuck in traffic on their way to work.

Residents of a retirement home.

Employees at a bank.

5 Marine biologists want to determine the number of dolphins left along a stretch of coast.

They tag 72 dolphins. After a couple of years they come back to the same area and check

84 dolphins, 12 of which have tags.

a

b

Find the proportion of dolphins that had tags upon checking.

Find x, the expected population of dolphins along that coast.


6 Conservationists previously captured, tagged and released 400 whales in a sanctuary.

Some time later a large number of whales were observed, and 34% were found to be

tagged. Estimate n, the population of the whales in the sanctuary.

Let’s practice

7 For a statistical survey, the population is deemed to be all people who own a pet dog.

Ariel decides to find her sample by talking to every person at her local dog park.

Describe the pros and cons of her sampling method.

8 An oil spill has spread over an area of 1650 km 2 . A team of biologists scan an area of

150 km 2 , and find 272 dead marine animals. Create and solve an equation to find y,

the estimated number of dead marine animals over the entire area of the oil spill.

9 A television station wants to estimate the number of viewers it had for a new show.

They know that the country’s population is 13 620 000. When they randomly selected

5000 people and asked them if they had watched the show, they found that 400 of them

had. Create and solve an equation to find p, the estimated number of people who watched

the show in the entire population.

10 A recent survey of 2265 elementary students found that 28% of them could be classified

as obese with a margin of error of ±2%. Using this survey, determine how many of the

3.7 million kindergarteners in the United States could be classified as obese.


11 For a statistical survey, the population is deemed to be all of the athletes at an Orlando

Magic game.

a

b

Describe two different samples that could be taken at the game to represent

this population.

Describe two samples that could be taken at this game that do not represent

this population. Explain why these samples aren’t appropriate.

12 Explain the difference between estimating a population total given the proportion from the

sample and estimating a population total given the sample statistic and margin of error.

Which result do you think is more accurate and why?

13 The average number of children per family in the U.S. is 1.9 with a margin of error of

0.3 children.

a

b

If there were about 128 million households in the U.S. in 2019, estimate the number of

children in the U.S. in 2019.

In reality, there were about 50 million children in the U.S. in 2019. Identify a source of

error in the estimation from part ( a ).

438


14 It is a common belief that the ability to roll your tongue is a genetically inherited trait.

Isaiah would like to determine what proprotion of students at his univeristy have the ability

to roll their tongues. In a random sample of 100 students, Isaiah determines that 70 can roll

their tongue.

The dot plot shows the results of a simulation used to predict the proportion of people who

can roll their tongue. 500 random samples of 100 students each from a population where

70% can roll their tongue were simulated.

a

b

c

If the sample mean is 0.701 and the margin of error is 0.096, interpret what this means

in the context of tongue rolling.

Isaiah’s Biology 101 professor claims that 75% of students at the university can roll their

tongue. Is her claim plausible? Explain your reasoning.

Explain how could Isaiah decrease the margin of error.

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Simulated Sample Proportion Who Can Roll Tongue


9.02 Classifying data

Concept summary

Data can be classified in a variety of ways.

Data which can be measured, such as the heights of a group of people, or counted, such as the

number of siblings, is numerical data. Many calculations can be performed on measurable data,

such as finding an average value, or a minimum or maximum value.

Data which is divided into groups (or categories) rather than measured, such as the type of pet( s )

owned by a group of people, is categorical data.

Numerical data

Data that can be measured or counted using numbers.

Categorical data

Data which is divided into groups.

Data can then also be classified by how many characteristics are being measured or counted.

If one characteristic is being measured, such as people’s heights, the data is univariate.

If two characteristics are being compared, such as people’s heights vs. their weights,

the data is bivariate.

Univariate data

Data that measures only one characteristic of a population.

Bivariate data

Data that measures two characteristics of a population.

A data distribution is a graphical or organized display of all data in the data set.

440


Worked examples

Example 1

Determine if the data distribution is univariate or bivariate.

a Data describing the number of years an actor has been working for and their annual earnings.

Approach

To determine whether a data distribution is univariate or bivariate consider whether it describes

one or two characteristics of the group its taken from.

Solution

This data measures two characteristics: number of years the actor has worked for and

annual earnings.

Therefore, this is an example of bivariate data.

b The shoe sizes of an entire football team.

Approach

To determine whether a data distribution is univariate or bivariate consider whether it describes

one or two characteristics of the group its taken from.

Solution

This data measures only one characteristic: shoe size.

Therefore, this is an example of univariate data.

Example 2

Classify the data set as categorical or numerical.

a The time spent, in minutes, driving to work of 100 randomly selected employees.

Solution

Time spent in minutes can take on any numerical value greater than zero. In particular,

we can measure the time taken and perform calculations on the data.

Therefore, this is an example of numerical data.


b The eye color classification of 100 students.

Solution

Eye color classification has qualitative values ( blue, brown, green, hazel, black ) that we can’t

measure or perform calculations on.

Therefore, this is an example of categorical data.

c Number of questions in a test.

Solution

Test questions can take on whole number values, such as 5 or 20, so we can meaningfully count

the number of test questions and perform calculations on the data.

Therefore, this is an example of numerical data.


1 State whether the following are examples of univariate data or bivariate data:

a

b

c

d

e

f

g

A school collects data on the weight of each student on the wrestling team.

A scientist collects data on iron levels in soil and growth of a type of weed in order to

investigate the relationship between them.

A school collects data on the shoe size of each student in the school.

A political group collects data on the taxable income and the addresses of the local

electoral constituents in order to investigate if there is an association.

A researcher collects data on the number of days a group of children spend in

daycare and the number of days spent home sick in order to investigate the relationship

between them.

John wishes to investigate whether there is an association between the amount of

natural sunlight in a classroom and students’ exam results.

Emma wishes to analyze the distribution of test scores from her hockey team.

2 Classify the following examples of data as either numerical or categorical:

a Number of siblings b Letter grades A-F

c Country of birth d Hourly rate of pay

e Length of pencils in millimeters f Weight of dogs in kilograms

g Hair color ( black, red, blonde, etc ) h Time taken to get to school in minutes

i Time spent playing games each day j Satisfaction rating on a 3-point scale:

dissatisfied, neutral, satisfied

442


Let’s practice

3 You’ve decided to collect and analyze data regarding the cafeteria lunches served at

your school.

Describe an example of data you could collect that would be classified as:

a Univariate b Bivariate c Categorical d Numerical

4 Classify the data presented in each display as categorical/numerical and univariate/bivariate:

a

c

Boxplot of Exam Scores

20 30 40 50 60 70 80 90 100

7

6

5

Exam Scores

Average Daily Commute

b

Number of Balls

8

7

6

5

4

3

2

1

0

Soccer

Football Tennis Volleyball

Frequency

4

3

2

1

d

0

16 18 20 22 24 26 28 30

Time (minutes)

Titanic Survival Rates

Survived Did Not Survive Total

First Class Passengers 201 123 324

Second Class Passengers 118 166 284

Third Class Passengers 181 528 709

Total 500 817 1317


e

f

Favorite Color

13%

12%

12%

0 1 2 3 4 5 6 7 8 9 10 11 12

Minutes to Eat Breakfast

19%

6%

8%

30%

Blue

Green

Red

Yellow

Pink

Purple

Orange

g

Stem

3

4

5

6

7

8

9

Leaf

3 0 1 7

2 9

8 1

0 7 9

8

0

h

Accidents per 100 drivers

27

25

23

21

19

17

15

13

11

9

16 17 18 19 20 21 22 23 24 25

Age (years)


5 A principal wants to compare the times spent on homework from students in each grade

level at her high school. Identify the variables and classify the data as either categorical

or numerical. Explain your reasoning.

6 Hurricane wind speeds are measured in miles per hour and then the hurricane is given a

ranking on a scale of 1 to 5. Identify and classify the data types for measuring hurricanes as

either categorical or numerical.

7 Ella is collecting data regarding the change people keep on hand. Describe the difference in

data types among the “type of coins”, “number of coins”, and “total value of pocket change”

and explain your reasoning.

444


9.03 Displaying and

interpreting univariate

data

Concept summary

Data displays are useful to aid with summarizing,

analyzing, and interpreting a data distribution.

The best data displays make useful information

easy to read for the intended audience.

Frequency tables can be used to organize both

categorical and numerical data.

Favorite color Frequency

red 5

blue 3

green 10

yellow 20

Categorical data can be displayed using circle

graphs or bar graphs.

In a circle graph, each piece represents the

percentage or proportion of a category and all

pieces add to 100%.

3%

Methods of Traveling School

11%

7%

9%

Car

Bus

Walk

Bicycle

Train

70%

In a bar graph, the height of each bar represents

the frequency of that category.

8

What type of pet do you have?

6

4

2

0

Fish

Dog Cat Lizard


Numerical data can be displayed using histograms, box plots, line plots, and stem-and-leaf plots.

Histograms and box plots are good for large quantities of data. The shape, center, and spread of

the data is easy to read, but the individual data values are lost in these types of displays.

Histograms display the frequency of data as either a count or relative proportion along the y-axis

and divide the numerical data into bins of equal width along the x-axis.

When constructing a histogram:

• There should be no gaps between the bars

• The bin range includes the value on the left side and excludes the value on the right side

• All bins should be the same width visually and numerically

60

Annual Rainfall (inches)

50

Number of Cities

40

30

20

10

0

1

5 9 13 17 21 25

Rainfall (in)

Generally we exclude the lower bound and include the upper bound, so the equivalent labels for

the x-axis would be:

60

Annual Rainfall (inches)

50

Number of Cities

40

30

20

10

0

[1,5]

[5,9] [9,13] [13,17] [17,21] [21,25]

Rainfall (in)

446


Box plots divide data into four equal quartiles using the five-number summary: minimum,

lower quartile, median, upper quartile, and maximum.

When constructing a boxplot:

• The box in the center represents the middle half and spread of the data

• The line in the box represents the median

• The two endpoints represent the minimum and maximum value

• The total count of data is not included in the box plot

Ages at the Movie Theater

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Since every individual data point must be included in a line plot and stem-and-leaf plot these data

displays are best for smaller data sets.

Line plots display the frequency of data by the number of dots at each value. This display is best

used for discrete values with a small range.

When constructing a line plot:

• The number line must count by equal amounts with no gaps or “breaks”

• If a value on the number line is empty it means no data exist for that value

• Dots or X’s represent each individual data point and should be stacked with equal size.

1 1 1 4

1 1 2

1 3 4

1 2 1 4

2 1 2

2 3 4

3

Stem-and-leaf plots organize data by place value to

compare frequencies and are best for data values with

a larger range.

When constructing a stem-and-leaf plot:

• In this case the “stem” represents the hundreds and tens

digits together and the “leaf” represents the ones digit

• Both the stem column and each leaf row should be in

increasing order

• No stem values can be skipped. If there is no data for that

particular stem, the leaf column will have an empty row.

Stem

10

11

12

13

14

15

16

Leaf

1 4 5 8

0 1 3

7 7 8

0 1 1

0 4

3 6

Key 10 1 = 101


Worked examples

Example 1

Determine the best type of data display( s ) for each set of data.

a In order to determine the popularity of each meal, the number of classmates who like tacos,

pizza, or cheeseburgers the most was collected.

Approach

First consider if the data is categorical or numerical, then consider the information being displayed.

Solution

Circle graph or bar chart would be the best display.

Reflection

Categorical data can be displayed with frequency tables, circle graphs, and bar graphs.

Since we want to see the relative popularity of each meal we can rule out a frequency table as the

best display. Note that a circle graph may not indicate to a viewer the total number of responses or

even the count of each response depending on how its labeled.

b The height all 196 Olympic gymnasts in the most recent summer games.

Approach

First consider if the data is categorical or numerical. Since height is numerical we want to decide

which graphs are easiest to construct based on the size of the data set, and also the range of

its values.

Solution

Histogram or box plot

Reflection

This is a very large data set. It would be inefficient to try and construct a line plot with a dot

representing every individual athlete or even a stem-and-leaf plot with each leaf representing

every individual athlete.

448


Example 2

Use the data display to answer each question.

Mrs. Sanchez’s Period 1 Math Quiz Results

0 10 20 30 40 50 60 70 80 90 100

a Estimate the median score from the math quiz.

Approach

The median is represented by the vertical line in the box of the box plot.

Solution

Around 63%

Reflection

The median is used to describe the center of the data.

b Estimate the range and interquartile range of the data.

Approach

The range of the data is the difference between the maximum and minimum values.

The interquartile range is the difference between the upper and lower quartiles.

Solution

The range is approximately 93% − 20% = 73% and the interquartile range is approximately

70% − 30% = 40%.

Reflection

The range and interquartile range are used to describe the spread of the data.


c What are the disadvantages of this box plot?

Solution

We don’t know how many students are in Mrs. Sanchez’s class and we can’t see any individual

scores from the quiz.

d What are the advantages of a box plot?

Solution

Box plots visually summarize large sets of data although they can be used for small sets too.

In a box plot it is easy to see and estimate the shape, center, and spread of data.


1 The frequency table shows the data distribution

for the length of leaves collected from a

species of tree in the botanical gardens.

Determine whether the following statements

are correct:

a

b

c

d

35 leaves less than 60 mm were collected.

11 leaves of less than 40 mm were

collected.

The most common leaf length is between

60 and 80 mm.

Leaves are more likely to be at least 60 mm in length.

Leaf length, ( mm )

Frequency

0 ≤ x < 20 5

20 ≤ x < 40 11

40 ≤ x < 60 19

60 ≤ x < 80 49

80 ≤ x < 84 43

2 The stem-and-leaf plot below shows the age of people to

enter through the gates of a concert in the first 5 seconds.

a

b

c

d

Find the number of people who passed through the gates

in the first 5 seconds.

Calculate the range of ages represented in the plot.

Find the mean and median age.

Find the proportion of concert-goers who were under

20 years old.

1

2

3

4

5

Leaf

1 2 4 5 6 6 7 9 9

2 3 5 5 7

1 3 8 9

8

Key 1 2 = 12 years old

450


3 Sophia is a nurse and picks up shifts when

they are available. She used a line plot to keep

track of the number of shifts she did each

week for a number of weeks.

a

b

In this line plot, what does each

dot represent?

What was the most frequently occurring

number of shifts per week?

3 4 5 6 7 8

Shifts per Week

4 20 people were asked how many hours of sleep they had gotten the previous night.

The numbers below are each person’s response:

1, 6, 9, 8, 7, 9, 7, 10, 2, 3, 8, 7, 7, 3, 7, 3, 3, 7, 10, 9

Select the appropriate frequency table to represent the given set of data:

A

Frequency

Hours of Sleep

B

Hours of Sleep

Frequency

1 1

2 2

3 6

6 7

7 13

8 15

9 18

10 20

1 1

2 1

3 4

6 1

7 6

8 2

9 3

10 2

C

Hours of Sleep

Frequency

D

Frequency

Hours of sleep

1 1

1 2

4 3

1 6

6 7

2 8

3 9

2 10

1 1

1 3

4 6

1 12

6 19

2 27

3 36

2 46


5 Noah asked his students to choose their favorite method of traveling. 9 students picked car,

13 people picked plane, 2 picked train, and 5 people picked boat. Select the appropriate

frequency table to represent the given set of data:

A

Mode of Transport

Frequency

B

Frequency

Mode of Transport

Car 9

Plane 13

Train 2

Boat 5

Car 9

Plane 13

Train 2

Boat 5

C

Mode of Transport

Frequency

D

Frequency

Mode of Transport

Car 9

Plane 22

Train 24

Boat 29

Car 9

Plane 22

Train 24

Boat 29

6 The table shows the team points earned by four different houses at their swimming competition:

Team Yellow Blue Red Green

Points 15 25 40 60

Select the appropriate graph to represent the given set of data:

A Stem-and-leaf plot B Histogram

C Circle graph D Scatter plot

7 The amount of snowfall ( in centimeters ) is recorded at the base of a mountain each day. The

results are given below:

6, 2, 0, 3, 2, 2, 3, 4, 2, 0, 3, 2, 3, 4, 6, 4, 3, 0, 5, 3

Select the appropriate graph to represent the given set of data:

A Circle graph B Scatter plot

C Segmented bar charts D Histogram

8 A group of people were surveyed on

the number of websites they visit on a daily basis, and

the results are displayed in the given line plot:

Construct a frequency table for this data.

Number of Websites Visited Daily

0 1 2 3 4 5 6

452


9 The CDC surveyed some random people, asking them to estimate approximately how much

time they spent in the sun during the previous summer. The frequency histogram shows

their responses.

Frequency

10

9

8

7

6

5

4

3

2

1

0

10

20 30 40 50 60 70 80 90

Hours in the Sun

a

b

Find the number of people who were surveyed.

Find the proportion of responders who spent at least 60 hours in the sun.

10 Mr. Price recorded his students’ favorite book genre in the bar graph shown.

Number of Students

12

11

10

9

8

7

6

5

4

3

2

1

0

Drama

Favorite Book Genre

Action Sci-Fi Detective

Genre

a

b

c

Which book genre is most popular?

Which book genre is least popular?

How many more students prefer Detective books than Drama books?


Let’s practice

11 Aviation glass is manufactured to be 12 mm thick. If the glass is too thin ( less than 11.5 mm )

or too thick ( more than 12.5 mm ) the aircraft will fail its glass safety check.

A regional airport conducts a glass safety check of all aircraft landing on their runways over a

one month period and displays the results in the line plot.

Glass Thickness (in mm)

10.5 11.0 11.5 12.0 12.5 13.0

Find the number of aircrafts that failed their glass safety check. Explain.

12 64 people are surveyed to find their shoe sizes, and the results are shown in the table.

Shoe Size 9 10 11 12

Number 3 8 6 4 1 7 7

Construct a line plot for this information.

13 John had 7 blue marbles, 4 black marbles, 8 yellow marbles, 13 white marbles, and 6 red

marbles. Construct a bar chart for this information.

14 Describe the differences between a histogram and a bar graph.

15 Mr. Rodriguez recorded the number of pets of his students. He found that 15 students

had no pets, 19 students had only one pet, 3 students had two pets and 8 students had

three pets.

a

Decide the best type of display for the data and justify your choice.

b Construct the display chosen in part ( a ).

454


16 This stem-and-leaf plot records the ages of customers at a

beachside cafe last Sunday. Use the stem-and-leaf plot to

construct a histogram for the data.

1

2

3

4

5

Leaf

1 5 9

2 3 5 6 7 9

0 1 2 4 6 7

1 3 4 5 8 9

1 4 7

6 2 4 5 6 7

Key: 1 2 = 12

17 Miss Yen recorded her students’ favorite pets

in the bar graph.

Number of Students

12

11

10

9

8

7

6

5

4

3

2

1

0

Student’s Favorite Pets

Hamster Dog Cat Fish Bird

Pets

Create a circle graph that displays the same data.


18 Consider the data in the frequency table.

a

Decide the best type of display for the data in

the frequency table and justify your choice.


Score Frequency

11 12

12 9

13 10

14 9

15 6


19 Bob asked his students to choose their favorite toy. 35% students picked cars, 10% of the

students picked planes, 30% picked balls and 25% picked video games.

a

Decide the best type of display for the data and justify your choice.


20 An online booking service allows customers to book tickets to see performers in concert.

Over the last month the service has been tracking how many tickets are sold in the first five

seconds after they become available, and the results were:

10, 10, 12, 12, 13, 14, 16, 18, 20, 24, 26, 27, 28, 56

Determine if each plot type could be used to display this data and describe the advantages

or disadvantages of each.

a Line plot b Stem-and-leaf plot

c Histogram d Circle graph

21 The cooking show Iron Hams has a panel of 30

judges. Each judge gives a rating from 1 and 5.

The line plot shows the results given to the winning

contestant from last year.

a

b

Explain the advantages of using a line plot to

display the data.

Explain the advantages or disadvantages of

using a stem-and-leaf plot to display this data.

Contestant Rating

1 2 3 4 5

22 The circle graph represents the popularity of each chip brand amongst high schoolers.

a Interpret the graph. List all information you know to be true.

b What are the disadvantages to using a circle graph for categorical data?

Tortillos

Snakis

Hot Cheezos

Wrinkles

456


9.04 Two-way

frequency tables

Concept summary

A two-way frequency table, or joint frequency table, can be used to summarize bivariate

categorical data. It is a data display that shows how many data values fit into multiple categories.

Marginal frequency

How often a particular category of data occurred, found in the “total” row and column of

a two-way frequency table

Joint frequency

In a two-way table, joint frequency is the number of times a combination of two

conditions occurs

Child

Adult

Total

Drawing

22 8 30

Painting

Graphics

18 7 25

9 36 45

Add

down

Total

49 51 100

Joint frequencies

Add across

Marginal frequencies


Worked examples

Example 1

The two-way frequency table displays the number of people at Chili Fest with which kind of chili

they bought and whether they added extra spice or not.

Added spice Did not add spice Total

Meat 331 401 732

Vegetarian 125 43 168

Total 456 444

a Describe the group of people which has exactly 401 people in it.

Approach

We should find 401 in the table and look up and across to identify the corresponding headings.

Solution

The number 401 lies in the row for people who chose meat and in the column for people who did

not add spice. This means that there were 401 people who chose meat chili and did not add spice.

b Identify whether the number of people who chose vegetarian chili is a marginal or joint

frequency and state its value.

Approach

It is not specified whether this group added extra spice or not, so we are looking at all people who

chose the vegetarian chili.

Solution

Since we are looking at the total number of people who chose vegetarian chili,

this is a marginal frequency.

There are 168 people who chose the vegetarian chili.

458


c Calculate the total number of people who ate chili at Chili Fest.

Approach

We can find this total in a variety of ways:

• Add all the joint frequencies

• Add the marginal frequencies for the type of chili

• Add the marginal frequencies whether they added spice or not

Solution

Let’s add the marginal frequencies for the type of chili.

Meat + Vegetarian = 732 + 168 = 900

There were 900 people who ate chili at Chili Fest.

Reflection

We can check this by adding up all of the joint frequencies.

331 + 401 + 125 + 43 = 900

Example 2

A scientist recorded some data on the flowering pattern and type of soil for a variety of

coreopsis plants:

• 1000 plants were observed

• 136 plants did not flower

• 394 plants that did flower were planted in peat soil

• 30 plants that were planted in sandysoil did not flower

The data can be organized into a joint frequency table:

Peat Soil

Sandy soil

Total

Flowered Did not flower Total


a Complete the table based on the given information about coreopsis plants.

Approach

We can first fill in the given information and then subtract joint frequencies from the marginal

frequencies to get the missing joint frequencies and subtract marginal frequencies from the total to

get the missing marginal frequencies.

Solution

Peat Soil 394


Sandy soil 30

Total 136 1000

Based on the information we have been given, we can already determine:

• Marginal frequency for number that flowered: 1000 − 136 = 864

• Joint frequency: 136 − 30 = 106


Peat Soil 394 106

Sandy Soil 30

Total 864 136 1000

Based on the new calculated information, we can determine:

• Marginal frequency for number planted in peat soil: 394 + 106 = 500

• Then the marginal frequency for number planted in peat soil: 1000 − 500 = 500

• Joint frequency for plants that flowered in sandy soil: 864 − 394 = 470


Peat Soil 394 106 500

Sandy soil 470 30 500

Total 864 136 1000

Reflection

We can always check by adding across each row and down each column to ensure the marginal

frequency is the sum of the joint frequencies.

460


b Keshawn said because there were a total of 1000 plants observed and 30 plants that were

planted in sandy soil did not flower, this means 1000 − 30 = 970 were planted in peat soil and

did flower. Identify and correct his error.

Solution

Keshawn seems to have a misconception around how joint frequencies relate to one another. He

has stated that

while it should be

sandy and flowered + peat and not flowered = Total

sandy and flowered + sandy and not flowered + peat and flowered +

peat and not flowered = Total

We need to consider all possible cases, not just opposite cases.

He should have said that because there were a total of 1000 plants observed and 30 plants that

were planted in sandy soil did not flower, this means 1000 − 30 = 970 were planted in peat soil

or were planted in sandy soil did not flower.


1 The two-way frequency table displays the number of deaf-blind children in each state and

their age group.

0-5 years 6-17 years Total

Florida 63 388 451

Alabama 52 143 195

Georgia 56 193 249

Total 171 724 895

For each of the following groups:

i

ii

Classify as either a joint frequency or a marginal frequency.

Find the value.

a

b

c

d

The number of deaf-blind children in Florida

The number of deaf-blind 0 to 5 year olds

The number of deaf-blind 6 to 17 year olds in Alabama

The number of deaf-blind children in Florida who are more than 5 years old.


2 Use the information given in the table to determine whether the following statements are

true or false:

Eighth Grade Ninth Grade Total

1.5 Hours or More 7 28 35

Less than 1.5 Hours 18 6 24

Total 25 34 59

a

b

c

d

Most students in general do homework for less than 1.5 hours per night.

Most eighth graders do homework for 1.5 hours or more per night.

There were more students surveyed who do homework for 1.5 hours or more per night

than there were who do less than 1.5 hours per night.

There were more eighth graders surveyed than ninth graders.

3 The following table gives information on the treatment plans prescribed to cats with feline

diabetes. Depending on the severity of the case, cats are either prescribed a special diet or

insulin injections.

Special Diet Insulin Injections Total

Ventian Pet Clinic 14 14 28

Mohawk Animal Clinic 11 9 20

Total 25 23 48

a

b

c

d

Find the number of cats from Venetian Pet Clinic that were prescribed with insulin.

Find the number of cats from Mohawk Animal Clinic that were prescribed with insulin.

Interpret the 28 in this context.

Interpret the 11 in this context.

Let’s practice

4 Use the given information to complete the two-way frequency table:

a

The following table gives information about the number of sales of each type of shoe at

Foot Swag, a shoe store in the mall.

Running

Shoes

Basketball

Slides

Work Boots

Total

Teenagers 84

Adults 43 13 29 85

Total 85 36 48 169

462


b

Laura asked some people in her community whether they were vegetarian or not.

27 responders said they were vegetarian, of which 8 were children. 10 children said

they were not vegetarian, and 19 adults said they are not vegetarians.

Not vegetarian vegetarian Total

Children

Adults

Total

c

Some students in Ms. Rochelle’s class were polled on how many siblings they have.

The results are provided in the following table:

1st Period 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3

2nd Period 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 7

3rd Period 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 7

Less Than Two

Siblings

Two or More

Siblings

Total

Period 1 Period 2 Period 3 Total

d

Students were asked if they prefer chocolate ( C ) or vanilla ( V ) ice cream. Each of the

students was also asked if they prefer sprinkles ( S ) or hot fudge ( F ) as an ice cream

topping. The results are displayed in the tree diagram:

12

S

C

30

18

F

41

29

S

V

12

F


Sprinkles Hot Fudge Total

Chocolate

Vanilla

Total

5 A scientist recorded some data on the diet of two types of squirrels observed in

the Rocky Mountains:

• 200 squirrels were observed

• 34 Red Squirrels observed eat a diet of seeds

• 128 squirrels observed were Pine Squirrels

• 106 squirrels observed eat acorns

a

Complete the following table based on the given information about squirrels:

Seeds

Acorns

Total

Red Squirrel Pine Squirrel Total

b

c

d

Compare the total number of Red squirrels to the number of Pine squirrels.

Compare the number of squirrels who were observed eating seeds to the number

of squirrels eating acorns.

Compare the number of Red squirrels observed eating seeds to the number of Pine

squirrels observed eating acorns.

6 A poll was given where people were asked: “Do you like pineapple on your pizza?”

The following information was recorded:

• 40 people responded to the poll

• 14 adults responded “yes”

• 16 of the people who responded were adults

• The number of people who responded “yes” was four times the number of people who

responded “no”

Find and explain the error made in each statement.

a

b

c

Since 14 adults responded yes, there were 40 − 14 = 26 adults who responded no.

There were 14 adults who said yes and 16 of the people were adults so the total of

adults was 30

The number of people who responded “yes” was four times the number of people who

responded “no” so we can take the total number of people, 40, and divide by 4 to find

the number of people who said no.

464



7 Determine if each statement is true or false. Justify your response.

a

b

The marginal frequencies for the columns must have the same sum as the marginal

frequencies for the rows.

Each of the joint frequencies in a single row can be equal to the marginal frequency of

that row.

8 Lucinda sets out to see how many of her classmates use the term “soccer” or “football” to

describe her favorite sport. Lucinda surveys her classmates from her first three periods.

Period 1

Period 2

Period 3

Total

Football Soccer Total

Complete the frequency table so that it meets the conditions:

• None of her class periods have the same number of students.

• Overall, most students use the term “soccer”.

• One of Lucinda’s class periods is equally divided between “soccer” and “football”.

• The number of students who prefer “football” in one class period was equal to the number

of students who prefer “soccer” in a different class period.


9.045 Honors: Two-way

relative frequency tables

Concept summary

The joint relative frequency is the ratio of a joint frequency to the total number of data points

and the marginal relative frequency is the ratio of the marginal frequency to the total number of

data points. When these relative frequencies are displayed in a two-way table we call it a two-way

relative frequency table.

Relative frequency table

Child

Adult

Total

Drawing

22% 8% 30%

Painting

Graphics

18% 7% 25%

9% 36% 45%

Add

down

Total

49% 51% 100%

Joint frequencies

Add across

Marginal frequencies

A two-way relative frequency table shows the proportion or percentage of each entry out of

the total number of data points.

Conditional relative frequency is the ratio of a joint relative frequency to a marginal

relative frequency.

Row-relative frequency

A relative frequency table in which all entries are divided by their row total

466


Column-relative frequency

A relative frequency table in which all entries are divided by their column total

Child

Adult

Drawing 44.9% 17.3%

Painting 36.7% 13.5%

Graphics 18.4% 69.2%

Total 100% 100%

We can represent a row-conditional relative or column-conditional relative frequency table as a

segmented bar chart for a more visual display of the same information.

Graphics

Painting

Drawing

100%

80%

60%

40%

20%

0%

Child

Adult

Using a tree diagram or relative frequency table, we can look at the different proportions of

a population that test positive and negative for a characteristic or condition compared to those

who actually have the characteristic or condition. In particular, we can identify the likelihood of

a false positive or false negative.

False positive

A test result which incorrectly says that a particular condition or attribute is present, when it is

should have said it was negative

False negative

A test result which incorrectly says that a particular condition or attribute is absent, when it is

should have said it was positive


Worked examples

Example 1

This two-way table represents survey results from a sample of college students regarding their

preferred place to study:

Library

Student

Union

Coffee Shop Home Total

Sociology Majors 43 21 59 22 145

Math Majors 34 16 15 41 106

Total 77 37 74 63 251

a Construct the relative frequency table for the data.

Approach

Relative frequency is found by dividing each entry in the table by the total population, which in this

case is 251 students.

Solution

Library

Student

Union

Coffee Shop Home Total

Sociology Majors

Math Majors

Total

Reflection

The joint relative frequencies may not appear to equal the marginal relative frequencies due to

rounding error.

468


b Construct the row-relative frequency table for the data.

Approach

Row-relative frequency is a conditional frequency found by dividing each entry in the table by

its row marginal frequency.

Solution

Library Student Union Coffee Shop Home Total

Sociology Majors

Math Majors

Total

Reflection

In row-relative frequency tables the final column will always be 100%.

c Construct the column-relative frequency table for the data.

Approach

Column-relative frequency is a conditional frequency found by dividing each entry in the table by

its column marginal frequency.

Solution

Sociology Majors

Library Student Union Coffee Shop Home Total

Math Majors

Total

Reflection

In column-relative frequency tables the final column will always be 100%.


Example 2

This two-way table represents the number of traffic violations in Portland, Oregon during the past

three weeks:

Speeding Texting While Driving Total

Teenagers Drivers 16 80 96

Adult Drivers 49 15 64

Total 65 95 160

a Determine which traffic violation is committed most by teenage drivers and what proportion of

the tickets for teenagers were of that type.

Approach

Teenage drivers represents a row category of data so we need to divide the joint frequencies in

the first row by the marginal frequency of the first row. This is called row-relative frequency.

Solution

Speeding violations represent

represents

teenage drivers.

Reflection

of teenage traffic violations and texting while driving

of teenage traffic violations. Texting while driving is committed the most by

Carefully consider whether the question is answered by a relative frequency, row-relative

frequency, or column-relative frequency before answering.

b Find the percentage of all speeding violations that are commited by adult drivers.

Approach

Speeding violations represent a column category of data so we need to divide the joint frequency

of an adult committing a speeding violation by the marginal frequency of speeding. This is called

the column-relative frequency.

Solutions

Adults commit

of speeding violations.

470


c Explain whether or not there is a relationship between age and type of traffic violation.

Justify your conclusion.

Approach

We can use the row-conditional or column-conditional relative frequency tables to help with this.

However, a segemented bar chart can help us to justify in a more visual way.

Solution

There is a relationship between age and traffic violation.

We can see that in both the row-relative and column-relative

frequency tables that the values for the same variable are

quite different. However, we can see this more easily in

the two segmented bar charts as we can see that the the

proportions are significantly different for the different ages

and violations.

Speeding

Texting While

Driving

Total

Teenage Drivers 16.7% 83.3% 100%

Adult Drivers 34.0% 66% 100%

100%

80%

60%

40%

20%

0%

Texting while driving

Teenage

Drivers

Adult

Drivers

Speeding

Speeding

Texting While

Driving

Teenage Drivers 24.6% 45.7%

Adult Drivers 75.4% 54.3%

Total 100% 100%

We can see that teenagers are more likely to be ticketed

for texting than speeding, and adults are more likely to

be ticketed for speeding than texting.

Adult Drivers

100%

80%

60%

40%

20%

Teenage Drivers

0%

Speeding

Texting while

Driving



1 A group of visitors to Busch Gardens were asked whether they brought a reusable water

bottle or purchased plastic water bottles during their visit. The table summarizes the results:

FL Resident Non-FL Resident Total

Brought Reusable Water Bottle 20 28 48

Purchased Plastic Water Bottles 11 26 37

Total 31 54 85

Identify each as marginal frequency, joint frequency, marginal relative frequency or joint

relative frequency.

a

b

c

d

The number of FL residents who purchased plastic water bottles.

The number of FL residents in the group.

The proportion of people in the group who brought resuable water bottles.

The percentage of non-FL residents who brought a reusable water bottle on the trip.

2 The following relative frequency table shows the percentage of overall sales given by style

of shoe and age category of customer.

Running Shoes

Basketball

Slides

Work Boots

Total

Teenagers 24% 15% 12% 51%

Adults 25% 7% 17% 49%

Total 49% 22% 29% 100%

a

b

Find each relative frequency:

i The proportion of sales that went to adults.

ii The probability that a randomly selected sale will be slides.

iii The most common sale by age and style.

Interpret each relative frequency in context.

i 29% ii 15% iii 51%

472


3 William started a cupcake business and has been taking orders for events in his area.

The two-way frequency table shows his orders ( in dozens of cupcakes ) for the next month.

Birthday

Parties

Weddings

Family

Reunions

Business

Events

Tres Leches Cupcakes 29 18 21 7 75

Chocolate Strawberry

Cupcakes

Total

24 6 26 6 62

Total 53 24 47 13 137

Find the relative frequencies:

a

b

c

d

The relative frequency of each event type.

The proportion of events that are weddings with tres leches cupcakes.

The percentage of chocolate strawberry cupcakes that will be going to birthday parties

rounded to the nearest percent.

The probability that the cupcakes ordered were tres leches, given that they were

ordered for a family reunion.

4 The following two-way table represents percent frequencies from a survey of students on

their favorite science experiment.

Bacteria

Culture Lab

Frog Dissection

Lab

DNA Testing

Lab

Total

9th Grade Students 28% 41% 31% 100%

10th Grade Students 56% 3% 41% 100%

Total 51% 9% 40% 100%

State whether the following statements are true or false:

a

b

c

More than 50% of all students favor the bacteria culture lab.

More than 20% of all students favor the DNA testing lab.

More than 60% of all students favor the bacteria culture lab.

5 Consider the table showing the testing results for an infectious disease:

Has the disease

Does not have

the disease

Total

Positive Test 7 4 11

Negative Test 3 86 89

Total 10 90 100

a

b

c

Calculate the percent of the population that actually have the disease.

Calculate the percent of people who are receiving accurate test results from this test.

Mei has been suffering symptoms typical of this disease and decides to get tested.

If her test result came out negative, what is the probability that she still has the disease?


Let’s practice

6 The following two-way table represents survey results from a sample of people regarding

their favorite burger at a local fast food restaurant:

Classic

Hamburger

Bacon

Cheeseburger

California

Burger

Veggie

Burger

Total

High School Students 29 16 17 22 84

College Students 34 28 21 22 105

Total 63 44 38 44 189

i

ii

a

b

c

Find the calculation that justifies each conclusion.

Classify the percentage as relative frequency, row-relative frequency, or column-relative

frequency.

46% of people who prefer a classic hamburger are high school students.

15% of people surveyed are college students who chose the bacon cheeseburger as

their favorite burger.

20% of all participants prefer the California Burger.

7 Luke and Gwen each asked their classmates, “What is your favorite club on campus?”

The results of the survey are in the two-way table below.

Anime Club Math Club Dance Club Total

Eleventh Grade 21 17 8 46

Twelfth Grade 17 8 21 46

Total 38 25 29 92

a

Complete the following table by providing the relative frequencies of each row:

Anime Club Math Club Dance Club Total

Eleventh Grade 100%

Twelfth Grade 100%

Total 100%

b

c

Whose observation is correct based on the relative frequency of row table?

• Gwen says that 17% of people in eleventh grade prefer dance club.

• Luke says that 17% of people who prefer dance club are in eleventh grade.

Whose observation is correct based on the relative frequency of row table?

• Gwen says that 37% of people who prefer anime club are in twelfth grade.

• Luke says that 37% of people in twelfth grade prefer anime club.

474


8 Neville’s class is having a party to celebrate the end of the term. He decided to poll his

classmates on what kind of party they wanted, as well as what time of day to have the party.

The results are given in the two-way table.

Pizza Party Ice Cream Party Total

During Lunch 8 5 13

After School 7 13 20

Total 15 18 33

a

Complete the following table by providing the relative frequencies of each column:

Pizza Party Ice Cream Party Total

During Lunch

After School

Total 100% 100% 100%

b

c

Find the percentage of students who want to have a pizza party after school.

State whether the following observations can be made based on the column-relative

frequency table:

i

ii

iii

iv

28% of students who want the party during lunch would like an ice cream party.

61% of all students want the party to be after school.

39% of all students want the party to be after school.

62% of students who want the party during lunch would like a pizza party.

9 Aaron made a game where he flips a coin and then pulls

a marble from a bag. When he flips the coin he either gets

heads ( H ) or tails ( T ). When he pulls a marble from the

bag he can either get green ( G ) or orange ( O ). Aaron ran

300 simulations of his game and recorded the following:

• Given that he flipped heads,

pulled an orange marble

• He pulled a green marble in

of the time he

of all trials

H

G

O

G

a

b

• He flipped heads in of all trials

Fill in the missing values of the tree diagram.

Complete the row-conditional relative frequency table.

T

O

Green Orange Total

Heads 1

Tails 1

Total 1


c

Complete the column-conditional relative frequency table.

Green Orange Total

Heads

Tails

Total 1 1 1

10 The segmented bar graph shows the number of Democrats and Republicans who support

and oppose increasing the minimum wage. Use the data in the segmented bar graph to

complete the row-conditional relative frequency table.

100

90

80

70

60

50

40

30

20

29

28

26

4

10

27

59

Strongly oppose

Oppose

Favour

Strongly favor

10 17

0

Republican

Group

Democrat

Strongly Oppose Oppose Favor Strongly Favor Total

Republican 100%

Democrat 100%


11 The following two-way table represents survey results from a sample of people regarding

their preferred method of streaming shows:

Television Laptop Cell Phone Tablet Total

Teenagers 22 44 64 30 160

Adults 48 29 18 30 125

Total 70 73 82 60 285

State whether the following statements are true or false and explain your reasoning:

a

b

c

d

The percent of adults surveyed is greater than the percent of teenagers surveyed.

The percent of adults who stream shows on tablets out of all adults is less than the

percent of teenagers who stream shows on tablets out of all teenagers.

Teenagers make up a greater proportion of laptop users than they do of tablet users.

The percent of teenagers who prefer to stream shows on televisions is less than the

percent of adults who prefer to stream shows on televisions.

476


12 The following two-way table represents a survey taken by dog owners about what kind of

food they give their dogs.

Dry

Kibble

Canned Dog

Food

Cooked Meat and

Vegetables

Total

Puppy 24 10 8 42

Adult Dog 36 30 52 118

Total 60 40 60 160

a

Explain the meaning of each relative frequency in context.

i

ii

b

Remy is analyzing the table for trends and concludes that adult dogs receive cooked

meat and vegetables over two times more than puppies receive dry kibble because

52 is more than double 24. Explain the error in their reasoning.

13 Determine if the statement is true or false. Justify each response.

a

b

If two groups have equal relative frequencies, their joint frequencies must be equal.

In a row-conditional relative frequency table, each element in the column must add up to

the proportion in the column total.

c The joint relative frequencies in a relative frequency table will always add up to 100%.

14 Although useful, allergy tests are not always 100% accurate. Suppose that a dust allergy test

gives false negatives 20% of the time and false positives 10% of the time.

Opehlia decides to take the test and gets a positive result. If 6% of the population have

a dust allergy what is the probability that Ophelia actually has the dust allergy? Construct

a two-way frequency table to justify your response.

15 The segmented bar graph

displays the sentencing type

for drug-related offenses of

white men and black men in

2006. Analyze the data shown

in the graph and use relative

frequencies to investigate

whether there are significant

differences in sentencing

by race.

Relative Frequency

100.0%

90.0%

80.0%

70.0%

60.0%

50.0%

40.0%

30.0%

20.0%

10.0%

0.0%

Sentence Type for Drug Offenses by Race

3%

4%

34%

26%

30%

30%

40%

33%

White

Black

Group

Other

Probation

Jail

Prison


9.05 Associations in

bivariate data

Concept summary

When looking at bivariate data, it can often appear that the two variables are correlated.

Correlation

A relationship between two variables.

It is important to be able to distinguish between causal relationships ( when changes in one

variable cause changes in the other variable ) and non-causal relationships.

Causation

A relationship between two events where one event causes the other.

To do so, we make sure to look at all aspects of an association between two variabels.

Association

A way to describe the form, direction or strength of the relationship between the two variables

in a bivariate data set.

For categorical data, we can describe an association as positive or negative, as well as whether the

association is strong or weak ( or if there is no association ).

There are more ways to describe an association for numerical data, since we can measure and

perform calculations on this type of data. Descriptions can include whether a relationship is linear

or nonlinear, whether a relationship is positive or negative ( whether one variable is increasing or

decreasing compared to the other ), as well as whether an association is strong or weak ( or if there

is no association ).

Worked examples

Example 1

Determine whether the following statement is true or false:

“There is a causal relationship between number of cigarettes a person smokes and

their life expectancy”

478


Approach

It is generally understood that smoking cigarettes can cause disease such as cancer, which is know

to have an effect on life expectancy.

Solution

True

Reflection

The evidence of a causal relationship usually comes from generally accepted truths or

verified research studies. A causal relationship is not confirmed by finding an association

between variables.

Example 2

Mayra surveyed some people on their way into Adventure Island Water Park. She asked the

participants whether they arrived in a carpool or drove alone to the park, and whether they

preferred the Lazy River or Raging Rapids.

Lazy River Raging Rapids Total

Carpooled to water park 25 16 41

Drove alone to water park 25 20 45

Total 50 36 86

Describe the association, if any, between how people arrived to the water park and their preferred

water park attraction activity. Justify your response.

Approach

The ratio of values in each column of the table can indicate the type of association

( either positive or negative ) and the strength of the association ( weak, strong, or no association ).

Solution

The ratio of joint frequencies for transportion of people who prefer the Lazy River is 1 : 1 and

the ratio of joint frequencies for people who prefer Raging Rapids is 4 : 5 indicating that there is

a weak positive association between carpooling to the water park and having a preference for

the Lazy River.



1 Describe the association between the following pairs of variables as positive,

negative or no association:

a

b

c

d

e

f

Height and weight

Rainfall and traffic accidents

Temperature and ice cream sales

Time spent studying and exam performance

Temperature and the number of heaters sold

Age of a person and how funny they are

2 Determine whether the following statements are true or false:

a

b

c

d

There is a causal relationship between the number of times a coin has landed on heads

previously, and the likelihood that it lands on heads on the next flip.

There is a causal relationship between the amount of weight training a person does and

their strength.

There is a causal relationship between whether a person vapes and their risk of

respiratory disease.

There is a causal relationship between the number of wins of a professional football

team and fluctiations in the stock market.

3 For the following pairs of variables, state whether one variable has a direct effect on

the other:

a

b

c

Hair length and gender

Marital status and film preference

Height and number of children

d Number of pets and time spent caring for pets ( hours )

e

f

Time spent to finish novel and number of pages in the novel

Time spent training and athletic performance

4 In a study on houses, it was found that an explanatory variable was the amount of

money spent on renovations. State whether each of the following could be a possible

response variable:

a

b

c

d

The number of home renovations in the same suburb.

The amount of money the house sells for.

The number of people living in the home.

The work salary of the home owners.

480


Let’s practice

5 A study found a strong positive association between the temperature and the number of

drownings. Bradyn claims that the rise in temperature is what causes the number of beach

drownings to rise. Explain Bradyn’s error.

6 Debbie is a truck driver who drives long distances to deliver products across the country.

She worries about the negative effects of driving long distances and decides to collect data

on the average monthly mileage driven and the blood pressure for a sample of truck drivers.

Debbie found a strong positive association between average mileage and blood pressure.

a

b

c

Determine what conclusions Debbie can make based on this association.

Explain what conclusions Debbie should not make based on this association.

Provide a reasonable explanation for the strong association that Debbie found in

her data.

7 The following data was taken from a sample of American adults:

Diabetes No Diabetes Total

Excessive Thirst 51 49 100

No excessive thirst 34 66 100

Total 85 115 200

a

b

c

Out of the Americans who experience excessive thirst, find the percent that also

have diabetes.

Out of the Americans who experience excessive thirst, find the percent that do not

have diabetes.

Based on the survey, is there an association between experiencing excessive thirst and

having diabetes? Explain.

8 The following data shows the proportion of people who tested positive or negative on a drug

test and whether or not they were a drug user.

Drug User

Not a Drug User

Positive Drug Test 0.86 0.10

Negative Drug Test 0.14 0.90

Total 1.00 1.00

a

b

c

Find the proportion of people who are not drug users that still test positive.

Find the proportion of people who are not drug users that test negative.

Based on the survey, is there an association between not using drugs and testing

positive? Explain.



9 Quiana surveyed some people on their way into Yosemite National Park on whether they

prefer hiking or rock climbing.

41 responders entered the park by car, of which 25 prefer hiking. 45 people entered the

park by bus, of which 20 said they prefer rock climbing.

a

Complete the following two-way table based on the results of Quiana’s survey:

Hiking

Rock

Climbing

Total

Arrived by Car

Arrived by Bus

Total

b

Describe the association, if any, between transportation mode and preferred activity. Use

data in the table to justify your response.

10

Allergic to

Nuts

Not Allergic

to Nuts

Total

Allergic to Diary

Not Allergic to Diary

Total 30

If 30 students were asked whether or not they were allergic to nuts and dairy, construct a

two-way frequency table for each condition:

a

b

c

No association between dairy and nut allergies.

A strong positive association between having a nut allergy and having a dairy allergy.

A weak negative association between having a nut allergy and having a dairy allergy.

11 Determine whether each statement is true or false. Provide an example to support

your claim.

a

b

c

If event A has a causal relationship with event B, then there is a correlation between

event A and event B.

If there is a correlation between event A and event B, then there is a causal relationship

between event A and event B.

If event A is positively associated with event B, then there is a causal relationship

between event A and event B

482


9.06 Scatter plots and

lines of fit

Concept summary

Functions can be used to model real-world events and interpret data from those events. Data that

measures or compares two characteristics of a population is known as bivariate data.

When analyzing and interpreting data, we often look for a relationship between two variables

called an association. We can describe an association according to the form, direction, or strength

of the relationship between the two variables.

For numerical data, descriptions include:

• linear or nonlinear

• positive or negative

• strong or weak

For categorical data, descriptions include:

• strong or weak

To more easily analyse a set of data and determine if there is an association between the variables,

we often construct a graph of the data, known as a scatter plot. On a scatter plot, we can then draw

a line that best estimates the relationship between two sets of data, called a line of fit or trend line.

Worked examples

Example 1

A scatter plot showing the median value of property sales made in the United States is shown below.

400

Value (thousands of $)

350

300

250

200

150

2010

2012 2014 2016 2018 2020 2022

Year


a Sketch an approximate line of best fit for the scatter plot.

Approach

The line of best fit should go through the data points in such a way that there are roughly the

same amount of data points on side of the line and the distance between the points and the line

is minimized.

Solution


400

350

300

250

200

150

2010

2012 2014 2016 2018 2020 2022

Year

b Use the line of best fit to predict the median housing price in 2022.

Solution

Looking at the graph we sketched in part ( a ),

the line of best fit approximately passes through

the point (2022, 365 ).

400

350


300

250

200

150

So we can predict that the median housing price in

the U.S. in 2022 will be around $365 000. 2010

2012 2014 2016 2018 2020 2022

Year

484



1 The average monthly temperature and the average wind speed in a particular location was

plotted over several months. The graph shows the points for each month’s data and their line

of best fit.

Use the line of best fit to approximate the wind speed on a day when the temperature is 5 ºC.

8

Wind speed (knots)

7

6

5

4

3

2

1

2

Temperature( 0 C)

4 6 8 10

2 Scientists conducted a study to see people’s reaction times after they’ve had different

amounts of sleep. The data is shown in the table.

Number of hours sleep ( x ) 1.1 1.5 2.1 2.5 3.5 4

Reaction time in seconds ( y ) 4.66 4.1 4.66 3.7 3.6 3.4

The data is shown with the line of best fit.

a

b

What would be the reaction time for

someone who has slept 5 hours?

Predict the number of hours someone

has slept if they have a reaction time of

4 seconds.

5

4

3

y

2

1

1

2 3 4 5

x


3 Consider the scatter plot:

a

Will the slope of the line of best fit be

positive or negative?

50

45

y

b

c

Estimate the y-intercept of the line of

best fit.

Estimate the slope of the line of best fit.

40

35

30

25

20

15

10

5

x

2

4 6 8 10

4 Consider the scatter plot:

a

b

c

Estimate the slope of the line of best fit.

Estimate the y-intercept of the line of

best fit.

Approximate the value of y when

x = 6.9.

12

11

10

9

8

7

6

5

4

3

2

1

y

x

2

4 6 8 10

Let’s practice

5 The table shows the price of various used Toyota Camrys.

Age ( years ) 1 2 0 5 7 4 3 4 8 2

Price ( thousands

of dollars )

16 13 21.99 10 8.6 12.5 11 11 4.5 14.5

a Construct the line of best fit.

b Find the y-intercept of the line of best fit.

c What does the y-intercept represent?

d At what age would a Camry be worth $0?

486


Price (thousands of dollars)

24

20

16

12

8

4

2

4 6 8 10

Age (years)

6 The table shows the number of eggs laid versus the number of ducks.

Ducks ( x ) 1 2 3 4 5 6 7 8

Eggs ( y ) 4 11 14 21 24 31 34 41

45

y

40

35

30

25

20

15

10

5

1

2

3 4 5 6 7 8 9

x

a

b

c

d

e

Construct the line of best fit.

Find the slope of the line of best fit.

Find the y-intercept of the line of best fit.

Find the equation of the line of best fit.

Estimate the number of eggs laid by 30 ducks.


7 The table shows the temperature of a cooling metal versus time.

Minutes ( x ) 1 2 3 4 5 6

Temperature ( y ) 27 27 23 23 19 19

a

b

c

d

Find the slope of the line of best fit.

Find the y-intercept of the line of best fit.


Estimate the number of minutes required

to reach the temperature of 15 ºC.

30

25

20

y

15

10

5

1

2

3 4 5 6 7

x

8 Consider the data shown in the table.

x 7 12 17 22 27 32 37

y 29 28 27 31 15 14 3

a Find the equation of the line of best fit.

b Use the equation to estimate the value of y when x = 23.

9 Consider the data shown in the table.

x 7 13 19 25 31 37 43

y 3 2 11 25 19 28 17

a

b


Use the equation to estimate the value

of y when x = 26.

488


10 Consider the scatter plot.

a

b

Graph the line of best fit.

Estimate the equation of the line

of best fit.

30

y

20

10

2

4 6 8 10

x


11 The table shows the number of people who went to watch a movie x weeks after

it was released.

Weeks ( x ) 1 2 3 4 5 6 7

Number of people ( y ) 17 17 13 13 9 9 5

a

b

c

d

Plot the points on the coordinate plane.

Determine the equation of the line of best fit.


Estimate the number of people who went to watch the movie 10 weeks after it was

released.

12 The table shows data on the number of kilograms of litter collected each week in a national

park x weeks after the park managers started an anti-littering campaign:

Weeks ( x ) 1 2 3 4 5 6 7

Kilograms of litter

collected ( y )

2.9 2.5 2.5 2.3 2.1 1.9 1.7

a

b

c

d

Plot the points on the coordinate plane.

Determine the equation of the line of best fit.


Estimate the number of kilograms of litter collected 12 weeks after the start of

the anti-littering campaign.


9.07 Analyzing lines of fit

using residuals

Concept summary

From a scatter plot and a line of fit, we can further analyze an association between two variables by

comparing the data points to the line of fit.

The difference between the observed value ( the data point ) and the predicted value ( the value of

the trend line at that point ) is called a residual.

80

y

75

70

Large positive residual

65

Predicted value

60

Actual value

55

Small negative residual

18 19 20 21 22

x

By taking the residuals of each point in the data set and plotting them at their corresponding

x-values, we form a residual plot for the data.

By observing the original scatter plot and the residual plot, we can analyze the data to see if there

is a correlation between the two variables.

We can also use the scatter plot and residual plot to look for outliers in the data set - data points

which stand out by being far away from their predicted value.

Following are some example scatter plots with the line of best fit and residuals, and their

corresponding residual plots.

490


Scatter plot and residual plot of weak positive linear association with no outliers:

80

75

y

12

Residual y

70

8

65

4

60

x

55

16 17 18 19 20 21

22 23 24

50

−4

45

x

−8

16

17 18 19 20 21 22 23 24

Scatter plot with residuals

Residual plot

Scatter plot and residual plot of strong negative linear association with an outlier:

10

y

Residual y

9

3

8

7

2

6

5

1

4

3

2

0.1 0.2 0.3 0.4 0.5 0.6

0.7

x

1

x

−1

0.1 0.2 0.3 0.4 0.5 0.6

0.7


Residual plot


Scatter plot and residual plot of non-linear association:

y

4

25

3

25

2

Residual y

15

1

x

10

−1

1 2 3 4 5 6 7 8

9

5

−2

x

−3

1 2 3 4 5 6 7 8

9

−4


Residual plot

Worked examples

Example 1

Describe the strength and direction of the correlation

for the scatterplot shown.

15

y

10

5

10 15 20 25

30

x

−5

Approach

The slope of the line of fit can be used to determine the direction of the correlation and the

residuals can be used to describe the strength.

Solution

Since the slope of the line of fit is negative, the correlation is also negative. The data points mostly

appear close to the line with relatively small residuals so we will describe this correlation as a

strong negative correlation.

492


Example 2

Consider the data set shown in the table of values and scatter plot below with the line of fit

y = −0.57x + 16.74

x 10 11 13 14 18 19 21 23 25 28 29 31

y 12 13 9 2 8 7 7 4 2 3 −1 −2

y

15

10

5

10 15 20 25

30

x

−5

a Create a residual plot for this data set.

Approach

The residual for each point is its vertical distance from the line of fit. We want to find this for each

point, and plot it against the same x-axis scale.

Solution

A residual can be calculated by finding the difference between the actual output of each data point

and the output predicted by the line of fit. This can be written as the formula:

residual = actual − predicted

For example, the point at (10, 12 ) has a predicted value of:

y = −0.57x + 16.74

Line of fit

= −0.57(10 ) + 16.74 Substitute into the line of fit

= −5.7 + 16.74 Simplify the product

= 11.04 Add

To find the residual we take residual = actual − predicted = 12 − 11.04 = 0.96


The residual can also be found by estimating the vertical distance between each data point and

the line of fit.

6

4

y

2

−2

−4

10 15 20 25

30

x

−6

b Identify a possible outlier in the scatterplot and calculate its residual.

Approach

We can identify outliers by finding data points that have a larger residual.

Solution

Looking at the residual plot, the point at x = 14 has a relatively larger residual compared to all of

the other points. So the point (14, 2 ) is an outlier.

To calculate the residual, start by finding the output predicted by the line of fit.

y = −0.57x + 16.74

Line of fit

= −0.57(14 ) + 16.74 Substitute into the line of fit

= −7.98 + 16.74 Simplify the product

= −8.76 Add

To find the residual we take residual = actual − predicted = 2 − 8.76 = −6.76

494



1 Determine whether each statement is true or false.

a

b

c

d

If there is a weak correlation between two variables, then there is a downward

linear trend.

A negative correlation implies a downward linear trend in a data set.

A weak correlation for a data set means that every data point is far from the line of

best fit.

The smaller the residual is for a given data point, the closer it is to the line of best fit.

e If a data set has perfect correlation then the sum of the residuals is 0.

2 The first graph shows the scatter plot of the time taken to get to school, in minutes, and daily

time spent watching TV, in hours, for 9 students. The second graph shows the residual plot

of the data.

Time watching TV (hours)

Graph 1 Graph 2

TV time residuals (hours)

4

0.3

0.2

3

0.1

Time to school (minutes)

2

−0.1

10 20 30

40 50 60 70 80 90

1

−0.2

Time to school (minutes)

−0.3

10

20 30 40 50 60 70 80 90

a

b

c

Determine if the data have a positive or negative correlation.

Describe the strength of the correlation.

Explain what this correlation implies in terms of the context.


3 For each of the following scatter plots, determine the strength and direction of

the correlation.

a

200

180

160

140

120

100

80

60

40

20

y

20 40 60 80 100 120 140 160

x

b

80

70

60

50

40

30

20

10

y

20 40 60 80 100120140160180

x

c

22

20

18

16

14

12

10

8

6

4

2

y

x

d

220

200

180

160

140

120

100

80

60

40

20

y

x

10 20 30 40 50 60 70 80

20 40 60 80 100 120

4 The graph shows a residual plot of

the electricity consumption of a residental

area for different average daily temperatures

for a linear line of fit. Does this indicate that a

linear model an appropriate choice for this

data? Explain your reasoning.

0.3

0.2

0.1

−0.1

−0.2

Consumption residuals (MWh)

Average temperature ( 0 F)

10 20 30 40 50 60 70 80 90

−0.3

496


5 The graph shows the scatter plot of

the batting average and adjusted yearly

salary, in thousands of US dollars,

for 10 minor league baseball players.

The residuals are also shown.

a

b

Describe the correlation of

the relationship.

Explain what this correlation implies in

terms of the context.

260

259

258

257

256

255

Salary (thousand dollars)

254

0.2

Batting average

0.3

6 The graph shows the residual plot of

the batting average and adjusted yearly salary,

in thousands of US dollars, for 10 minor league

baseball players.

Determine if a linear model is an appropriate

choice for the data given in the residual plot.


2

1.5

1

0.5

−0.5

−1

−1.5

−2

Salary residual (thousand dollars)

Batting average

0.2 0.3

Let’s practice

7 The scatter plot shows the number

of crashes in 2020 that involved cars of a

given year of manufacture.

Identify and interpret the strength and

direction of the correlation.

14000

13000

12000

11000

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

Number of crashes

Year of manufacture

1990 1995 20002005 2010 2015 2020


8 The scatter plot shows the heights of bean

plants at 3 weeks given a certain amount of

water each day.

Identify and interpret the strength and

direction of the correlation.

24 in

22 in

20 in

18 in

16 in

14 in

12 in

10 in

8 in

6 in

4 in

2 in

Height

Daily water

1 floz2 floz3 floz4 floz5 floz6 floz7 floz

9 Two sprinters, Luigi and Noah, run a quarter-mile nine times under various temperatures.

The residuals are plotted for both data sets.

8

Luigi

Residual time (Seconds)

8

Noah

Residual time (Seconds)

6

6

4

4

2

Temp ( 0 F)

2

Temp ( 0 F)

−2

30

35 40 45

50

−2

30

35 40 45

50

−4

−4

−6

−6

−8

−8

Identify which runner had a stronger correlation between their quarter-mile sprint time and

the temperature when they were running. Explain your reasoning.

10 The table lists the number of push-ups an athlete could do in different temperatures.

Temperature (ºF ) 33 35 37 39 41 43 45 47 49

Number of push-ups 90 84 82 85 81 80 78 97 76

The first graph shows the scatter plot with the line of fit while the second graph shows the

residual plot of the data.

498


Graph 1 Graph 2

Number of push-ups

Number of push-ups residual

100

95

90

85

80

15

10

5

30 35 40 45

Temp ( 0 F)

50

75

30

35 40 45

Temp ( 0 F)

50

−5

a

b

c

List the temperatures where the number of push-ups completed were more than what

was predicted by the line of fit.

Identify the outlier( s ) in the data set.

Determine what conclusion can be made about the outlier( s ) based on

the residual graph.

11 To determine whether the presence of alligators in the Everglades is influenced by the

number of airboat tours, the number of tours and the number of alligator sightings was

recorded each month over several months.

The first graph shows the scatter plot while the second graph shows the residual plot of

the data.

Graph 1 Graph 2

Alligator sightings

Residual of alligator sightings

100

90

80

10

5

−5

2

Number of airboat tours

4 6 8

10

70

60

50

2

Number of airboat tours

4 6 8

10

−10

−15

−20

−25

−30

a

Identify the outlier( s ) in the data set. Explain what the outlier( s ) represent in terms of

the context.

b Explain how you can use the residual plot to justify your answer to part ( a ).


12 Draw an example of a scatter plot and line of fit that demonstrates the following types

of correlation:

a Weak b Strong c Positive d Negative

e No correlation f Strong, negative correlation with one outlier

13 Construct residual plots for the following scatter plots and lines of fit shown for two different

data sets.

a

8

4

−4

−8

−12

−16

−20

−24

y

2 4 6 8 10 12 14 16 18 20

x

b

30

20

10

−10

−20

−30

−40

−50

y

1 2 3 4 5 6 7 8 9 10

x

14 The following tables show the residual values after a regression line has been fitted to

a data set.

i

ii

Construct a residual plot for the data.

Describe the suitability of a linear model for the data set.

a

x 12 20 10 18 9 20

Residual −4 −2 5 2 3 −1

b

x 5 3 15 16 8 11

Residual 0.5 1.2 −0.5 0.5 −0.9 −1.3

500


15 The following tables show a set of data and the corresponding y-values predicted by the

regression line at those x-coordinates.

i Complete the table by finding the residuals.

ii Construct a residual plot for the data.

iii State the more suitable for the data set: linear or a non-linear model.

a

x

y

2

14

4

10.5

6

12

8

8.5

10

7

Predicted y 12 11 10 9 8

Residuals

b

x 2 4 6 8 10 12

y 26 20 18 20 26 36

Predicted y 4 10 16 22 28 34

Residuals


16 Consider the residual plot for a set of data.

5

4

3

2

1

−1

−2

−3

−4

−5

Residual

1 2 3 4 5 6 7 8 9 10

x

Construct a scatter plot that could have been used to create the residual plot.

17 Finn considers the residuals of the scatter

plot shown and concludes that the data is

correlated because the residuals are all

between about −5 and 5 units. Is Finn’s

conclusion well founded? Give reasons

why or why not.

55

50

45

40

35

30

y

25

20

15

10

5

x

10

20 30 40 50 60 70

80

90


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