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©Curriculum Associates, LLC Copying is not permitted. 91Lesson 9 Analyze Linear Functions

Name: Analyze Linear Functions

Lesson 9

Vocabularylinear function a

function with a graph

that is a non-vertical

straight line, which can

be represented by a

linear equation in the

form y 5 mx 1 b .

x

y

1

O 1 2 3 4

2

4

3y 5 x 1 1

y 5 x 1 1 is a linear

function .

Prerequisite: Identify Linear Functions

Study the example problem showing a linear function. Then solve problems 1–6.

1 In the example problem, what is the relationship between the perimeter and the side length of an equilateral triangle?

2 What is the rate of change of the function? What does it represent?

3 The equation y 5 3x is in the form y 5 mx . Jessie says that equations of the form y 5 mx have no rate of change . Is Jessie correct?

Example

An equation for the perimeter y of an equilateral triangle with side length x is y 5 3x . Does the equation y 5 3x represent a linear function?

Complete the table and graph the equation .

Side Length (units) 0 1 2 3 4 5

Perimeter (units) 0 3 6 9 12 15

The graph is a straight line, so the equation y 5 3x represents a linear function .

Peri

met

er (u

nits

)Side Length (units)

2 4 61 3 5Ox

y

63

12

18

9

15

©Curriculum Associates, LLC Copying is not permitted.92 Lesson 9 Analyze Linear Functions

Solve.

4 Tell whether each equation is a linear function . Then graph each equation on the same coordinate grid . Describe the results .

a. y 5 2x2

b. y 5 x 1 2

5 Write an equation that is a linear function . Then write an equation that is not a linear function . Justify your answers .

6 The weekly salary of a store manager includes a $30 bonus plus the number of hours the manager works multiplied by the manager’s earnings per hour . Is this situation defined by a linear function? Write and graph an equation to check your prediction .

21

1

21 3 4

22

21222324

23

24

3

2

4

x

y

O

2 4 6 8 91 3 5 7Ox

y

2010

40

60

Earn

ings

($)

Number of Hours

30

50

708090


Name: Lesson 9

1 Check the equation in the example problem by substituting for x using one of the x-values from the table . Do you get the y-value shown in the table?

2 What do the slope and the y-intercept mean in the context of the problem?

3 Use the table to find the slope and the y-intercept of the function represented .

x 0 1 2 3 4

y 6 9 12 15 18

Example

Chuck’s Appliance Repair charges a $25 service fee plus $35 for each hour the repair takes . Write an equation that relates the total cost y of a repair and the number of hours the repair takes x .

Use a table of values to find the slope and y-intercept . When x 5 0, y 5 25, so the y-intercept is 25 . As x increases by 1, y increases by 35, so the

slope is 35 ·· 1 , or 35 .

You can write an equation for this function by substituting values for the slope m and the y-intercept b into the equation y 5 mx 1 b .

The equation is y 5 35x 1 25 .

Write an Equation Using Slope and y-Intercept

Study the example problem showing how to write an equation using the slope and the y-intercept. Then solve problems 1–7.

Vocabularyslope the ratio rise ··· run ,

which tells you how

many units a line goes up

for every unit that it goes

over .

y-intercept the

y-coordinate of the point

where a graph intersects

the y-axis .

x

y

1

O 1 2 3 4

2

4

5

3y 5 2x 1 1

The slope is 2 . The

y-intercept is 1 .

Hours, x Total Cost ($), y

0 25

1 60

2 95

3 130

4 165


Solve.

4 What are the slope and the y-intercept of the equation y 5 0 .5x 1 3?

5 Write an equation for the table of values . Explain how you got your answer .

x 0 1 2 3 4

y 1 5 9 13 17

6 An amusement park charges $8 for admission and $2 for each ride . Use the graph to find the slope and the y-intercept . Then write an equation for the function that relates the total cost to the number of rides .

Show your work.

Solution:

7 Write an equation for the table of values . Explain how you got your answer .

x 1 3 3 .5 6 7

y 4 10 11 .5 19 22

Tota

l Cos

t ($)

Number of Rides2 4 6 8 9 101 3 5 7O

x

y

84

16

24

12

20

283236


Name: Lesson 9

Use an Equation to Find Slope and y-Intercept

Study the example showing how to use an equation to find the slope and the y-intercept of a linear function. Then solve problems 1–8.

1 Use the table of values in the example problem to graph the function . Explain how to find the slope and the y-intercept from the graph .

2 Explain how the equation y 5 7 .5x 1 50 in the example problem shows the slope and the y-intercept .

3 What do the slope and the y-intercept mean in the context of the problem?

Example

A fitness club charges its members a sign-up fee and a weekly fee . The cost y of membership at the fitness club is given by the equation y 5 7 .5x 1 50, where x is the number of weeks of membership . Make a table of values for the equation and find its slope and y-intercept .

Use the equation to make a table of values by substituting values of x into the equation and solving for y . Then find the slope and the y-intercept .

When x 5 0, y 5 50 . The y-intercept is 50 . As x increases by 10, y increases by 75, so the slope

is 75 ·· 10 , or 7 .5 .

Number of Weeks, x

Total Cost ($), y

0 50

10 125

20 200

30 275

40 350

50 425

Tota

l Co

st ($

)

Number of Weeks10 20 30 40 50O

x

y

10050

200

300

150

250

350400450


Solve.

4 A taxi service charges a pick-up fee plus a charge for each mile driven . The equation y 5 1 .8x 1 5 gives the total cost y to travel x miles in the taxi . Complete the table . Explain how to use the table to find the slope and the y-intercept for this function .

x 0 10 20 30 40

y

5 A different taxi service charges a pick-up fee of $4 plus a charge of $1 .75 per mile driven . Write an equation for this function, and identify the slope and the y-intercept .

6 Enrico is filling his pool . The pool has 3,000 gallons of water in it now . The water hose that Enrico uses puts 500 gallons per hour into the pool . Write an equation for the number of gallons y of water in the pool after x hours . Identify the slope and the y-intercept .

7 The Peach Festival charges $12 for admission and $2 .25 for each pound of peaches picked . Write an equation for the total cost y if you pick x pounds of peaches . Use your equation to find the total cost of attending the festival and picking 5 pounds of peaches .

Show your work.

Solution:

8 Write an equation for the function that passes through

the points 1 1, 3 ·· 2 2 and 1 3 ·· 2 , 2 2 .


Name: Lesson 9

Example

Paolo is going to take part in a 20-kilometer walk . He makes this graph to show the relationship between time and remaining distance if he maintains his planned speed . Write an equation for this function .

Make a table to find the slope and the y-intercept .

The table shows that when x 5 0, y 5 20, so the y-intercept is 20 . As each x-coordinate increases by 1, each y-coordinate

decreases by 4, so the slope is 24 ··· 1 , or 24 .

The equation is y 5 24x 1 20 .

Write Equations with Negative Slope

Study the example problem showing an equation with a negative slope. Then solve problems 1–7.

1 What do the y-intercept and the slope mean in the context of the example problem?

2 What is Paolo’s planned speed?

3 Explain why the slope of the function in the example problem is negative and not positive .

4 Describe how the function in the example problem would have to change to show a positive slope .

x 0 1 2 3 4 5

y 20 16 12 8 4 0

Dis

tanc

eRe

mai

ning

(km

)

Time (hr)2 41 3 5O

x

y

84

1612

20


5 Sasha is driving her car at an average rate of 60 miles per hour . She is driving directly to Atlanta and is 400 miles away . The equation y 5 400 2 60x can be used to represent the distance y Sasha is from Atlanta after x hours . Identify the slope and the y-intercept, and explain what they represent .

6 A restaurant has a container that holds 25 gallons of lemonade . They sell lemonade at a rate of about 2 .5 gallons per hour . Suppose that the container is full . Write an equation that shows how much lemonade y (in gallons) is in the container after x hours . Identify the slope and the y-intercept .

Show your work.

Solution:

7 Write an equation for the function shown in the graph . Identify the slope and the y-intercept . Then graph a different linear function that has the same slope as the function shown . Write an equation for your function .

Solve.

1

21 3 4 5 6 7

22

212324252627

23

24

25

3

2

4

6

5

7

x

y

O2221


Name: Lesson 9

1 Martin has an 18-cup container of flour that he uses for muffins only . He uses 3 cups of the flour for every batch of muffins he makes . Write an equation to show how much flour is left in the container after x batches of muffins . Then graph the function .

Show your work.

Solution:

Analyze Linear Functions

Solve the problems.

2 Which equation describes the function shown in the table?

x 22 21 0 1 2

y 21 2 5 8 11

A y 5 x 1 5

B y 5 3x 1 5

C y 5 1 ·· 3 x 1 5

D y 5 5x 1 3

Jacob chose D as the correct answer . How did he get that answer?

Is the slope positive or negative?

How can you use a table to find the slope of a function?

Flou

r in

Cont

aine

r (c)

Flour Use

Number of Mu�n Batches2 4 61 3 5O

x

y

63

12

18

9

15


Solve.

3 The cost of having a package delivered by Quick Bicycle Delivery is a function of the weight of the package . The graph of this function is shown .

2 4 6 8 91 3 5 7Ox

y

21

4

6

3

5

Weight (lb)

Cost

of D

eliv

ery

($)

Tell whether each statement is True or False .

a. When the weight is greater than 15 pounds, the cost will be greater than $10 . u True u False

b. An equation that represents this graph is y 5 x 1 0 .6 . u True u False

c. The slope is 1 . u True u False

d. The cost of delivery decreases by $0 .60 for each pound the weight decreases . u True u False

4 The lines passing through which pairs of points have a positive y-intercept? Select all that apply .

A (23, 10) and (1, 2)

B (21, 6) and (0, 2)

C (1, 2) and (3, 4)

D (0, 24) and (3, 22)

E (3, 1) and (5, 2)

Remember that you can use the formula for the slope when you know two points.

What does the slope represent in the function?

analyze linear functions name - edl · chuck’s appliance repair charges a $25 service fee plus...

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