Intercepts of Graphs

Lesson 3-4

Intercepts of Graphs

Learn Intercepts of Graphs of Functions

The intercepts of graphs are points where the graph intersects an axis.

The x-intercept is the x-coordinate of a point where a graph crosses the x-axis.

The y-intercept is the y-coordinate of a point where a graph crosses the y-axis.

A function is positive when its graph lies above the x-axis.

A function is negative when its graph lies below the x-axis.

y

O

x

Example 1 Intercepts of the Graph of a Linear Function

Use the graph to estimate the x- and y-intercepts of the function and describe where the function is positive and negative.

The x-intercept is the point where the

y

graph crosses the x-axis, ( 3 , 0 ).

The y-intercept is the point where the graph crosses the y-axis, ( 0 , 6 ). A function is positive when its graph lies above the x-axis, or when x < 3 .

x O

A function is negative when its graph lies below the x-axis, or when x > 3 .

Check

Use the graph to estimate the x- and y-intercepts of the function and describe where the function is positive and negative. C

A.x-intercept: (-2, 0); y-intercept: (0, -6); positive: x > -2; negative: x < -2

B.x-intercept: (0, -6); y-intercept: (-2, 0); positive: x < -2; negative: x > -2

y x

O

C.x-intercept: (-2, 0); y-intercept: (0, -6); positive: x < -2; negative: x > -2

D.x-intercept: (0, -6); y-intercept: (-2, 0); positive: x > -2; negative: x < -2

Go OnlineYou can complete an Extra Example online.

Today's Goals

Identify the intercepts of functions and intervals where functions are positive and negative.

Solve equations by graphing.

Today's Vocabulary

x-intercept y-intercept positive negative root zero

Study Tip:

Notice that intercept can be used to refer to either the point where the graph intersects the axis or the nonzero coordinate of the point where the graph intersects the axis.

Think About It!

Explain why this function is linear.

Sample answer: The function is linear because its graph forms a straight line.

Study Tip

To help remember the difference between the x- and y-intercepts, remember that the x-intercept is where the graph intersects the x-axis, and the y-intercept is where the graph intersects the y-axis.

Copyright ? McGraw-Hill Education

Lesson 3-4 ? Intercepts of Graphs167

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

Your Notes

Think About It! Explain why this function is nonlinear.

Sample answer: The function is nonlinear because its graph does not form a straight line.

Watch Out! The graphs of nonlinear functions can have more than one x-intercept.

Example 2 Intercepts of the Graph of a Nonlinear Function

Use the graph to estimate the x- and y-intercepts of the function and describe where the function is positive and negative.

4y

2

x

4 3 2 1O 1 2 3 4

4 6 8 10 12

x-intercepts: -4 and 3 . y-intercept: -12 . positive: when x < -4 and when x > 3 . negative: x is between -4 and 3 .

Check

Use the graph of the function to determine key features.

y

x O

Copyright ? McGraw-Hill Education

Part ADetermine whether each ordered pair represents an x-intercept, a y-intercept, or neither.

(1, 0) x-intercept

(0, 1)

neither

Part BDescribe where the function is positive and negative. D

A.positive: x < 1 and x > 4; negative: x is between 1 and 4

B. positive: x > 1 and x < 4; negative: x is between 1 and 4

C. positive: x is between 1 and 4; negative: x > 1 and x < 4

D. positive: x is between 1 and 4; negative: x < 1 and x > 4

Go OnlineYou can complete an Extra Example online.

168Module 3 ? Relations and Functions

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

Example 3 Find Intercepts from a Graph

SPORTSThe graph shows the height of a ball for each second x that it is airborne. Use the graph to estimate the x- and y-intercepts of the function, where the function is positive and negative, and interpret the meanings in the context of the situation.

The x-intercept is 9 . That means that the ball will hit the ground after 9 seconds.

Height (ft)

Height of the Ball

10 y 9

8

7

6

5

4

3

2

1

x

0 1 2 3 4 5 6 7 8 9 10

Time (s)

The y-intercept is 4 . This means that at time 0 , the ball was at a height of 4 feet.

The function is positive when x is between 0 and 9 , which means that the ball is in the air for 9 seconds.

No portion of the graph shows that the function is negative .

Check

FITNESSThe graph shows the number of people y at a gym x hours after the gym opens.

200 y 160

Gym Occupancy

120

80

40

x

0

2 4 6 8 10 12

Time (hours)

Part AUse the graph to estimate the x- and y-intercepts. x-intercept: ( 12 , 0 ) y-intercept: ( 0 , 20 )

Part BWhich statements describe the meaning of the x- and y-intercepts in the context of the situation? Select all that apply. A, C

A. There were 20 people at the gym when it opened.

B. The gym closed after 20 hours.

C. The gym closed after 12 hours.

D. There were 12 people at the gym when it opened.

Think About It!

The function is only graphed from 0 to 9 seconds. What can you assume about the function when x > 9? Interpret this meaning. Does it make sense in the context of the situation?

Sample answer: I assume that the graph continues to follow the same path, so the function is negative when x > 9. This means that the ball would be at a negative height when x > 9. This does not make sense in the context of the situation because it is impossible for a ball to be at a negative height.

Copyright ? McGraw-Hill Education

Number of People

Go OnlineYou can complete an Extra Example online.

Lesson 3-4 ? Intercepts of Graphs169

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

Example 4 Find Intercepts from a Table

LUNCHViolet starts the semester with $150 in her student lunch account. Each day she spends $3.75 on lunch. The table shows the function relating the amount of money remaining in her lunch account to the number of days Violet has purchased lunch.

Time (Days) Balance ($)

x

y

0

150

2

142.50

5

131.25

10

112.50

Part A Find the intercepts.

15

93.75

The x-intercept is where y = 0 ,

30

37.50

so the x-intercept is 40 .

40

0

The y-intercept is where x = 0 , so the y-intercept is 150 .

Part B Describe what the intercepts mean in the context of the situation.

The x-intercept means that after buying lunch for 40 days, Violet will have $ 0 left in her lunch account, or it will take Violet 40 days to use all of the money in her lunch account. The y-intercept means that Violet's lunch account has $ 150 after buying lunch for 0 days, or the beginning balance of her lunch account is $ 150 .

Copyright ? McGraw-Hill Education

Check

MOVIESAshley received a gift card to the movie theater for her birthday. The table shows the amount of money remaining on her gift card y after x trips to the movie theater.

Number of Trips Balance ($)

x

y

0

90

1

81

2

72

3

63

5

45

7

27

10

0

Part A Find the y-intercept.( 0 , 90 )

Part BFind the x-intercept and describe what it means in the context of the situation. C

A. (10, 0); The initial balance on the gift card was $10.

B. (90, 0); The initial balance on the gift card was $90.

C.(10, 0); After 10 trips to the movies, there will be no money left on the gift card.

D.(90, 0); After 90 trips to the movies, there will be no money left on the gift card.

Go OnlineYou can complete an Extra Example online.

170Module 3 ? Relations and Functions

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

Learn Solving Equations by Graphing

The solution, or root, of an equation is any value that makes the equation true. A zero is an x-intercept of the graph of the function.

For example, the root of 3x = 6 is 2. A linear equation, like 3x = 6, has at most one root, while a nonlinear equation, like x2 + 4x - 5 = 0, may have more than one.

Equation

Related Function

3x = 6

f(x) = 3x - 6 or y = 3x - 6

x2 + 4x - 5 = 0 f(x) = x2 + 4x - 5 or y = x2 + 4x - 5

The graph of the related function can be used to find the solutions of an equation. The related function is formed by solving the equation for 0 and then replacing 0 with f(x) or y.

Values of x for which f(x) = 0 are located at the x-intercepts of the graph of a function and are called the zeros of the function f. The roots of an equation are the same as the zeros of its related function. The solutions and roots of an equation are the same value as the zeros and x-intercepts of its related function. For the equation 3x = 6:

? 2 is the solution of 3x = 6.

? 2 is the root of 3x = 6.

? 2 is the zero of f(x) = 3x - 6.

? 2 is the x-intercept of f(x) = 3x - 6.

Think About It!

What is the difference between a root and a zero?

Sample answer: A root is the solution of an equation, while a zero is the x-intercept of the related function.

Copyright ? McGraw-Hill Education

Example 5 Solve a Linear Equation by Graphing

Solve -2x + 7 = 1 by graphing. Check your solution.

Find the related function.

-2x + 7 = 1 -2x + 7 - 1 = 1 - 1

-2x + 6 = 0

Original equation Subtract 1 from each side. Simplify.

Graph the left side of the equation. The related function is f(x) = -2x + 6 , which can be graphed.

y

O

x

The graph intersects the x-axis at 3 . This is the x-intercept, or zero, which is also the root of the equation. So, the solution of the equation is 3 .

Check your solution by solving the equation algebraically.

Go OnlineYou can complete an Extra Example online.

Go Online An alternate method is available for this example.

Think About It!

Suppose you first solved the equation algebraically. How could you use your solution to graph the zero of the related function?

Sample answer: The solution of the equation is the same as the x-intercept, or zero, of the function. So, I can graph the zero by plotting a point at 3 on the x-axis.

Lesson 3-4 ? Intercepts of Graphs171

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

Go Online

You can watch a video to see how to use a graphing calculator with this example.

Example 6 Solve a Nonlinear Equation by Graphing

Solve x2 - 4x = -3 by graphing. Check your solution.

Find the related function. x2 - 4x = -3

x2 - 4x + 3 = -3 + 3 x2 - 4x + 3 = 0

Original equation Add 3 to each side. Simplify.

Graph the left side of the equation. The related function is f(x) = x2 - 4x = -3 , which can be graphed.

y

The graph intersects the x-axis

at 1 and 3 . These are

the x-intercepts, or zeros, which

are also the roots of the

equation. So, the solutions of

O

x

the equation are 1 and

3.

Copyright ? McGraw-Hill Education

Talk About It!

Does solving the equation algebraically give a different solution? Explain your reasoning.

Sample answer: No; solving algebraically results in the false statement 8 = 0, which means there is no solution.

Example 7 Solve an Equation of a Horizontal Line by Graphing

Solve 4x + 3 = 4x - 5 by graphing. Check your solution.

Find the related function.

4x + 3 = 4x - 5

Original equation

4x + 3 + 5 = 4x - 5 + 5

Add 5 to each side.

4x + 8 = 4xS implify.

4x - 4x + 8 = 4x - 4x

Subtract 4x from each side.

8 = 0 S implify.

Graph the left side of the equation. The related function is f(x) = 8 , which can be graphed.

y

The graph does not intersect

the x-axis. This means that there is no x-intercept and, therefore, there is no

solution.

O

x

Go OnlineYou can complete an Extra Example online.

172Module 3 ? Relations and Functions

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

Check

Equations and the graphs of their related functions are shown. Write the related function and its zero(s) under the appropriate graph.

y

y

2

=

1 2

x

x O

3x + 7 = 3x + 2

O

x

related function: f(x) = -_21_ x + 2 zeros: 4

related function: f(x) = 5 zeros: no solution

y 8 6 4 2

-8-6-4-2O 2 4 6 8 x

-4 -6 -8 x2 6 = x

related function: f(x) = x2 - x + 6 zeros: -2 and 3

Copyright ? McGraw-Hill Education

Apply Example 8 Estimate Solutions by Graphing

PARTY Haley is ordering invitations for her graduation party. She has $40 to spend and each invitation costs $0.96. The function m = 40 - 0.96p represents the amount of money m Haley has left after ordering p party invitations Find the zero of the function. Describe what this value means in the context of this situation.

1What is the task? Describe the task in your own words. Then list any questions that you may have. How can you find answers to your questions?

Sample answer: I need to find the zero of the function and describe what it means. How can I determine the meaning of the zero from a graph of the function? I can review graphing linear functions and labeling axes.

2How will you approach the task? What have you learned that you can use to help you complete the task?

Sample answer: I will graph the function by making a table of values. I will

estimate the x-intercept of the graph to find the zero. I will then check my

solution by solving the equation algebraically. I will use the axes labels to

help me interpret my solution. (continued on the next page)

Go OnlineYou can complete an Extra Example online.

Lesson 3-4 ? Intercepts of Graphs173

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

3What is your solution? Use your strategy to solve the problem.

Graph the function. Graduation Party

50 m 40

Estimate the solution. __4_2__ invitations Check the solution.

Amount of Money ($)

30

20

10

p

0

10 20 30 40 50

Number of Invitations

___4_1_.6_7_ invitations.

What does your solution mean in the context of the situation? Sample answer: Haley can order 41 invitations with the amount of money she has to spend.

4How can you know that your solution is reasonable?

Write About It!Write an argument that can be used to defend your solution. Sample answer: This amount is close to the estimated zero of 42 invitations from the graph.

Copyright ? McGraw-Hill Education

Check

DATABlair's cell phone plan allows her to use 3 GB of data, and she uses approximately 0.14 GB of data each day. The function g = 3 - 0.14d represents the amount of data g in GB she has left after d days.

Part AExamine the graph of the function to estimate its zero to the nearest day.

The graph appears to intersect the x-axis at 22 .

Amount of Data (GB)

Data Usage

g 6 5 4 3 2 1

d 0 5 10 15 20 25 30

Number of Days

Part BSolve algebraically to check your answer. Round to the nearest tenth.

x = 21.4

Part CDescribe what your answer to Part B means in this context. After 21.4 days, Blair has 0 GB left.

Go OnlineYou can complete an Extra Example online.

174Module 3 ? Relations and Functions

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download