Learn Graphing Linear Functions by Using Tables€¦ · x = a are vertical lines. The graph of x = −2 is a vertical line through (−2, y ) for all real values of y. Graph ordered

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Lesson 4-1

Graphing Linear Functions

Learn Graphing Linear Functions by Using TablesA table of values can be used to graph a linear equation. Every ordered pair that makes the equation true represents a point on its graph. So, the graph of an equation represents all its solutions.

Example 1 Graph by Making a TableGraph -2x - 3 = y by making a table.

Step 1 Choose any values of x from the domain and make a table.

Step 2 Substitute each x-value into the equation to find the corresponding y-value. Then, write the x- and y-values as an ordered pair.

Step 3 Graph the ordered pairs in the table and connect them with a line.

y

xO−4 −2−8−6

8642

4 62 8

−4−6−8

Talk About It!

What values of x might be easiest to use when graphing a linear equation when the x-coefficient is a whole number? Justify your argument.

Go Online You can complete an Extra Example online.

x -2x - 3 y (x, y)

-4

-2

0

1

3

Study Tip

Exactness Although only two points are needed to graph a linear function, choosing three to five x-values that are spaced out can verify that your graph is correct.

Explore Points on a Line

Online Activity Use an interactive tool to complete an Explore.

INQUIRY How is the graph of a linear equation related to its solutions?

Today’s Goals● Graph linear functions

by making tables of values.

● Graph linear functions by using the x- and y-intercepts.

Lesson 4-1 • Graphing Linear Functions 209

Sample answer: -2, -1, 0, 1, and 2 would be easiest to use because they are small and therefore easy to substitute into the equation to find the y-values.

-2(-4) -3 5 (-4, 5)-2(-2) -3 1 (-2, 1)-2(0) -3 -3 (0, -3)-2(1) -3 -5 (1, -5)-2(3) -3 -9 (3, -9)

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

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Check Graph y = 2x + 5 by using a table.

x y

-5

-3

-1

0

2

Example 2 Choose Appropriate Domain ValuesGraph y = 1 __ 4 x + 3 by making a table.

Step 1 Make a table.

Step 2 Find the y-values.

Step 3 Graph the ordered pairs in the table and connect them with a line.

Check Graph y = 3 __ 5 x - 2 by making a table.

x y

-10

-5

0

5

10


Think About It!

What are some values of x that you might choose in order to graph y = 1 _ 7 x - 12?

Watch Out!

Equivalent Equations Sometimes, the variables are on the same side of the equal sign. Rewrite these equations by solving for y to make it easier to find values for y.

x 1 __ 4 x + 3 y (x, y)

-8

-4

0

4

8

y

xO−4−6 −2−8

8

46

2

4 62 8

−4−6−8

Your Notes

y

xO−4−8 −2−6

8

46

2

4 82 6

−4

−8−6

−8 −4−2−6 84 62

8

46

2

−8

−4−6

y

xO

210 Module 4 • Linear and Nonlinear Functions

Sample answer: {-14, -7, 0, 7, 14}

1 __ 4 (-8) + 3 1 (-8, 1)

1 __ 4 (-4) + 3 2 (-4, 2)

1 __ 4 (0) + 3 3 (0, 3)

1 __ 4 (4) + 3 4 (4, 4)

1 __ 4 (8) + 3 5 (8, 5)

-5-1359

-8-5-2

14


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Example 3 Graph y = aGraph y = 5 by making a table.

Step 1 Rewrite the equation. y = 0x + 5

Step 2 Make a table.

Step 3 Graph the line.

The graph of y = 5 is a horizontal line through (x, 5) for all values of x in the domain.

Example 4 Graph x = aGraph x = -2.

You learned in the previous example that the graphs of functions of the form y = a are horizontal lines. Graphs of functions of the form x = a are vertical lines.

The graph of x = −2 is a vertical line through (−2, y ) for all real values of y. Graph ordered pairs that have x-coordinates of -2 and connect them with a vertical line.

Check Graph x = 6.

y

xO

Think About It!

In general, what does the graph of an equation of the form y = a, where a is any real number, look like?

Think About It!

Is the graph of x = a a function? Why or why not?


x 0x + 5 y (x, y)

-2

-1

0

1

2

y

xO

y

xO−4−8 −2−6

8

46

2

4 82 6

−4

−8−6


Sample answer: a horizontal line through (x, a) for all values of x in the domain.

Sample answer: The element -2 in the domain is paired with more than one element of the range.

0(-2) + 5 5 (-2, 5)0(-1) + 5 5 (-1, 5)0(0) + 5 5 (0, 5)0(1) + 5 5 (1, 5)0(2) + 5 5 (2, 5)


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Think About It!

Why are the x- and y-intercepts easy to find?

Think About It!

What does a line that only has an x-intercept look like? a line that only has a y-intercept?

Learn Graphing Linear Functions by Using the InterceptsYou can graph a linear equation given only two points on the line. Using the x- and y-intercepts is common because they are easy to find. The intercepts provide the ordered pairs of two points through which the graph of the linear equation passes.

Example 5 Graph by Using InterceptsGraph -x + 2y = 8 by using the x- and y-intercepts.

To find the x-intercept, let .

Original equation

Replace y with 0.

Simplify.

Divide.

This means that the graph intersects the x-axis at .

To find the y-intercept, let .

Original equation

Replace x with 0.

Simplify.

Divide.

This means that the graph intersects the y-axis at .

Graph the equation.

Step 1 Graph the x-intercept.

Step 2 Graph the y-intercept.

Step 3 Draw a line through the points.

y

xO−6−8−9 −5 −4 −3 −2 −1 1−7

45

23

1

−2

−4−5

−3

Study Tip

Tools When drawing lines by hand, it is helpful to use a straightedge or a ruler.

Explore Lines Through Two Points

Online Activity Use graphing technology to complete an Explore.

INQUIRY How many lines can be formed with two given points?

Go OnlineYou can watch a video to see how to graph linear functions.


Sample answer: Because either the x- or y-value of an intercept is 0.

Sample answer: A line that only has an x-intercept is a vertical line. A line that only has a y-intercept is a horizontal line.

y = 0

(-8, 0)

(0, 4)

x = 0

-x + 2y = 8 -x + 2(0) = 8 -x = 8 x = -8

-x + 2y = 8 -0 + 2y = 8 2y = 8 y = 4


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(continued on the next page)

Check Graph 4y = -12x + 36 by using the x- and y- intercepts.

x-intercept:

y-Intercept:

Example 6 Use InterceptsPETS Angelina bought a 15-pound bag of food for her dog. The bag contains about 60 cups of food, and she feeds her dog 2 1 __ 2 or 5 __ 2 cups of food per day. The function y + 5 __ 2 x = 60 represents the amount of food left in the bag y after x days. Graph the amount of dog food left in the bag as a function of time.

Part A Find the x- and y-intercepts and interpret their meaning in the context of the situation.

To find the x-intercept, let .

Original equation

Replace y with 0.

Simplify.

Multiply each side by 2 __ 5 .

The x-intercept is 24. This means that the graph intersects the x-axis at . So, after 24 days, there is no dog food left in the bag.

To find the y-intercept, let .

Original equation

Replace x with 0.

Simplify.

The y-intercept is 60. This means that the graph intersects the y-axis at . So, after 0 days, there are 60 cups of food in the bag.

−4 −2 −1−3 42 31

8

46

2

−8

−4−6

y

xO

Think About It!

Find another point on the graph. What does it mean in the context of the problem?

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


y = 0

x = 0

(24, 0)

(0, 60)

3

9

y + 5 __ 2 x = 60

0 + 5 __ 2 x = 60

5 __ 2 x = 60

x = 24

y + 5 __ 2 x = 60

y + 5 __ 2 (0) = 60

y = 60

Sample answer: (10, 35); After 10 days, there are 35 cups of dog food left in the bag.


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Part B Graph the equation by using the intercepts.

y

xO

10

5 10 15 20 25 30 35 4540

2030405060708090

CheckPEANUTS A farm produces about 4362 pounds of peanuts per acre. One cup of peanut butter requires about 2 __ 3 pound of peanuts. If one acre of peanuts is harvested to make peanut butter, the function y = - 2 __ 3 x + 4362 represents the pounds of peanuts remaining y after x cups of peanut butter are made.

x-intercept:

y-intercept:

Which graph uses the x- and y-intercepts to correctly graph the equation?

A.

20000 4000 6000

Am

ount

of

Pean

uts

(lb) 6000

4000

2000

0

Amount ofPeanut Butter (c)

Peanut FarmingB.

20000 4000 6000

Am

ount

of

Pean

uts

(lb) 6000

4000

2000

0


Peanut Farming

C.

20000 4000 6000

Am

ount

of

Pean

uts

(lb) 6000

4000

2000

0


Peanut FarmingD.

20000 4000 6000

Am

ount

of

Pean

uts

(lb) 6000

4000

2000

0


Peanut Farming


Think About It!

What assumptions did you make about the amount of food Angelina feeds her dog each day?

Go OnlineYou can watch a video to see how to graph a linear function using a graphing calculator.


6543

4362

C

Sample answer: I assumed that the bag of food contains exactly 60 cups and that Angelina feeds her dog the exact same amount each day.


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Learn Graphing Linear Functions by Using Tables€¦ · x = a are vertical lines. The graph of x = −2 is a vertical line through (−2, y ) for all real values of y. Graph ordered