Graphs of Linear Functions
from Intercepts
Brenda Meery
Kaitlyn Spong
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Printed: January 29, 2015
AUTHORS
Brenda Meery
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Graphs of Linear Functions from Intercepts
1
Graphs of Linear Functions
from Intercepts
Here you will learn how to graph a linear function by first finding the x and y intercepts.
What are the intercepts of 4x + 2y = 8? How could you use the intercepts to quickly graph the function?
Watch This
Khan Academy X and Y Intercepts
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/58470
Guidance
To graph a linear function, you need to plot only two points. These points can then be lined up with a straight edge
and joined to graph the straight line. While any two points can be used to graph a linear function, two points in
particular that can be used are the x-intercept and the y-intercept. Graphing a linear function by plotting the x− and
y− intercepts is often referred to as the intercept method.
The x-intercept is where the graph crosses the x-axis. Its coordinates are (x, 0). Because all x-intercepts have a
y-coordinate equal to 0, you can find an x-intercept by substituting 0 for y in the equation and solving for x.
The y-intercept is where the graph crosses the y-axis. Its coordinates are (0, y). Because all y-intercepts have a
x-coordinate equal to 0, you can find an y-intercept by substituting 0 for x in the equation and solving for y.
Example A
Identify the x− and y-intercepts for each line.
(a) 2x + y − 6 = 0
(b) 21 x − 4y = 4
Solution:
(a)
1
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Let y = 0. Solve for ‘x’.
Let x = 0. Solve for ‘y’.
2x + y − 6 = 0
2x + y − 6 = 0
2x + (0) − 6 = 0
2(0) + y − 6 = 0
2x − 6 = 0
y−6 = 0
2x − 6 + 6 = 0 + 6
y−6+6 = 0+6
2x = 6
2x 6
=
2
2
x=3
y=6
The y-intercept is (0, 6)
The x-intercept is (3, 0)
(b)
Let y = 0. Solve for ‘x’.
1
x − 4y = 4
2
1
x − 4(0) = 4
2
1
x−0 = 4
2
1
x=4
2
1 1
2
x = 2(4)
2
x=8
Let x = 0. Solve for ‘y’.
1
x − 4y = 4
2
1
(0) − 4y = 4
2
4
−4y
=
−4
−4
y = −1
The x-intercept is (8, 0)
The y-intercept is (0, −1)
0 − 4y = 4
− 4y = 4
Example B
Use the intercept method to graph 2x − 3y = −12.
Solution:
2
Let y = 0. Solve for ‘x’.
Let x = 0. Solve for ‘y’.
2x − 3y = −12
2x − 3y = −12
2x − 3(0) = −12
2(0) − 3y = −12
2x − 0 = −12
0 − 3y = −12
2x = −12
2x −12
=
2
2
x = −6
− 3y = −12
−3y −12
=
−3
−3
y=4
The x-intercept is (−6, 0)
The y-intercept is (0, 4)
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Chapter 1. Graphs of Linear Functions from Intercepts
Example C
Use the x− and y-intercepts of the graph to identify the linear function that matches the graph.
a) y = 2x − 8
b) x − 2y + 8 = 0
c) 2x + y − 8 = 0
The x-intercept is (–8, 0) and the y-intercept is (0, 4).
3
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Solution: Find the x and y intercepts for each equation and see which matches the graph.
a) x intercept: 0 = 2x − 8 → x = 4
y intercept: y = 2(0) − 8 → y = −8
b) x intercept: x − 2(0) + 8 = 0 → x = −8
y intercept: 0 − 2y + 8 = 0 → y = 4
c) x intercept: 2x + 0 − 8 = 0 → x = 4
y intercept: 2(0) + y − 8 = 0 → y = 8
The x and y intercepts match for x − 2y + 8 = 0 so this is the equation of the line.
Concept Problem Revisited
The linear function 4x + 2y = 8 can be graphed by using the intercept method.
To determine the x-intercept, let y = 0.
To determine the y-intercept, let x = 0.
Solve for ‘x’.
Solve for ‘y’.
4x + 2y = 8
4x + 2y = 8
4x + 2(0) = 8
4(0) + 2y = 8
4x + 0 = 8
0 + 2y = 8
4x = 8
4x 8
=
4
4
x=2
2y = 8
2y 8
=
2
2
y=4
The x-intercept is (2, 0)
The y-intercept is (0, 4)
Plot the x-intercept on the x-axis and the y-intercept on the y-axis. Join the two points with a straight line.
4
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Chapter 1. Graphs of Linear Functions from Intercepts
Vocabulary
Intercept Method
The intercept method is a way of graphing a linear function by using the coordinates of the x− and yintercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and joining them with a
straight line.
x-intercept
An x-intercept of a relation is the x−coordinate of the point where the relation intersects the x-axis.
y-intercept
A y-intercept of a relation is the y−coordinate of the point where the relation intersects the y-axis.
Guided Practice
1. Identify the x− and y-intercepts of the following linear functions:
(i) 2(x − 3) = y + 4
(ii) 3x + 23 y − 3 = 0
2. Use the intercept method to graph the following relation:
(i) 5x + 2y = −10
3. Use the x− and y-intercepts of the graph, to match the graph to its function.
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(i) 2x + y = 6
(ii) 4x − 3y − 12 = 0
(iii) 5x + 3y = 15
Answers:
1. (i)
2(x − 3) = y + 4
Simplify the equation
2(x − 3) = y + 4
2x − 6 = y + 4
2x − 6 + 6 = y + 4 + 6
2x = y + 10
You may leave the function in this form.
2x − y = y − y + 10
2x − y = 10
If you prefer to have both variables on the same side of the equation, this form may also be used. The choice is your
preference.
6
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Chapter 1. Graphs of Linear Functions from Intercepts
Let y = 0. Solve for x.
Let x = 0. Solve for y.
2x − y = 10
2x − y = 10
2x − (0) = 10
2(0) − y = 10
2x = 10
2x 10
=
2
2
x=5
0 − y = 10
−y
10
=
−1 −1
y = −10
The x-intercept is (5, 0)
The y-intercept is (0, −10)
(ii)
2
3x + y − 3 = 0
3
2
y − 3(3) = 3(0)
3(3x) + 3
3
2
3(3x) + 3
y − 3(3) = 3(0)
3
9x + 2y − 9 = 0
Simplify the equation.
Multiply each term by 3.
9x + 2y − 9 + 9 = 0 + 9
9x + 2y = 9
Let y = 0. Solve for x.
Let x = 0. Solve for y.
9x + 2y = 9
9x + 2y = 9
9x + 2(0) = 9
9(0) + 2y = 9
9x + 0 = 9
9x 9
=
9
9
x=1
0 + 2y = 9
2y 9
=
2
2
y = 4.5
The x-intercept is (1, 0)
The y-intercept is (0, 4.5)
Let y = 0. Solve for x.
Let x = 0. Solve for y.
5x + 2y = −10
5x + 2y = −10
5x + 2(0) = −10
5(0) + 2y = −10
5x + 0 = −10
5x −10
=
5
5
x = −2
0 + 2y = −10
2y −10
=
2
2
y = −5
The x-intercept is (−2, 0)
The y-intercept is (0, −5)
2.
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3. Identify the x− and y-intercepts from the graph.
The x-intercept is (3, 0)
The y-intercept is (0, -4)
Determine the x− and y-intercept for each of the functions. If the intercepts match those of the graph, then the linear
function will be the one that matches the graph.
(i)
Let y = 0. Solve for x.
Let x = 0. Solve for y.
2x + y = 6
2x + y = 6
2x + (0) = 6
2(0) + y = 6
2x = 6
2x 6
=
2
2
x=3
0+y = 6
The x-intercept is (3, 0)
The y-intercept is (0, 6)
This matches the graph.
This does not match the graph.
2x + y = 6 is not the linear function for the graph.
(ii)
8
y=6
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Chapter 1. Graphs of Linear Functions from Intercepts
Let y = 0. Solve for x.
Let x = 0. Solve for y.
4x − 3y − 12 = 0
4x − 3y − 12 = 0
4x − 3y − 12 + 12 = 0 + 12
4x − 3y − 12 + 12 = 0 + 12
4x − 3y = 12
4x − 3y = 12
4x − 3(0) = 12
4(0) − 3y = 12
4x − 0 = 12
0 − 3y = 12
4x = 12
4x 12
=
4
4
x=3
− 3y = 12
−3y
12
=
−3
−3
y = −4
The x-intercept is (3, 0)
The y-intercept is (0, −4)
This matches the graph.
This matches the graph.
4x − 3y − 12 = 0 is the linear function for the graph.
(iii)
Let y = 0. Solve for x.
Let x = 0. Solve for y.
5x + 3y = 15
5x + 3y = 15
5x + 3(0) = 15
5(0) + 3y = 15
5x + 0 = 15
0 + 3y = 15
5x = 15
5x 15
=
5
5
x=3
3y = 15
3y 15
=
3
3
y=5
The x-intercept is (3, 0)
The y-intercept is (0, 5)
This matches the graph.
This does not match the graph.
5x + 3y = 15 is not the linear function for the graph.
Explore More
For 1-10, complete the following table:
TABLE 1.1:
Function
7x − 3y = 21
8x − 3y + 24 = 0
y
x
4−2 =3
7x + 2y − 14 = 0
2
1
3 x − 4 y = −2
x-intercept
1.
3.
5.
7.
9.
y-intercept
2.
4.
6.
8.
10.
Use the intercept method to graph each of the linear functions in the above table.
9
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11. 7x − 3y = 21
12. 8x − 3y + 24 = 0
13. 4x − 2y = 3
14. 7x + 2y − 14 = 0
15. 32 x − 14 y = −2
Use the x− and y-intercepts to match each graph to its function.
a. 7x + 5y − 35 = 0
b. y = 5x + 10
c. 2x + 4y + 8 = 0
d. 2x + y = 2
16. .
17. .
10
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Chapter 1. Graphs of Linear Functions from Intercepts
18. .
19. .
11
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12