<meta http-equiv="refresh" content="1; url=/nojavascript/">
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Algebra I Concepts - Honors Go to the latest version.

3.5: Graphs of Linear Functions from Intercepts

Difficulty Level: Advanced Created by: CK-12
%
Progress
Practice Intercepts by Substitution
Practice
Progress
%
Practice Now

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

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) \frac{1}{2}x-4y=4

Solution:

(a)

&\text{Let} \ y = 0. \ \text{Solve for} \ `x\text{'}. && \text{Let} \ x = 0. \ \text{Solve for} \ `y\text{'}.\\& 2x+y-6=0 && 2x+y-6=0\\& 2x+({\color{red}0})-6=0 && 2({\color{red}0})+y-6=0\\& 2x-6=0 && y-6=0\\& 2x-6+6=0+6 && y-6+6=0+6\\& 2x=6 && y=6\\& \frac{2x}{2}=\frac{6}{2} && \text{The} \ y \text{-intercept is} \ (0, 6)\\& x=3\\& \text{The} \ x \text{-intercept is} \ (3, 0)

(b)

& \text{Let} \ y = 0. \ \text{Solve for} \ `x\text{'}. && \text{Let} \ x = 0. \ \text{Solve for} \ `y\text{'}.\\& \frac{1}{2}x-4y=4 && \frac{1}{2}x-4y=4\\& \frac{1}{2}x-4({\color{red}0})=4 && \frac{1}{2}({\color{red}0})-4y=4\\& \frac{1}{2}x-0=4 && 0-4y=4\\& \frac{1}{2}x=4 && -4y=4\\& \overset{1}{\cancel{2}}\left(\frac{1}{\cancel{2}}\right)x=2(4) && \frac{-4y}{-4}=\frac{4}{-4}\\& x=8 && y=-1\\& \text{The} \ x \text{-intercept is} \ (8, 0) && \text{The} \ y \text{-intercept is} \ (0, -1)

Example B

Use the intercept method to graph 2x-3y=-12 .

Solution:

& \text{Let} \ y = 0. \ \text{Solve for} \ `x\text{'}. && \text{Let} \ x = 0. \ \text{Solve for} \ `y\text{'}.\\& 2x-3y=-12 && 2x-3y=-12\\& 2x-3({\color{red}0})=-12 && 2({\color{red}0})-3y=-12\\& 2x-0=-12 && 0-3y=-12\\& 2x=-12 && -3y=-12\\& \frac{2x}{2}=\frac{-12}{2} && \frac{-3y}{-3}=\frac{-12}{-3}\\& x=-6 && y=4\\& \text{The} \ x \text{-intercept is} \ (-6, 0) && \text{The} \ y \text{-intercept is} \ (0, 4)

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).

Solution: Find the x and y intercepts for each equation and see which matches the graph.

a) x intercept: 0=2x-8 \rightarrow x=4

y intercept: y=2(0)-8 \rightarrow y=-8

b) x intercept: x-2(0)+8=0 \rightarrow x=-8

y intercept: 0-2y+8=0 \rightarrow y=4

c) x intercept: 2x+0-8=0 \rightarrow x=4

y intercept: 2(0)+y-8=0 \rightarrow 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.

& \text{To determine the }x \text{-intercept, let } y=0. && \text{To determine the } y \text{-intercept, let } x=0.\\& \text{Solve for} \ `x\text{'}. && \text{Solve for} \ `y\text{'} .\\& 4x+2y=8 && 4x+2y=8\\& 4x+2({\color{red}0})=8 && 4({\color{red}0})+2y=8\\& 4x+{\color{red}0}=8 && {\color{red}0}+2y=8\\   & 4x=8 && 2y=8\\& \frac{4x}{4}=\frac{8}{4} && \frac{2y}{2}=\frac{8}{2}\\& x=2 && y=4\\& \text{The} \ x \text{-intercept is} \ (2, 0) && \text{The} \ y \text{-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.

Vocabulary

Intercept Method
The intercept method is a way of graphing a linear function by using the coordinates of the x- and y -intercepts. 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+\frac{2}{3}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.

(i) 2x+y=6
(ii) 4x-3y-12=0
(iii) 5x+3y=15

Answers:

1. (i)

2(x-3)&=y+4 && \text{Simplify the equation}\\2(x-3)&=y+4\\2x-6&=y+4\\2x-6+6&=y+4+6\\2x&=y+10 && \text{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.

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 2x-y=10 && 2x-y=10\\& 2x-({\color{red}0})=10 && 2({\color{red}0})-y=10\\& 2x=10 && 0-y=10\\& \frac{2x}{2}=\frac{10}{2} && \frac{-y}{-1}=\frac{10}{-1}\\& x=5 && y=-10\\& \text{The} \ x \text{-intercept is} \ (5, 0) && \text{The} \ y \text{-intercept is} \ (0, -10)

(ii)

3x+\frac{2}{3}y-3&=0 && \text{Simplify the equation.}\\3(3x)+3\left(\frac{2}{3}\right)y-3(3)&=3(0) && \text{Multiply each term by 3.}\\3(3x)+\cancel{3}\left(\frac{2}{\cancel{3}}\right)y-3(3)&=3(0)\\9x+2y-9&=0\\9x+2y-9+9&=0+9\\9x+2y&=9

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 9x+2y=9 && 9x+2y=9\\& 9x+2({\color{red}0})=9 && 9({\color{red}0})+2y=9\\& 9x+0=9 && 0+2y=9\\& \frac{9x}{9}=\frac{9}{9} && \frac{2y}{2}=\frac{9}{2}\\& x=1 && y=4.5\\& \text{The} \ x \text{-intercept is} \ (1, 0) && \text{The} \ y \text{-intercept is} \ (0, 4.5)

2.

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 5x+2y=-10 && 5x+2y=-10\\& 5x+2({\color{red}0})=-10 && 5({\color{red}0})+2y=-10\\& 5x+0=-10 && 0+2y=-10\\& \frac{5x}{5}=\frac{-10}{5} && \frac{2y}{2}=\frac{-10}{2}\\& x=-2 && y=-5\\& \text{The} \ x \text{-intercept is} \ (-2, 0) && \text{The} \ y \text{-intercept is} \ (0, -5)

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)

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 2x+y=6 && 2x+y=6\\& 2x+({\color{red}0})=6 && 2({\color{red}0})+y=6\\& 2x=6 && 0+y=6\\& \frac{2x}{2}=\frac{6}{2} && y=6\\& x=3\\& \text{The} \ x \text{-intercept is} \ (3, 0) && \text{The} \ y \text{-intercept is} \ (0, 6)\\& \text{This matches the graph.} && \text{This does not match the graph.}

2x+y=6 is not the linear function for the graph.

(ii)

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{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({\color{red}0})=12 && 4({\color{red}0})-3y=12\\& 4x-0=12 && 0-3y=12\\& 4x=12 && -3y=12\\& \frac{4x}{4}=\frac{12}{4} && \frac{-3y}{-3}=\frac{12}{-3}\\& x=3 && y=-4\\& \text{The} \ x \text{-intercept is} \ (3, 0) && \text{The} \ y \text{-intercept is} \ (0, -4)\\& \text{This matches the graph.} && \text{This matches the graph.}

4x-3y-12=0 is the linear function for the graph.

(iii)

& \text{Let} \ y = 0. \ \text{Solve for} \ x. && \text{Let} \ x = 0. \ \text{Solve for} \ y.\\& 5x+3y=15 && 5x+3y=15\\& 5x+3({\color{red}0})=15 && 5({\color{red}0})+3y=15\\& 5x+0=15 && 0+3y=15\\& 5x=15 && 3y=15\\& \frac{5x}{5}=\frac{15}{5} && \frac{3y}{3}=\frac{15}{3}\\& x=3 && y=5\\& \text{The} \ x \text{-intercept is} \ (3, 0) && \text{The} \ y \text{-intercept is} \ (0, 5)\\& \text{This matches the graph.} && \text{This does not match the graph.}

5x+3y=15 is not the linear function for the graph.

Practice

For 1-10, complete the following table:

Function x -intercept y -intercept
7x-3y=21 1. 2.
8x-3y+24=0 3. 4.
\frac{x}{4}-\frac{y}{2}=3 5. 6.
7x+2y-14=0 7. 8.
\frac{2}{3}x-\frac{1}{4}y=-2 9. 10.

Use the intercept method to graph each of the linear functions in the above table.

  1. 7x-3y=21
  2. 8x-3y+24=0
  3. \frac{x}{4}-\frac{y}{2}=3
  4. 7x+2y-14=0
  5. \frac{2}{3}x-\frac{1}{4}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
  1. .

  1. .

  1. .

  1. .

Vocabulary

x-intercept

x-intercept

An x-intercept is a location where a graph crosses the x-axis. As a coordinate pair, this point will always have the form (x, 0). x-intercepts are also called solutions, roots or zeros.
y-intercept

y-intercept

A y-intercept is a location where a graph crosses the y-axis. As a coordinate pair, this point will always have the form (0, y).
Intercept

Intercept

The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.
Intercept Method

Intercept Method

The intercept method is a way of graphing a linear function by using the coordinates of the x and y-intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and then joining them with a straight line.

Image Attributions

Description

Difficulty Level:

Advanced

Grades:

Date Created:

Apr 30, 2013

Last Modified:

Feb 26, 2015
Files can only be attached to the latest version of Modality

Reviews

Help us create better content by rating and reviewing this modality.
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALG.453.L.2

Original text