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3.5: Graphs of Linear Functions from Intercepts

Difficulty Level: At Grade Created by: CK-12

What are the intercepts of \begin{align*}4x+2y=8\end{align*}? How could you use the intercepts to quickly graph the function?

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 \begin{align*}x\end{align*}-intercept and the \begin{align*}y\end{align*}-intercept. Graphing a linear function by plotting the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*} intercepts is often referred to as the intercept method.

The \begin{align*}x\end{align*}-intercept is where the graph crosses the \begin{align*}x\end{align*}-axis. Its coordinates are \begin{align*}(x, 0)\end{align*}. Because all \begin{align*}x\end{align*}-intercepts have a \begin{align*}y\end{align*} coordinate equal to 0, you can find an \begin{align*}x\end{align*}-intercept by substituting 0 for \begin{align*}y\end{align*} in the equation and solving for \begin{align*}x\end{align*}.

The \begin{align*}y\end{align*}-intercept is where the graph crosses the \begin{align*}y\end{align*}-axis. Its coordinates are \begin{align*}(0, y)\end{align*}. Because all \begin{align*}y\end{align*}-intercepts have a \begin{align*}x\end{align*} coordinate equal to 0, you can find an \begin{align*}y\end{align*}-intercept by substituting 0 for \begin{align*}x\end{align*} in the equation and solving for \begin{align*}y\end{align*}.

Example A

Identify the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts for each line.

(a) \begin{align*}2x+y-6=0\end{align*}

(b) \begin{align*}\frac{1}{2}x-4y=4\end{align*}

Solution:

(a)

(b)

Example B

Use the intercept method to graph \begin{align*}2x-3y=-12\end{align*}

Solution:

Example C

Use the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts of the graph to identify the linear function that matches the graph.

The \begin{align*}x\end{align*}-intercept is (-8, 0) and the \begin{align*}y\end{align*}-intercept is (0, 4).

Solution:

This does not match the graph. This does not match the graph. Confirm the \begin{align*}y\end{align*}-intercept.

The \begin{align*}y\end{align*}-intercept is (0, 4). This matches the graph.

The linear function that matches the graph is

Concept Problem Revisited

The linear function \begin{align*}4x+2y=8\end{align*} can be graphed by using the intercept method.

Plot the \begin{align*}x\end{align*}-intercept on the \begin{align*}x\end{align*}-axis and the \begin{align*}y\end{align*}-intercept on the \begin{align*}y\end{align*}-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 \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts. The graph is drawn by plotting these coordinates on the Cartesian plane and joining them with a straight line.
\begin{align*}x\end{align*}-intercept
A \begin{align*}x\end{align*}-intercept of a relation is the \begin{align*}x-\end{align*}coordinate of the point where the relation intersects the \begin{align*}x\end{align*}-axis.
\begin{align*}y\end{align*}-intercept

A \begin{align*}y\end{align*}-intercept of a relation is the \begin{align*}y-\end{align*}coordinate of the point where the relation intersects the \begin{align*}y\end{align*}-axis.

Guided Practice

1. Identify the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts of the following linear functions:

(i) \begin{align*}2(x-3)=y+4\end{align*}
(ii) \begin{align*}3x+\frac{2}{3}y-3=0\end{align*}

2. Use the intercept method to graph the following relation:

(i) \begin{align*}5x+2y=-10\end{align*}

3. Use the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts of the graph, to match the graph to its function.

(i) \begin{align*}2x+y=6\end{align*}
(ii) \begin{align*}4x-3y-12=0\end{align*}
(iii) \begin{align*}5x+3y=15\end{align*}

1. (i)

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.

(ii)

2. \begin{align*}5x+2y=-10\end{align*} Determine the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts.

3. Identify the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts from the graph.

The \begin{align*}x\end{align*}-intercept is (3, 0)

The \begin{align*}y\end{align*}-intercept is (0, -4)

Determine the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-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)

(ii)

(iii)

Practice

For 1-10, complete the following table:

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

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

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

Use the \begin{align*}x-\end{align*} and \begin{align*}y\end{align*}-intercepts to match each graph to its function.

a.\begin{align*}7x+5y-35=0\end{align*}
b.\begin{align*}y=5x+10\end{align*}
c.\begin{align*}2x+4y+8=0\end{align*}
d.\begin{align*}2x+y=2\end{align*}

Date Created:

Dec 19, 2012

Apr 29, 2014
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