Graphing a Line using the x and y-Intercepts

Another method to graph a line in the $XY$ -plane is to use the intercepts. What are intercepts? These are points of the line that are found on the $\color{red}\large{x}$ and $\color{red}\large{y}$ axes. There are two kinds of intercepts.

The first one is called the x-intercept because it is the point of the line located on the horizontal axis ( $x$ -axis).

The second is the y-intercept which is the point of the line located on the vertical axis ( $y$ -axis).

Here is a quick diagram that gives you the idea.

x intercept described by the point (-3,0); y intercept described by the point (0,2)

Since the $x$ -intercept is a point where the line crosses the $x$ -axis, it is a point with a $y$ -value of zero.

for x-intercept, y is always equal to zero

In the same manner, since the $y$ -intercept is a point where the line crosses the $y$ -axis, it must be a point with an $x$ -value of zero.

for y-intercept, x is always equal to zero.

Using the informal definitions of $x$ and $y$ -intercepts above, it makes a lot of sense why the procedures below on how to find them work!

Rules on How to Find the Intercepts

To find the x-intercept:

Let $y = 0$ in the equation, then solve for $x$ .

To find the y-intercept:

Let $x = 0$ in the equation, then solve for $y$ .

Examples of How to Graph a Line using the x and y-intercepts

Example 1: Graph the equation of the line $2x-4y=8$ using its intercepts.

I hope you recognize that this is an equation of a line in Standard Form where both the $x$ and $y$ variables are found on one side of the equation opposite the constant term. It is a common practice in an algebra class to ask students to graph the line using the intercept method when the line is in Standard Form.

Here we go!

To find the x-intercept:

Let $y=0$ in the equation, then solve for $x$ .

The $x$ -intercept is (4, 0).

To find the y-intercept:

Let $x=0$ in the equation, then solve for $y$ .

The y-intercept is (0, –2).

Now we can plot the two points on the $xy$ axis and connect them using a straight edge ruler to show the graph of the line.

two points plotted on an xy axis. the x intercept located at (4,0) and y intercept located at (0,-2)

Example 2: Graph the equation of the line using its intercepts.

This equation of the line is in the Slope-Intercept Form. We can actually graph this using another technique which uses the slope and the $y$ -intercept taken directly from the equation. You can see a separate lesson on how to graph a line using slope and y-intercept.

Since this lesson is about intercepts, let’s work this out using this method.

To find the x-intercept:

Let $y=0$ in the equation, then solve for $x$ .

The $x$ -intercept is (–2, 0).

To find the y-intercept:

Let $x=0$ in the equation, then solve for $y$ .

The $y$ -intercept is (0, 3).

Plot the intercepts in the axes and draw a straight line passing through them using a ruler.

two intercepts plotted on an xy axis. x intercept at (-2,0) and y intercept at (0,3)

You might also be interested in:

Three Ways to Graph a Line
Graphing a Line using Slope and y-intercept
Graphing a Line using Table of Values