Graphing with Intercepts

Identify the Intercepts on a Graph

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line.

Let’s look at the graph of the lines shown in Figure 1.

First, notice where each of these lines crosses the x- axis:

Do you see a pattern?

For each row, the y- coordinate of the point where the line crosses the x- axis is zero. The point where the line crosses the x- axis has the form $(a,0)$; and is called the x-intercept of the line. The x- intercept occurs when y is zero.

Now, let's look at the points where these lines cross the y-axis.

Find the $x\text{- and}y\text{-intercepts}$ of each line:

This table displays linear equations and their respective graphical representations on an x-y coordinate plane, demonstrating how algebraic expressions translate visually.

ⓐ $x+2y=4$
ⓑ $3x-y=6$
ⓒ $x+y=\mathrm{-5}$

Solution

Solution

This table illustrates the relationship between a graph's axis crossings and their corresponding mathematical x and y-intercept terms.

ⓐ
The graph crosses the x-axis at the point (4, 0).	The x-intercept is (4, 0).
The graph crosses the y-axis at the point (0, 2).	The y-intercept is (0, 2).

This table illustrates the relationship between the visual description of graph intercepts and their formal definitions and coordinates.

ⓑ
The graph crosses the x-axis at the point (2, 0).	The x-intercept is (2, 0)
The graph crosses the y-axis at the point (0, −6).	The y-intercept is (0, −6).

Examples showing how to derive x- and y-intercepts from descriptions of a graph crossing the axes.

ⓒ
The graph crosses the x-axis at the point (−5, 0).	The x-intercept is (−5, 0).
The graph crosses the y-axis at the point (0, −5).	The y-intercept is (0, −5).

Find the Intercepts from an Equation of a Line

Recognizing that the $x\text{-intercept}$ occurs when $y$ is zero and that the $y\text{-intercept}$ occurs when $x$ is zero gives us a method to find the intercepts of a line from its equation. To find the $x\text{-intercept,}$ let $y=0$ and solve for $x.$ To find the $y\text{-intercept},$ let $x=0$ and solve for $y.$

Find the intercepts of $2x+y=6$

Solution

We'll fill in Figure 2.

To find the x- intercept, let $y=0$:

Step-by-step procedure for finding the x-intercept of an equation.

Substitute 0 for y.
Add.
Divide by 2.
The x-intercept is (3, 0).

To find the y- intercept, let $x=0$:

Steps demonstrating how to find the y-intercept by substituting x=0 and solving the equation.

Substitute 0 for x.
Multiply.
Add.
The y-intercept is (0, 6).

The intercepts are the points $(3,0)$ and $(0,6)$.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

Graph $4x\mathrm{-3}y=12$ using intercepts.

Solution

Solution

Find the intercepts and a third point.

We list the points and show the graph.

$4x\mathrm{-3}y=12$
$x$	$y$	$(x,y)$
$3$	$0$	$(3,0)$
$0$	$\mathrm{-4}$	$(0,\mathrm{-4})$
$6$	$4$	$(6,4)$

Graph $y=5x$ using the intercepts.

Solution

Solution

This line has only one intercept! It is the point $(0,0).$

To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for $x,$ so we’ll let $x$ be $1$ and $\mathrm{-1}.$

Organize the points in a table.

$y=5x$
$x$	$y$	$(x,y)$
$0$	$0$	$(0,0)$
$1$	$5$	$(1,5)$
$\mathrm{-1}$	$\mathrm{-5}$	$(\mathrm{-1},\mathrm{-5})$

Plot the three points, check that they line up, and draw the line.

Choose the Most Convenient Method to Graph a Line

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.

Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form $Ax+By=C$. We substituted $y=0$ and $x=0$ to find the x- and y- intercepts, and then found a third point by choosing a value for x or y.

This leads to the following strategy for choosing the most convenient method to graph a line.