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 graphs of four lines and see where each one crosses the axes.

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:

Example. Find the $x$- and $y$-intercepts of each line.

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

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

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

Find the intercepts from an equation of a line

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

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

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

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

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

The $y$-intercept is $(0, 6)$.

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

Example. Find the intercepts of $4x - 3y = 12$.

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

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

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

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

The intercepts are the points $(3, 0)$ and $(0, -4)$.

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.

Example. Graph $-x + 2y = 6$ using intercepts.

First, find the $x$-intercept. Let $y = 0$:

The $x$-intercept is $(-6, 0)$.

Now find the $y$-intercept. Let $x = 0$:

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

Find a third point. We’ll use $x = 2$:

A third solution to the equation is $(2, 4)$.

Summarize the three points in a table:

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

Example. Graph $4x - 3y = 12$ using intercepts.

Find the intercepts and a third point.

To find the $x$-intercept, let $y = 0$: $4x - 3(0) = 12$, so $4x = 12$ and $x = 3$. The $x$-intercept is $(3, 0)$.

To find the $y$-intercept, let $x = 0$: $4(0) - 3y = 12$, so $-3y = 12$ and $y = -4$. The $y$-intercept is $(0, -4)$.

For a third point, let $y = 4$: $4x - 3(4) = 12$, so $4x - 12 = 12$, $4x = 24$, and $x = 6$. A third solution is $(6, 4)$.

Example. Graph $y = 5x$ using the intercepts.

To find the $x$-intercept, let $y = 0$: $0 = 5x$, so $x = 0$. The $x$-intercept is $(0, 0)$.

To find the $y$-intercept, let $x = 0$: $y = 5(0)$, so $y = 0$. The $y$-intercept is $(0, 0)$.

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

To ensure accuracy, we still need to plot three points. Since the $x$-intercept and $y$-intercept are the same point, we need two more points to graph the line. As always, we can choose any values for $x$, so let’s use $x = 1$ and $x = -1$:

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 equations we’ve graphed in this chapter, and the methods we used to graph them.

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

Notice that in the first two equations, $y$ 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.

The next two equations 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 the last two equations, 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.

Example. Identify the most convenient method to graph each line: (a) $y = -3$ (b) $4x - 6y = 12$ (c) $x = 2$ (d) $y = \tfrac{2}{5}x - 1$.

(a) $y = -3$ has only one variable, $y$. Its graph is a horizontal line crossing the $y$-axis at $-3$.

(b) $4x - 6y = 12$ is of the form $Ax + By = C$. Find the intercepts and one more point.

(c) $x = 2$ has only one variable, $x$. The graph is a vertical line crossing the $x$-axis at $2$.

(d) $y = \tfrac{2}{5}x - 1$ has $y$ isolated on the left side of the equation, so it will be easiest to graph this line by plotting three points.

Key terms

intercept of a line — a point where a line crosses an axis. $x$-intercept — the point $(a, 0)$ where a line crosses the $x$-axis; it occurs when $y$ is zero. $y$-intercept — the point $(0, b)$ where a line crosses the $y$-axis; it occurs when $x$ is zero.

This section is adapted from Prealgebra 2e, Section 11.3: Graphing with Intercepts by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the labeled coordinate-grid figures as accessible inline graphics and the intercept summaries as tables; omitted the Be Prepared quiz, Self Check checklist, Manipulative Mathematics and media links, and end-of-section exercises; converted the practice problems (“Try Its”) into interactive exercises with instant feedback, including a graph-it-yourself exercise and a match-the-graph multiple-choice question.