Graph with Intercepts

Identify the $x$- and $y$-intercepts on a graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same line. One way to recognize that they are indeed the same line is to look at where the line crosses the $x$-axis and the $y$-axis. These points are called the intercepts of the line.

Let’s look at the graphs of the lines in the figure below.

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 a line. The $x$-intercept occurs when $y$ is zero.

Now let’s look at the points where these lines cross the $y$-axis:

What is the pattern here? In each row, the $x$-coordinate of the point where the line crosses the $y$-axis is zero. The point where the line crosses the $y$-axis has the form $(0, b)$ and is called the $y$-intercept of the line. The $y$-intercept occurs when $x$ is zero.

Example. Find the $x$- and $y$-intercepts on each graph.

(a) 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 $(0, 2)$.

(b) 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 $(0, -6)$.

(c) 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 $x$- and $y$-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 $2x + y = 6$.

We will let $y = 0$ to find the $x$-intercept, and let $x = 0$ to find the $y$-intercept. We fill in a table, which reminds us of what we need to find.

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 $(0, 6)$, as shown in the table below.

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 need to find three points whose coordinates are solutions to the equation. You can use the $x$- and $y$-intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up — then draw the line. This method is often the quickest way to graph a line.

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

Step 1. Find the $x$- and $y$-intercepts of the line. Let $y = 0$ and solve for $x$; let $x = 0$ and solve for $y$.

Find the $x$-intercept. Let $y = 0$:

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

Find the $y$-intercept. Let $x = 0$:

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

Step 2. Find another solution to the equation. We’ll use $x = 2$:

A third point is $(2, 4)$.

Step 3. Plot the three points. Check that the points line up.

Step 4. Draw the line.

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

Find the intercepts and a third point.

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

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

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

We list the points in the table below and show the graph.

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

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

$y$-intercept, let $x = 0$: $y = 5 \cdot 0$, so $y = 0$. The $y$-intercept is $(0, 0)$.

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

To ensure accuracy we need to plot three points. Since the $x$- and $y$-intercepts are the same point, we need two more points to graph the line.

Let $x = 1$: $y = 5(1) = 5$. Let $x = -1$: $y = 5(-1) = -5$.

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

When an equation has both $x$ and $y$ on the same side, as in $Ax + By = C$, finding the intercepts is often the fastest way to graph the line — but it’s not the only method. If $y$ is already isolated, plotting points directly is usually quicker; if the equation has only one variable, its graph is a vertical or horizontal line. Choosing the intercept method makes the most sense when the equation is already written with both variables on one side, since setting $x = 0$ or $y = 0$ in turn makes the arithmetic simple.

Key terms

intercepts of a line — the points where a line crosses the $x$-axis and the $y$-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 Elementary Algebra 2e, Section 4.3: Graph 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, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.