Graph Lines Using Intercepts

The x-intercept is the point where the graph intersects the x-axis and the y-intercept is the point where the graph intersects the y-axis. These points have the form (x, 0) and (0, y) respectively.

To find the x- and y-intercepts algebraically, we use the fact that all x-intercepts have a y-value of zero and all y-intercepts have an x-value of zero. For example,

Graph: 3x − 5y = 15

Tip 1: To find the y-intercept, set x = 0 and determine the corresponding y-value. Similarly, to find the x-intercept we set y = 0 and determine the corresponding x-value.

Keep in mind that the intercepts are ordered pairs and not numbers. In other words, the x-intercept is not x = 5 but rather (5, 0).

Two points determine a line. If we find the x- and y-intercepts, then we can use them to graph the line. As you can see, they are fairly easy to find. Plot the points and draw a line through them with a straightedge.

Done. Let’s do another one.

Graph: y = −x + 9

We begin by finding the x-intercept.

The x-intercept is (3, 0).

The y-intercept is (0, 9). Now graph the two points.

Tip 2: Use Desmos.com to check your answer – it’s totally free. Just type in the equation.

This is a nice and easy method for determining the two points you need for graphing a line. In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.

Graph −4x + 3y = 12 using the intercepts.

Graph −4x + 2y = −6 using the intercepts.

Graph y = −5x +15 using the intercepts.

Graph y = −3/4 x + 9 using the intercepts.

This brings us to one of the most popular questions in linear graphing. Do all lines have x- and y-intercepts? The answer is NO. Horizontal lines, of the form y = b, do not necessarily have x-intercepts. Vertical lines, of the form x = a, do not necessarily have y-intercepts.

Graph y = 3.