Finding x-intercepts and y-intercepts
The intercepts of a graph are points at which the graph crosses the axes. The x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero. To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation [latex]y=3x - 1[/latex]. To find the x-intercept, set [latex]y=0[/latex].
[latex]\begin{array}{ll}y=3x - 1\hfill & \hfill \\ 0=3x - 1\hfill & \hfill \\ 1=3x\hfill & \hfill \\ \frac{1}{3}=x\hfill & \hfill \\ \left(\frac{1}{3},0\right)\hfill & x\text{-intercept}\hfill \end{array}[/latex]
To find the y-intercept, set [latex]x=0[/latex].
[latex]\begin{array}{l}y=3x - 1\hfill \\ y=3\left(0\right)-1\hfill \\ y=-1\hfill \\ \left(0,-1\right)y\text{-intercept}\hfill \end{array}[/latex]
We can confirm that our results make sense by observing a graph of the equation as in Figure 10. Notice that the graph crosses the axes where we predicted it would.
Figure 12
How To: Given an equation, find the intercepts.
- Find the x-intercept by setting [latex]y=0[/latex] and solving for [latex]x[/latex].
- Find the y-intercept by setting [latex]x=0[/latex] and solving for [latex]y[/latex].
Example 4: Finding the Intercepts of the Given Equation
Find the intercepts of the equation [latex]y=-3x - 4[/latex]. Then sketch the graph using only the intercepts.Solution
Set [latex]y=0[/latex] to find the x-intercept.[latex]\begin{array}{l}y=-3x - 4\hfill \\ 0=-3x - 4\hfill \\ 4=-3x\hfill \\ -\frac{4}{3}=x\hfill \\ \left(-\frac{4}{3},0\right)x\text{-intercept}\hfill \end{array}[/latex]
Set [latex]x=0[/latex] to find the y-intercept.
[latex]\begin{array}{l}y=-3x - 4\hfill \\ y=-3\left(0\right)-4\hfill \\ y=-4\hfill \\ \left(0,-4\right)y\text{-intercept}\hfill \end{array}[/latex]
Plot both points, and draw a line passing through them as in Figure 11.
Figure 13
Try It 1
Find the intercepts of the equation and sketch the graph: [latex]y=-\frac{3}{4}x+3[/latex]. SolutionLicenses & Attributions
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- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.