# Finding x-intercepts and y-intercepts

The**intercepts**of a graph are points at which the graph crosses the axes. The

**is the point at which the graph crosses the**

*x-*intercept*x-*axis. At this point, the

*y-*coordinate is zero. The

**is the point at which the graph crosses the**

*y-*intercept*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]. Solution## Licenses & Attributions

### CC licensed content, Specific attribution

- College Algebra.
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