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Guías de estudio > College Algebra

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 y=3x−1y=3x - 1. To find the x-intercept, set y=0y=0.
y=3x−10=3x−11=3x13=x(13,0)x-intercept\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}
To find the y-intercept, set x=0x=0.
y=3x−1y=3(0)−1y=−1(0,−1)y-intercept\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}
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.
This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x – 1 is plotted on the coordinate plane Figure 12

How To: Given an equation, find the intercepts.

  1. Find the x-intercept by setting y=0y=0 and solving for xx.
  2. Find the y-intercept by setting x=0x=0 and solving for yy.

Example 4: Finding the Intercepts of the Given Equation

Find the intercepts of the equation y=−3x−4y=-3x - 4. Then sketch the graph using only the intercepts.

Solution

Set y=0y=0 to find the x-intercept.
y=−3x−40=−3x−44=−3x−43=x(−43,0)x-intercept\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}
Set x=0x=0 to find the y-intercept.
y=−3x−4y=−3(0)−4y=−4(0,−4)y-intercept\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}
Plot both points, and draw a line passing through them as in Figure 11.
This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4/3, 0) and (0, -4). Figure 13

Try It 1

Find the intercepts of the equation and sketch the graph: y=−34x+3y=-\frac{3}{4}x+3. Solution
 

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