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Study Guides > Intermediate Algebra

Graph Using Intercepts

Learning Outcomes

  • Recognize when an ordered pair is a y-intercept or an x-intercept
  • Graph a linear equation using x and y-intercepts

Intercepts

The intercepts of a line are the points where the line intersects or crosses the horizontal and vertical axes. To help you remember what “intercept” means, think about the word “intersect.” The two words sound alike and in this case mean the same thing. The straight line on the graph below intersects the two coordinate axes. The point where the line crosses the x-axis is called the x-intercept. The y-intercept is the point where the line crosses the y-axis. A line going through two points. One point is on the x-axis and is labeled the x-intercept. The other point is on the y-axis and is labeled y-intercept. The x-intercept above is the point [latex](−2,0)[/latex]. The y-intercept above is the point ([latex]0, 2[/latex]). Notice that the y-intercept always occurs where [latex]x=0[/latex], and the x-intercept always occurs where [latex]y=0[/latex]. To find the x and y-intercepts of a linear equation, you can substitute [latex]0[/latex] for y and for x respectively. For example, the linear equation [latex]3y+2x=6[/latex] has an x intercept when [latex]y=0[/latex], so [latex]3\left(0\right)+2x=6[/latex].

[latex]\begin{array}{r}2x=6\\x=3\end{array}[/latex]

The x-intercept is [latex](3,0)[/latex]. Likewise, the y-intercept occurs when [latex]x=0[/latex].

[latex]\begin{array}{r}3y+2\left(0\right)=6\\3y=6\\y=2\end{array}[/latex]

The y-intercept is [latex](0,2)[/latex].

Using Intercepts to Graph Lines

You can use intercepts to graph linear equations. Once you have found the two intercepts, draw a line through them. Do this with the equation [latex]3y+2x=6[/latex]. You figured out that the intercepts of the line this equation represents are [latex](0,2)[/latex] and [latex](3,0)[/latex]. That is all you need to know. A line drawn through the points (0,2) and (3,0). The point (0,2) is labeled y-intercept and the point (3,0) is labeled x-intercept. The line is labeled 3y+2x=6.

Example

Graph [latex]5y+3x=30[/latex] using the x and y-intercepts.

Answer: When an equation is in [latex]Ax+By=C[/latex] form, you can easily find the x- and y-intercepts and then graph. To find the y-intercept, set [latex]x=0[/latex] and solve for y.

[latex]\begin{array}{r}5y+3x=30\\5y+3\left(0\right)=30\\5y+0=30\\5y=30\\y=\,\,\,6\\y\text{-intercept}\,\left(0,6\right)\end{array}[/latex]

To find the x-intercept, set [latex]y=0[/latex] and solve for x.

[latex]\begin{array}{r}5y+3x=30\\5\left(0\right)+3x=30\\0+3x=30\\3x=30\\x=10\\x\text{-intercept}\left(10,0\right)\end{array}[/latex]

The graph can be seen below.

https://youtu.be/k8r-q_T6UFk

Example

Graph [latex]y=2x-4[/latex] using the x and y-intercepts.

Answer: First, find the y-intercept. Set x equal to zero and solve for y.

[latex]\begin{array}{l}y=2x-4\\y=2\left(0\right)-4\\y=0-4\\y=-4\\y\text{-intercept}\left(0,-4\right)\end{array}[/latex]

To find the x-intercept, set [latex]y=0[/latex] and solve for x.

[latex]\begin{array}{l}y=2x-4\\0=2x-4\\4=2x\\x=2\\x\text{-intercept}\left(2,0\right)\end{array}[/latex]

The graph can be seen below. Line passing through (0,-4) and (2,0)