Graphing and Analyzing Linear Equations: Learn It 3

Using Intercepts to Plot Lines in the Coordinate Plane

The intercepts of a graph are points where the graph crosses the axes. The [latex]x[/latex]–intercept is the point where the graph crosses the x-axis. At this point, the y-coordinate is zero. The [latex]y[/latex]–intercept is the point where the graph crosses the [latex]y[/latex]–axis. At this point, the [latex]x[/latex]–coordinate is zero.

intercepts

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

  • The [latex]x[/latex]-intercept is the point at which the graph crosses the [latex]x[/latex]-axis. At this point, the [latex]y[/latex]-coordinate is zero.
  • The [latex]y[/latex]-intercept is the point at which the graph crosses the [latex]y[/latex]-axis. At this point, the [latex]x[/latex]-coordinate is zero.
  • Find the [latex]x[/latex]-intercept by setting [latex]y=0[/latex] and solving for [latex]x[/latex].
  • Find the [latex]y[/latex]–intercept by setting [latex]x=0[/latex] and solving for [latex]y[/latex].
Find the intercepts of the equation [latex]y=3x - 1[/latex].

  • To find the [latex]x[/latex]–intercept, set [latex]y=0[/latex].
[latex]\begin{array}{llllll}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 [latex]y[/latex]–intercept, set [latex]x=0[/latex].
[latex]\begin{array}{lllll}y=3x - 1\hfill & \hfill \\ y=3\left(0\right)-1\hfill & \hfill \\ y=-1\hfill & \hfill \\ \left(0,-1\right)\hfill & y\text{-intercept}\hfill \end{array}[/latex]

Find the intercepts of the equation [latex]y=-3x - 4[/latex]. Then sketch the graph using only the intercepts.

Using a Graphing Utility to Plot Lines

Graphing utilities are powerful tools that allow you to visualize mathematical concepts and plot lines quickly and accurately. Whether you’re checking your work, exploring different equations, or just trying to understand how changes in variables affect a graph, these online tools can help. Below are some popular graphing utilities you can use to plot lines and analyze functions. Simply enter your equation, and the utility will generate the graph for you.

https://www.geogebra.org/graphing

https://www.desmos.com/calculator

https://www.mathway.com/Graph

https://www.symbolab.com/graphing-calculator

Try it now

These graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:

  1. Graph the line [latex]y=-\frac{2}{3}x-\frac{4}{3}[/latex].
  2. On the next line enter [latex]y=-a x-\frac{4}{3}[/latex]. You will see a button pop up that says “add slider: a”, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.
  3. What part of a line does the variable a represent? The slope or the y-intercept?