Choosing the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.

Here are six equations we graphed in this tutorial, and the method we used to graph each of them.

\(\begin{array}{cccccccc}& & & \mathbf{\text{Equation}}\hfill & & \phantom{\rule{5em}{0ex}}& & \mathbf{\text{Method}}\hfill \\ \text{#1}\hfill & & & x=2\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Vertical line}\hfill \\ \text{#2}\hfill & & & y=4\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Horizontal line}\hfill \\ \text{#3}\hfill & & & \text{−}x+2y=6\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Intercepts}\hfill \\ \text{#4}\hfill & & & 4x-3y=12\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Intercepts}\hfill \\ \text{#5}\hfill & & & y=4x-2\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Slope–intercept}\hfill \\ \text{#6}\hfill & & & y=\text{−}x+4\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Slope–intercept}\hfill \end{array}\)

Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #3 and #4, both \(x\) and \(y\) are on the same side of the equation. These two equations are of the form \(Ax+By=C\). We substituted \(y=0\) to find the x-intercept and \(x=0\) to find the y-intercept, and then found a third point by choosing another value for \(x\) or \(y\).

Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.

This leads to the following strategy.

Strategy for Choosing the Most Convenient Method to Graph a Line

Consider the form of the equation.

If it only has one variable, it is a vertical or horizontal line.
- \(x=a\) is a vertical line passing through the x-axis at \(a\).
- \(y=b\) is a horizontal line passing through the y-axis at \(b\).
If \(y\) is isolated on one side of the equation, in the form \(y=mx+b\), graph by using the slope and y-intercept.
- Identify the slope and y-intercept and then graph.
If the equation is of the form \(Ax+By=C\), find the intercepts.
- Find the x- and y-intercepts, a third point, and then graph.

Example

Determine the most convenient method to graph each line.

\(y=-6\)
\(5x-3y=15\)
\(x=7\)
\(y=\frac{2}{5}x-1\).

Solution

\(y=-6\)
This equation has only one variable,\(y\). Its graph is a horizontal line crossing the y-axis at \(-6\).
\(5x-3y=15\)
This equation is of the form \(Ax+By=C\). The easiest way to graph it will be to find the intercepts and one more point.
\(x=7\)
There is only one variable, \(x\). The graph is a vertical line crossing the x-axis at 7.
\(y=\frac{2}{5}x-1\)
Since this equation is in \(y=mx+b\) form, it will be easiest to graph this line by using the slope and y-intercept.

Choosing the Most Convenient Method to Graph a Line