2.2 Graphs of linear functions  (Page 7/15)

Writing equations of perpendicular lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the following function:

The slope of the line is 2, and its negative reciprocal is $-\frac{1}{2}.$ Any function with a slope of $-\frac{1}{2}$ will be perpendicular to $f(x).$ So the lines formed by all of the following functions will be perpendicular to $f(x).$

As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to $f(x)$ and passes through the point $\left(4,\text{ 0}\right).$ We already know that the slope is $-\frac{1}{2}.$ Now we can use the point to find the y -intercept by substituting the given values into the slope-intercept form of a line and solving for $b.$

The equation for the function with a slope of $-\frac{1}{2}$ and a y- intercept of 2 is

So $g(x)=-\frac{1}{2}x+2$ is perpendicular to $f\left(x\right)=2x+4$ and passes through the point $\left(4,\text{ 0}\right).$ Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.