CPM Homework Help : PC3 Problem 5-7

Compute without a calculator Use the graph of $f(x)=-x^2+9$ to sketch the graph of $y=\frac{1}{f(x)}$ without using a graphing calculator. State the equations of the asymptotes in the graph of $y=\frac{1}{f(x)}$ .

Note: This problem was updated from its original version, which was a repeat of a problem in the reciprocal functions lesson.

Step 1:

Graph the $y=$ function and identify the zeros ( $x$ -intercepts).
For the reciprocal function, $\frac{1}{0}$ is undefined, so these are the locations of the vertical asymptotes.
Sketch the vertical asymptotes.

Step 2:

Mark the approximate places where $y=1$ or $−1$ . These points do not change because $\frac{1}{1}=1$ and $\frac{1}{-1}=−1$ .

Step 3:

The reciprocal at the point $(0,9)$ is $\left(0,\frac{1}{9}\right)$ . Mark it.
Draw a smooth curve from it to where $y=−1$ is marked and to where $y=1$ is marked.

Step 4:

When the $y$ -values of the original function are small, $\frac{1}{(\pm \text{small number)}}$ is a ± large number. Show this on your graph.

Step 5:

When the $y$ -values of the original function are large, $\frac{1}{\text{large number}}$ is a small number. Show this on your graph.