Home > PC3 > Chapter 5 > Lesson 5.1.1 > Problem5-7

5-7.

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.

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.

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

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.

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.

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