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Sketch the graph of the following function:

\[ f(x) = \frac{x^2 - 16}{x^2 - 9} \]

To do so, carry out the following steps.

\begin{enumerate}
    \item Factor the numerator and denominator. Where is $f(x) = 0$? Where is $f(x)$ not defined?
    \item Take the limit of $f$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$. Are there any horizontal asymptotes? If so, what are they?
    \item Are there any vertical asymptotes? If so, where are they? Determine what the graph of $f$ looks like near the asymptotes by taking the one-sided limits of $f$ as $x$ approaches the asymptote.
    \item Take the derivative of $f$. Where is $f$ increasing and decreasing? Where does $f$ have critical points?
    \item Take the second derivative of $f$. Where is $f$ concave up and down? Where does $f$ have inflection points?
\end{enumerate}

Collect all of your answers in the following table. Use interval notation for all of the answers for which it is relevant.

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\begin{tabular}{|c|c|}\hline
     \textbf{Zeros} & \hspace{250pt}  \\\hline
     \textbf{Vertical Asymptotes} & \hspace{250pt} \\\hline
     \textbf{Horizontal Asymptotes} & \hspace{250pt} \\\hline
     \textbf{Increasing} & \hspace{250pt} \\\hline
     \textbf{Decreasing} & \hspace{250pt} \\\hline
     \textbf{Critical Points} & \hspace{250pt} \\\hline
     \textbf{Concave Up} & \hspace{250pt} \\\hline
     \textbf{Concave Down} & \hspace{250pt} \\\hline
     \textbf{Inflection Points} & \hspace{250pt} \\\hline
\end{tabular}
\end{center}

Now, use this information to sketch the graph of the function $y=f(x)$.
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