Answer
The graph of the function is:
Work Step by Step
Step 1: Substitute $x=-x$.
$\begin{align}
& f\left( x \right)=\frac{4{{x}^{2}}}{{{x}^{2}}-9} \\
& f\left( -x \right)=\frac{4{{\left( -x \right)}^{2}}}{{{\left( -x \right)}^{2}}-9} \\
& =\frac{4{{x}^{2}}}{{{x}^{2}}-9} \\
& =f\left( x \right)
\end{align}$
Therefore, the function $f\left( -x \right)$ is equal to $f\left( x \right)$. So, the graph of the function is either symmetrical about the $y$ axis or origin.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$.
$\begin{align}
& \frac{4{{x}^{2}}}{{{x}^{2}}-9}=0 \\
& x=0
\end{align}$
Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$
$\begin{align}
& f\left( 0 \right)=\frac{4\left( 0 \right)}{\left( 0 \right)-9} \\
& f\left( 0 \right)=0 \\
\end{align}$
Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is:
$y=4$.
Step 5: For the vertical asymptote, equate the denominator to 0.
$\begin{align}
& {{x}^{2}}-9=0 \\
& x=\pm 3
\end{align}$
