Answer
The graph of the function is:
Work Step by Step
Step 1: Substitute $x=-x$
$\begin{align}
& f\left( x \right)=\frac{2x}{{{x}^{2}}-4} \\
& f\left( -x \right)=\frac{2\left( -x \right)}{{{\left( -x \right)}^{2}}-4} \\
& =\frac{-2x}{{{x}^{2}}-4} \\
& =-f\left( x \right)
\end{align}$
Therefore, the function $f\left( -x \right)$ is equal to $-f\left( x \right)$. So, the graph of the function is symmetrical about the $y$ axis and origin.
Step 2: To calculate the x intercepts, equate $f\left( x \right)=0$.
$\begin{align}
& \frac{2x}{{{x}^{2}}-4}=0 \\
& x=0
\end{align}$
Step 3: To calculate the y intercepts, evaluate $f\left( 0 \right)$.
$\begin{align}
& f\left( 0 \right)=\frac{2\left( 0 \right)}{{{\left( 0 \right)}^{2}}-4} \\
& f\left( 0 \right)=0 \\
\end{align}$
Step 4: Since the degree of the numerator is less than the denominator, there is no horizontal asymptote.
Step 5: For the vertical asymptote, equate the denominator to 0.
$\begin{align}
& {{x}^{2}}-4=0 \\
& x=\pm 2
\end{align}$
