Answer
Intercepts: $(-2,0)$ and $($2, $0)$.
The graph has origin symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=\frac{x^{2}-4}{2x} & y=\frac{0^{2}-4}{2(0)}\\
x^{2}-4=0 & undefined\\
x^{2}=4 & \\
x=\pm 2 &
\end{array}\right]$
Intercepts: $(-2,0)$ and $($2, $0)$.
Test x-axis symmetry: Replace $y$ with $-y$
$-y=\displaystyle \frac{x^{2}-4}{2x}$ ... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$ y=\displaystyle \frac{(-x)^{2}-4}{2(-x)}$
$y=-\displaystyle \frac{x^{2}-4}{2x}$ ... different from initial equation
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y$.
$-y=\displaystyle \frac{(-x)^{2}-4}{2(-x)}$
$-y=\displaystyle \frac{x^{2}-4}{-2x}$
$y=\displaystyle \frac{x^{2}-4}{2x}$ ...same as initial equation