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