Answer
Intercepts: $(0,4)$.
The graph has y-axis symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=x^{2}+4 & y=0^{2}+4\\
x^{2}=-4 & y=4\\
none & \\
&
\end{array}\right]$
Intercepts: $(0,4)$.
Test x-axis symmetry: Replace $y$ with $-y$
$-y=x^{2}+4$ ... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$ y=(-x)^{2}+4$
$y=x^{2}+4$ ...same as initial equation.
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y$.
$-y=(-x)^{2}+4$
$-y=x^{2}+4$ ... different from initial equation