Answer
The function is only symmetric about the $y$-axis
Work Step by Step
$f(x)=x^{4}+5x^{2}-12$
$\textbf{Symmetry about the $x$-axis}$
This function has no symmetry about the $x$-axis because if it had, it would violate the Vertical Rule Test. Another way to realize this is that changing $f(x)$ by $-f(x)$ does not yield an equivalent function.
$\textbf{Symmetry about the $y$-axis}$
Substitute $x$ by $-x$ in $f(x)$ and simplify:
$f(-x)=(-x)^{4}+5(-x)^{2}-12=...$
$...=x^{4}+5x^{2}-12$
Since $f(-x)=f(x)$, this function is symmetric about the $y$-axis.
$\textbf{Symmetry about the origin}$
This function has no symmetry about the origin beacuse, as it was seen when testing for $y$-axis symmetry, $f(-x)$ is not equal to $-f(x)$.
The graph of this function is:
