Answer
Intercepts: $(-1,0), (1,0)$, and $(0,-1)$.
The graph has y-axis symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=x^{4}-1 & y=0^{4}-1\\
x^{4}=1 & y=-1\\
x=\pm 1 & \\
&
\end{array}\right]$
Intercepts: $(-1,0), (1,0)$, and $(0,-1)$.
Test x-axis symmetry: Replace $y$ with $-y$
$-y=x^{4}-1$ ... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$y=(-x)^{4}-1$
$y=x^{4}-1$ ...same as initial equation
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y$.
$-y=(-x)^{4}-1$
$-y=x^{4}-1$ ... different from initial equation