Answer
Intercepts: $(4, 0) , (-1,0)$ , and $(0,-4)$.
No symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=x^{2}-3x-4 & y=0^{2}-3(0)-4\\
0=(x-4)(x+1) & y=-4\\
x=4, x=-1 & \\
&
\end{array}\right]$
Intercepts: $(4, 0) , (-1,0)$ , and $(0,-4)$.
Test x-axis symmetry: Replace $y$ with $-y$
$-y=x^{2}-3x-4$ ... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$ y=(-x)^{2}-3(-x)-4$
$y=x^{2}+3x-4$ ... different from initial equation.
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y$.
$-y=(-x)^{2}-3(-x)-4$
$-y=x^{2}+3x-4$ ... different from initial equation