Answer
Intercepts: $(0,0)$.
The graph has origin symmetry.
Work Step by Step
$\left[\begin{array}{ll}
\text{x-intercepts:} & \text{y-intercepts:}\\
0=\frac{3x}{x^{2}+9} & y=\frac{3(0)}{0^{2}+9}\\
3x=0 & y=0\\
x=0 & \\
&
\end{array}\right]$
Intercepts: $(0,0)$.
Test x-axis symmetry: Replace $y$ with $-y$
$-y=\displaystyle \frac{3x}{x^{2}+9}$ ... different from initial equation
Test y-axis symmetry: Replace $x$ with $-x$
$ y=\displaystyle \frac{3(-x)}{(-x)^{2}+9}$
$y=-\displaystyle \frac{3x}{x^{2}+9}$ ... different from initial equation
Test origin symmetry: $x\rightarrow-x$ and $y\rightarrow-y$.
$-y=\displaystyle \frac{3(-x)}{(-x)^{2}+9}$
$-y=-\displaystyle \frac{3x}{x^{2}+9}$
$y=\displaystyle \frac{3x}{x^{2}+9}$ ...same as initial equation