Answer
Graph the function as:
Work Step by Step
Step 1: Substitute $x=-x.$
$\begin{align}
& f\left( x \right)=\frac{x-4}{{{x}^{2}}-x-6} \\
& f\left( -x \right)=\frac{\left( -x \right)-4}{{{\left( -x \right)}^{2}}-\left( -x \right)-6} \\
& =\frac{-\left( x+4 \right)}{{{x}^{2}}+x-6}
\end{align}$
and
$\begin{align}
& -f\left( x \right)=-\left( \frac{x-4}{{{x}^{2}}-x-6} \right) \\
& \ne f\left( -x \right)
\end{align}$
Hence, the graph of the function is symmetric about neither $y\text{-}$ axis nor origin.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$.
$\begin{align}
& \frac{x-4}{{{x}^{2}}-x-6}=0 \\
& x=4
\end{align}$
Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$.
$f\left( 0 \right)=\frac{2}{3}$
Step 4: Since the degree of the numerator is less than the denominator, there is no horizontal asymptote.
Step 5: For the vertical asymptote, equate the denominator to 0.
$x=-2,3$
