Answer
The graph is shown below:
Work Step by Step
To find the x-intercepts of the function, equate the function $f\left( x \right)={{x}^{2}}+2x-8$ to zero.
$\begin{align}
& {{x}^{2}}+2x-8=0 \\
& {{x}^{2}}+4x-2x-8=0 \\
& x\left( x+4 \right)-2\left( x+4 \right)=0 \\
& \left( x+4 \right)\left( x-2 \right)=0
\end{align}$
Therefore, $x=-4$ and $x=2$ are the x-intercepts.
To find the y-intercept of the function, find the value of $f\left( 0 \right)$.
$\begin{align}
& f\left( 0 \right)={{0}^{2}}+2\left( 0 \right)-8 \\
& =-8
\end{align}$
Thus, the y-intercept is -8.
Substitute x with $-x$ to check the symmetry of the function:
$\begin{align}
& f\left( -x \right)={{\left( -x \right)}^{2}}+2\left( -x \right)-8 \\
& ={{x}^{2}}-2x-8
\end{align}$
Since $f\left( x \right)\ne f\left( -x \right)$, the graph is not symmetric with respect to the y-axis and since $f\left( -x \right)\ne -f\left( x \right)$, the graph is not symmetric through the origin.
