Answer
The graph is shown below:
Work Step by Step
To find the x-intercepts of the function equate the function $f\left( x \right)={{x}^{3}}-4{{x}^{2}}-x+4$ to zero:
$\begin{align}
& {{x}^{3}}-4{{x}^{2}}-x+4=0 \\
& {{x}^{2}}\left( x-4 \right)-1\left( x-4 \right)=0 \\
& \left( {{x}^{2}}-1 \right)\left( x-4 \right)=0 \\
& \left( x+1 \right)\left( x-1 \right)\left( x-4 \right)=0
\end{align}$
$x=-1,1,4$ are the x-intercepts.
To find the y-intercept of the function, find the value of $f\left( 0 \right)$.
$\begin{align}
& f\left( 0 \right)=\left( 0+1 \right)\left( 0-1 \right)\left( 0-4 \right) \\
& f\left( 0 \right)=4 \\
\end{align}$
Thus, the y-intercept is 4.
Substitute x with $-x$ to check the symmetry of the function:
$f\left( x \right)={{x}^{3}}-4{{x}^{2}}-x+4$:
$\begin{align}
& f\left( -x \right)={{\left( -x \right)}^{3}}-4{{\left( -x \right)}^{2}}-\left( -x \right)+4 \\
& =-{{x}^{3}}-4{{x}^{2}}+x+4
\end{align}$
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.
