Answer
The function is $g=-{{\left( x+4 \right)}^{2}}.$
Work Step by Step
The graph of $g\left( x \right)$ is obtained by transforming the graph of $f\left( x \right)={{x}^{2}}.$
Consider the graph of the function $f\left( x \right)={{x}^{2}}$ that passes through the origin $\left( 0,0 \right)$.
The graph of the function $f\left( x \right)={{x}^{2}}$ has the vertex at the origin and to shift the vertex from the origin to $\left( -4,0 \right)$, x needs to be replaced. Now, the given function is replaced by ${{x}^{2}}={{\left( x+4 \right)}^{2}}$.
Then, the function becomes $f={{x}^{2}},$ where $c=4.$
So, the function is
$\begin{align}
& f={{\left( x+c \right)}^{2}} \\
& ={{\left( x+4 \right)}^{2}}
\end{align}$
In order to reflect the graph about the x axis, a negative sign needs to be multiplied by the function's values.
