Answer
The equation of the function $g\left( x \right)$ is $-\frac{1}{4}\sqrt{16-{{x}^{2}}}-1$.
Work Step by Step
First, plot the graph of the function $f\left( x \right)=\sqrt{16-{{x}^{2}}}$.
It can be observed from the given graph that the graph has undergone a reflection, vertical shrinking and a vertical downward shift.
In the provided graph, the graph of the function $f\left( x \right)$ is first reflected about the x-axis. Thus, the points $\left( -4,0 \right),\left( 0,4 \right)\text{, and }\left( 4,0 \right)$ will be converted to $\left( -4,0 \right),\left( 0,-4 \right)\text{, and }\left( 4,0 \right)$. The new equation after reflecting the graph about the x axis will be as follows:
$\begin{align}
& h\left( x \right)=-f\left( x \right) \\
& =-\sqrt{16-{{x}^{2}}}
\end{align}$
Then, it is vertically shrunk by 4 units. Thus, the points $\left( -4,0 \right),\left( 0,-4 \right)\text{, and }\left( 4,0 \right)$ will be converted to $\left( -4,0 \right),\left( 0,-1 \right)\text{, and }\left( 4,0 \right)$.The equation of the graph after vertical shrinking is as follows:
$\begin{align}
& h'\left( x \right)=\frac{h\left( x \right)}{4} \\
& =\frac{-\sqrt{16-{{x}^{2}}}}{4}
\end{align}$
Finally, it is shifted vertically 1 unit downward . Thus, the points $\left( -4,0 \right),\left( 0,-1 \right)\text{, and }\left( 4,0 \right)$ will be converted to $\left( -4,-1 \right),\left( 0,-2 \right)\text{, and }\left( 4,-1 \right)$.
Thus, we get the equation of the graph after vertical shifting as follows:
$\begin{align}
& g\left( x \right)=h'\left( x \right)-1 \\
& =\frac{-\sqrt{16-{{x}^{2}}}}{4}-1
\end{align}$
