Answer
The required equation of the parabola is \[f\left( x \right)=-2{{\left( x+3 \right)}^{2}}-1\]
Work Step by Step
Now, putting in the value of vertex, that is, $\left( -3,-1 \right)$ in the general form of parabola, we get:
$\begin{align}
& f\left( x \right)=a{{\left( x-\left( -3 \right) \right)}^{2}}+\left( -1 \right) \\
& f\left( x \right)=a{{\left( x+3 \right)}^{2}}-1 \\
\end{align}$
Since the graph of $f\left( x \right)$ passes through the point $\left( -2,-3 \right)$, thus, the point $\left( -2,-3 \right)$ satisfies the equation of $f\left( x \right)$.
Thus,
$\begin{align}
& -3=a{{\left( -2+3 \right)}^{2}}-1 \\
& =a{{\left( 1 \right)}^{2}}-1 \\
& =a-1
\end{align}$
Simplifying,
$\begin{align}
& a-1=-3 \\
& a=-2
\end{align}$
Now, putting in the value of a in $f\left( x \right)=a{{\left( x+3 \right)}^{2}}-1$.
Therefore, the equation of the parabola is $f\left( x \right)=-2{{\left( x+3 \right)}^{2}}-1$
