Answer
The standard form of the given parabola is \[g\left( x \right)=-3{{\left( x-5 \right)}^{2}}-7\]
Work Step by Step
We know that a quadratic function can be expressed in the standard form as $f\left( x \right)=a{{\left( x-h \right)}^{2}}+k$ corresponding to which the graph is a parabola whose vertex is the point $\left( h,k \right)$. The parabola attains its maximum value at -7 so it must be opening downwards. Thus $a<0$ , therefore, $f\left( h \right)=k$ is the maximum. Therefore, $k=-7$.
The parabola attains its maximum at 5, thus $h=5$
The parabola is downwards so the shape of the parabola is $g\left( x \right)=-3{{x}^{2}}$.
Now, put the obtained results in the standard form to get:
$\begin{align}
& g\left( x \right)=-3{{\left( x-h \right)}^{2}}+k \\
& =-3{{\left( x-5 \right)}^{2}}+\left( -7 \right) \\
& =-3{{\left( x-5 \right)}^{2}}-7
\end{align}$
Hence, the parabola is expressed in standard form as $g\left( x \right)=-3{{\left( x-5 \right)}^{2}}-7$.
