Answer
$f(x)=2(x-3)^2-2
$
Work Step by Step
Given the standard function, $
f(x)=x^2
$, the equation of the graph that is stretched vertically has a constant multiplier of $\dfrac{1}{c}$ in the $y-$variable. Hence, a vertical stretching of a factor of $2$ units has the equation,
\begin{array}{l}\require{cancel}
\dfrac{1}{2}f(x)=x^2
\\\\
f(x)=2x^2
.\end{array}
Given the function, $
f(x)=2x^2
$, the equation of the graph that shifts down has a negative constant added to the equation. Hence, a shift of $2$ units down has the equation
\begin{array}{l}\require{cancel}
f(x)=2x^2-2
.\end{array}
Given the function, $
f(x)=2x^2-2
$, the equation of the graph that shifts to the right has a negative constant added to the $x-$variable. Hence, a shift of $3$ units to the right has the equation
\begin{array}{l}\require{cancel}
f(x)=2(x-3)^2-2
.\end{array}
