Answer
$I$.
Work Step by Step
RECALL:
(1) The graph of $y=f(x-h)$ involves a horizontal shift of $|h|$ units (to the right when $h \gt 0$, to the left when $h\lt0$) of the parent function $f(x)$.
(2) The graph of $y=f(x)+k$ involves a vertical shift of $|k|$ units (upward when $k \gt 0$, downward when $k\lt0$) of the parent function $f(x)$.
(3) The graph of $y=a \cdot f(x-h)$ involves a vertical stretch or compression (stretch when $a\gt1$, compression when $0\lt a \lt1$) of the parent function $f(x)$.
(4) The graph of $y=-f(x)$ involves a reflection about the $x$-axis of the parent function $f(x)$.
The given graph is a parabola so its parent function is $f(x)=x^2$, the graph of which is a parabola that opens upward. (Refer to the attached image below for the graph of $f(x)=x^2$).
The given graph shows that the graph of the parent function was reflected about the $x$-axis and shifted $2$ units to the left.
Use the rules listed above to find the equation of the given graph.
(1) Here the graph is stretched by a factor larger than $1$, which from the options given, must be $2$. Hence by Rule (3), the function becomes $y=f(x)=2x^2$.
Therefore the answer is $I$.
