Answer
Option $(J)$
Work Step by Step
RECALL:
(i)
$y=-f(x)$ involves a reflection about the x-axis of the parent function $y=f(x)$.
(ii)
$y=af(x)$ involves either a vertical compression by a factor of $a$ of the parent function $f(x)$ when $a\gt 1$or a vertical stretch if $0\lt a \lt1$.
(iii)
$y=f(x) + k$ involves either a vertical shift of $k$ units upward of the parent function when $k\gt 0$ or $|k|$ units downward when $k \lt0$.
The graph involves parabola, so its parent function is $y=x^2$.
Note that compared to the parent function's graph whose vertex is at $(0, 0)$, opens upward, and contains the point $(1, 1)$, the given graph opens downward, has its vertex at $(0, 0)$, and contains the point $(1, -2)$.
This implies that the given graph involves the following transformations of the parent function $f(x)=|x|$:
(1) a reflection about the x-axis.
This means the tentative equation of the graph is $y=-x^2$.
(2) a vertical compression by a factor of 2 (since instead of containing the point $(1, -1)$, it contains $(1. -2)$.
This means the equation of the graph is $y=-2x^2$
Therefore, the answer is Option $(J)$.
