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 that opens downward, 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 $(-2, 0)$, and contains the point $(-1, -1)$. 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 horizontal shift of by 2 units to the left. This means the equation of the graph is $y=-(x+2)^2$ Therefore, the answer is Option $(F)$.