## Precalculus (10th Edition)

$J$.
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 then stretched by a number larger than $1$, which from the options given, must be $2$. Use the rules listed above to find the equation of the given graph. (1) Reflecting $f(x)=x^2$ about the $x$-axis (Rule (4) above) makes the equation of the resulting function $y=-f(x) = -x^2$. (2) Stretching the graph vertically by $2$ (Rule (3) above) makes the equation of the resulting function $y=-2x^2$. Therefore the answer is $J$. 