Answer
$f(x)=(x+4)^2-2$.
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 $f(x)$.
Use the rules listed above to find the equation of the given graph.
(1) Shifting the graph horizontally $4$ units to the left (Rule (1) above) makes the equation of the resulting function $f(x)=(x+4)^2$.
(2) Shifting the graph vertically $2$ units down (Rule (2) above) makes the equation of the resulting function $f(x)=(x+4)^2-2$.
