Answer
The graph of $y=-(2)^{x+2}$ involves (i) a $2$-unit shift to the left and (ii) a reflection about the $x$-axis of the parent function $y=2^x$.
The given function has:
domain: $(-\infty, +\infty)$;
range: $y\lt 0$; and
horizontal asymptote: $y=0$
Work Step by Step
Recall:
(1) The graph of the function $y=f(x+h)$ involves a horizontal shift ($h$ units to the left when $h \gt0$, $|h|$ units to the right when $h\lt0$) of the parent function $y=f(x)$.
(2) The graph of the function $y=-f(x)$ involves a reflection about the $x$-axis of the parent function $y=f(x)$.
The given function has $f(x)=2^x$ as its parent function and can be written as $y=-f(x+2)$.
Thus, the graph of the given function involves (i) a $2$-unit shift to the left and (ii) a reflection about the $x$-axis of the parent function $f(x)=\left(\frac{1}{2}\right)^x$.
Recall:
The function is $y=a^x$ has:
(1) domain: $(-\infty, +\infty)$;
(2) range: $y\gt 0$; and
(3) horizontal asymptote: $y=0$
With a horizontal shift $1$ unit to the left and a reflection about the $x$-axis of the parent function $f(x)=2^x$, the function $y=-2^{x+2}$ has:
(1) domain: $(-\infty, +\infty)$;
(2) range: $y\lt 0$; and
(3) horizontal asymptote: $y=0$
