Answer
$\color{blue}{y=-\sqrt{-x}-2}$
Work Step by Step
RECALL:
(i)
$y=-f(x)$ involves a reflection about the x-axis of the parent function $y=f(x)$.
(ii)
$y=f(-x)$ involves a reflection about the y-axis of the parent function $y=f(x)$.
(iii)
$y=af(x)$ involves either a vertical compression by a factor of $a$ of the parent function $f(x)$ when $a\gt 1$ or a vertical stretch when $0 \lt a \lt 1$.
(iv)
$y=f(x-h)$ involves a horizontal shift of either $h$ units to the right of the parent function $f(x)$ when $h \gt 0$ or $|h|$ units to the left when $h \lt0$.
(v)
$y=f(x) + k$ 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 given function involves multiple transformations of the parent function $y=\sqrt{x}$:
(1) Shift up 2 units.
Using rule (v) above gives:
$y=\sqrt{x} + 2$.
(2) Reflection about the x-axis.
Using rule (i) above gives:
$y=-f(x)
\\y= -(\sqrt{x}+2)$
(3) Reflected about the y-axis.
Using rule (ii) above gives:
$y=-(\sqrt{-x} + 2)
\\y=-\sqrt{-x} - 2$
Therefore, after all the transformations were applied to the parent function, the resulting function is:
$\color{blue}{y=-\sqrt{-x}-2}$
