## College Algebra (10th Edition)

$\color{blue}{y=-\sqrt{-x}-2}$
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: $\color{blue}{y=-\sqrt{-x}-2}$