## College Algebra (10th Edition)

$y=-\sqrt{x-3}-2$
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 graph involves the following transformations of the parent function $f(x)=\sqrt{x}$: (1) Reflection about the x-axis: Using (1) above gives: $y=-f(x) \\y=-\sqrt{x}$ (2) Shifted 3 units to the right. Using (iv) above gives: $y=f(x-h)$ where $h=3$. Thus, the equation is: $\\y=-\sqrt{x-3}$ (3) Shifted down $2$ units: Using (v) above gives: $y=f(x) +k$ where $k=-2$. Thus, the equation is: $\\y=-\sqrt{x-3}-2$ Therefore, the function is: $y=-\sqrt{x-3}-2$