## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$y=\sqrt {-x-3}+2$
In order to find the answer, we will have to recall the following about the graph of $y=f(x)$. a) The graph of the function $y=-f(x)$ involves a reflection about the $x$-axis of the original function $f(x)$. b) The graph of the function $y=f(-x)$ involves a reflection about the $y$-axis of the original function $f(x)$. c) The graph of the function $y=f(x)+a$ defines a vertical shift of $|a|$ units upward when $a \gt 0$, and downward side when $a\lt 0$ of the original function $f(x)$. d) The graph of $y=f(x-p)$ defines a horizontal shift of $|p|$ units to the right when $p \gt 0$, and to the left when $p \lt 0$ of the original function $f(x)$. e) The graph of $y=k\cdot f(x)$ can be obtained a vertical stretch when $k\gt 1$ or compression when $0\lt k \lt1$) of the original function $f(x)$. (1) As mentioned in point $(c)$, the resulting graph involves a $2$ units shift upward. That is, $y=\sqrt {x}+2$. (2) As mentioned in point $(b)$, the resulting graph involves a reflection about the $y$-axis of the original function $f(x)$. That is, $y=f(-x)\\ y =\sqrt {-x}+2$ (3) As mentioned in point $(d)$, the resulting graph involves a $3$ units shift towards the left and $p=-3$. That is, $y= f(x-p) =\sqrt {-(x+3)}+2$ Finally, after using multiple transformations, we have new function as: $y=\sqrt {-x-3}+2$