Answer
$y=-\sqrt {x-3}-2$
Work Step by Step
In order to find the answer, we will have to recall the following some point 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 $f(x)$.
a) The graph of the function $y=f(-x)$ involves a reflection about the $y$-axis of the original fbunction $f(x)$.
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 $y=k \ f(x-h)$ 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 $(a)$, the resulting graph involves a reflection about the $x$-axis of the original function $f(x)=\sqrt x$. That is, $y=-f(x)\\ y =-\sqrt x$
(2) As mentioned in point $(d)$, the resulting graph involves a $3$ units shift towards the right of the original function $y=f(x)=\sqrt x$ and $p=3$. That is, $y=-\sqrt {x-3}$
(3) As mentioned in point $(c)$, the resulting graph involves a $2$ units shift downward of the original function $y=f(x)=\sqrt x$ and $a=3$. That is, $y=-\sqrt {x-3}-2$
Finally, after using multiple transformations, we have new function as: $y=-\sqrt {x-3}-2$
