Exercise 1 Page 580

We want to use the given graph to fill in the blanks in the given radical function.

We can see that the function represented by the graph is a transformation of the parent function $g(x)=\sqrt{x}.$ Let's recall some of the possible transformations of radical functions.

Transformations of $g(x)$
Vertical Translations	$$\begin{array}{c}\text{Translation up }k\text{ units, }k>0\\ y=g(x)+k\end{array}$$
	$$\begin{array}{c}\text{Translation down }k\text{ units, }k>0\\ y=g(x)-k\end{array}$$
Horizontal Translations	$$\begin{array}{c}\text{Translation right }h\text{ units, }h>0\\ y=g(x-h)\end{array}$$
	$$\begin{array}{c}\text{Translation left }h\text{ units, }h>0\\ y=g(x+h)\end{array}$$
Reflections	$$\begin{array}{c}\text{In the }x\text{-axis}\\ y=\text{-}g(x)\end{array}$$
	$$\begin{array}{c}\text{In the }y\text{-axis}\\ y=g(\text{-}x)\end{array}$$

Now, let's figure out which transformations we can use to obtain the graph of the given function from the parent function.

We can see that the graph of the given function is decreasing, while the graph of the parent function is increasing. This tells us that we need to reflect the parent function in the $x\text{-}$axis. Now, notice that the point $(0,0)$ on the graph of $y=\text{-}\sqrt{x}$ is translated to $(1,2).$ Therefore, we should translate the reflected parent function $1$ unit right and $2$ units up. We see that reflecting the parent function in the $x\text{-}$axis, translating it $1$ unit right, and translating it $2$ units up results in the given graph. This allows us to fill in the blanks in the function rule.