Let’s approach this problem with two different stretches in mind.

Method 1: Using a Vertical Stretch

No horizontal stretch will be used, so the function’s equation will be of the form $y=a\sqrt{\left(x-h\right)}+k$.

Refer to the graph of $y=\sqrt{x}$.
The graph passes through the point $\left(0\mathrm{,}0\right)$, which is unaffected by stretches or reflections, so this point can be easily used to determine the translations.

Note: If the original graph does not pass through $\left(0\mathrm{,}0\right)$, you may need to determine the stretches and reflections before determining the translations.

The point $\left(0\mathrm{,}0\right)$ corresponds to $\left(2\mathrm{,}0\right)$ on the transformed graph, so there is a horizontal translation $2$ units to the right. There is no vertical translation. As such, $h=2$ and $k=0$. So, our equation is now $y=a\sqrt{x-2}+0$ or $y=a\sqrt{x-2}$.

To determine the $a$-value, select a point on the graph of the transformed function that was affected by the stretch, say $\left(6\mathrm{,}-6\right)$, substitute these values into the equation, and solve for $a$.


So, the equation of the function is $y=-3\sqrt{x-2}$. Notice the $-3$ suggests a vertical reflection in the $x$-axis and a vertical stretch about the $x$-axis by a factor of $3$. This matches the original graph shown.

Method 2: Using a Horizontal Stretch

This method is similar to the previous one, but because there is a vertical reflection, an $a$-value of $-1$ will be used. So, $y=-\sqrt{b\left(x-h\right)}+k$.

The $h$ and $k$-values can still be determined by examining how the point $\left(0\mathrm{,}0\right)$ was translated. Again, the $h$- and $k$-values are $2$ and $0$, respectively, so $y=-\sqrt{b\left(x-2\right)}$.

The point $\left(6\mathrm{,}-6\right)$ is on the graph of the transformed function, so it can be used to determine $b$.


So, the equation, using a horizontal stretch, is $y=-\sqrt{9\left(x-2\right)}$.