Question

Find the function that is finally graphed after each of the following transformations is applied to the graph of $y=\sqrt{x}$in the order stated.

(1) Reflect about the $x$-axis

(2) Shift right $3$units

(3) Shift down $2$units

Short answer

The function after all the transformation is$y=-\sqrt{x-3}-2$

Step-by-step solution

Step 1. Given information

We have been given a function $y=\sqrt{x}$

Step 2. Use the first transformation

For a function $y=f\left(x\right)$, the reflection about the $x$-axis of the graph of the function is given by $y=-f\left(x\right)$.

So,

$y=-\sqrt{x}$

Step 3. Use the second transformation

If the graph of $y=f\left(x\right)$is shifted right by $3$units then $y=f(x-3)$

So,

$y=-\sqrt{x-3}$

Step 4. Use the third transformation 

If the graph of $y=f\left(x\right)$is shifted down by $2$units then $y=f\left(x\right)-2$

So,

$y=-\sqrt{x-3}-2$