Q. 27 - Precalculus Enhanced with Graphing Utilities

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) Shift up $2$ units

(2) Reflect about the $x$-axis

(3) Reflect about the $y$-axis

Step-by-Step Solution

Answer

The function after all the transformation is $y=-\left(\sqrt{-x}+2\right)$

Step 1. Given information

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

Step 2. Use the first transformation

If the graph of $y=f\left(x\right)$ is shifted up by $2$ units then add $2$ to $f\left(x\right)$.

So,

$y=\sqrt{x}+2$

Step 3. Use the second 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,

$y=-\left(\sqrt{x}+2\right)$

Step 4. Use the third transformation

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

So,

$y=-\left(\sqrt{-x}+2\right)$