Q. 29 - 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) Reflect about the x-axis

(2) Shift up 2 units

(3) Shift left 3 units

Step-by-Step Solution

Answer

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

Step 1. Given information

We have been given a 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 $y=f\left(x\right)+2$.

So,

$y=-\sqrt{x}+2$ 

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 left by $3$ units then $y=f(x+3)$

So,

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

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

So,

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

Step 4. Use the third transformation

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

So,

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