Chapter 2.5, Problem 29AYU

In Problems $29-32,$ 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)Vertical stretch by a factor of $3$

(2)Shift up $4$ units

(3) Shift left $5$ units

To determine

The function that is finally graphed after the following transformations applied on the graph of $y=\sqrt{x}$

$1)$ Vertical stretched by a factor 3

$2)$ Shift up 4 units and

$3)$ Shift left 5 units.

Answer to Problem 29AYU

Solution:

The function is $y=3\sqrt{x+5}+4$

Explanation of Solution

Given Information:

The function is $y=\sqrt{x}$.

Explanation:

The graph of function $y=f(x)$ is vertically stretched by factor $h$, then the function is $y=hf(x)$

To stretch $y=\sqrt{x}$ vertically, multiply $\sqrt{x}$ by 3

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

The graph of function $y=f(x)$ is shift up by $h$ units, then the function is $y=f(x)+h$

To shift up $y=3\sqrt{x}$ by 4 units, add 4 to $3\sqrt{x}$

$\Rightarrow y=3\sqrt{x}+4$

The graph of function $y=f(x)$ is shift left by $h$ units, then the function is $y=f(x+h)$

To shift graph of $y=3\sqrt{x}+4$ to the left by 5 units, replace $x$ by $x+5$

$\Rightarrow y=3\sqrt{x+5}+4$

Thus, $y=3\sqrt{x+5}+4$ is the function that is finally graphed from the function $y=\sqrt{x}$