Chapter 2.5, Problem 43E

To determine

To graph: the function $f\left(x\right)=\frac{1}{2}\sqrt{x+4}-3$ using the graph of a standard function and then applying transformations.

Explanation of Solution

Given information:

the function is $f\left(x\right)=\frac{1}{2}\sqrt{x+4}-3$

Graph:

Start with the standard function graph $f(x)=\sqrt{x}$ .

To plot the graph of $f(x)=\sqrt{x}$ ,use the plotting points method, substitute different value of $x$ on $f(x)=\sqrt{x}$ to determine corresponding value of $f(x)$ .

The table is thus obtained as:

x	0	1	4	9
y	0	1	2	3

now plot each point of $x$ along the horizontal axis and the corresponding value of $f(x)=y$ along vertical axis and connect each point to form a smooth curve.

The graph of $f(x)=\sqrt{x}$ is obtained as:

  

Now, the graph of $f\left(x\right)=\frac{1}{2}\sqrt{x+4}-3$ is the graph of $f(x)=\sqrt{x}$ shifted $4$ unitleft, then shrink the graph of $f\left(x\right)=\sqrt{x+4}$ vertically by a factor $\frac{1}{2}$ , then shifted the graph of $f\left(x\right)=\frac{1}{2}\sqrt{x+4}$ $3$ unit downwards.

The graphof $f\left(x\right)=\frac{1}{2}\sqrt{x+4}-3$ is obtained as:

  

Interpretation:

From the above graph it can be obtained that the graph of $f\left(x\right)=\frac{1}{2}\sqrt{x+4}-3$ is obtained by shifting the graph of $f(x)=\sqrt{x}$ $4$ unit left, shrinking the graph of $f\left(x\right)=\sqrt{x+4}$ vertically by a factor $\frac{1}{2}$ , then shifting the graph of $f\left(x\right)=\frac{1}{2}\sqrt{x+4}$ $3$ unit downwards.