Chapter 1.4, Problem 58E

(a)

To determine

To identify: the parent function f.

Explanation of Solution

Given information:

The given equation is $g(x)=\frac{1}{2}|x-2|-3$

The parent function of the given equation is $f(x)=\left|x\right|$ .

(b)

To determine

To describe: the sequence of transformations from f to h.

Explanation of Solution

Given information:

The given equation is $g(x)=\frac{1}{2}|x-2|-3$

The parent function of the given equation is $f(x)=\left|x\right|$ .

The sequence of transformations from f to g

Vertical compression by the factor $0.5$

  $\Rightarrow \frac{1}{2}f(x)=\frac{1}{2}\left|x\right|$

Right shift of $2$ unit $\Rightarrow \frac{1}{2}f(x-2)=\frac{1}{2}|x-2|$

Down shift of $3$ units. $\Rightarrow \frac{1}{2}f(x-2)-3=\frac{1}{2}|x-2|-3$

(c)

To determine

To sketch: the graph of the g by hand.

Explanation of Solution

Given information:

The given equation is $g(x)=\frac{1}{2}|x-2|-3$

The graph of the g is given below

  

(d)

To determine

To write: the function g in terms of f.

Explanation of Solution

Given information:

The given equation is $g(x)=\frac{1}{2}|x-2|-3$

The parent function of the given equation is $f(x)=\left|x\right|$ .

  $\begin{array}{l}g(x)=\frac{1}{2}|x-2|-3\\ g(x)=\frac{1}{2}f(x-2)-3\end{array}$

Hence,

The function f in terms of f is $g(x)=\frac{1}{2}f(x-2)-3$