Chapter 1.7, Problem 36E

(a)

To determine

To identify: the parent function $f(x)$ of $g(x)$

Explanation of Solution

Given information:

Consider $g(x)=-\frac{1}{2}{x}^2-2$

Parent function is the simplest function of family of functions, which has the same shape

as that of entire family.

We have $g(x)=-\frac{1}{2}{x}^2-2$

The function $f(x)=x^2$ is the simplest function and have same shape as that of $g(x).$

Hence, the parent function is $f(x)=x^2$ .

(b)

To determine

To explain: the sequence of transformations from $f(x)$ to $g(x).$

Explanation of Solution

Given information:

Consider $g(x)=-\frac{1}{2}{x}^2-2$

The parent function is $f(x)=x^2$ .

If the function $f(x)=x^2$ is vertical compression by the factor $0.5$ units , then reflected

about x- axis and shifted right by two units, then $g(x)=-\frac{1}{2}{x}^2-2$ is obtained.

Hence,

The transformation of $f\text{ to }g$ is vertical compression, reflection across x- axis and shifted

Right two units.

(c)

To determine

To sketch: the graph of $g(x)$

Explanation of Solution

Given information:

Consider $g(x)=-\frac{1}{2}{x}^2-2$

Graph:

The graph of $g(x)=-\frac{1}{2}{x}^2-2$

  

(d)

To determine

To write: the function $g(x)$ in terms of $f(x)$

Explanation of Solution

Given information:

Consider $g(x)=-\frac{1}{2}{x}^2-2$

Parent function is $f(x)=x^2$

We have

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

Hence, the parent function is $g(x)=-\frac{1}{2}f(x)-2$ .