Chapter 1.4, Problem 52E

(a)

To determine

To identify: the parent function f.

Explanation of Solution

Given information:

The given equation is $g(x)=3+2{(x-4)}^2$

The parent function of the given equation is $f(x)=x^2$ .

(b)

To determine

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

Explanation of Solution

Given information:

The given equation is $g(x)=3+2{(x-4)}^2$

The parent function of the given equation is $f(x)=x^2$ .

The sequence of transformations from f to g

Vertical stretch by the factor two $\Rightarrow 2f(x)=2x^2$

right shift of $4$ units $\Rightarrow 2f(x-4)=2{(x-4)}^2$

up shift of $3$ units. $\Rightarrow 3+2f(x-4)=3+2{(x-4)}^2$

(c)

To determine

To sketch: the graph of the g by hand.

Explanation of Solution

Given information:

The given equation is $g(x)=3+2{(x-4)}^2$

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 $g(x)=3+2{(x-4)}^2$

The parent function of the given equation is $f(x)=x^2$ .

  $\begin{array}{l}g(x)=3+2{(}^x\\ g(x)=3+2f{(}^x\end{array}$

Hence,

The function f in terms of f is $g(x)=3+2f(x-4)$