Chapter 1, Problem 80RE

a.

To determine

The parent function of f

Answer to Problem 80RE

.Parent function will be $x^3$

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-\frac{1}{3}{x}^3$

Rule for shifting the function horizontally is;

  $h\left(x\right)=f\left(x+c\right)$ ( forward )

  $h\left(x\right)=f\left(x-c\right)$ ( backward )

Now, rule for shifting the function vertically is;

  $h\left(x\right)=f\left(x\right)+c$ ( upward)

  $h\left(x\right)=f\left(x\right)-c$ ( downward )

Here, function is given;

  $h\left(x\right)=-\frac{1}{3}{x}^3$

It can be found by multiplying $-\frac{1}{3}$ with $x^3$

Therefore parent function will be $x^3$

b.

To determine

Sequence of transformation from f to h.

Answer to Problem 80RE

Transformation from f to h takes place by multiplying $\frac{1}{3}$ with $f\left(x\right)=x^3$ and mirror it around x axis.

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-\frac{1}{3}{x}^3$

Rule for shifting the function horizontally is;

  $h\left(x\right)=f\left(x+c\right)$ ( forward )

Here, c is real constant.

Now, rule for shifting the function vertically is;

  $h\left(x\right)=f\left(x\right)+c$ ( upward )

Here, c is real constant.

Here, function is given;

  $h\left(x\right)=-\frac{1}{3}{x}^3$

It can be found by multiplying $\frac{1}{3}$ with $f\left(x\right)=x^3$ and mirror it around x axis.

c.

To determine

To sketch :

The graph of h.

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-\frac{1}{3}{x}^3$

For finding h multiply $\frac{1}{3}$ with $f\left(x\right)=x^3$ and mirror it around x axis.

Now plot of h is ;

d.

To determine

Function notation to write h in terms of f

Answer to Problem 80RE

h in terms of f will be;

  $h\left(x\right)=-\frac{1}{3}f\left(x\right)$

Explanation of Solution

Given:

The function is given as:

  $h\left(x\right)=-\frac{1}{3}{x}^3$

h in terms of f can be written as;

  $h\left(x\right)=-\frac{1}{3}f\left(x\right)$