Chapter 1.7, Problem 33E

(a)

To determine

Find the parent function $f$ .

Answer to Problem 33E

For the function $g(x)=3{(x-2)}^3$ , the parent function is $f(x)={(x)}^3$ .

Explanation of Solution

Given information: $g(x)=3{(x-2)}^3$

Concept used:

To identify the parent function, strip all the arithmetic operations on the function to leave behind one higher order operation in just x.

Conclusion:

So, remove the arithmetic operation of multiplication by 3 from the given funcion to get the parent function.

(b)

To determine

Find the sequence of transformation from f to g.

Answer to Problem 33E

The shape of $f(x)={(x)}^3$ is drawn reflected in the x-axis and then shifted right by 2 units and stretched by 3 units.

Explanation of Solution

Given information: $g(x)=3{(x-2)}^3$ and $f(x)={(x)}^3$

Concept Used:

The sequence of transformations from f to g depicts the steps followed and the transformations used to reach from the parent function f to g .

Conclusion:

The shape of $f(x)={(x)}^3$ is drawn reflected in the x-axis and then shifted by 2 units and then stretched is the required sequence of transformations from $f$ to $g$ .

(c)

To determine

To sketch the graph of $g$ .

Explanation of Solution

Given information: $g(x)=3{(x-2)}^3$

Concept Used:

Use the sequence of transformation to plot the graph of the function

Conclusion:

Obtain the graph of $g(x)=3{(x-2)}^3$ .

The graph can be obtained using the graphing calculator as shown below:

  

(d)

To determine

To write $g$ in terms of $f$

Answer to Problem 33E

  $g(x)=3f(x)$

Explanation of Solution

Given information:

  $f(x)={(x)}^3$

  $g(x)=3{(x-2)}^3$

Conclusion:

Multiply 3 to $f(x)$ to get $g(x)$ in terms of $f(x)$

  $g(x)=3f(x)$