Chapter 1.7, Problem 34E

(a)

To determine

Find the parent function $f$ .

Answer to Problem 34E

For the function, $g(x)=-\frac{1}{2}{(x+1)}^3$ the parent function is $f(x)=x^3$ .

Explanation of Solution

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

So, remove the arithmetic operation of multiplication by -1/2 from the given funcion to get the parent function.

(b)

To determine

Find the sequence of transformation from f to g.

Answer to Problem 34E

The shape of $g(x)=-\frac{1}{2}{(x+1)}^3$ is drawn reflected in the y-axis and then stretched by by 1/2 units.

Explanation of Solution

Given information: $g(x)=-\frac{1}{2}{(x+1)}^3$ and $f(x)=x^3$

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 .

The shape of is drawn reflected in the x-axis and then shifted by 10 units is the required sequence of transformations from f to g .

(c)

To determine

To sketch the graph of $g$ .

Answer to Problem 34E

Explanation of Solution

Given $g(x)={(x+3)}^3-10$

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

Obtain the graph of $g(x)={(x+3)}^3-10$ .

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 34E

  $g(x)=-\frac{1}{2}f(x)$

Explanation of Solution

Given information: $g(x)=-\frac{1}{2}{(x+1)}^3$ and $f(x)=x^3$

Conclusion:

Multiply $f(x)$ by -1/2to get $g(x)$ in terms of $f(x)$