Chapter 1.7, Problem 56E

(a)

To determine

To find: The function for the given graph.

Answer to Problem 56E

Therequired function is $f\left(x\right)=\frac{1}{4}{x}^3$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=x^3$ .

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The parent function for the above graph is,

  $f\left(x\right)=x^3$

Consider the form for the equation of the graph is,

  $f\left(x\right)=a{x}^3+b$

The graph that passes through the points $\left(2,2\right)$ and $\left(0,0\right)$ is,

  $\begin{array}{c}2=a{\left(2\right)}^3+b\\ =8a+b\end{array}$

The other point is $\left(0,0\right)$ then,

  $\begin{array}{c}0=a{\left(0\right)}^3+b\\ b=0\end{array}$

Then,

  $\begin{array}{l}2=8a+0\\ a=\frac{1}{4}\end{array}$

Then, the equation of the graph is,

  $f\left(x\right)=\frac{1}{4}{x}^3$

(b)

To determine

To find: The function for the given graph.

Answer to Problem 56E

The required function is $f\left(x\right)=-2x^3$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=x^3$ .

The given graph is shown in Figure 2

  

Figure 2

Calculation:

The parent function for the above graph is,

  $f\left(x\right)=x^3$

Consider the form for the equation of the graph is,

  $f\left(x\right)=a{x}^3+b$

The graph that passes through the points $\left(1,-2\right)$ and $\left(0,0\right)$ is,

  $\begin{array}{c}-2=a{\left(1\right)}^3+b\\ =a+b\end{array}$

The other point is $\left(0,0\right)$ then,

  $\begin{array}{c}0=a{\left(0\right)}^3+b\\ b=0\end{array}$

Then,

  $\begin{array}{l}-2=a+0\\ a=-2\end{array}$

Then, the equation of the graph is,

  $f\left(x\right)=-2x^3$