Chapter 1.7, Problem 70E

(a)

To determine

To find: The transformation of the parent function $f\left(x\right)=x^2$ and then graph the function.

Answer to Problem 70E

The transformation of the parent function $f\left(x\right)=x^2$ is described and the required function is shown in Figure 1

Explanation of Solution

Given:

The number $N$ is,

  $N=-0.068{\left(x-13.68\right)}^2+119$

The year $t$ that corresponds to 2003 is $t=3$ .

Calculation:

Consider the parent function $f\left(x\right)=x^2$ .

Shift the parent function horizontally to the right by 13.68 units and then shrink it vertically by 0.068 units and then reflect the graph on $x$ axis. At last shift the graph vertically in the upward direction by 119 units. The equation obtained is,

  $N=-0.068{\left(x-13.68\right)}^2+119$

The graph for the above function is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The average rate of change of the function from 2003 to 2010 and interpret the answer in the context of the problem.

Answer to Problem 70E

The average rate of change of function from 2003 to 2010 is $0.98$ . This suggest that the given number $N$ of household in the United States from 2003 to 2010 increases every year.

Explanation of Solution

Given:

The number $N$ is,

  $N=-0.068{\left(x-13.68\right)}^2+119$

The year $t$ that corresponds to 2003 is $t=3$ .

Calculation:

Consider the average rate of change of the function from 2003 to 2010 is,

  $\frac{N\left(2010\right)-N\left(2003\right)}{2010-2003}$

Then, for $x=3$ .

  $\begin{array}{c}N=-0.068{\left(3-13.68\right)}^2+119\\ =111.24\end{array}$

Then, for $x=10$ .

  $\begin{array}{c}N=-0.068{\left(10-13.68\right)}^2+119\\ =118.08\end{array}$

Then from the average rate of change equation.

  $\begin{array}{c}\frac{N\left(2010\right)-N\left(2003\right)}{2010-2003}=\frac{118.08-111.24}{2010-2003}\\ =0.98\end{array}$

This suggest that the given number $N$ of household in the United States from 2003 to 2010 increases every year.

(c)

To determine

To find: The model that predicts the number of household in the United States in 2010 and explain how the answer is reasonable.

Answer to Problem 70E

The number of households in the United States in $2018$ is approximately $117.73$ and this is reasonable.

Explanation of Solution

Given:

The number $N$ is,

  $N=-0.068{\left(x-13.68\right)}^2+119$

Calculation:

Consider the given functions,

  $N=-0.068{\left(x-13.68\right)}^2+119$

Then, for $x=18$ ,

  $\begin{array}{c}N=-0.068{\left(18-13.68\right)}^2+119\\ =117.73\end{array}$

This, shows that the number of households in the United States in $2018$ is approximately $117.73$ and this is reasonable.