Chapter 9, Problem 63SGR

To determine

To find: An equation for the function that describes the data.

Answer to Problem 63SGR

Exponential model

Explanation of Solution

Given Information: A table of values

First of all we find the difference by subtracting each value from the next value of y.

  $\begin{array}{l}2-1=1\\ 4-2=2\\ 8-4=4\\ 16-8=8\end{array}$

x	0	1	2	3	4
y	1	2	4	8	16

Here the difference is not the same therefore there is not a linear model.

Now find the second difference

  $\begin{array}{l}2-1=1\\ 4-2=2\\ 8-4=4\end{array}$

Thus the difference is also not the same therefore this does not describe the quadratic model.

Now find the ratio of each team to the previous team.

  $\begin{array}{l}\frac{2}{1}=2,\frac{8}{4}=2\\ \frac{4}{2}=2\frac{16}{8}=2\end{array}$

Here the ratio is same therefore this describes an exponential model of the form

  $y=a{b}^x$

A = y-intercept b = common ratio

From the table y-intercept is (0,1)

So $\begin{array}{l}a=1andb=2\\ $ theny=1{(2)}^x=2^x$\end{array}$

Then the model is exponential $y=2^x$