Chapter 6.7, Problem 22E

To determine

Whether the data show an exponential relationship, then write a function tht models the data.

Answer to Problem 22E

The exponential function that models the data is $y=\left(14.3\right){\left(1.68\right)}^x$ .

Explanation of Solution

Given information :

The given data is $\begin{array}{cccccc}x& -3& -1& 1& 3& 5\\ y& 2& 7& 24& 68& 194\end{array}$ .

Calculation :

Make a scatter plot of the data. The scatter plot of the data is drawn below:

  

Therefore, the data appears exponential.

Choose any two points to write a model, such as $\left(1,\text{ 24}\right)$ and $\left(3,\text{ 68}\right)$ . Substitute the coordinate of these points into $y=a{b}^x$ . Therefore,

  $24=a{b}^1$ …… (1)

  $68=a{b}^3$ …… (2)

Divide equation (2) by equation (1) and solve:

  $\begin{array}{l}\frac{68}{24}=\frac{a{b}^3}{a{b}^1}\\ \frac{68}{24}=b^2\text{ (Take square root on both sides) }\\ 1.68=b\text{ (Take positive square root as }b>0\text{)}\end{array}$

Substitute $b=1.68$ in equation (1) and solve for a:

  $\begin{array}{l}24=a{\left(1.68\right)}^1\\ \frac{24}{1.68}=a\\ 14.3=a\end{array}$

So, the exponential function that models the data is

  $y=\left(14.3\right){\left(1.68\right)}^x$

Hence,

The exponential function that models the data is $y=\left(14.3\right){\left(1.68\right)}^x$ .