Chapter 7.6, Problem 8PPE

To determine

To examinewhether the table represents a linear or exponential function.

Answer to Problem 8PPE

The given relationship is not linear but exponential.

Explanation of Solution

Given Information:The table with the following data is given.

$x$	$1$	$2$	$3$	$4$
$y$	$2$	$8$	$32$	$128$

Formula used: The slope between the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is calculated as follows.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculate the slope between the points $A\left(x_1,y_1\right)=\left(1,2\right)$ and $B\left(x_2,y_2\right)=\left(2,8\right)$

  $\begin{array}{c}{m}_1=\frac{8-2}{2-1}\\ =\frac{6}{1}\\ =6\end{array}$

The slope of line through the given points is $6$ .

Calculate the slope between the points $B\left(x_2,y_2\right)=\left(2,8\right)$ and $C\left(x_3,y_3\right)=\left(3,32\right)$

  $\begin{array}{c}{m}_2=\frac{32-8}{3-2}\\ =\frac{24}{1}\\ =24\end{array}$

The slope of line through the given points is $24$ .

Calculate the slope between the points $C\left(x_3,y_3\right)=\left(3,32\right)$ and $D\left(x_4,y_4\right)=\left(4,128\right)$

  $\begin{array}{c}{m}_3=\frac{128-32}{4-3}\\ =\frac{96}{1}\\ =96\end{array}$

The slope of line through the given points is $96$ .

The slope signifies the rate of change.

The relationship is linear if the rate of change is constant.

Hence, the given relationship is not linear but exponential.

Plot the graph for the given data.

  

The graph of the given relation.