Chapter 5.3, Problem 32AYU

To determine

To find: The function is linear, exponential or neither.

Answer to Problem 32AYU

Solution:

Exponential function.

Explanation of Solution

Given:

x	-1	0	1	2	3
f ( x )	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{1}{16}$	$\frac{1}{32}$

  

From the above table, the average rate of change from -1 to 0 is -0.25, and the average rate of change from 0 to 1 is -0.125. Because the average rate of change is not constant, the function is not a linear function. The ratio of consecutive output for a 1-unit increase in the input is a constant $\frac{1}{2}$ . Because the ratio of consecutive output from is constant, the function is an exponential function with growth factor $a=\frac{1}{2}$ . The initial value of the exponential function is $C=\frac{1}{4}.$ Therefore the exponential function that models the data is $F\left(x\right)= C{a}^x=\frac{1}{4}{\left(\frac{1}{2}\right)}^x$ .