The table represents an exponential function.
| x | y |
|---|---|
| 1 | 6 |
| 2 | 4 |
| 3 | 8/3 |
| 4 | 16/9 |
What is the multiplicative rate of change of the function?
A. 1/3
B. 2/3
C. 2
D. 9
Answer (1)
Divide consecutive y-values to find the multiplicative rate of change.
Calculate 6 4 = 3 2 .
Calculate 4 8/3 = 3 2 .
The multiplicative rate of change is 3 2 .
Explanation
Understanding the Problem We are given a table of values for an exponential function and asked to find the multiplicative rate of change. The multiplicative rate of change is the factor by which the y-value changes when x increases by 1.
Calculating the Multiplicative Rate of Change To find the multiplicative rate of change, we can divide consecutive y-values. Let's divide the y-value when x=2 by the y-value when x=1: 6 4 = 3 2 Now, let's divide the y-value when x=3 by the y-value when x=2: 4 3 8 = 3 8 × 4 1 = 12 8 = 3 2 Finally, let's divide the y-value when x=4 by the y-value when x=3: 3 8 9 16 = 9 16 × 8 3 = 72 48 = 3 2 Since the ratio of consecutive y-values is constant, the multiplicative rate of change is 3 2 .
Final Answer The multiplicative rate of change of the function is 3 2 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, the multiplicative rate of change is 2. If you invest money in an account that earns compound interest, the multiplicative rate of change is related to the interest rate. Understanding the multiplicative rate of change helps us predict how these quantities will change over time.
