The table represents an exponential function.
| x | y |
|---|---|
| 1 | 2 |
| 2 | \frac{2}{5} |
| 3 | \frac{2}{25} |
| 4 | \frac{2}{125} |
What is the multiplicative rate of change of the function?
A. \frac{1}{5}
B. \frac{2}{5}
C. 2
D. 5
Answer (1)
Calculate the ratio between y ( 2 ) and y ( 1 ) : y ( 1 ) y ( 2 ) = 2 5 2 = 5 1 .
Calculate the ratio between y ( 3 ) and y ( 2 ) : y ( 2 ) y ( 3 ) = 5 2 25 2 = 5 1 .
Calculate the ratio between y ( 4 ) and y ( 3 ) : y ( 3 ) y ( 4 ) = 25 2 125 2 = 5 1 .
The multiplicative rate of change is 5 1 .
Explanation
Understanding the Problem We are given a table representing 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. To find this, we can divide any y value by the y value that precedes it.
Calculating the Ratio To find the multiplicative rate of change, we will calculate the ratio between consecutive y values. Let's calculate the ratio between y ( 2 ) and y ( 1 ) : y ( 1 ) y ( 2 ) = 2 5 2 = 5 2 × 2 1 = 5 1
Verifying the Ratio Now, let's calculate the ratio between y ( 3 ) and y ( 2 ) : y ( 2 ) y ( 3 ) = 5 2 25 2 = 25 2 × 2 5 = 5 1
Confirming the Ratio Finally, let's calculate the ratio between y ( 4 ) and y ( 3 ) : y ( 3 ) y ( 4 ) = 25 2 125 2 = 125 2 × 2 25 = 5 1
Determining the Multiplicative Rate of Change Since the ratio between consecutive y values is consistently 5 1 , the multiplicative rate of change of the function is 5 1 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. In finance, the multiplicative rate of change is crucial for understanding investment growth. For example, if an investment grows by a factor of 5 1 each year, it means the investment is decreasing in value. Understanding this rate helps investors make informed decisions about their portfolios.
