The table represents an exponential function.
| x | y |
|---|---|
| 1 | 2 |
| 2 | [tex]\frac{2}{5}[/tex] |
| 3 | [tex]\frac{2}{25}[/tex] |
| 4 | [tex]\frac{2}{125}[/tex] |
What is the multiplicative rate of change of the function?
Answer (2)
Calculate the ratio between y values when x=2 and x=1: 2 5 2 = 5 1 .
Calculate the ratio between y values when x=3 and x=2: 5 2 25 2 = 5 1 .
Calculate the ratio between y values when x=4 and x=3: 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. In other words, it is the ratio between consecutive y values.
Calculating the Multiplicative Rate of Change To find the multiplicative rate of change, we can calculate the ratio between consecutive y values. Let's calculate the ratio between the y values when x = 2 and x = 1 . This is given by 2 5 2 = 5 2 × 2 1 = 5 1 . Now, let's calculate the ratio between the y values when x = 3 and x = 2 . This is given by 5 2 25 2 = 25 2 × 2 5 = 5 1 . Finally, let's calculate the ratio between the y values when x = 4 and x = 3 . This is given by 25 2 125 2 = 125 2 × 2 25 = 5 1 . Since the ratio between consecutive y values is constant, the multiplicative rate of change is 5 1 .
Final Answer 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. The multiplicative rate of change represents the factor by which the quantity changes over a fixed period. For example, if a population grows at a rate of 1/5 per year, it means that the population increases by 20% each year. Understanding the multiplicative rate of change helps us predict future values and make informed decisions.
The multiplicative rate of change of the exponential function, calculated from the given table, is 5 1 . This indicates that for each unit increase in x, the y-value is multiplied by 5 1 .
;
