What is the rate of change of the function described in the table?
| x | y |
| --- | ------ |
| -1 | 1/10 |
| 0 | 1/2 |
| 1 | 5/2 |
| 2 | 25/2 |
| 3 | 125/2 |
A. 12/5
B. 5
C. 25/2
D. 25
Answer (1)
Calculate the ratios of consecutive y-values: y ( − 1 ) y ( 0 ) = 5 , y ( 0 ) y ( 1 ) = 5 , y ( 1 ) y ( 2 ) = 5 , y ( 2 ) y ( 3 ) = 5 .
Since the ratios are constant, the function is exponential.
The common ratio is the rate of change.
The rate of change of the function is 5 .
Explanation
Analyzing the Problem We are given a table of x and y values and asked to find the rate of change of the function described by the table. The rate of change can be found by examining the ratio or difference between consecutive y values for equally spaced x values. If the ratio is constant, the function is exponential, and the common ratio is the rate of change. If the difference is constant, the function is linear, and the common difference is the rate of change.
Calculating Ratios Let's calculate the ratios of consecutive y -values:
y ( − 1 ) y ( 0 ) = 10 1 2 1 = 2 1 × 1 10 = 5 y ( 0 ) y ( 1 ) = 2 1 2 5 = 2 5 × 1 2 = 5 y ( 1 ) y ( 2 ) = 2 5 2 25 = 2 25 × 5 2 = 5 y ( 2 ) y ( 3 ) = 2 25 2 125 = 2 125 × 25 2 = 5
Since the ratio between consecutive y values is constant and equal to 5, the function is exponential, and the rate of change is 5.
Conclusion The rate of change of the function is 5.
Examples
Understanding rates of change is crucial in various real-world scenarios. For instance, consider a bacterial population that doubles every hour. The rate of change, in this case, is 2, indicating exponential growth. Similarly, in finance, the rate of change can represent the interest rate on an investment, determining how quickly your money grows. Recognizing and calculating rates of change helps in making informed decisions, whether it's predicting population growth, managing finances, or understanding scientific phenomena.
