What is the rate of change of the function represented by the table?
| x | y |
|---|---|
| 1 | -8.5 |
| 2 | -6 |
| 3 | -3.5 |
| 4 | -1 |
A. -2.5
B. -1
C. 1
D. 2.5
Answer (1)
Calculate the rate of change between the first two points: m 1 = 2 − 1 − 6 − ( − 8.5 ) = 2.5 .
Calculate the rate of change between the second and third points: m 2 = 3 − 2 − 3.5 − ( − 6 ) = 2.5 .
Calculate the rate of change between the third and fourth points: m 3 = 4 − 3 − 1 − ( − 3.5 ) = 2.5 .
Since the rate of change is constant, the rate of change of the function is 2.5 .
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the rate of change of the function represented by the table. The rate of change, also known as the slope, can be found by calculating the difference in y values divided by the difference in x values between consecutive points.
Calculating the Rate of Change (1st pair) Let's calculate the rate of change between the first two points ( 1 , − 8.5 ) and ( 2 , − 6 ) . The rate of change m 1 is given by: m 1 = 2 − 1 − 6 − ( − 8.5 ) = 1 − 6 + 8.5 = 1 2.5 = 2.5
Calculating the Rate of Change (2nd pair) Now, let's calculate the rate of change between the second and third points ( 2 , − 6 ) and ( 3 , − 3.5 ) . The rate of change m 2 is given by: m 2 = 3 − 2 − 3.5 − ( − 6 ) = 1 − 3.5 + 6 = 1 2.5 = 2.5
Calculating the Rate of Change (3rd pair) Next, let's calculate the rate of change between the third and fourth points ( 3 , − 3.5 ) and ( 4 , − 1 ) . The rate of change m 3 is given by: m 3 = 4 − 3 − 1 − ( − 3.5 ) = 1 − 1 + 3.5 = 1 2.5 = 2.5
Conclusion Since m 1 = m 2 = m 3 = 2.5 , the rate of change of the function is constant and equal to 2.5.
Examples
Understanding the rate of change is crucial in many real-world applications. For instance, if you're tracking the distance a car travels over time, the rate of change represents the car's speed. Similarly, in economics, it can represent the rate of inflation or the growth rate of a company's revenue. By calculating the rate of change, we can make predictions and informed decisions based on the trend observed. For example, if a company's revenue is growing at a constant rate of 2.5 per quarter, we can estimate its revenue in future quarters.
