The table below represents the number of math problems Jana completed as a function of the number of minutes since she began doing her homework. Does this situation represent a linear or non-linear function, and why?
| Minutes | Math Problems Completed |
|---|---|
| 1 | 3 |
| 2 | 7 |
| 3 | 12 |
| 4 | 16 |
| 5 | 19 |
A. It represents a linear function because there is a constant rate of change.
B. It represents a linear function because there is not a constant rate of change.
C. It represents a non-linear function because there is a constant rate of change.
D. It represents a non-linear function because there is not a constant rate of change.
Answer (2)
Calculate the rate of change between consecutive data points.
Observe that the rates of change are 4, 5, 4, and 3.
Since the rates of change are not constant, the function is non-linear.
Therefore, the situation represents a non-linear function because there is not a constant rate of change. It represents a non-linear function because there is not a constant rate of change.
Explanation
Understanding the Problem We are given a table that shows the number of math problems Jana completed as a function of the number of minutes she spent on her homework. We need to determine if this relationship is linear or non-linear and provide a reason for our answer. A function is linear if the rate of change between any two points is constant.
Calculating Rate of Change To determine if the function is linear, we need to calculate the rate of change between consecutive points in the table. The rate of change is calculated as the change in the number of math problems completed divided by the change in time (minutes).
Rate of Change between (1,3) and (2,7) Let's calculate the rate of change between the first two points (1, 3) and (2, 7): 2 − 1 7 − 3 = 1 4 = 4 So, the rate of change between the first two points is 4.
Rate of Change between (2,7) and (3,12) Now, let's calculate the rate of change between the second and third points (2, 7) and (3, 12): 3 − 2 12 − 7 = 1 5 = 5 So, the rate of change between the second and third points is 5.
Rate of Change between (3,12) and (4,16) Next, let's calculate the rate of change between the third and fourth points (3, 12) and (4, 16): 4 − 3 16 − 12 = 1 4 = 4 So, the rate of change between the third and fourth points is 4.
Rate of Change between (4,16) and (5,19) Finally, let's calculate the rate of change between the fourth and fifth points (4, 16) and (5, 19): 5 − 4 19 − 16 = 1 3 = 3 So, the rate of change between the fourth and fifth points is 3.
Determining Linearity We have calculated the rates of change between consecutive points as 4, 5, 4, and 3. Since the rates of change are not constant, the function is non-linear.
Conclusion Therefore, the situation represents a non-linear function because there is not a constant rate of change.
Examples
Imagine you're tracking the growth of a plant each week. If the plant grows by a consistent amount every week, that's a linear relationship. But if the growth varies—some weeks it grows a lot, other weeks not so much—that's a non-linear relationship. Understanding whether a relationship is linear or non-linear helps you make predictions about future growth or changes. This concept applies to many real-world situations, from population growth to financial investments.
The relationship between the number of minutes Jana spent on homework and the math problems completed is non-linear because the rates of change between data points are not constant. After calculating the rates, we found values of 4, 5, 4, and 3, which vary. Thus, the correct answer is D: "It represents a non-linear function because there is not a constant rate of change."
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