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 (1)
Calculate the rate of change between consecutive data points.
Observe that the rates of change are 4, 5, 4, and 3.
Since the rate of change is not constant, the function is non-linear.
Conclude that 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
Analyzing the Data Let's analyze the given data to determine if the function is linear or non-linear. A linear function has a constant rate of change. We need to calculate the rate of change between consecutive points in the table.
Calculating Rates of Change The rate of change between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: x 2 − x 1 y 2 − y 1 Let's calculate the rates of change between the given points:
Between (1, 3) and (2, 7): 2 − 1 7 − 3 = 1 4 = 4
Between (2, 7) and (3, 12): 3 − 2 12 − 7 = 1 5 = 5
Between (3, 12) and (4, 16): 4 − 3 16 − 12 = 1 4 = 4
Between (4, 16) and (5, 19): 5 − 4 19 − 16 = 1 3 = 3
Determining Linearity The rates of change are 4, 5, 4, and 3. Since the rate of change is not constant, the function is non-linear.
Conclusion Therefore, the correct answer is: It 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 the same amount every week, that's a linear relationship. But if the growth varies—some weeks it grows more, some less—that's a non-linear relationship. Understanding whether a relationship is linear or non-linear helps you make predictions. For example, in business, you might analyze sales data to see if sales are increasing at a steady rate (linear) or if the rate changes over time (non-linear), which could influence your business strategies.
