To determine if a data set can be represented by a linear function, we need to understand the inherent property of linear functions. Let's analyze each option provided:
Option 1: The set must have a constant additive rate of change.
- A linear function is defined by a straight line on a graph. This implies that for every one-unit increase in the independent variable (x), the dependent variable (y) changes by a constant amount, known as the slope. This constant change is referred to as the constant additive rate of change. If the change is consistent across the data set, the relationship between the variables is linear.
Option 2: The set must have a constant multiplicative rate of change.
- This option describes exponential functions rather than linear functions. An exponential function involves multiplication by a constant rate rather than addition. For example, if each value in the data set is doubled, the set exhibits exponential growth, not linear growth.
Option 3: The values in the set must be positive.
- There is no requirement for the values in a data set representing a linear function to be positive. Linear functions can have negative, positive, or zero values depending on the slope and intercept.
Option 4: The values in the set must be increasing.
- This isn't necessarily true for a linear function. A linear relationship between variables can also describe decreasing trends if the slope is negative. The key aspect is the consistency of the rate of change, not the direction of change.
From the above analysis, we conclude that the correct condition for a data set to be represented by a linear function is that it must have a constant additive rate of change.
Thus, the correct answer is:
The set must have a constant additive rate of change.
