A data set contains an independent and a dependent variable. Which must be true of the data set if a linear function can be used to represent the data?
A. The set must have a constant additive rate of change.
B. The set must have a constant multiplicative rate of change.
C. The values in the set must be positive.
D. The values in the set must be increasing.

A linear function has a constant additive rate of change.
A constant additive rate of change means that for every unit increase in the independent variable, the dependent variable increases by a constant amount.
The values in the set do not necessarily have to be positive or increasing.
Therefore, the set must have a constant additive rate of change. The set must have a constant additive rate of change. ​

Explanation

Understanding the Problem We are given a data set with an independent and a dependent variable. We need to determine which statement must be true if a linear function can be used to represent the data. The options are:


The set must have a constant additive rate of change.
The set must have a constant multiplicative rate of change.
The values in the set must be positive.
The values in the set must be increasing.


Linear Function Recall the definition of a linear function: f ( x ) = m x + b , where m is the slope and b is the y-intercept. The slope m represents the rate of change of the function. For a linear function, the rate of change is constant and additive.

Constant Rate of Change A constant additive rate of change means that for every unit increase in the independent variable, the dependent variable increases by a constant amount. A constant multiplicative rate of change corresponds to an exponential function, not a linear function.

Values in the Set The values in the set do not necessarily have to be positive, as a linear function can have negative values. The values in the set do not necessarily have to be increasing, as a linear function can have a negative slope, resulting in decreasing values.

Conclusion Therefore, the set must have a constant additive rate of change.


Examples
Linear functions are used in many real-world applications, such as calculating the cost of a taxi ride based on the distance traveled. The initial fee is the y-intercept, and the cost per mile is the slope. Understanding linear functions helps you predict and analyze these types of relationships.

Answered by GinnyAnswer | 2025-07-03

For a data set to be represented by a linear function, it must exhibit a constant additive rate of change, meaning for every unit increase in the independent variable, the dependent variable changes by a fixed amount. Other characteristics, such as positivity or the requirement that values must increase, are not necessary. Thus, the correct answer is option A.
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Answered by Anonymous | 2025-07-04