Question

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?

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.

Explanation:

To determine which statement is true for a data set represented by a linear function, we analyze each option:

1. Option 1 (Constant additive rate of change): A linear function has the form \( y = mx + b \), where \( m \) (the slope) represents the rate of change. The slope \( m=\frac{\Delta y}{\Delta x} \), meaning the change in \( y \) (dependent variable) for a unit change in \( x \) (independent variable) is constant. This is an additive rate of change (e.g., for each 1 - unit increase in \( x \), \( y \) increases by \( m \) units).
2. Option 2 (Constant multiplicative rate of change): A constant multiplicative rate of change is characteristic of exponential functions (e.g., \( y = ab^x \), where the rate of change is multiplicative, \( b \) is the base). Linear functions do not have a constant multiplicative rate of change.
3. Option 3 (Values must be positive): Linear functions can have negative values. For example, \( y=-2x + 3 \) will have negative \( y \) - values when \( x>\frac{3}{2} \). So the values in the data set do not have to be positive.
4. Option 4 (Values must be increasing): Linear functions can be decreasing (when the slope \( m<0 \)). For example, \( y=-x + 5 \) is a decreasing linear function. So the values in the data set do not have to be increasing.

Answer:

A. The set must have a constant additive rate of change.