Question

pauley graphs the change in temperature of a glass of hot tea over time. he sees that the function appears to decrease quickly at first, then decrease more slowly as time passes. which best describes this function?
○ it is linear because the graph decreases over time.
○ it is linear because there is both an independent and a dependent variable.
○ it is nonlinear because linear functions are increasing functions.
○ it is nonlinear because linear functions increase or decrease at the same rate.

Explanation:

To determine the type of function, we analyze the properties of linear and nonlinear functions:

1. A linear function has a constant rate of change (slope), meaning it increases or decreases at the same rate throughout.
2. The temperature of the hot tea decreases quickly at first and then more slowly, indicating the rate of change is not constant.
3. Let's evaluate each option:

• Option 1: A linear function's decrease is at a constant rate, not changing speed. So this is incorrect.
• Option 2: Having independent and dependent variables doesn't make a function linear (e.g., quadratic functions also have them). Incorrect.
• Option 3: Linear functions can decrease (e.g., \(y = -2x + 3\)). Incorrect.
• Option 4: Since the temperature's rate of decrease changes (not constant), the function is nonlinear, and linear functions have a constant rate. Correct.

Answer:

It is nonlinear because linear functions increase or decrease at the same rate.