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 nonlinear because linear functions increase or decrease at the same rate.
it is linear because there is both an independent and a dependent variable.
it is nonlinear because linear functions are increasing functions.
it is linear because the graph decreases over time.

Explanation:

To determine the type of function, we recall the definition of a linear function: it has a constant rate of change (slope). The tea's temperature decreases quickly first, then more slowly, meaning the rate of change is not constant. Let's analyze each option:

• Option 1: Correct. Linear functions have a constant rate (increase/decrease at the same rate). Since the tea's temperature change rate isn't constant, it's nonlinear.
• Option 2: Wrong. Having independent (time) and dependent (temperature) variables doesn't make a function linear; linearity requires constant rate.
• Option 3: Wrong. Linear functions can be increasing or decreasing (e.g., \(y = -x\) is linear and decreasing), so this reasoning is incorrect.
• Option 4: Wrong. A graph decreasing over time doesn't mean it's linear; the rate of decrease must be constant for linearity, which it isn't here.

Answer:

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