Question

linear functions and applications
which of the following statements are true? check all that apply.
average rate of change describes how quickly function outputs increase or decrease over an interval
average rate of change describes how much function outputs increase or decrease over an interval
average rate of change = $\frac{x_2 - x_1}{y_2 - y_1}$ = $\frac{change in x}{change in y}$ = $\frac{delta x}{delta y}$
average rate of change is another term for slope of the secant line over a given interval.
average rate of change of a function over a specified interval is the ratio: $\frac{change in output}{change in input}$

Explanation:

Step1: Define average rate of change

The average rate of change of a function over an interval measures how much the function's output (y - values) changes with respect to the change in the input (x - values) over that interval. It is not about how quickly in a non - numerical sense but about the amount of change. So, the statement "Average Rate of Change describes HOW QUICKLY function outputs increase or decrease over an interval" is false.

Step2: Correct description of average rate of change

The average rate of change describes how much function outputs increase or decrease over an interval. So, the statement "Average Rate of Change describes HOW MUCH function outputs increase or decrease over an interval" is true.

Step3: Formula of average rate of change

The formula for the average rate of change of a function is $\frac{y_2 - y_1}{x_2 - x_1}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{\Delta y}{\Delta x}$, not $\frac{x_2 - x_1}{y_2 - y_1}$. So, the statement with the incorrect formula is false.

Step4: Relationship with secant line

The average rate of change of a function over a given interval is indeed the slope of the secant line connecting the two points on the function corresponding to the endpoints of the interval. So, the statement "Average Rate of Change is another term for Slope of the Secant Line over a given interval" is true.

Step5: Ratio definition

The average rate of change of a function over a specified interval is the ratio $\frac{\text{Change in Output}}{\text{Change in Input}}$. So, this statement is true.

Answer:

• Average Rate of Change describes HOW MUCH function outputs increase or decrease over an interval
• Average Rate of Change is another term for Slope of the Secant Line over a given interval
• Average Rate of Change of a function over a specified interval is the ratio: $\frac{\text{Change in Output}}{\text{Change in Input}}$