Question

determine whether the statement is true or false, and explain why. the average rate of change between two points is equal to the slope of the line segment connecting the two points. choose the correct answer below. a. the statement is false because the slope of a line segment gives the reciprocal of the average rate of change. b. the statement is true because the slope is equal to the change in x over the change in f(x), which gives the average rate of change. c. the statement is true because the slope is equal to the change in f(x) over the change in x, which gives the average rate of change. d. the statement is false because the slope of a line segment gives the negative reciprocal of the average rate of change.

Explanation:

Step1: Recall slope and average - rate - of - change formulas

The slope $m$ of a line segment connecting two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. If $y = f(x)$, the average rate of change of the function $y = f(x)$ from $x_1$ to $x_2$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Compare the two concepts

Since the slope of the line segment connecting two points $(x_1,f(x_1))$ and $(x_2,f(x_2))$ and the average rate of change of the function $y = f(x)$ from $x_1$ to $x_2$ are both calculated as $\frac{\text{change in }y}{\text{change in }x}=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$, the statement is true.

Answer:

C. The statement is true because the slope is equal to the change in f(x) over the change in x, which gives the average rate of change.