Question

1. what does a positive average rate of change indicate about a function over an interval?

a. the function is decreasing.
b. the function is undefined.
c. the function is constant.
d. the function is increasing.

Explanation:

Step1: Recall the concept of average rate of change

The average rate of change of a function \(y = f(x)\) over an interval \([x_1,x_2]\) is given by \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).

Step2: Analyze the sign of the average rate of change

If the average rate of change \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}>0\), then \(f(x_2)-f(x_1)\) and \(x_2 - x_1\) have the same sign. When \(x_2>x_1\) (i.e., \(x_2 - x_1>0\)), then \(f(x_2)>f(x_1)\). This means that as \(x\) increases, \(y = f(x)\) also increases.

Answer:

d. The function is increasing.