Question

The x-axis ranges from 0 to 6 in increments in unit increments. The vertical axis represents f of x ranging from negative 2 to 3 in unit increments. A line passes through the coordinates left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 1 comma 1 right parenthesis, left parenthesis 3 comma 0 right parenthesis, left parenthesis 4 comma negative 0 point 5 right parenthesis, left parenthesis 5 comma negative 0 point 5 right parenthesis, and left parenthesis 6 comma 0 right parenthesis.

Given the graph of f(x) , on which interval is the average rate of change the greatest?

(1 point)
Responses

The function has the greatest average rate of change over the interval [0,1] .
The function has the greatest average rate of change over the interval left bracket 0 comma 1 right bracket .

The function has the greatest average rate of change over the interval [4,5] .
The function has the greatest average rate of change over the interval left bracket 4 comma 5 right bracket .

The function has the greatest average rate of change over the interval [1,4] .
The function has the greatest average rate of change over the interval left bracket 1 comma 4 right bracket .

The function has the greatest average rate of change over the interval [5,6] .
The function has the greatest average rate of change over the interval left bracket 5 comma 6 right bracket .

Answer 1

Compute the average rate of change (slope) on each interval:

- [0,1]: f(0) = 2/3 (interpolating between (-2,0) and (1,1)), f(1)=1 → slope = (1 − 2/3)/(1 − 0) = 1/3 ≈ 0.333
- [4,5]: f(4)=−0.5, f(5)=−0.5 → slope = 0
- [1,4]: f(1)=1, f(4)=−0.5 → slope = (−0.5 − 1)/(4 − 1) = −1.5/3 = −0.5
- [5,6]: f(5)=−0.5, f(6)=0 → slope = (0 − (−0.5))/(6 − 5) = 0.5

Greatest average rate of change is 0.5 on the interval [5,6].