Question

The x-axis ranges from 0 to 7 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 1 comma 0 right parenthesis, left parenthesis 1 comma 1 right parenthesis, left parenthesis 3 comma 0 right parenthesis, left parenthesis 5 comma 2 right parenthesis, left parenthesis 6 comma 2 right parenthesis, left parenthesis 7 comma 0 right parenthesis.

Describe the graph of f(x) by selecting the correct statement.

(1 point)
Responses

The average rate of change is negative only on the interval [6,7] .
The average rate of change is negative only on the interval left bracket six comma seven right bracket .

The average rate of change is negative on the interval [0,1] and on the interval [3,4] .
The average rate of change is negative on the interval left bracket 0 comma 1 right bracket and on the interval left bracket 3 comma 4 right bracket .

The average rate of change is negative on the interval [1,3] and also on the interval [6,7] .
The average rate of change is negative on the interval left bracket 1 comma 3 right bracket and also on the interval left bracket 6 comma 7 right bracket .

The average rate of change is negative only on the interval [1,3] .
The average rate of change is negative only on the interval left bracket 1 comma 3 right bracket .

Answer 1

Compute average rates on the unit intervals using the given points (piecewise linear):

- [0,1]: f(0)=0.5 (on segment (-1,0)→(1,1)), f(1)=1 ⇒ AROC = (1−0.5)/1 = +0.5 (positive)
- [1,3]: f(1)=1, f(3)=0 ⇒ AROC = (0−1)/2 = −0.5 (negative)
- [3,4]: f(3)=0, f(4)=1 (on segment (3,0)→(5,2)) ⇒ AROC = +1 (positive)
- [6,7]: f(6)=2, f(7)=0 ⇒ AROC = (0−2)/1 = −2 (negative)

Therefore the average rate of change is negative on [1,3] and on [6,7].