Question

The main points Are (0,-1) (1,1) (3,0) (5,2) (6,2) (7,0)

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

(1 point)
Responses

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 .

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 only on the interval [6,7]
.

Answer 1

To analyze the average rate of change of the function based on the given points, we will calculate the average rate of change between the relevant intervals using the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

We process the specified intervals one by one:

1. **Interval \([1,3]\)**:
- Points: (1, 1) and (3, 0)
- Calculation:
\[
\text{Average Rate of Change} = \frac{0 - 1}{3 - 1} = \frac{-1}{2} = -0.5
\]
Since this is negative, the average rate of change is negative on \([1,3]\).

2. **Interval \([6,7]\)**:
- Points: (6, 2) and (7, 0)
- Calculation:
\[
\text{Average Rate of Change} = \frac{0 - 2}{7 - 6} = \frac{-2}{1} = -2
\]
This is also negative, so the average rate of change is negative on \([6,7]\).

3. **Interval \([0,1]\)**:
- Points: (0, -1) and (1, 1)
- Calculation:
\[
\text{Average Rate of Change} = \frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2
\]
This is positive.

4. **Interval \([3,6]\)**:
- Points: (3, 0) and (6, 2)
- Calculation:
\[
\text{Average Rate of Change} = \frac{2 - 0}{6 - 3} = \frac{2}{3} \approx 0.67
\]
This is positive.

5. **Interval \([3,4]\)**:
- This interval does not contain any points based on the data provided.

Now summarizing the findings:
- The average rate of change is negative on \([1,3]\) and also on \([6,7]\).
- The average rate of change is positive on \([0,1]\) and \([3,6]\).

Based on this analysis, the correct statement is:

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