Question

The average rate of change is negative over which of the following intervals? Points on the graph are as follows: (0, -1), (1, 0), (2, -1), (3, 1), and (5, 1)

Answer 1

The average rate of change between two points on a graph can be determined by the formula:

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

where \( f(a) \) and \( f(b) \) are the function values at points \( a \) and \( b \).

Now, let's analyze the points provided:

- \( (0, -1) \) → \( f(0) = -1 \)
- \( (1, 0) \) → \( f(1) = 0 \)
- \( (2, -1) \) → \( f(2) = -1 \)
- \( (3, 1) \) → \( f(3) = 1 \)
- \( (5, 1) \) → \( f(5) = 1 \)

Now, we can check each interval between the points to find where the average rate of change is negative:

1. **Interval \([0, 1]\)**:
\[
\text{Average Rate of Change} = \frac{f(1) - f(0)}{1 - 0} = \frac{0 - (-1)}{1 - 0} = \frac{1}{1} = 1
\]
(Not negative)

2. **Interval \([1, 2]\)**:
\[
\text{Average Rate of Change} = \frac{f(2) - f(1)}{2 - 1} = \frac{-1 - 0}{2 - 1} = \frac{-1}{1} = -1
\]
(Negative)

3. **Interval \([2, 3]\)**:
\[
\text{Average Rate of Change} = \frac{f(3) - f(2)}{3 - 2} = \frac{1 - (-1)}{3 - 2} = \frac{2}{1} = 2
\]
(Not negative)

4. **Interval \([3, 5]\)**:
\[
\text{Average Rate of Change} = \frac{f(5) - f(3)}{5 - 3} = \frac{1 - 1}{5 - 3} = \frac{0}{2} = 0
\]
(Not negative)

5. **Interval \([0, 2]\)**:
\[
\text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0} = \frac{-1 - (-1)}{2 - 0} = \frac{0}{2} = 0
\]
(Not negative)

6. **Interval \([1, 3]\)**:
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{1 - 0}{3 - 1} = \frac{1}{2}
\]
(Not negative)

7. **Interval \([2, 5]\)**:
\[
\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{1 - (-1)}{5 - 2} = \frac{2}{3}
\]
(Not negative)

From our analysis, the only interval where the average rate of change is negative is:

**Interval \([1, 2]\)**.