Question

Describe the behavior of the function by determining over which interval the function has a negative average rate of change. Consider only intervals with consecutive endpoints shown on the graph. The points on the graph are as follows: (0, 0), (1, 4), (2, 4), (3, 0), (4, 4), (5, 5)

Answer 1

To determine the intervals over which the function has a negative average rate of change, we first need to calculate the average rate of change between consecutive points on the given graph. The average rate of change of a function \( f \) between the points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is given by the formula:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Now, let's calculate the average rate of change for each pair of consecutive points from the graph:

1. **Interval between (0, 0) and (1, 4)**:
\[
\text{Average Rate of Change} = \frac{4 - 0}{1 - 0} = \frac{4}{1} = 4
\]

2. **Interval between (1, 4) and (2, 4)**:
\[
\text{Average Rate of Change} = \frac{4 - 4}{2 - 1} = \frac{0}{1} = 0
\]

3. **Interval between (2, 4) and (3, 0)**:
\[
\text{Average Rate of Change} = \frac{0 - 4}{3 - 2} = \frac{-4}{1} = -4
\]

4. **Interval between (3, 0) and (4, 4)**:
\[
\text{Average Rate of Change} = \frac{4 - 0}{4 - 3} = \frac{4}{1} = 4
\]

5. **Interval between (4, 4) and (5, 5)**:
\[
\text{Average Rate of Change} = \frac{5 - 4}{5 - 4} = \frac{1}{1} = 1
\]

Now, let's summarize the average rates of change we calculated:

- From (0, 0) to (1, 4): **4** (positive)
- From (1, 4) to (2, 4): **0** (neither negative nor positive)
- From (2, 4) to (3, 0): **-4** (negative)
- From (3, 0) to (4, 4): **4** (positive)
- From (4, 4) to (5, 5): **1** (positive)

Based on the above calculations, the only interval where the function has a negative average rate of change is:

**Interval: (2, 3)**

This means that between the points \( (2, 4) \) and \( (3, 0) \), the function decreases, leading to a negative average rate of change.