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 average rate of change is negative over the interval ___,____

Answer 1

To determine the intervals where the average rate of change of a function is negative, you will need to look at the behavior of the function over different intervals and assess how the function values (y-values) are changing as you move from the left endpoint to the right endpoint of each interval.

The average rate of change of a function \( f \) over the interval \([a, b]\) can be calculated using the formula:

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

If \( f(b) < f(a) \), then the average rate of change is negative, indicating that the function is decreasing over that interval.

To identify specific intervals, follow these steps:

1. **Identify intervals** based on the endpoints shown on the graph. For example, if your graph shows intervals like [a,b], [b,c], [c,d], etc., record these intervals.

2. **Evaluate the function values** at the endpoints:
- For each interval, determine \( f(a) \) and \( f(b) \).

3. **Calculate the average rate of change** for each interval:
- Apply the formula mentioned above to determine if the average rate of change is negative.

4. **Determine the intervals with a negative average rate of change** by checking the condition \( f(b) < f(a) \).

Once you have completed this analysis, you can fill in the statement accordingly. For example:

"The average rate of change is negative over the interval \([x_1, x_2]\), where \( f(x_1) > f(x_2) \)."

Make sure to specify the actual intervals based on your observations from the graph.