Question

Describe the behavior of the function by determining over which intervals the functthas a negative average rate of change
The average rate of change is negative over the ______ interval and over the interval ___
Points on this graph are (1,0), (2,1) , (3,-1) and (5,-1)

Answer 1

To determine the intervals where the average rate of change of the function is negative, we need to analyze the provided points.

The points given are:
1. (1, 0)
2. (2, 1)
3. (3, -1)
4. (5, -1)

The average rate of change of a function between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

We can calculate the average rate of change over the intervals defined by the provided points:

1. **Between (1, 0) and (2, 1):**
\[
\text{Average Rate of Change} = \frac{1 - 0}{2 - 1} = \frac{1}{1} = 1 \quad \text{(Positive)}
\]

2. **Between (2, 1) and (3, -1):**
\[
\text{Average Rate of Change} = \frac{-1 - 1}{3 - 2} = \frac{-2}{1} = -2 \quad \text{(Negative)}
\]

3. **Between (3, -1) and (5, -1):**
\[
\text{Average Rate of Change} = \frac{-1 - (-1)}{5 - 3} = \frac{0}{2} = 0 \quad \text{(Neutral)}
\]

4. **Between (1, 0) and (3, -1):**
\[
\text{Average Rate of Change} = \frac{-1 - 0}{3 - 1} = \frac{-1}{2} = -0.5 \quad \text{(Negative)}
\]

5. **Between (2, 1) and (5, -1):**
\[
\text{Average Rate of Change} = \frac{-1 - 1}{5 - 2} = \frac{-2}{3} \quad \text{(Negative)}
\]

From this analysis, we find that the average rate of change is negative over the following intervals:
- From \(x = 2\) to \(x = 3\)
- From \(x = 1\) to \(x = 3\)
- From \(x = 2\) to \(x = 5\)

Therefore, we can summarize:

The average rate of change is negative over the **interval \( (2, 3) \)** and over the interval **\( (1, 3) \)**, and also \( (2, 5) \). If you seek only explicit intervals as defined between two points, the most distinct interval is \( (2, 3) \).