Question

Tho e first point of the parabola is at (0, -1), the second is at (1, 0), the third is at (2, -1), the fourth at (3, 1), and the fifth one at (5, 1).

Describe the behavior of the function by determining over which interval the function has a negative average rate of change.

The average rate of change is negative over the interval [_, _]

Answer 1

To determine the intervals where the average rate of change is negative for the given points on the parabola, we need to look at the change in \( y \) values over corresponding changes in \( x \) values between the points. The average rate of change between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

We will calculate the average rate of change between each pair of consecutive points.

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

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

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

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

Now, we examine the calculated average rates:

- The average rate of change is **1** from (0, -1) to (1, 0) (positive).
- The average rate of change is **-1** from (1, 0) to (2, -1) (negative).
- The average rate of change is **2** from (2, -1) to (3, 1) (positive).
- The average rate of change is **0** from (3, 1) to (5, 1) (neither positive nor negative).

Thus, the function has a negative average rate of change over the interval:

**[1, 2]**