To determine the average rate of change of the function based on the provided table, we can follow these steps:
1. Identify the data points: The table gives us the following pairs [tex]\((x, y)\)[/tex]:
- [tex]\((-2, 7)\)[/tex]
- [tex]\((-1, 6)\)[/tex]
- [tex]\((0, 5)\)[/tex]
- [tex]\((1, 4)\)[/tex]
2. Calculate the slopes (rate of change) between consecutive points:
- Between points [tex]\((-2, 7)\)[/tex] and [tex]\((-1, 6)\)[/tex]:
[tex]\[
\text{slope} = \frac{6 - 7}{-1 - (-2)} = \frac{-1}{1} = -1
\][/tex]
- Between points [tex]\((-1, 6)\)[/tex] and [tex]\((0, 5)\)[/tex]:
[tex]\[
\text{slope} = \frac{5 - 6}{0 - (-1)} = \frac{-1}{1} = -1
\][/tex]
- Between points [tex]\((0, 5)\)[/tex] and [tex]\((1, 4)\)[/tex]:
[tex]\[
\text{slope} = \frac{4 - 5}{1 - 0} = \frac{-1}{1} = -1
\][/tex]
3. List the slopes: The slopes for each segment are:
- [tex]\(-1\)[/tex]
- [tex]\(-1\)[/tex]
- [tex]\(-1\)[/tex]
4. Calculate the average of these slopes to find the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{-1 + (-1) + (-1)}{3} = \frac{-3}{3} = -1
\][/tex]
Thus, the function has an average rate of change of [tex]\(-1\)[/tex].
