To determine if the tables show a constant rate of change, we first need to calculate the rate of change between each pair of points in each table.
### First Table
The first table has the following values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
| [tex]$x$[/tex] | [tex]$y$[/tex] |
|-----|-----|
| 1 | 3 |
| 2 | 1 |
| 3 | -1 |
| 4 | -3 |
| 5 | -5 |
| 6 | -7 |
We will calculate the rate of change (slope) between each consecutive pair of points. The rate of change is calculated using the formula:
[tex]\[
\text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
1. Between (1, 3) and (2, 1):
[tex]\[
\frac{1 - 3}{2 - 1} = \frac{-2}{1} = -2
\][/tex]
2. Between (2, 1) and (3, -1):
[tex]\[
\frac{-1 - 1}{3 - 2} = \frac{-2}{1} = -2
\][/tex]
3. Between (3, -1) and (4, -3):
[tex]\[
\frac{-3 - (-1)}{4 - 3} = \frac{-2}{1} = -2
\][/tex]
4. Between (4, -3) and (5, -5):
[tex]\[
\frac{-5 - (-3)}{5 - 4} = \frac{-2}{1} = -2
\][/tex]
5. Between (5, -5) and (6, -7):
[tex]\[
\frac{-7 - (-5)}{6 - 5} = \frac{-2}{1} = -2
\][/tex]
So, the constant rate of change for the first table is [tex]\(-2\)[/tex] for all intervals.
### Second Table
The second table has the following values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
| [tex]$x$[/tex] | [tex]$y$[/tex] |
|-----|------|
| 1 | 5 |
| 2 | 5.25 |
| 3 | 5.5 |
| 4 | 5.75 |
| 5 | 6 |
| 6 | 6.25 |
We will calculate the rate of change between each consecutive pair of points using the same formula.
1. Between (1, 5) and (2, 5.25):
[tex]\[
\frac{5.25 - 5}{2 - 1} = \frac{0.25}{1} = 0.25
\][/tex]
2. Between (2, 5.25) and (3, 5.5):
[tex]\[
\frac{5.5 - 5.25}{3 - 2} = \frac{0.25}{1} = 0.25
\][/tex]
3. Between (3, 5.5) and (4, 5.75):
[tex]\[
\frac{5.75 - 5.5}{4 - 3} = \frac{0.25}{1} = 0.25
\][/tex]
4. Between (4, 5.75) and (5, 6):
[tex]\[
\frac{6 - 5.75}{5 - 4} = \frac{0.25}{1} = 0.25
\][/tex]
5. Between (5, 6) and (6, 6.25):
[tex]\[
\frac{6.25 - 6}{6 - 5} = \frac{0.25}{1} = 0.25
\][/tex]
So, the constant rate of change for the second table is [tex]\(0.25\)[/tex] for all intervals.
### Conclusion
Both tables show a constant rate of change between all their respective pairs of points. Specifically:
- The first table has a constant rate of change of [tex]\(-2\)[/tex].
- The second table has a constant rate of change of [tex]\(0.25\)[/tex].
