IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Join our interactive Q&A platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
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].
### 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].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.
