To determine the rate of change for the given table, we need to find the rate at which [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] increases. Here is the detailed step-by-step process to achieve that:
1. Identify the Data Points:
The table provides the following data points:
[tex]\[
(1, -4), (2, -1), (3, 2), (4, 5)
\][/tex]
2. Calculate the Rate of Change Between Each Pair of Points:
The rate of change (or slope) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Let's apply this formula to each consecutive pair of points:
- For [tex]\((1, -4)\)[/tex] and [tex]\((2, -1)\)[/tex]:
[tex]\[
\text{Rate of change} = \frac{-1 - (-4)}{2 - 1} = \frac{-1 + 4}{1} = \frac{3}{1} = 3.0
\][/tex]
- For [tex]\((2, -1)\)[/tex] and [tex]\((3, 2)\)[/tex]:
[tex]\[
\text{Rate of change} = \frac{2 - (-1)}{3 - 2} = \frac{2 + 1}{1} = \frac{3}{1} = 3.0
\][/tex]
- For [tex]\((3, 2)\)[/tex] and [tex]\((4, 5)\)[/tex]:
[tex]\[
\text{Rate of change} = \frac{5 - 2}{4 - 3} = \frac{5 - 2}{1} = \frac{3}{1} = 3.0
\][/tex]
3. Consistent Rate of Change:
We observe that the rate of change between each consecutive pair of points is the same, equal to 3.0.
4. Conclusion:
Since the rate of change is consistent and equal to 3.0, we conclude that the rate of change for the table is:
[tex]\[
\boxed{3}
\][/tex]
Therefore, the correct answer is:
[tex]\(3\)[/tex].
