Examine the table and determine the rate of change.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & -4 \\
\hline
2 & -1 \\
\hline
3 & 2 \\
\hline
4 & 5 \\
\hline
\end{tabular}

What is the rate of change?

A. [tex]$-4$[/tex]
B. [tex]$-3$[/tex]
C. [tex]$1$[/tex]
D. [tex]$3$[/tex]



Answer :

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].