To find the rate of change represented by the data in the table, let's follow these steps:
### Step 1: Understand the Data
The given table provides pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{c|ccccc}
x & -3 & -1 & 1 & 3 & 5 \\
\hline
y & 2 & 14 & 26 & 38 & 50
\end{array}
\][/tex]
### Step 2: Calculate the Rate of Change for Each Interval
The rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[
\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
We will calculate the rate of change for each consecutive pair of points.
Between [tex]\((-3, 2)\)[/tex] and [tex]\((-1, 14)\)[/tex]:
[tex]\[
\text{Rate of Change} = \frac{14 - 2}{-1 - (-3)} = \frac{12}{2} = 6.0
\][/tex]
Between [tex]\((-1, 14)\)[/tex] and [tex]\((1, 26)\)[/tex]:
[tex]\[
\text{Rate of Change} = \frac{26 - 14}{1 - (-1)} = \frac{12}{2} = 6.0
\][/tex]
Between [tex]\((1, 26)\)[/tex] and [tex]\((3, 38)\)[/tex]:
[tex]\[
\text{Rate of Change} = \frac{38 - 26}{3 - 1} = \frac{12}{2} = 6.0
\][/tex]
Between [tex]\((3, 38)\)[/tex] and [tex]\((5, 50)\)[/tex]:
[tex]\[
\text{Rate of Change} = \frac{50 - 38}{5 - 3} = \frac{12}{2} = 6.0
\][/tex]
### Step 3: Summarize the Results
We have found that the rate of change for each interval is as follows:
[tex]\[
6.0, \, 6.0, \, 6.0, \, 6.0
\][/tex]
### Step 4: Calculate the Average Rate of Change
Since the rate of change is constant across all intervals, the average rate of change is simply the common value of the rates we calculated:
[tex]\[
\text{Average Rate of Change} = 6.0
\][/tex]
### Final Answer
The rate of change represented by the data in the table is [tex]\( 6.0 \)[/tex].
[tex]\[
\boxed{6.0}
\][/tex]
