To determine which relationship has a zero slope, we need to calculate the slopes of the two given data sets.
### Data Set 1
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 2 \\
\hline
-1 & 2 \\
\hline
1 & 2 \\
\hline
3 & 2 \\
\hline
\end{array}
\][/tex]
### Data Set 2
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 3 \\
\hline
-1 & 1 \\
\hline
1 & -1 \\
\hline
3 & -3 \\
\hline
\end{array}
\][/tex]
#### Slope Calculation for Data Set 1
The slope is calculated using the formula:
[tex]\[
\text{slope} = \frac{\Delta y}{\Delta x}
\][/tex]
For any two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], [tex]\(\Delta y = y_2 - y_1\)[/tex] and [tex]\(\Delta x = x_2 - x_1\)[/tex].
Choosing the first two points from Data Set 1:
[tex]\[
(x_1, y_1) = (-3, 2)
\][/tex]
[tex]\[
(x_2, y_2) = (-1, 2)
\][/tex]
[tex]\[
\Delta y = 2 - 2 = 0
\][/tex]
[tex]\[
\Delta x = -1 - (-3) = -1 + 3 = 2
\][/tex]
Substitute these into the slope formula:
[tex]\[
\text{slope} = \frac{0}{2} = 0
\][/tex]
Thus, the slope for Data Set 1 is [tex]\(0\)[/tex].
#### Slope Calculation for Data Set 2
Choosing the first two points from Data Set 2:
[tex]\[
(x_1, y_1) = (-3, 3)
\][/tex]
[tex]\[
(x_2, y_2) = (-1, 1)
\][/tex]
[tex]\[
\Delta y = 1 - 3 = -2
\][/tex]
[tex]\[
\Delta x = -1 - (-3) = -1 + 3 = 2
\][/tex]
Substitute these into the slope formula:
[tex]\[
\text{slope} = \frac{-2}{2} = -1
\][/tex]
Thus, the slope for Data Set 2 is [tex]\(-1\)[/tex].
### Conclusion
- The slope for Data Set 1 is [tex]\(0\)[/tex].
- The slope for Data Set 2 is [tex]\(-1\)[/tex].
A zero slope indicates a horizontal line. Therefore, the relationship represented by Data Set 1 has a zero slope.
