To determine which relationship has a zero slope, we need to analyze the two sets of data points given.
First, let's recall that the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For a line to have a zero slope, the change in [tex]\(y\)[/tex]-values ([tex]\(y_2 - y_1\)[/tex]) must be zero, meaning that all [tex]\(y\)[/tex]-values in the dataset are the same.
Let's apply this to the given data sets.
### Dataset 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]
Here, the [tex]\(y\)[/tex]-values are:
[tex]\[ y = 2, 2, 2, 2 \][/tex]
Since all [tex]\(y\)[/tex]-values are identical, the slope between any two points in this dataset will be:
[tex]\[ \text{slope} = \frac{2 - 2}{\text{any } x_2 - x_1} = 0 \][/tex]
Thus, Dataset 1 has a zero slope.
### Dataset 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]
The [tex]\(y\)[/tex]-values are:
[tex]\[ y = 3, 1, -1, -3 \][/tex]
Here, the [tex]\(y\)[/tex]-values are not identical. Therefore, the slope between different points will not be zero. To see this more clearly, let's calculate one of the slopes:
Using points [tex]\((-3, 3)\)[/tex] and [tex]\((-1, 1)\)[/tex]:
[tex]\[ \text{slope} = \frac{1 - 3}{-1 - (-3)} = \frac{-2}{2} = -1 \][/tex]
So, Dataset 2 does not have a zero slope.
Based on the analysis, the relationship that has a zero slope is represented by:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline -3 & 2 \\
\hline -1 & 2 \\
\hline 1 & 2 \\
\hline 3 & 2 \\
\hline
\end{array}
\][/tex]
Therefore, the correct answer is that the relationship with a zero slope is represented by Dataset 1.
