To determine which relationship has a zero slope, we need to calculate the slope for each of the sets of points given in the two tables.
### Relationship 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]
We can calculate the slope between any two points using the formula for the slope \( m \):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
#### Slope calculations:
1. Between points \((-3, 2)\) and \((-1, 2)\):
[tex]\[
m_1 = \frac{2 - 2}{-1 - (-3)} = \frac{0}{2} = 0
\][/tex]
2. Between points \((-3, 2)\) and \( (1, 2)\):
[tex]\[
m_2 = \frac{2 - 2}{1 - (-3)} = \frac{0}{4} = 0
\][/tex]
3. Between points \((-3, 2)\) and \((3, 2)\):
[tex]\[
m_3 = \frac{2 - 2}{3 - (-3)} = \frac{0}{6} = 0
\][/tex]
The slope for each pair of points in Relationship 1 is \(0\).
### Relationship 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 calculations:
1. Between points \((-3, 3)\) and \((-1, 1)\):
[tex]\[
m_1 = \frac{1 - 3}{-1 - (-3)} = \frac{-2}{2} = -1
\][/tex]
2. Between points \((-3, 3)\) and \( (1, -1)\):
[tex]\[
m_2 = \frac{-1 - 3}{1 - (-3)} = \frac{-4}{4} = -1
\][/tex]
3. Between points \((-3, 3)\) and \((3, -3)\):
[tex]\[
m_3 = \frac{-3 - 3}{3 - (-3)} = \frac{-6}{6} = -1
\][/tex]
The slope for each pair of points in Relationship 2 is \(-1\).
### Conclusion:
Since the first relationship has a consistent slope of [tex]\(0\)[/tex], the relationship with a zero slope is Relationship 1.
