To determine which of the given relationships has a zero slope, we need to inspect the rates of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] in each table. Let's break down the steps in detail.
### Analyzing the First Table:
[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 have points [tex]\((-3, 2)\)[/tex], [tex]\((-1, 2)\)[/tex], [tex]\( (1, 2)\)[/tex], and [tex]\( (3, 2)\)[/tex].
1. Calculate the slope between consecutive points:
- Between [tex]\((-3, 2)\)[/tex] and [tex]\((-1, 2)\)[/tex]:
[tex]\[
\text{slope} = \frac{2 - 2}{-1 - (-3)} = \frac{0}{2} = 0
\][/tex]
- Between [tex]\((-1, 2)\)[/tex] and [tex]\( (1, 2)\)[/tex]:
[tex]\[
\text{slope} = \frac{2 - 2}{1 - (-1)} = \frac{0}{2} = 0
\][/tex]
- Between [tex]\( (1, 2)\)[/tex] and [tex]\( (3, 2)\)[/tex]:
[tex]\[
\text{slope} = \frac{2 - 2}{3 - 1} = \frac{0}{2} = 0
\][/tex]
Since all slopes between consecutive points are 0, the first table represents a relationship with a zero slope.
### Analyzing the Second Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 3 \\
\hline
-1 & 1 \\
\hline
1 & -1 \\
\hline
3 & -3 \\
\hline
\end{array}
\][/tex]
We have points [tex]\((-3, 3)\)[/tex], [tex]\((-1, 1)\)[/tex], [tex]\( (1, -1)\)[/tex], and [tex]\( (3, -3)\)[/tex].
1. Calculate the slope between consecutive points:
- Between [tex]\((-3, 3)\)[/tex] and [tex]\((-1, 1)\)[/tex]:
[tex]\[
\text{slope} = \frac{1 - 3}{-1 - (-3)} = \frac{-2}{2} = -1
\][/tex]
- Between [tex]\((-1, 1)\)[/tex] and [tex]\( (1, -1)\)[/tex]:
[tex]\[
\text{slope} = \frac{-1 - 1}{1 - (-1)} = \frac{-2}{2} = -1
\][/tex]
- Between [tex]\( (1, -1)\)[/tex] and [tex]\( (3, -3)\)[/tex]:
[tex]\[
\text{slope} = \frac{-3 - (-1)}{3 - 1} = \frac{-2}{2} = -1
\][/tex]
In this case, all slopes between consecutive points are [tex]\(-1\)[/tex], indicating a consistent, negative slope.
### Conclusion:
From our analysis:
- The first table has a zero slope because all slopes between points are 0.
- The second table does not have a zero slope because all slopes between points are [tex]\(-1\)[/tex].
Thus, the relationship in the first table has a zero slope.
