Answer :
To determine which relationship has a zero slope, we need to analyze the given data tables and calculate the slope for each.
### Table 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]
### Table 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]
#### Finding the Slope of Table 1:
- Slope is calculated as \(\frac{\Delta y}{\Delta x}\).
For Table 1:
- Between \((-3, 2)\) and \((-1, 2)\):
[tex]\[ \frac{2 - 2}{-1 - (-3)} = \frac{0}{2} = 0 \][/tex]
- Between \((-1, 2)\) and \( (1, 2)\):
[tex]\[ \frac{2 - 2}{1 - (-1)} = \frac{0}{2} = 0 \][/tex]
- Between \( (1, 2)\) and \( (3, 2)\):
[tex]\[ \frac{2 - 2}{3 - 1} = \frac{0}{2} = 0 \][/tex]
Since the slope is 0 for all intervals, the entire relationship in Table 1 has a zero slope.
#### Finding the Slope of Table 2:
For Table 2:
- Between \((-3, 3)\) and \((-1, 1)\):
[tex]\[ \frac{1 - 3}{-1 - (-3)} = \frac{-2}{2} = -1 \][/tex]
- Between \((-1, 1)\) and \( (1, -1)\):
[tex]\[ \frac{-1 - 1}{1 - (-1)} = \frac{-2}{2} = -1 \][/tex]
- Between \( (1, -1)\) and \( (3, -3)\):
[tex]\[ \frac{-3 - (-1)}{3 - 1} = \frac{-2}{2} = -1 \][/tex]
Since the slope is -1 for all intervals, the slope is not zero for Table 2.
### Conclusion:
The relationship in Table 1 has a zero slope.
### Table 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]
### Table 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]
#### Finding the Slope of Table 1:
- Slope is calculated as \(\frac{\Delta y}{\Delta x}\).
For Table 1:
- Between \((-3, 2)\) and \((-1, 2)\):
[tex]\[ \frac{2 - 2}{-1 - (-3)} = \frac{0}{2} = 0 \][/tex]
- Between \((-1, 2)\) and \( (1, 2)\):
[tex]\[ \frac{2 - 2}{1 - (-1)} = \frac{0}{2} = 0 \][/tex]
- Between \( (1, 2)\) and \( (3, 2)\):
[tex]\[ \frac{2 - 2}{3 - 1} = \frac{0}{2} = 0 \][/tex]
Since the slope is 0 for all intervals, the entire relationship in Table 1 has a zero slope.
#### Finding the Slope of Table 2:
For Table 2:
- Between \((-3, 3)\) and \((-1, 1)\):
[tex]\[ \frac{1 - 3}{-1 - (-3)} = \frac{-2}{2} = -1 \][/tex]
- Between \((-1, 1)\) and \( (1, -1)\):
[tex]\[ \frac{-1 - 1}{1 - (-1)} = \frac{-2}{2} = -1 \][/tex]
- Between \( (1, -1)\) and \( (3, -3)\):
[tex]\[ \frac{-3 - (-1)}{3 - 1} = \frac{-2}{2} = -1 \][/tex]
Since the slope is -1 for all intervals, the slope is not zero for Table 2.
### Conclusion:
The relationship in Table 1 has a zero slope.
