https://file.aiquickdraw.com/mathbot/file/1726138504155pfcvbphb.jpg

Let's analyze the tables in the image:

1. **Table 6**:
   \[
   \begin{array}{|c|c|c|c|c|c|c|}
   \hline
   x & -4 & -2 & 0 & 2 & 4 \\
   \hline
   y & 5 & -3 & -3 & 5 & 21 \\
   \hline
   \end{array}
   \]

2. **Table 8**:
   \[
   \begin{array}{|c|c|c|c|c|}
   \hline
   x & -4 & -2 & 0 & 2 & 4 \\
   \hline
   y & 41 & 5 & -7 & 5 & 41 \\
   \hline
   \end{array}
   \]

To determine whether these tables represent linear or quadratic relationships, we can perform a difference test. This involves calculating the differences between successive \(y\)-values:

### Table 6:
\[
\begin{aligned}
\Delta y_1 &= -3 - 5 = -8 \\
\Delta y_2 &= -3 - (-3) = 0 \\
\Delta y_3 &= 5 - (-3) = 8 \\
\Delta y_4 &= 21 - 5 = 16 \\
\end{aligned}
\]

First differences: \([-8, 0, 8, 16]\).

Next, calculate the second differences:
\[
\begin{aligned}
\Delta^2 y_1 &= 0 - (-8) = 8 \\
\Delta^2 y_2 &= 8 - 0 = 8 \\
\Delta^2 y_3 &= 16 - 8 = 8 \\
\end{aligned}
\]

The second differences are constant, which means **Table 6 represents a quadratic relationship**.

### Table 8:
\[
\begin{aligned}
\Delta y_1 &= 5 - 41 = -36 \\
\Delta y_2 &= -7 - 5 = -12 \\
\Delta y_3 &= 5 - (-7) = 12 \\
\Delta y_4 &= 41 - 5 = 36 \\
\end{aligned}
\]

First differences: \([-36, -12, 12, 36]\).

Next, calculate the second differences:
\[
\begin{aligned}
\Delta^2 y_1 &= -12 - (-36) = 24 \\
\Delta^2 y_2 &= 12 - (-12) = 24 \\
\Delta^2 y_3 &= 36 - 12 = 24 \\
\end{aligned}
\]

Since the second differences are constant, **Table 8 also represents a quadratic relationship**.

### Conclusion:
Both tables represent quadratic relationships, as the second differences are constant in both cases.

Would you like further clarification on this, or should I dive into more details? Here are five questions that build on this information:

1. How do we determine if a function is linear or quadratic?
2. What are higher-order differences, and how do they help classify functions?
3. Can you explain how the second difference test works?
4. How can we derive the equation of a quadratic function from a table?
5. How does the behavior of quadratic functions differ from linear ones?

**Tip**: Always check the consistency of second differences to quickly identify quadratic functions from tabular data.

Learn how to identify quadratic functions by examining the second differences in tabular data. This example includes two tables where second differences reveal quadratic relationships, suitable for students in Grades 7-10.

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