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To determine which representation shows a **proportional relationship** between \( x \) and \( y \), we need to identify where \( y \) is directly proportional to \( x \). This means that for each pair \((x, y)\), the ratio \( \frac{y}{x} \) should be constant.

Let's analyze each option:

### Option A: Table
\[
\begin{array}{c|c}
x & y \\
\hline
2 & 1 \\
4 & 3 \\
17 & 16 \\
29 & 28 \\
\end{array}
\]

For a proportional relationship, \( \frac{y}{x} \) should be constant:
1. \( \frac{y}{x} = \frac{1}{2} = 0.5 \)
2. \( \frac{y}{x} = \frac{3}{4} = 0.75 \)
3. \( \frac{y}{x} = \frac{16}{17} \approx 0.941 \)
4. \( \frac{y}{x} = \frac{28}{29} \approx 0.966 \)

The ratios are not the same, so **Option A does not represent a proportional relationship**.

### Option B: Table
\[
\begin{array}{c|c}
x & y \\
\hline
9 & 28 \\
12 & 34 \\
16 & 52 \\
28 & 82 \\
\end{array}
\]

Let's calculate \( \frac{y}{x} \) for each pair:
1. \( \frac{y}{x} = \frac{28}{9} \approx 3.11 \)
2. \( \frac{y}{x} = \frac{34}{12} \approx 2.83 \)
3. \( \frac{y}{x} = \frac{52}{16} = 3.25 \)
4. \( \frac{y}{x} = \frac{82}{28} \approx 2.93 \)

The ratios are not the same, so **Option B does not represent a proportional relationship**.

### Option C: Graph
The graph shows a straight line passing through the origin (0,0) with a positive slope. 

- A straight line through the origin indicates a proportional relationship where \( y = kx \) for some constant \( k \).

Thus, **Option C represents a proportional relationship**.

### Option D: Graph
The graph shows a straight line, but it does **not** pass through the origin (0,0).

- Since it does not go through the origin, **Option D does not represent a proportional relationship**.

### Conclusion:
The correct answer is **Option C**, as it represents a proportional relationship between \( x \) and \( y \).

Would you like more details on any part of this solution or have any questions?

Here are 5 related questions you might consider:
1. What is the definition of a proportional relationship in mathematics?
2. How do you determine if a graph represents a proportional relationship?
3. What are some examples of real-world proportional relationships?
4. How can you convert a non-proportional relationship into a proportional one?
5. What is the significance of the slope in a proportional relationship graph?

**Tip:** When checking for proportional relationships in a table, always look for a constant ratio \( \frac{y}{x} \).

Learn how to identify proportional relationships in mathematics with detailed examples and analysis. This page discusses the concept of proportional relationships and provides methods for determining them using tables and graphs. Suitable for Grades 6-8.

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