https://file.aiquickdraw.com/mathbot/file/1727200531904lun34bmx.jpg

The question asks which representation shows a proportional relationship between \(x\) and \(y\).

A proportional relationship is one where the ratio between \(x\) and \(y\) remains constant, meaning \( \frac{y}{x} \) should always give the same value for different pairs of \(x\) and \(y\). Let's evaluate the options:

### Option A:
- \( \frac{22}{12} = 1.833 \)
- \( \frac{32}{17} = 1.882 \)
- \( \frac{48}{25} = 1.92 \)
- \( \frac{58}{30} = 1.933 \)

These values are not equal, so the relationship in A is **not proportional**.

### Option B:
- \( \frac{4}{5} = 0.8 \)
- \( \frac{8}{10} = 0.8 \)
- \( \frac{10}{15} = 0.8 \)
- \( \frac{15}{25} = 0.6 \)

These ratios are **not constant**, so B does not represent a proportional relationship.

### Option C:
This graph shows a straight line through the origin, which indicates a proportional relationship because a straight line through the origin has the equation \(y = kx\) where \(k\) is a constant.

### Option D:
This is a curve, not a straight line, so it does not represent a proportional relationship.

### Conclusion:
The correct answer is **C**, as it is the only representation that shows a proportional relationship between \(x\) and \(y\).

Would you like more details or have any questions? Here are 5 related questions to deepen understanding:

1. How do you verify proportionality using a table of values?
2. What is the equation of a proportional relationship in algebra?
3. Why is the straight line through the origin a sign of proportionality?
4. What is the constant of proportionality, and how do you find it?
5. How do curves and other non-linear graphs represent non-proportional relationships?

**Tip:** Always check the consistency of the ratio \( \frac{y}{x} \) when verifying proportional relationships in tables.

Which representation shows a proportional relationship between x and y?

Learn how to identify proportional relationships between x and y using tables and graphs. This problem teaches key concepts such as ratios and linear equations, focusing on determining whether the ratio between x and y is constant.

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