To determine which graph shows a proportional relationship between $x$ and $y$, we should recall the following key property:

A proportional relationship between $x$ and $y$ means that $y = kx$, where $k$ is a constant. This relationship produces a straight line passing through the origin $(0, 0)$.

Analysis of the graphs:

• Graph A: Shows a step-like behavior. This is not a straight line and doesn't pass through the origin, so it cannot represent a proportional relationship.
• Graph B: Displays a horizontal line. A horizontal line indicates a constant $y$ value regardless of $x$, which is not proportional since $y$ would not increase as $x$ increases.
• Graph C: Depicts a straight vertical line. This means $x$ is constant while $y$ changes, which is also not proportional.

Conclusion:

None of the graphs provided in the image represent a proportional relationship between $x$ and $y$. For a graph to show a proportional relationship, it must be a straight line that passes through the origin.

Would you like further clarification or an explanation of proportional relationships?

Here are 5 questions to expand this topic:

1. What are examples of real-life proportional relationships?
2. Can you identify proportional relationships in tables or equations?
3. How does the slope of a line relate to proportionality?
4. What happens if a line is straight but doesn’t pass through the origin?
5. Can you describe how proportional graphs differ from non-proportional ones?

Tip: To identify proportionality quickly, check if the graph forms a straight line through the origin or if the ratio $\frac{y}{x}$ remains constant.