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Which characteristics of a graph tell you that it represents a proportional relationship?
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GPT-5 mini
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A graph represents a proportional relationship when:
- It is a straight line (linear).
- The line passes through the origin (0, 0) — equivalently the y‑intercept is 0.
- The ratio y/x is constant for every point on the graph; that constant is the constant of proportionality (slope k in y = kx).
- Doubling (or scaling) x scales y by the same factor (y changes in direct proportion to x).
How to check quickly:
- Pick any two nonzero points (x1,y1) and (x2,y2). If y1/x1 = y2/x2 (or equivalently (y2−y1)/(x2−x1) = y1/x1) and one point is (0,0), the relationship is proportional.
- Or verify the y‑intercept is 0 and the graph is a straight line.
Example: y = 4x is proportional (line through origin, slope 4). y = 4x + 2 is linear but not proportional (y‑intercept ≠ 0).
- It is a straight line (linear).
- The line passes through the origin (0, 0) — equivalently the y‑intercept is 0.
- The ratio y/x is constant for every point on the graph; that constant is the constant of proportionality (slope k in y = kx).
- Doubling (or scaling) x scales y by the same factor (y changes in direct proportion to x).
How to check quickly:
- Pick any two nonzero points (x1,y1) and (x2,y2). If y1/x1 = y2/x2 (or equivalently (y2−y1)/(x2−x1) = y1/x1) and one point is (0,0), the relationship is proportional.
- Or verify the y‑intercept is 0 and the graph is a straight line.
Example: y = 4x is proportional (line through origin, slope 4). y = 4x + 2 is linear but not proportional (y‑intercept ≠ 0).
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