Proportional Graphs - Pre Algebra

1. A graph passes through the points $(-4, -6)$, $(0, 0)$, and $(8, 12)$. A student claims this represents a proportional relationship because 'it goes through zero and the points make a straight line.' Evaluate this student's reasoning.
   Answer: The student's reasoning is correct because passing through the origin and forming a straight line are the two essential criteria for proportional relationships The student's reasoning is mathematically sound. A proportional relationship requires exactly two conditions: the graph must be linear (straight line) and pass through the origin (0,0). This graph satisfies both requirements. The constant of proportionality is $k = \frac{3}{2}$. Choice A incorrectly restricts proportional relationships to positive slopes. Choice C incorrectly excludes negative values. Choice D incorrectly states the origin requirement.
2. A proportional relationship is graphed on a coordinate plane. If the point $(6, 15)$ lies on this graph, which of the following points must also lie on the graph, and why?
   Answer: The point $(0, 0)$ because proportional relationships always pass through the origin, establishing the constant of proportionality as $\frac{5}{2}$ Proportional relationships must pass through the origin (0,0) by definition. The constant of proportionality is $k = \frac{15}{6} = \frac{5}{2}$, so the relationship is $y = \frac{5}{2}x$. Choice A shows a correct proportional point but gives an incomplete explanation of why it must be there. Choice C incorrectly suggests adding the same value maintains proportionality. Choice D incorrectly describes symmetry properties that don't apply to proportional relationships.
3. A student claims that the relationship $y = 3x + 0$ will produce a proportional graph because 'adding zero doesn't change anything.' Based on the graph of this equation, what can you conclude about the student's reasoning?
   Answer: The student is correct about the outcome but wrong about the reasoning since $y = 3x + 0$ simplifies to $y = 3x$ which is proportional The student reaches the correct conclusion that this represents a proportional relationship, but for the wrong reasons. The equation $y = 3x + 0$ simplifies to $y = 3x$, which is indeed proportional because it passes through the origin and maintains a constant ratio. The student's reasoning about 'adding zero doesn't change anything' is mathematically sound but misses the key point about proportional form. Choice A accepts flawed reasoning. Choice B incorrectly generalizes about plus signs. Choice C is wrong because $y = 3x + 0$ is equivalent to $y = 3x$.

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