Distinguish proportional relationships from linear but non-proportional relationships.
Decide whether a linear pattern is proportional
Problem
For \(x\): \(0,~1,~2,~3\) and \(y\): \(0,~3,~6,~9\), is the relationship proportional or linear but non-proportional? Does it pass through \((0,~0)\)?
Big Picture
What this problem is really about
Proportionality requires more than a constant additive change. Pair the aligned values, check that y divided by x is constant for every nonzero input, and separately inspect what happens at x equals zero. Keeping the origin test separate avoids the undefined zero-over-zero ratio while still confirming whether the relationship has a nonzero start.
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