Variation Investigator

Pre-Algebra · Axiom Academy

Two quantities can be tied together in opposite ways. When one changes, does the other rise with it, or trade off against it? Two quantities, two ways to be connected All over math, science, and money, two quantities move together: buy more slices, pay more; drive faster, arrive sooner. But "connected" hides two opposite stories. In direct variation they grow together; in inverse variation one rises only as the other falls. Watch both stories draw themselves side by side, then drive each one yourself. As the sweep line moves right (x growing from 0), one point climbs a perfectly straight line through the origin — that's direct variation, y = kx . A second point rides a curve that dives toward the axes — that's inverse variation, xy = k , so y = k/x . Same x, two completely different shapes. One straight, one curved — two relationships hiding inside the same word, "varies." Direct variation: double the input, double the output Slide k , the constant of variation. The line y = kx always passes through the origin, and its steepness IS k . The two dots sit at some x and at 2x : read their heights — the second is always exactly twice the first. That's the signature of direct variation: double x and you double y . Steeper or shallower, the line stays straight and pinned to the origin — and doubling x always doubles y. Inverse variation: double the input, halve the output

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