Inverse Variation Calculator

Start with three points: (2, 12), (3, 8) and (4, 6). Multiply each pair: 2 × 12 = 24, 3 × 8 = 24, 4 × 6 = 24. All three agree, so the relationship is y = 24 ÷ x. Any further pair follows: at x = 8, y = 3; at y = 48, x = 1/2. The curve passes through none of the axes — at x = 100 the value is 0.24, at x = 10,000 it is 0.0024, and it never becomes zero however far you go. Now three points that also fall neatly: (1, 9), (2, 8) and (3, 7). The products are 9, 16 and 21. They do not agree, so this is not an inverse variation. But the points are not random — the slope between any two is −1 and the line through them is y = 10 − x. Every point sits on that line, so y genuinely falls as x rises. That is a real relationship and a different one. The line reaches zero at x = 10 and goes negative after it; an inverse curve would still be positive at x = 10, at x = 1,000 and at every x you care to name. Try the inverse square with (1, 100), (2, 25) and (5, 4). The products x² × y come to 100, 100 and 100, so y = 100 ÷ x². Notice the scaling: going from x = 1 to x = 2 does not halve y from 100 to 50, it quarters it to 25. This is why moving twice as far from a lamp leaves a quarter of the light rather than half, and it is the single most useful thing to remember about inverse-square laws. The square root runs the other way. Points (1, 6), (4, 3) and (9, 2) give √x × y of 6, 6 and 6, so y = 6 ÷ √x. Quadrupling x halves y — a much gentler fall than the plain case. Comparing these exactly matters: the test is done on k² = x × y², which stays rational, so k comes out as exactly 6 rather than 5.999999999. Finally, a warning about pairs. Take (2, 6) and (3, 4). Both products are 12, so as far as two points can tell this is a perfect inverse variation — and the two points also lie on a falling straight line. Two points are consistent with both stories at once. Add a third and at most one of them survives.