inverse variation

Inverse variation describes two quantities that move in opposite directions in a precise way: as one grows, the other shrinks so that their product stays fixed. The everyday picture is speed and travel time — drive twice as fast and the trip takes half as long, because distance (the product) is constant.

Algebraically, y varies inversely as x means y = k/x for some nonzero constant k, called the constant of variation. Equivalently, x·y = k stays the same for every pair (x, y) in the relationship. Because of the division by x, the value x = 0 is excluded, and the relation lives on a rational expression.

Contrast this with direct variation, y = kx, where the two quantities rise and fall together. Inverse variation is the reciprocal pattern: its graph is a hyperbola, sweeping toward the axes but never touching them, precisely because neither x nor y is ever allowed to be zero.