Inverse Variation
Inverse variation describes a relationship where one quantity increases as the other decreases, with their product always remaining constant.
Formula
y = \frac{k}{x} \text{ or } xy = k
Definition
Inverse variation means that when one quantity goes up, the other goes down, and their product always stays the same, they move in opposite directions. Formally, inverse variation is $y = k/x$, or equivalently $xy = k$, where $k$ is the nonzero constant of variation: $xy = k$ for all pairs, the graph is a hyperbola that never passes through the origin, and doubling $x$ halves $y$. This is a degree-$(-1)$ homogeneous function, the composition of direct variation with the reciprocal function; more generally, $y$ varies inversely as $x^n$ means $y = k/x^n$, a negative-degree monomial that shows up as inverse-square laws throughout physics.
Example
Traveling $120$ miles, faster speed means less time: $60$ mph takes $2$ hours, $30$ mph takes $4$ hours, $120$ mph takes $1$ hour, and in each case speed $\times$ time $= 120$. Checking data $(3, 8)$, $(4, 6)$, $(6, 4)$, $(12, 2)$: the products are all $24$, confirming $y = 24/x$. Boyle's Law states $PV = k$ (pressure times volume is constant at fixed temperature), so doubling pressure halves volume, inverse variation with $k = nRT$.
Key Insight
"Inverse" means opposite: as one value doubles, the other halves, and the product, multiplying them together, is always the same constant, the key to inverse variation. The graphs of inverse variation are hyperbolas that live in opposite quadrants (both positive or both negative), bending away from both axes but never crossing them. Inverse square laws ($y = k/x^2$) appear throughout physics, gravitational and electrostatic forces, light and sound intensity, radiation, arising from the geometry of spreading through 3D space: the surface area of a sphere of radius $r$ is $4\pi r^2$, so intensity per unit area falls as $1/r^2$ as the sphere expands.