Inverse Variation
Fractions & DecimalsInverse 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.
Example
If you travel $120$ miles, faster speed means less time: $60$ mph takes $2$ hours, $30$ mph takes $4$ hours, $120$ mph takes $1$ hour. In each case, speed $\times$ time $= 120$. Speed and time vary inversely.
Key Insight
"Inverse" means opposite. As one value doubles, the other halves. The product (multiply them together) is always the same constant - that constant is the key to inverse variation.
Definition
Inverse variation (inverse proportion) is $y = k/x$, or equivalently $xy = k$, where $k$ is the nonzero constant of variation. Properties: (1) $xy = k$ for all pairs; (2) the graph is a hyperbola (never passing through the origin); (3) doubling $x$ halves $y$. To verify, check that $xy$ is constant.
Example
Data: $(3, 8)$, $(4, 6)$, $(6, 4)$, $(12, 2)$. Products: $3 \times 8=24$, $4 \times 6=24$, $6 \times 4=24$, $12 \times 2=24$. Constant product $k=24$, so $y=24/x$. Compare with direct: in inverse variation, the product is constant; in direct, the ratio is constant.
Key Insight
Inverse variation graphs are hyperbolas that live in opposite quadrants (both positive or both negative). They never cross the axes. The graph is always "bending away" from both axes, approaching but never reaching them.
Definition
Inverse variation $y = k/x$ is a degree-$(-1)$ homogeneous function: $f(cx) = c^{-1} f(x)$. It is the composition of direct variation with the reciprocal function. The graph $y = k/x$ is a rectangular hyperbola with asymptotes at $x=0$ and $y=0$. In general, $y$ varies inversely as $x^n$ means $y = k/x^n$ (inverse variation of degree $n$), a negative-degree monomial.
Example
Boyle's Law: $PV = k$ (pressure times volume is constant at fixed temperature). Doubling pressure halves volume. This is inverse variation with $k = nRT$ (ideal gas constant times moles times temperature). The law fails near phase transitions, where the gas becomes liquid.
Key Insight
Inverse square laws ($y = k/x^2$) appear throughout physics: gravitational and electrostatic forces, light and sound intensity, radiation. They arise 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.