Direct Variation
Direct variation describes a relationship where two quantities increase or decrease together at a constant rate, expressed as y = kx.
Formula
y = kx \text{ (}k\text{ is the constant of variation)}
Definition
Direct variation means that when one quantity goes up, the other goes up by the same proportion, and when one goes down, the other goes down too, they move together in the same direction. Formally, direct variation (direct proportion) is a relationship $y = kx$ where $k$ (the constant of variation) is nonzero: $y/x = k$ for all nonzero $x$, the graph is a line through the origin with slope $k$, and doubling $x$ doubles $y$. This is a one-dimensional linear map, a degree-$1$ homogeneous function satisfying both additivity and $f(cx) = cf(x)$; more generally, $y$ varies directly as $x^n$ means $y = kx^n$ (direct variation of degree $n$), a monomial power-law relationship common throughout physics.
Example
If you earn $\$12$ per hour, your pay varies directly with hours worked: $1$ hr is $\$12$, $2$ hrs is $\$24$, $3$ hrs is $\$36$, doubling the hours doubles the pay. Checking data $(2, 8)$, $(5, 20)$, $(7, 28)$: the ratio $y/x$ is $4$ in every case, so $y=4x$ is direct variation with $k=4$. The period $T$ of a simple pendulum varies directly as the square root of its length $L$: $T = 2\pi\sqrt{L/g}$, direct variation in the form $T = k\sqrt{L}$, so doubling $L$ multiplies $T$ by $\sqrt{2} \approx 1.414$, not $2$.
Key Insight
"Direct" means they go the same direction together: more hours, more pay, less hours, less pay, and the ratio pay/hours never changes. Direct variation graphs always pass through the origin, which is why a linear relationship like Fahrenheit-to-Celsius conversion, related linearly but not proportionally, is different from Hooke's law ($F=kx$), which is genuine direct variation. Power-law direct variation $y \sim x^n$ appears throughout physics (gravitational force $\sim 1/r^2$, kinetic energy $\sim v^2$), and log-log plots linearize power laws ($\log y = n\log x + \log k$), making the exponent $n$ the slope, a key tool for identifying power-law relationships in experimental data.