Variation, Transformations, and Other Models

Direct and inverse variation

Two quantities are in direct variation (direct proportion) when one is a constant multiple of the other: $y=kx$. Doubling $x$ doubles $y$, and the graph is a straight line through the origin with gradient $k$, the constant of proportionality. The cost of fuel is directly proportional to liters purchased; the circumference of a circle is directly proportional to its diameter.

Two quantities are in inverse variation when their product is constant: $y=\dfrac{k}{x}$. As one grows the other shrinks proportionally, and the graph is a hyperbola with the axes as asymptotes. If a job takes $T$ hours with $n$ workers such that $T=\frac{240}{n}$, then more workers means proportionally less time — 4 workers take 60 hours, 8 workers take 30 hours. Physical laws such as "pre