Direct Variation

Two quantities vary directly when one is always the same multiple of the other: y = kx. The number k is the constant of variation.

If 4 tickets cost $30, then k = 30/4 = 7.5 dollars per ticket, and y = 7.5x for any number of tickets.

The test is whether y/x stays constant. If it does, it is direct variation.

The graph is a straight line through the origin, and that matters: zero tickets must cost zero dollars. A situation with a booking fee is still linear, but it is not direct variation.

Direct variation is the special case of a linear function with zero intercept: y = kx, equivalently y/x = k for all x ≠ 0. It is exactly the proportional relationships from the ratio unit, expressed as functions.

The distinguishing feature is not linearity but the origin. y = 3x + 5 is linear yet not proportional, since doubling x does not double y — that scaling property, f(cx) = c·f(x), holds only when b = 0.

The companion relationship is inverse variation, y = k/x or xy = k, where the product rather than the quotient stays constant. Its graph is a hyperbola with the axes as asymptotes rather than a line. Distinguishing these two by asking whether the quotient or the product is fixed handles most variation problems, and both generalise to joint variation with several variables.