Direct Variation Calculator

Direct variation (or direct proportionality) describes a relationship between two variables where an increase in one causes a proportional increase in the other. Mathematically, it is written as y = kx, where k is the constant of variation. The graph of a direct variation always passes through the origin.

To find k, rearrange the direct variation formula: k = y / x (for linear variation). Simply divide the known y value by the corresponding x value. For example, if y = 12 and x = 4, then k = 12 / 4 = 3.

A relationship is a direct variation if the ratio y/x (or y/f(x) for other types) is constant for all data points. On a graph, a direct variation always produces a straight line passing through the origin (0, 0). If the line does not pass through the origin, it is not a direct variation.

Using the formula y = kx, substitute k = 3 and x = 8: y = 3 × 8 = 24. So y equals 24 when x is 8 in a direct variation with constant k = 3.

In direct variation (y = kx), when x increases, y increases proportionally. In inverse variation (y = k/x), when x increases, y decreases. Direct variation graphs are straight lines through the origin, while inverse variation graphs are hyperbolas.

This means y = kx², so y is proportional to x squared rather than x itself. Doubling x will quadruple y. This type of variation is common in physics — for example, the kinetic energy of an object varies directly as the square of its velocity.

A classic example is the relationship between distance and time at constant speed: distance = speed × time. Here, speed acts as the constant of variation k. Other examples include the cost of items (total cost = price per item × quantity) and gravitational force proportional to mass.

Yes, k can be negative. A negative k means the dependent variable y decreases as x increases, but the rate of change is still constant. For example, y = -2x means for every 1 unit increase in x, y decreases by 2 units.