quadratically

Quadratically is an adverb derived from quadratic, used to describe a relationship, growth, or change that is proportional to the square of a variable. In mathematics and the sciences, a quantity that varies quadratically with respect to a variable x changes in proportion to x^2. The term often appears in descriptions of functions, equations, and models that depend on second-degree terms.

Common usage includes describing how a quantity responds to changes in a variable. For example, in physics the distance traveled under constant acceleration grows quadratically with time, expressed as s = (1/2) a t^2. In numerical methods, one says that the error decreases quadratically with the step size, meaning the error is proportional to the square of the step. In computer science and mathematics, statements such as “the running time grows quadratically with the input size” or “the model varies quadratically with a parameter” convey that the rate of change or growth is governed by a square relationship.

Quadratically contrasted with linear or cubic relations helps illustrate different rates of change. It is also used in the context of quadratic forms and second-degree polynomials, where the dominant term involves x^2. Etymologically, the term reflects the geometric idea of a square, which underlies many physical and mathematical phenomena.

See also quadratic function, quadratic equation, quadratic form, and asymptotic notation, which provide related concepts for analyzing relationships that are second-degree in one or more variables.