Cubic function

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Cubic function" – news · newspapers · books · scholar · JSTOR (September 2019)

In mathematics, a cubic function is a function of the form ${\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,}$ that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.

Setting f(x) = 0 produces a cubic equation of the form

${\displaystyle ax^{3}+bx^{2}+cx+d=0,}$

whose solutions are called roots of the function. The derivative of a cubic function is a quadratic function.

A cubic function with real coefficients has either one or three real roots (which may not be distinct);^[1] all odd-degree polynomials with real coefficients have at least one real root.

The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only three possible graphs for cubic functions.

Cubic functions are fundamental for cubic interpolation.