Editing Cubic function (section)

{{short description|Polynomial function  of degree 3}}
{{distinguish|Cubic equation}}
{{one source|date=September 2019}}
[[Image:Polynomialdeg3.svg|thumb|right|210px|Graph of a cubic function with 3 [[real number|real]] [[root of a function|roots]] (where the curve crosses the horizontal axis—where {{math|''y'' {{=}} 0}}). The case shown has two [[critical point (mathematics)|critical points]]. Here the function is {{math|''f''(''x'') {{=}} (''x''<sup>3</sup> + 3''x''<sup>2</sup> − 6''x'' − 8)/4}}.]]

In [[mathematics]], a '''cubic function''' is a [[function (mathematics)|function]] of the form <math>f(x)=ax^3+bx^2+cx+d,</math> that is, a [[polynomial function]] of degree three. In many texts, the ''coefficients'' {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|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 number]]s 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 of a function|domain]] is restricted to the real numbers.

Setting {{math|''f''(''x'') {{=}} 0}} produces a [[cubic equation]] of the form
:<math>ax^3+bx^2+cx+d=0,</math>
whose solutions are called [[root of a function|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 ([[Multiplicity (mathematics)|which may not be distinct]]);<ref>{{Cite book|last1=Bostock|first1=Linda|url=https://books.google.com/books?id=e2C3tFnAR-wC&q=A+cubic+function+has+either+one+or+three+real+roots&pg=PA462|title=Pure Mathematics 2|last2=Chandler|first2=Suzanne|last3=Chandler|first3=F. S.|date=1979|publisher=Nelson Thornes|isbn=978-0-85950-097-5|pages=462|language=en|quote=Thus a cubic equation has either three real roots... or one real root...}}</ref> all odd-degree polynomials with real coefficients have at least one real root.

The [[graph of a function|graph]] of a cubic function always has a single [[inflection point]]. It may have two [[critical point (mathematics)|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]].