Editing Quadratic function (section)

{{Short description|Polynomial function of degree two}}
In [[mathematics]], a '''quadratic function''' of a single [[variable (mathematics)|variable]] is a [[function (mathematics)|function]] of the form<ref name="wolfram">{{cite web |last=Weisstein |first=Eric Wolfgang |title=Quadratic Equation |url=https://mathworld.wolfram.com/QuadraticEquation.html |access-date=2013-01-06 |website=[[MathWorld]]}}</ref>
:<math>f(x)=ax^2+bx+c,\quad a \ne 0,</math>
where {{tmath|x}} is its variable, and {{tmath|a}}, {{tmath|b}}, and {{tmath|c}} are [[coefficient]]s. The [[mathematical expression|expression]] {{tmath|\textstyle ax^2+bx+c}}, especially when treated as an [[mathematical object|object]] in itself rather than as a function, is a '''quadratic polynomial''', a [[polynomial]] of degree two. In [[elementary mathematics]] a polynomial and its associated [[polynomial function]] are rarely distinguished and the terms ''quadratic function'' and ''quadratic polynomial'' are nearly synonymous and often abbreviated as ''quadratic''.

[[Image:Polynomialdeg2.svg|thumb|right|A quadratic polynomial with two [[real number|real]] roots (crossings of the {{mvar|x}} axis).]]

The [[graph of a function|graph]] of a [[function of a real variable|real]] single-variable quadratic function is a [[parabola]]. If a quadratic function is [[equation|equated]] with zero, then the result is a [[quadratic equation]]. The solutions of a quadratic equation are the [[zero of a function|zero]]s (or ''roots'') of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the [[quadratic formula]].

A quadratic polynomial or quadratic function can involve more than one variable. For example, a two-variable quadratic function of variables {{tmath|x}} and {{tmath|y}} has the form
:<math> f(x,y) = a x^2 + bx y+ cy^2 + d x+ ey + f ,</math>
with at least one of {{tmath|a}}, {{tmath|b}}, and {{tmath|c}} not equal to zero. In general the zeros of such a quadratic function describe a [[conic section]] (a [[circle]] or other [[ellipse]], a [[parabola]], or a [[hyperbola]]) in the {{tmath|x}}–{{tmath|y}} plane. A quadratic function can have an arbitrarily large number of variables. The set of its zero form a [[quadric]], which is a [[surface (geometry)|surface]] in the case of three variables and a [[hypersurface]] in general case.