Quadratic function: Difference between revisions

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{{Short description|Polynomial function of degree two}}

{{for|the zeros of a quadratic function|Quadratic equation|Quadratic formula}}

ForIn example[[mathematics]], a '''quadratic function''' is typically a [[univariate]]function (single-variablemathematics) quadratic |function]] hasof the form<ref name="wolfram">{{cite web |last=Weisstein |first=Eric Wolfgang |title=Quadratic Equation |url=https://mathworld.wolfram.com/QuadraticEquation.html |url-status=live |archive-url=https://web.archive.org/web/20200312030923/https://mathworld.wolfram.com/QuadraticEquation.html |archive-date=2020-03-12 |access-date=2013-01-06 |website=[[MathWorld]]}}</ref>

In [[mathematics]], a '''quadratic polynomial''' is a [[polynomial]] of degree two in one or more variables. A '''quadratic function''' is the [[polynomial function]] defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".

:<math>f(x)=ax^2+bx+c,\quad a \ne 0,</math>

where {{mvar|x}} is its variable

More generally, a ''quadratic function'' is a function defined by a '''quadratic polynomial''', that is, a [[polynomial]] of degree two in one or more variables.

[[Image:Polynomialdeg2.svg|thumb|right|A quadratic polynomial with two [[real number|real]] [[root of a polynomial|roots]] (crossings of the ''x'' axis) and hence no [[complex number|complex]] roots. Some other quadratic polynomials have their [[minimum]] above the ''x'' axis, in which case there are no real roots and two complex roots.]]

In [[mathematics]], a '''quadratic polynomial''' is a [[polynomial]] of degree two in one or more variables. A '''quadratic function''' is the [[polynomial function]] defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".

For example, a [[univariate]] (single-variable) quadratic function has the form<ref name="wolfram">{{cite web |last=Weisstein |first=Eric Wolfgang |title=Quadratic Equation |url=https://mathworld.wolfram.com/QuadraticEquation.html |url-status=live |archive-url=https://web.archive.org/web/20200312030923/https://mathworld.wolfram.com/QuadraticEquation.html |archive-date=2020-03-12 |access-date=2013-01-06 |website=[[MathWorld]]}}</ref>

[[Image:Polynomialdeg2.svg|thumb|right|A quadratic polynomial with two [[real number|real]] [[root of a polynomial|roots]] (crossings of the ''x'' axis) and hence no [[complex number|complex]] roots. Some other quadratic polynomials have their [[minimum]] above the ''x'' axis, in which case there are no real roots and two complex roots.]]

:<math>f(x)=ax^2+bx+c,\quad a \ne 0,</math>

where {{mvar|x}} is its variable. The [[graph of a function|graph]] of a univariate quadratic function of a single variable is a [[parabola]], a [[curve]] that has an [[axis of symmetry]] parallel to the {{math|''y''}}-axis.

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 of the corresponding quadratic function.

The [[bivariate function|bivariate]]In case inof terms ofrwo variables {{math|''x''}} and {{math|''y''}}, a quadrtic function has the form

:<math> f(x,y) = a x^2 + bx y+ cy^2 + d x+ ey + f ,</math>

with at least one of {{math|''a, b, c''}} not equal to zero. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), a [[conic section]] (a [[circle]] or other [[ellipse]], a [[parabola]], or a [[hyperbola]]).