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}}

In [[algebra]], a '''quadratic function''', a '''quadratic polynomial''', a '''polynomial of degree 2''', or simply a '''quadratic''', is a [[polynomial function]] with one or more variables in which the highest-degree term is of the second degree.

[[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.]]

For example, a ''univariate'' (single-variable) quadratic function has the form<ref name="wolfram">{{cite web | url=http://mathworld.wolfram.com/QuadraticEquation.html | title=Quadratic Equation from Wolfram MathWorld | access-date=January 6, 2013}}</ref>

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

in the single variable ''x''. The [[graph of a function|graph]] of a univariate quadratic function is a [[parabola]] whose axis of symmetry is parallel to the {{math|''y''}}-axis, as shown at right.

If the quadratic function is set equal to zero, then the result is a [[quadratic equation]]. The solutions to the univariate equation are called the [[root of a function|root]]s of the univariate function.

The bivariate case in terms of variables ''x'' and ''y'' has the form

:<math> f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\!</math>

with at least one of ''a, b, c'' not equal to zero, and an equation setting this function equal to zero gives rise to a [[conic section]] (a [[circle]] or other [[ellipse]], a [[parabola]], or a [[hyperbola]]).

A quadratic function in three variables ''x'', ''y,'' and ''z'' contains exclusively terms ''x''², ''y''², ''z''², ''xy'', ''xz'', ''yz'', ''x'', ''y'', ''z'', and a constant: