Quadratic function: Difference between revisions

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[[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{{mvar|x}} theis singleits variable ''x''. The [[graph of a function|graph]] of a univariate quadratic function 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]] case in terms of variables {{math|''x''}} and {{math|''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 {{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]]).

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

:<math>f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz +j,</math>

withwhere at least one of the [[coefficient]]s {{math|''a, b, c, d, e,'' or ''f''}} of the second-degree terms beingis not non-zero.

In general there can be an arbitrarily large number of variables, in which case the resulting [[surface (geometry)|surface]] of setting a quadratic function to zero is called a [[quadric]], but the highest degree term must be of degree 2, such as {{math|''x''², ''xy'', ''yz'',}} etc.

==Etymology==