Quadratic function: Difference between revisions

Revision as of 12:04, 27 November 2022

Revision as of 12:35, 27 November 2022

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

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

In [[algebramathematics]], a '''quadratic function''', a '''quadratic polynomial''', is a '''[[polynomial]] of degree 2''',two in one or simplymore avariables. A '''quadratic function''', isit athe [[polynomial function]] ofdefined [[degreeby ofa quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial|degree]] twoand its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in onemany orelementary morecourses, variableswhere both terms are often abbreviated as "quadratic".

[[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.]]

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:<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 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.

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

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

InA generalquadratic therefunction can behave an arbitrarily large number of variables,. inThe whichset caseof theits resultingzero form a [[quadric]], which is a [[surface (geometry)|surface]] ofin settingthe acase quadraticof functionthree to zero isvariables calledand a [[quadrichypersurface]], butin thegeneral highest degree term must be of degree 2, such as {{math|''x''², ''xy'', ''yz'',}} etccase.

==Etymology==