Quadratic function: Difference between revisions

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{{Short description|Polynomial function of degree two}}
{{not to be confused with|Quartic function}}
{{for|the zeros of a quadratic function|Quadratic equation|Quadratic formula}}
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>
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 {{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]] [[root of a polynomial|roots]] (crossings of the ''{{mvar|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.]]
 
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]].
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>
 
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)=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.
 
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 {{mathtmath|''a}}, {{tmath|b}}, and {{tmath|c''}} not equal to zero. The zeros of this quadratic function is, inIn general (thatthe is, if a certain expressionzeros of thesuch coefficientsa isquadratic notfunction equal to zero),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.
 
A quadratic function in three variables {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} contains exclusively terms {{math|''x''<sup>2</sup>}}, {{math|''y''<sup>2</sup>}}, {{math|''z''<sup>2</sup>}}, {{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>
 
where at least one of the [[coefficient]]s {{math|''a, b, c, d, e, f''}} of the second-degree terms is not zero.
 
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.
 
==Etymology==
 
The adjective ''quadratic'' comes from the [[Latin]] word ''[[wikt:en:quadratum#Latin|quadrātum]]'' ("[[square (geometry)|square]]"). A term raised to the second power like {{mathtmath|''\textstyle x''<sup>^2</sup>}} is called a [[square (algebra)|square]] in algebra because it is the area of a ''square'' with side {{mathtmath|''x''}}.
 
==Terminology==
 
===Coefficients===
The [[coefficients]] of a quadricquadratic function are often taken to be [[real number|real]] or [[complex number]]s, but they may be taken in any [[ring (mathematics)|ring]], in which case the [[___domain of a function|___domain]] and the [[codomain]] are this ring (see [[polynomial evaluation]]).
 
===Degree===
When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "[[Degeneracy (mathematics)|degenerate case]]". Usually the context will establish which of the two is meant.
 
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial. However, where the "[[degree of a polynomial]]" refers to the ''largest'' degree of a non-zero term of the polynomial, more typically "order" refers to the ''lowest'' degree of a non-zero term of a [[power series]].
 
===Variables===
 
A quadratic polynomial may involve a single [[Variable (mathematics)|variable]] ''x'' (the [[univariate]] case), or multiple variables such as ''x'', ''y'', and ''z'' (the multivariate case).
 
====The one-variable case====
 
Any single-variable quadratic polynomial may be written as
:<math>ax^2 + bx + c,</math>
where ''x'' is the variable, and ''a'', ''b'', and ''c'' represent the [[coefficient]]s. Such polynomials often arise in a [[quadratic equation]] <math>ax^2 + bx + c = 0.</math> The solutions to this equation are called the [[Root of a function|roots]] and can be expressed in terms of the coefficients as the [[quadratic formula]]. Each quadratic polynomial has an associated quadratic function, whose [[graph of a function|graph]] is a [[parabola]].
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====Bivariate and multivariate cases====
 
Any quadratic polynomial with two variables may be written as
:<math>a x^2 + b y^2 + cxy + dx+ e y + f,</math>
where {{math|''x''}} and {{math|''y''}} are the variables and {{math|''a'', ''b'', ''c'', ''d'', ''e'', ''f''}} are the coefficients, and one of {{mvar|a}}, {{mvar|b}} and {{mvar|c}} is nonzero. Such polynomials are fundamental to the study of [[conic section]]s, as the [[implicit equation]] of a conic section is obtained by equating to zero a quadratic polynomial, and the [[zero of a function|zeros]] of a quadratic function form a (possibly degenerate) conic section.
 
Similarly, quadratic polynomials with three or more variables correspond to [[quadric]] surfaces or [[hypersurface]]s.
 
Quadratic polynomials that have only terms of degree two are called [[quadratic form]]s.
 
==Forms of a univariate quadratic function==
A univariate quadratic function can be expressed in three formats:<ref>{{Cite book |last1=Hughes Hallett |first1=Deborah J. |author-link1=Deborah Hughes Hallett |title=College Algebra |last2=Connally |first2=Eric |author-link2=Eric Connally |last3=McCallum |first3=William George |author-link3=William G. McCallum |publisher=[[Wiley (publisher)|John Wiley & Sons Inc.]] |year=2007 |isbn=9780471271758 |page=205}}</ref>
A univariate quadratic function can be expressed in three formats:<ref>{{citation
|title=College Algebra
|first1=Deborah
|last1=Hughes-Hallett | author1-link = Deborah Hughes Hallett
|first2=Eric
|last2=Connally
|first3=William G.
|last3=McCallum | author3-link = William G. McCallum
|publisher=John Wiley & Sons Inc.
|year=2007
|isbn=9780471271758
|page=205
}}
</ref>
 
* <math>f(x) = a x^2 + b x + c</math> is called the '''standard form''',
* <math>f(x) = a(x - r_1)(x - r_2)</math> is called the '''factored form''', where {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are the roots of the quadratic function and the solutions of the corresponding quadratic equation.
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Regardless of the format, the graph of a univariate quadratic function <math>f(x) = ax^2 + bx + c</math> is a [[parabola]] (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation <math>y = ax^2 + bx + c</math>.
 
* If {{math|''a'' &gt; 0}}, the parabola opens upwards.
* If {{math|''a'' &lt; 0}}, the parabola opens downwards.
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The coefficient {{math|''a''}} controls the degree of curvature of the graph; a larger magnitude of {{math|''a''}} gives the graph a more closed (sharply curved) appearance.
 
The coefficients {{math|''b''}} and {{math|''a''}} together control the ___location of the axis of symmetry of the parabola (also the {{math|''x''}}-coordinate of the vertex and the ''h'' parameter in the vertex form) which is at
:<math>x = -\frac{b}{2a}.</math>
 
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\end{align}</math>
so the vertex, {{math|(''h'', ''k'')}}, of the parabola in standard form is
: <math> \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right). </math><ref>{{Citationcite neededbook|datetitle=OctoberCo-ordinate 2022Geometry|first=Percy|last=Coleman|publisher=Oxford University Press|year=1914|page=[https://books.google.com/books?id=TJU5AQAAMAAJ&pg=PA137 137]}}</ref>
If the quadratic function is in factored form
:<math>f(x) = a(x - r_1)(x - r_2)</math>
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: <math>\begin{align}
f(x) &= ax^2+bx+c \\
&= a(x-r_1)(x-r_2), \\
\end{align}</math>
 
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===Upper bound on the magnitude of the roots===
 
The [[absolute value|modulus]] of the roots of a quadratic <math>ax^2+bx+c</math> can be no greater than <math>\frac{\max(|a|, |b|, |c|)}{|a|}\times \phi, </math> where <math>\phi</math> is the [[golden ratio]] <math>\frac{1+\sqrt{5}}{2}.</math><ref>{{Cite journal |last=Lord, |first=Nick, "|date=2007-11-01 |title=Golden boundsBounds for the rootsRoots of quadraticQuadratic equations",Equations |url=https://doi.org/10.2307/40378441 |journal=[[The ''Mathematical Gazette'']] |volume=91, November|issue=522 2007,|pages=549 549|doi=10.<1017/ref>{{importanceS0025557200182324 |jstor=40378441 inline|<!url--Formulaaccess=subscription doesn't scale under scale of ''x''; a realistic formula should scale by α when b ↦ bα and c ↦cα<sup>2}}</supref>-->}}
 
==The square root of a univariate quadratic function==
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:<math>x_{n}=\sin^{2}(2^{n} \theta \pi)</math>
 
where the initial condition parameter <math>\theta</math> is given by <math>\theta = \tfrac{1}{\pi}\sin^{-1}(x_0^{1/2})</math>. For rational <math>\theta</math>, after a finite number of iterations <math>x_n</math> maps into a periodic sequence. But almost all <math>\theta</math> are irrational, and, for irrational <math>\theta</math>, <math>x_n</math> never repeats itself &ndash; it is non-periodic and exhibits [[sensitive dependence on initial conditions]], so it is said to be chaotic.
 
The solution of the logistic map when ''r''=2 is
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A '''bivariate quadratic function''' is a second-degree polynomial of the form
:<math> f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F,</math>
where ''A, B, C, D'', and ''E'' are fixed [[coefficient]]s and ''F'' is the [[constant term]].
Such a function describes a quadratic [[Surface (mathematics)|surface]]. Setting <math>f(x,y)</math> equal to zero describes the intersection of the surface with the plane <math>z=0,</math> which is a [[locus (mathematics)|locus]] of points equivalent to a [[conic section]].
 
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==References==
{{Reflist}}
* {{Cite book |last=Glencoe |first=McGraw-Hill |title=Algebra 1 |date=2003 |publisher=Glencoe/McGraw Hill |isbn=9780078250835}}
*Algebra 1, Glencoe, {{isbn|0-07-825083-8}}
* {{Cite book |last=Saxon |first=John H. |title=Algebra 2 |date=May 1991 |publisher=Saxon Publishers, Incorporated |isbn=9780939798629}}
*Algebra 2, Saxon, {{isbn|0-939798-62-X}}
 
==External links==
* {{MathWorld|title=Quadratic|urlname=Quadratic}}
 
{{Polynomials}}