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 [[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.

:<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>

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

with at least one of {{tmath|a}}, {{tmath|b}}, and {{tmath|c}} not equal to zero. In general the zeros of such a quadratic function 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.

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:

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

with at least one of the [[coefficient]]s ''a, b, c, d, e,'' or ''f'' of the second-degree terms being 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 ''x''², ''xy'', ''yz'', etc.

==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''^{^2}}} 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 polynomialquadratic function are often taken to be [[real number|real]] or [[Complex quadratic polynomial|complex number]]s, but in fact, a polynomialthey may be definedtaken overin any [[ring (mathematics)|ring]].{{Citation, neededin which case the [[___domain of a function|date=October___domain]] 2022}}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. In [[elementary algebra]], suchSuch polynomials often arise in the form of a [[quadratic equation]] <math>ax^2 + bx + c = 0.</math>. The solutions to this equation are called the [[Root of a function|roots]] of the quadratic polynomial, and maycan be foundexpressed throughin [[factorization]], [[completing the square]], [[Graphterms of a function|graphing]], [[Newton's method]], or through the usecoefficients ofas the [[quadratic formula]]. Each quadratic polynomial has an associated quadratic function, whose [[graph of a function|graph]] is a [[parabola]].

====Bivariate caseand multivariate cases====

Any quadratic polynomial with two variables may be written as

====Bivariate case====

:<math> f(x,y) = 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'', and ''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, whichas arethe characterized[[implicit equation]] of a conic section is obtained by equating theto expressionzero fora ''f''quadratic (''x''polynomial, ''y'')and tothe [[zero. of a function|zeros]] of a quadratic function form a (possibly degenerate) conic section.

AnySimilarly, quadratic polynomialpolynomials with twothree or more variables maycorrespond to [[quadric]] besurfaces writtenor as[[hypersurface]]s.

:<math> f(x,y) = a x^2 + b y^2 + cxy + dx+ e y + f, \,\!</math>

Quadratic polynomials that have only terms of degree two are called [[quadratic form]]s.

where ''x'' and ''y'' are the variables and ''a'', ''b'', ''c'', ''d'', ''e'', and ''f'' are the coefficients. Such polynomials are fundamental to the study of [[conic section]]s, which are characterized by equating the expression for ''f'' (''x'', ''y'') to zero.

Similarly, quadratic polynomials with three or more variables correspond to [[quadric]] surfaces and [[hypersurface]]s. In [[linear algebra]], quadratic polynomials can be generalized to the notion of a [[quadratic form]] on a [[vector space]].

==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

* <math>f(x) = a x^2 + b x + c \,\!</math> is called the '''standard form''',

|title=College Algebra

* <math>f(x) = a(x - r_1)(x - r_2)\,\!</math> is called the '''factored form''', where {{math|''r''₁}} and {{math|''r''₂}} are the roots of the quadratic function and the solutions of the corresponding quadratic equation.

|first1=Deborah

* <math>f(x) = a(x - h)^2 + k \,\!</math> is called the '''vertex form''', where {{math|''h''}} and {{math|''k''}} are the {{math|''x''}} and {{math|''y''}} coordinates of the vertex, respectively.

|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

|url=https://books.google.com/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22}}, [https://books.google.com/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22 Search result]

</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''₁}} and {{math|''r''₂}} are the roots of the quadratic function and the solutions of the corresponding quadratic equation.

* <math>f(x) = a(x - h)^2 + k \,\!</math> is called the '''vertex form''', where {{math|''h''}} and {{math|''k''}} are the {{math|''x''}} and {{math|''y''}} coordinates of the vertex, respectively.

The coefficient {{math|''a''}} is the same value in all three forms. To convert the '''standard form''' to '''factored form''', one needs only the [[quadratic formula]] to determine the two roots {{math|''r''₁}} and {{math|''r''₂}}. To convert the '''standard form''' to '''vertex form''', one needs a process called [[completing the square]]. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

==Graph of the univariate function==

[[Image:Function ax^2.svg|thumb|350px|<math>f(x) = ax^2 |_{a=\{0.1,0.3,1,3\}} \!</math>]]

[[Image:Function x^2+bx.svg|thumb|350px|<math>f(x) = x^2 + bx |_{b=\{1,2,3,4\}} \!</math>]]

[[Image:Function x^2-bx.svg|thumb|350px|<math>f(x) = x^2 + bx |_{b=\{-1,-2,-3,-4\}} \!</math>]]

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.

Line 91 ⟶ 66:

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>

Line 99 ⟶ 74:

The '''vertex''' of a parabola is the place where it turns; hence, it is also called the '''turning point'''. If the quadratic function is in vertex form, the vertex is {{math|(''h'', ''k'')}}. Using the method of completing the square, one can turn the standard form

:<math>f(x) = a x^2 + b x + c \,\!</math>

into

: <math>\begin{align}

Line 107 ⟶ 82:

\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>

the average of the two roots, i.e.,

: <math>\frac{r_1 + r_2}{2} \,\!</math>

is the {{math|''x''}}-coordinate of the vertex, and hence the vertex {{math|(''h'', ''k'')}} is

: <math> \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).\!</math>

The vertex is also the maximum point if {{math|''a'' &lt; 0}}, or the minimum point if {{math|''a'' &gt; 0}}.

Line 126 ⟶ 101:

Using [[calculus]], the vertex point, being a [[minima and maxima|maximum or minimum]] of the function, can be obtained by finding the roots of the [[derivative]]:

:<math>f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b \,\!.</math>

{{math|''x''}} is a root of {{math|''f'' '(''x'')}} if {{math|''f'' '(''x'') {{=}} 0}}

resulting in

:<math>x=-\frac{b}{2a}</math>

with the corresponding function value

:<math>f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = c-\frac{b^2}{4a} \,\!,</math>

so again the vertex point coordinates, {{math|(''h'', ''k'')}}, can be expressed as

:<math> \left (-\frac {b}{2a}, c-\frac {b^2}{4a} \right). </math>

Line 145 ⟶ 120:

: <math>\begin{align}

f(x) &= ax^2+bx+c \\

&= a(x-r_1)(x-r_2), \\

\end{align}</math>

Line 159 ⟶ 134:

===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 inline|<!jstor=40378441 |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==

The [[square root]] of a univariate quadratic function gives rise to one of the four conic sections, [[almost always]] either to an [[ellipse]] or to a [[hyperbola]].

If <math>a>0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the [[ordinate]] of the [[minimum]] point of the corresponding parabola <math> y_p = a x^2 + b x + c \,\!.</math>. If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.

If <math>a<0\,\!</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either a circle or other ellipse or nothing at all. If the ordinate of the [[maximum]] point of the corresponding parabola

<math> y_p = a x^2 + b x + c \,\!</math> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an [[Empty set|empty]] locus of points.

==Iteration==

Line 179 ⟶ 154:

one has

:<math>f(x)=a(x-c)^2+c=h^{(-1)}(g(h(x))),\,\!</math>

where

:<math>g(x)=ax^2\,\!</math> and <math>h(x)=x-c.\,\!</math>

So by induction,

:<math>f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!</math>

can be obtained, where <math>g^{(n)}(x)</math> can be easily computed as

:<math>g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.\,\!</math>

Finally, we have

:<math>f^{(n)}(x)=a^{2^n-1}(x-c)^{2^n}+c\,\!</math>

as the solution.

Line 201 ⟶ 176:

:<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

Line 212 ⟶ 187:

{{Further|Quadric|Quadratic form}}

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]].

===Minimum/maximum===

If <math> 4AB-E^2 <0 \,</math> the function has no maximum or minimum; its graph forms a hyperbolic [[paraboloid]].

If <math> 4AB-E^2 >0 \,</math> the function has a minimum if both {{nowrap|''A'' > 0}} and {{nowrap|''B'' > 0}}, and a maximum if both {{nowrap|''A'' < 0}} and {{nowrap|''B'' < 0}}; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at <math> (x_m, y_m) \,</math> where:

:<math>x_m = -\frac{2BC-DE}{4AB-E^2},</math>

Line 226 ⟶ 201:

:<math>y_m = -\frac{2AD-CE}{4AB-E^2}.</math>

If <math> 4AB- E^2 =0 \,</math> and <math> DE-2CB=2AD-CE \ne 0 \,</math> the function has no maximum or minimum; its graph forms a parabolic [[cylinder (geometry)|cylinder]].

If <math> 4AB- E^2 =0 \,</math> and <math> DE-2CB=2AD-CE =0 \,</math> the function achieves the maximum/minimum at a line—a minimum if ''A''>0 and a maximum if ''A''<0; its graph forms a parabolic cylinder.

==See also==

Line 240 ⟶ 215:

==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}}