Quadratic function: Difference between revisions

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

{{not to be confused with|Quartic function}}

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>

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

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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====

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==Forms of a univariate quadratic function==

A univariate quadratic function can be expressed in three formats:<ref>{{Cite book |lastlast1=Hughes Hallett |firstfirst1=Deborah J. |author-linklink1=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>

* <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.

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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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===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 Bounds for the Roots of Quadratic Equations |url=https://doi.org/10.2307/40378441 |journal=[[The Mathematical Gazette]] |volume=91 |issue=522 |pages=549 |doi=10.1017/S0025557200182324 |jstor=40378441 |url-access=subscription }}</ref>

==The square root of a univariate quadratic function==

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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}}

* {{Cite book |last=Saxon |first=John H. |title=Algebra 2 |date=May 1991 |publisher=Saxon Publishers, Incorporated |isbn=9780939798629}}

{{Polynomials}}