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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\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==