Linear function
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In [[mathematics]], the term '''linear function''' can refer to either of two different but related concepts.
==Usage in elementary mathematics==
[[Image:Linear functions2.PNG|300px|thumb|Three geometric linear functions — the red and blue ones have the same slope (''m''), while the red and green ones have the same y-intercept (''b'').]]
{{main|linear equation}}
In elementary [[algebra]] and [[analytic geometry]], the term ''linear function'' is sometimes used to mean a first degree [[polynomial]] [[function (mathematics)|function]] of one [[variable]]. These functions are called "linear" because they are precisely the functions whose [[graph of a function|graph]] in the [[Cartesian coordinate plane]] is a straight line.
Such a function can be written as
: <math>f(x) = mx + b</math>
(called slope-intercept form), where <math>m</math> and <math>b</math> are [[real number|real]] [[constant]]s and <math>x</math> is a real variable. The constant <math>m</math> is often called the [[slope]] or gradient, while <math>b</math> is the [[y-intercept]], which gives the point of intersection between the graph of the function and the <math>y</math>-axis. Changing <math>m</math> makes the line steeper or shallower, while changing <math>b</math> moves the line up or down.
Examples of functions whose graph is a line include the following:
* <math>f_{1}(x) = 2x+1</math>
* <math>f_{2}(x) = x/2+1</math>
* <math>f_{3}(x) = x/2-1</math>
The graphs of these are shown in the image at right.
==Usage in advanced mathematics==
In advanced mathematics, a ''linear function'' often means a [[function (mathematics)|function]] that is a [[linear map]], that is, a map between two [[vector space]]s that preserves vector addition and [[scalar multiplication]].
For example, if <math>x</math> and <math>f(x)</math> are represented as [[coordinate vector]]s, then the linear functions are those functions that can be expressed as
:<math>f(x) = \mathrm{M}x</math>, where M is a [[matrix]].
A function <math>f(x) = mx + b</math> is a linear map if and only if <math>b = 0</math>. For other values of <math>b</math> this falls in the more general class of [[affine map]]s.
==External links==
*[http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/linearFunctions.html Linear Functions on Id Mind]
*[http://www.mathopenref.com/linearexplorer.html Interactive tool to explore linear functions]
[[Category:Polynomials]]
[[als:Lineare Funktion]]
[[bg:Линейна функция]]
[[ca:Funció lineal]]
[[de:Lineare Funktion]]
[[es:Función lineal]]
[[fr:Fonction linéaire]]
[[it:funzione lineare]]
[[he:פונקציה לינארית]]
[[ms:Fungsi lelurus]]
[[nl:Lineaire functie]]
[[ja:一次関数]]
[[pl:Funkcja liniowa]]
[[pt:Função linear]]
[[pt:Função Polinomial#Função do Primeiro Grau]]
[[ru:Линейная функция]]
[[sl:Linearna funkcija]]
[[sr:Линеарна функција]]
[[fi:Lineaarinen funktio]]
[[th:ฟังก์ชันเส้นตรง]]
[[uk:Лінійна функція]]
[[zh:線性函數]]