Linear function 152111 217327063 2008-06-05T15:31:09Z 62.108.160.100 interwiki 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]]