- Linear function
In

mathematics , the term**linear function**can refer to either of two different but related concepts.**Analytic geometry**In

analytic geometry , the term "linear function" is sometimes used to mean a first degreepolynomial function of onevariable . These functions are called "linear" because they are precisely the functions whose graph in theCartesian coordinate plane is a straight line.Such a function can be written as

: $f(x)\; =\; mx\; +\; b$

(called slope-intercept form), where $m$ and $b$ are real

constant s and $x$ is a real variable. The constant $m$ is often called theslope or gradient, while $b$ is they-intercept , which gives the point of intersection between the graph of the function and the $y$-axis. Changing $m$ makes the line steeper or shallower, while changing $b$ moves the line up or down.Examples of functions whose graph is a line include the following:

* $f\_\{1\}(x)\; =\; 2x+1$

* $f\_\{2\}(x)\; =\; x/2+1$

* $f\_\{3\}(x)\; =\; x/2-1.$The graphs of these are shown in the image at right.

**Vector spaces**In advanced mathematics, a "linear function" often means a function that is a

linear map , that is, a map between twovector space s that preserves vector addition andscalar multiplication .For example, if $x$ and $f(x)$ are represented as

coordinate vector s, then the linear functions are those functions that can be expressed as:$f(x)\; =\; mathrm\{M\}x$, where M is a matrix.

A function $f(x)\; =\; mx\; +\; b$ is a linear map if and only if $b\; =\; 0$. For other values of $b$ this falls in the more general class of

affine map s.**ee also***

Function (mathematics) **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*]

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