Linear function
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In mathematics, the term linear function can refer to either of two different but related concepts.
[edit] Usage in elementary mathematics
In elementary algebra and analytic geometry, the term linear function is often used to mean a first degree polynomial function of one variable. These functions are called "linear" because they are precisely the functions whose 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 constants and <math>x</math> is a real variable. The constant <math>m</math> is often called the slope 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.
[edit] Usage in advanced mathematics
In advanced mathematics, a linear function often means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.
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 maps.
[edit] External links
- Linear function - Easy math
- Linear Functions on Id Mindals:Lineare Funktion
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