# Linear function

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

## Usage in elementary mathematics

File:Linear functions2.PNG
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).

In elementary algebra and analytic geometry, the term linear function is sometimes 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

$f(x)=mx+b$ (called slope-intercept form), where $m$ and $b$ are real constants and $x$ is a real variable. The constant $m$ is often called the slope or gradient, while $b$ is the y-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.

For example, if $x$ and $f(x)$ are represented as coordinate vectors, 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 maps. 