# Linear function

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

• a first-degree polynomial function of one variable;
• a map between two vector spaces that preserves vector addition and scalar multiplication.

## Contents

### Analytic geometry

In analytic geometry, the term linear function is sometimes used to mean a first-degree polynomial function of one variable. These functions are known as "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

(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:

• f1(x) = 2x + 1
• f2(x) = x / 2 + 1
• f3(x) = x / 2 − 1.

The graphs of these are shown in the image at right.

### Vector spaces

In advanced mathematics, a linear function means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.

For example, if x and f(x) are represented as coordinate vectors, then the linear functions are those functions f that can be expressed as

where M is a matrix. A function

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.