# Linear function

In mathematics, the term **linear function** refers to two distinct but related notions:^{[1]}

In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero).

where *a* and *b* are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. *a* is frequently referred to as the slope of the line, and *b* as the intercept.

A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.

In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.

In linear algebra, a linear function is a map *f* between two vector spaces s.t.

Here *a* denotes a constant belonging to some field *K* of scalars (for example, the real numbers) and **x** and **y** are elements of a vector space, which might be *K* itself.

In other terms the linear function preserves vector addition and scalar multiplication.

Some authors use "linear function" only for linear maps that take values in the scalar field;^{[6]} these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) *f*(0, ..., 0) = 0, or, equivalently, when the above constant b equals zero. Geometrically, the graph of the function must pass through the origin.