Editing Linear function (section)

== As a polynomial function ==
{{main article|Linear function (calculus)}}
[[File:Linear Function Graph.svg|thumb|Graphs of two linear functions.]]

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).

When the function is of only one [[variable (mathematics)|variable]], it is of the form
:<math>f(x)=ax+b,</math>
where {{mvar|''a''}} and {{mvar|''b''}} are [[constant (mathematics)|constant]]s, often [[real number]]s. The [[graph of a function|graph]] of such a function of one variable is a nonvertical line. {{mvar|''a''}} is frequently referred to as the slope of the line, and {{mvar|''b''}} as the intercept.

If ''a > 0'' then the [[Slope|gradient]] is positive and the graph slopes upwards.

If ''a < 0'' then the [[Slope|gradient]] is negative and the graph slopes downwards.

For a function <math>f(x_1, \ldots, x_k)</math> of any finite number of variables, the general formula is
:<math>f(x_1, \ldots, x_k) = b + a_1 x_1 + \cdots + a_k x_k ,</math>
and the graph is a [[hyperplane]] of dimension {{nowrap|''k''}}.

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 function|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 map]]s.