Linear function (calculus): Difference between revisions

Revision as of 13:15, 28 June 2013

Revision as of 13:17, 28 June 2013

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[[Image:wiki_linear_function.png|thumb|right|Graph of the linear function: {{math|1=''y''(''x'') = −''x'' + 2}}]]<!-- people, find an SVG image please instead of this abomination -->

In [[calculus]] and related areas of mathematics, a '''linear function''' from the real numbers to the real numbers is a function whose graph is a [[line (geometry)|line]] in the plane.<ref>Stewart 2012, p. 23</ref> Their characteristic property that when the value of the input variable is changed, the change in the output is a constant multiple, known as the [[slope (mathematics)|slope]], of the change in the input variable. Linear functions can be obtained from [[linear equation]]s, which are also closely related with lines.

== Properties ==

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A the graph of a nonconstant linear function has exactly one intersection point with the {{mvar|x}}-axis. This point is {{math|({{sfrac|−''b''|''a''}}, 0)}}. From this, it follows that a nonconstant linear function has exactly one [[zero of a function|zero]] or root. That is, there is exactly one solution to the equation {{math|1=''ax'' + ''b'' = 0}}. The zero is {{math|1=''x'' =}}&nbsp;{{sfrac|−''b''|''a''}}.

==Slope==

== Relationship with linear equations ==

<nowiki>The [[slope (mathematics)|slope]] of a linear function tells how steeply the graph of the function is slanted. </nowiki>

== Relationship with linear equations ==

[[Image: wiki_linearna_funkcija_eks1.png|thumb||right]]<!-- are PNG and a translit from a foreign language necessary? -->

The points on a line have coordinates which can also be thought of as the solutions of [[linear equation]]s in two variables (the equation of the line). These solution sets define functions which are linear functions. This connection between linear equations and linear functions provides the most common way to produce linear functions.