Linear function (calculus): Difference between revisions

Revision as of 04:59, 20 June 2013

Revision as of 20:55, 20 June 2013

Line 5:

[[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 -->

A linear function is a [[polynomial]] function with one independent variable {{mvar|x}} of degree at most one, i.e. <math>f(x)=ax+b</math>.<ref>Stewart 2012, p. 24</ref> Here {{mvar|x}} is the independent variable. The [[graph of a function|graph]] of a linear function, that is, the set of all points whose coordinates have the form ({{mvar|x}}, {{mvar|f(x)}}), is a line, which is why this type of [[Function (mathematics)|function]] is called ''linear''. Some authors, for various reasons, also require that the coefficient of the variable (the {{mvar|a}} in {{mvar|ax + b}}) should not be zero.<ref>{{harvnb|Swokowski|1983|loc=pg. 34}} is but one of many well known references that could be cited.</ref> This requirement can also be expressed by saying that the degree of the polynomial in this polynomial function is exactly one, or by saying that the line which is the graph of a linear function is a ''slanted'' line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, <math>f(x) = b</math>, will be considered to be linear functions (their graphs are horizontal lines).

A linear function is a [[polynomial]] function with one independent variable {{mvar|x}}, i.e. <math>y(x)=ax+b</math>.<ref>Stewart 2012, p. 24</ref> Here {{mvar|x}} is the independent variable and {{mvar|y}} is the dependent variable.

The [[Domain_of_a_function|___domain]] or set of allowed values for {{mvar|x}} of a linear function is the entire set of [[real number]]s {{math|'''R'''}}. This means that any real number can be substituted for {{mvar|x}}.

The set of points {{math|(''x'', ''y''(''x''))}} is the line that is the [[graph of a function|graph]] of the function. Because two different points determine a line, it is enough to substitute two different values for {{mvar|x}} in the linear function and determine {{mvar|yf(x)}} for each of these values. This will give the coordinates of two different points that lie on the line. Because {{mathmvar|''y''(''x'')f}} is a function, thethis line cannotwill not be vertical.

BecauseSince the graph of a linear function is a nonvertical line, athis linear functionline has exactly one intersection point with the {{mvar|y}}-axis. This point is {{math|(0, ''b'')}}.

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''}}.

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.

There are three standard forms for linear functions.

There are many standard forms of linear equations, but only three will be examined to see how to obtain linear functions:

*[[#General form|General form]]

*[[#Slope-intercept form|Slope-intercept form]]