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{{Short description|Polynomial function of degree at most one}}
{{
[[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 -->▼
{{incomplete|the case of multivariate functions and vector valued functions, which must be considered, as this article is linked to from [[Jacobian matrix]]|date=February 2020}}
In [[calculus]] and related areas of mathematics, a '''linear function''' from the real numbers to the real numbers is a function whose graph (in [[Cartesian coordinates]] with uniform scales) 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 of the change in the input variable.▼
▲[[Image:wiki linear function.png|thumb|right|Graph of the linear function:
▲In [[calculus]] and related areas of mathematics, a '''linear function''' from the [[real
The characteristic property of linear functions is that when the input variable is changed, the change in the output is [[Proportionality (mathematics)|proportional]] to the change in the input.
Linear functions are related to [[linear equation]]s.
== Properties ==
A linear function is a [[polynomial function]] in which the [[variable (mathematics)|variable]] {{mvar|x}} has degree at most one
:
Such a function is called ''linear'' because its [[graph of a function|graph]], the set of all points <math>(x,f(x))</math> in the [[Cartesian plane]], is a [[line (geometry)|line]]. The coefficient ''a'' is called the ''slope'' of the function and of the line (see below).
If the slope is <math>a=0</math>, this is a ''constant function'' <math>f(x)=b</math> defining a horizontal line, which some authors exclude from the class of linear functions.{{sfn|Swokowski|1983|loc=p. 34}} With this definition, the degree of a linear polynomial would be exactly one, and its graph would be a line that is neither vertical nor horizontal. However, in this article, <math>a\neq 0</math> is not required, so constant functions will be considered linear.
If <math>b=0</math> then the linear function is said to be ''homogeneous''. Such function defines a line that passes through the origin of the coordinate system, that is, the point <math>(x,y)=(0,0)</math>. In advanced mathematics texts, the term ''linear function'' often denotes specifically homogeneous linear functions, while the term [[affine function]] is used for the general case, which includes <math>b\neq0</math>.
The natural [[Domain of a function|___domain]] of a linear function <math>f(x)</math>, the set of allowed input values for {{math|''x''}}, is the entire set of [[real number]]s, <math>x\in \mathbb R.</math> One can also consider such functions with {{math|''x''}} in an arbitrary [[field (mathematics)|field]], taking the coefficients {{math|''a, b''}} in that field.
The graph <math>y=f(x)=ax+b</math> is a non-vertical line having exactly one intersection with the {{math|''y''}}-axis, its {{math|''y''}}-intercept point <math>(x,y)=(0,b).</math> The {{math|''y''}}-intercept value <math>y=f(0)=b</math> is also called the ''initial value'' of <math>f(x).</math> If <math>a\neq 0,</math> the graph is a non-horizontal line having exactly one intersection with the {{math|''x''}}-axis, the {{math|''x''}}-intercept point <math>(x,y)=(-\tfrac ba,0).</math> The {{math|''x''}}-intercept value <math>x=-\tfrac ba,</math> the solution of the equation <math>f(x)=0,</math> is also called the ''root'' or [[zero of a function|''zero'']] of <math>f(x).</math>
==Slope==
[[File:Slope picture.svg|thumb|right|128px|The slope of a line is the ratio <math>\tfrac{\Delta y}{\Delta x}</math> between a change in {{mvar|x}}, denoted
The [[slope (mathematics)|slope]] of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run).
The slope measures the constant rate of change of <math>f(x)</math> per unit change in ''x'': whenever the input {{mvar|x}} is increased by one unit, the output changes by {{mvar|a}} units: <math>f(x{+}1)=f(x)+a</math>, and more generally <math>f(x{+}\Delta x)=f(x)+a\Delta x</math> for any number <math>\Delta x</math>. If the slope is positive, <math>a > 0</math>, then the function <math>f(x)</math> is increasing; if <math>a < 0</math>, then <math>f(x)</math> is decreasing
In [[differential calculus|calculus]], the derivative of a general function measures its rate of change.
The fundamental idea of differential calculus is that any [[differentiable function|smooth]] function <math>f(x)</math> (not necessarily linear) can be closely [[linear approximation|approximated]]
==Slope-intercept, point-slope, and two-point forms==
A given linear function <math>
:<math>
from which one can immediately see the slope ''
Given a slope ''
:<math>
In graphical terms, this gives the line <math>y=
The ''two-point form'' starts with two known values <math>
:<math>
Its graph <math>y=
:<math>\frac{y-
==Relationship with linear equations==
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where we denote <math>a=-\tfrac{A}{B}</math> and <math>b=\tfrac{C}{B}</math>. That is, one may consider ''y'' as a dependent variable (output) obtained from the independent variable (input) ''x'' via a linear function: <math>y = f(x) = ax+b</math>. In the ''xy''-coordinate plane, the possible values of <math>(x,y)</math> form a line, the graph of the function <math>f(x)</math>. If <math>B=0</math> in the original equation, the resulting line <math>x=\tfrac{C}{A}</math> is vertical, and cannot be written as <math>y=f(x)</math>.
The features of the graph <math>y = f(x) = ax+b</math> can be interpreted in terms of the variables ''x'' and ''y''. The ''y''-intercept is the
For example, the linear function <math>y = -2x + 4</math> has slope <math>a=-2</math>, ''y''-intercept point <math>(0,b)=(0,4)</math>, and ''x''-intercept point <math>(2,0)</math>.
===Example===
Suppose salami and sausage cost
Note that
Also, we could choose ''y'' as the independent variable, and compute ''x'' by the [[inverse function|inverse]] linear function: <math>x = g(y) = -\tfrac12 y +2</math> over the ___domain <math>0\le y \le 4</math>.
== Relationship with other classes of functions ==
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:<math>\log_r y = a \log_r x + b \quad\Rightarrow\quad y = r^b\cdot x^a</math>
[[File:Archimedean-Spiral.png|thumb|Archimedean spiral defined by the polar equation r = {{frac|1|2}}θ + 2]]
On the other hand, the graph of a linear function in terms of [[polar coordinates]]:
:<math>r =f(\
is an [[Archimedean spiral]] if <math>a \neq 0</math> and a [[circle]] otherwise.
== Notes ==▼
== See also ==
* [[Affine map]], a generalization
* [[Arithmetic progression]], a linear function of integer argument
▲== Notes ==
{{Reflist}}
== References ==
* {{citation
| last = Stewart | first = James
| year = 2012
| title = Calculus: Early Transcendentals
| edition = 7E
| publisher = Brooks/Cole
| isbn = 978-0-538-49790-9
}}
* {{citation
| last = Swokowski | first = Earl W.
| year = 1983
| title = Calculus with analytic geometry
| edition = Alternate
| publisher = Prindle, Weber & Schmidt
| place = Boston
| isbn = 0871503417
| url-access = registration
| url = https://archive.org/details/calculuswithanal00swok
}}
== External links ==
* https://web.archive.org/web/20130524101825/http://www.math.okstate.edu/~noell/ebsm/linear.html
*
{{Polynomials}}
{{Authority control}}
[[Category:Calculus]]
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