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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 the graph includes points with negative values of ''x'' or ''y'', which have no meaning in terms of the original variables (unless we imagine selling meat to the butcher). Thus we should restrict our function <math>f(x)</math> to the ___domain <math>0\le x\le 2</math>.
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>.
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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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