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{{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}}
[[Image:wiki linear function.png|thumb|right|Graph of the linear function: <math>y(x) = -x + 2</math>]]
In [[calculus]] and related areas of mathematics, a '''linear function''' from the [[real number]]s to the real numbers is a function whose graph (in [[Cartesian coordinates]]) is a non-vertical [[line (geometry)|line]] in the plane.
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.
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== Properties ==
A linear function is a [[polynomial function]] in which the [[variable (mathematics)|variable]] {{mvar|x}} has degree at most one:
:<math>f(x)=ax+b</math>.
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.
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>.
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In [[differential calculus|calculus]], the derivative of a general function measures its rate of change. A linear function <math>f(x)=ax+b</math> has a constant rate of change equal to its slope {{mvar|a}}, so its derivative is the constant function <math>f\,'(x)=a</math>.
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]] near a given point <math>x=c</math> by a unique linear function. The [[derivative]] <math>f\,'(c)</math> is the slope of this linear function, and the approximation is: <math>f(x) \approx f\,'(c)(x{-}c)+f(c)</math> for <math>x\approx c</math>. The graph of the linear approximation is the [[tangent line]] of the graph <math>y=f(x)</math> at the point <math>(c,f(c))</math>. The derivative slope <math>f\,'(c)</math> generally varies with the point ''c''. Linear functions can be characterized as the only real functions whose derivative is constant: if <math>f\,'(x)=a</math> for all ''x'', then <math>f(x)=ax+b</math> for <math>b=f(0)</math>.
==Slope-intercept, point-slope, and two-point forms==
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== 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}}
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