Linear function (calculus): Difference between revisions

Revision as of 04:25, 9 April 2023

Revision as of 09:25, 9 April 2023

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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 [[Realreal number|real numbers]]s to the [[Real number|real numbers]] is a function whose graph (in [[Cartesian coordinates]]) is a non-vertical [[line (geometry)|line]] in the plane.<ref>Stewart 2012, p. 23</ref>

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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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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== Relationship with other classes of functions ==

If the [[coefficient]] of the variable is not zero ({{math|''a'' ≠ 0}}), then a linear function is represented by a [[degree of a polynomial|degree]] 1 [[polynomial]] (also called a ''linear polynomial''), otherwise it is a [[constant function]] – also a polynomial function, but of zero degree.

A straight line, when drawn in a different kind of coordinate system may represent other functions.