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{{distinguish|linear functional|linear map}}
{{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:
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]]) is a [[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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==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). If the line is the graph of the linear function
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
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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 =
On the other hand, the graph of a linear function in terms of [[polar coordinates]]:
:<math>r =f(\theta ) = a\theta + b</math>
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