Linear function (calculus): Difference between revisions

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{{Short description|Polynomial function of degree at most one}}

{{distinguishDistinguish|linear functional|linear map}}

[[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: {{<math|1=''>y''(''x'') = −''-x'' + 2}}</math>]]<!-- people, find an SVG image please instead of this abomination -->

In [[calculus]] and related areas of mathematics, a '''linear function''' from the [[real numbersnumber]]s to the real numbers is a function whose graph (in [[Cartesian coordinates]] with uniform scales) is a non-vertical [[line (geometry)|line]] in the plane.<ref>{{sfn|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.

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:<ref>{{sfn|Stewart |2012, |p. =24</ref>}}

:<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.<ref>{{harvnbsfn|Swokowski|1983|loc=p. 34}}</ref> With this definition, the degree of a linear polynomial would be exactly one, and its graph awould diagonalbe a line that is neither vertical nor horizontal. However, wein willthis not requirearticle, <math>a\neq 0</math> inis thisnot articlerequired, 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 ''x'', is the entire set of [[real number]]s, <math>x\in \mathbb R</math>. One can also consider such functions with ''x'' in an arbitrary [[field (mathematics)|field]], taking the coefficients ''a,b'' in that field.

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 ''y''-axis, its ''y''-intercept point <math>(x,y)=(0,b)</math>. The ''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 ''x''-axis, the ''x''-intercept point <math>(x,y)=(-\tfrac ba,0)</math>. The ''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>.

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 {{<math|&>\Delta;'' x''}}</math>, and the corresponding change in {{mvar|y}}, denoted {{<math|&>\Delta;'' y''}}</math>]]

The [[slope (mathematics)|slope]] of a nonvertical line is a number that measures how steeply the line is slanted (rise-over-run). The [[first derivative]] of a linear function, inIf the sense of calculus,line is exactly this slope of the graph of the linear function. For {{<math|1=''>f''(''x'') = ''ax'' + ''b''}}</math>, this slope and derivative is given by the constant {{mvar|a}}. Linear functions can be characterized as the only real-valued functions that are defined on the entire real line and have a [[constant function|constant]] derivative.

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. Because aA linear function {{<math|1=''>f''(''x'') = ''ax'' + ''b''}}</math> has a constant rate of change {{mvar|a}},equal itto hasits the property that whenever the inputslope {{mvar|xa}} is increased by one unit, theso outputits changes by {{mvar|a}} units. If {{mvar|a}}derivative is positive, this will cause the value of theconstant function to increase<math>f\, while if {{mvar|'(x)=a}} is negative it will cause the value to decrease. More generally, if the input increases by some other amount, {{mvar|c}}, the output changes by&nbsp;{{</math|''ca''}}>.

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]] by a unique linear function near a given point <math>x=c</math> by a unique linear function. The [[derivative]] <math>f\,'(ac)</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==

A given linear function <math>f(x)</math> can be written in several standard formulas displaying its various properties. The simplest is the ''slope-intercept form'':

:<math>f(x)= ax+b</math>,

from which one can immediately see the slope ''a'' and the initial value <math>f(0)=b</math>, which is the ''y''-intercept of the graph <math>y=\ellf(x)</math>.

Given a slope ''a'' and one known value <math>f(x_0)=y_0</math>, we write the ''point-slope form'':

:<math>f(x) = a(x{-}x_0)+y_0</math>.

In graphical terms, this gives the line <math>y=f(x)</math> with slope ''a'' passing through the point <math>(x_0,y_0)</math>.

The ''two-point form'' starts with two known values <math>f(x_0)=y_0</math> and <math>f(x_1)=y_1</math>. One computes the slope <math>a=\tfrac{y_2-y_1-y_0}{x_2-x_1-x_0}</math> and inserts this into the point-slope form:

:<math>f(x) = \tfrac{y_1-y_0}{x_1-x_0}(x{-}x_0\!) + y_0</math>.

Its graph <math>y=f(x)</math> is the unique line passing through the points <math>(x_0,y_0\!), (x_1,y_1\!)</math>. The equation <math>y=f(x)</math> may also be written to emphasize the constant slope:

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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 intialinitial value <math>y=f(0)=b</math> at <math>x=0</math>. The slope ''a'' measures the rate of change of the output ''y'' per unit change in the input ''x''. In the graph, moving one unit to the right (increasing ''x'' by 1) moves the ''y''-value up by ''a'': that is, <math>f(x{+}1) = f(x) + a</math>. Negative slope ''a'' indicates a decrease in ''y'' for each increase in ''x''.

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 &euro;6€6 and &euro;3€3 per kilogram, and we wish to buy &euro;12€12 worth. How much of each can we purchase? LettingIf ''x'' kilograms of salami and ''y'' be the weightskilograms of salami and sausage, thecosts a total costof is:€12 <math>6xthen, €6×''x'' + 3y€3×''y'' = 12</math>€12. Solving for ''y'' gives the point-slope form <math>y = -2x + 4</math>, as above. That is, if we first choose the amount of salami ''x'', the amount of sausage can be computed as a function <math>y = f(x) = -2x + 4</math>. Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos: <math>f(x{+}1) = f(x) - 2</math>, and the slope is &minus;2. The ''y''-intercept point <math>(x,y)=(0,4)</math> corresponds to buying only 4kg4&nbsp;kg of sausage; while the ''x''-intercept point <math>(x,y)=(2,0)</math> corresponds to buying only 2kg2&nbsp;kg of salami.

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(\varphitheta ) = a\varphitheta + b</math>

is an [[Archimedean spiral]] if <math>a \neq 0</math> and a [[circle]] otherwise.

== Notes ==

<references/>

== See also ==

* [[Affine map]], a generalization

* [[Arithmetic progression]], a linear function of integer argument

== Notes ==

{{Reflist}}

== References ==

* {{citation

* James Stewart (2012), ''Calculus: Early Transcendentals'', edition 7E, Brooks/Cole. {{isbn|978-0-538-49790-9}}

| last = Stewart | first = James

* {{citation|first=Earl W.|last=Swokowski|title=Calculus with analytic geometry|edition=Alternate|year=1983|publisher=Prindle, Weber & Schmidt|place=Boston|isbn=0871503417}}

| 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

* httphttps://wwwweb.archive.org/web/20180722042342/https://corestandards.org/assets/CCSSI_Math%20Standards.pdf

{{Polynomials}}

{{Authority control}}

[[Category:Calculus]]