Linear function (calculus): Difference between revisions

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

{{confuse|linear map}}

{{Distinguish|linear functional|linear map}}

In [[calculus]] and related areas of mathematics, a '''linear function''' from the real numbers to the real numbers is a function whose graph is a line in the plane.<ref>Stewart 2012, p. 23</ref>

{{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.{{sfn|Stewart|2012|p=23}}

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 of linear functions ==

[[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 -->

== Properties ==

A linear function is a [[polynomial]] function with one independent variable {{mvar|x}} of degree at most one, i.e. <math>f(x)=ax+b</math>.<ref>Stewart 2012, p. 24</ref> Here {{mvar|x}} is the independent variable. The [[graph of a function|graph]] of a linear function, that is, the set of all points whose coordinates have the form ({{mvar|x}}, {{mvar|f(x)}}), is a line, which is why this type of [[Function (mathematics)|function]] is called ''linear''. Some authors, for various reasons, also require that the coefficient of the variable (the {{mvar|a}} in {{mvar|ax + b}}) should not be zero.<ref>{{harvnb|Swokowski|1983|loc=pg. 34}} is but one of many well known references that could be cited.</ref> This requirement can also be expressed by saying that the degree of the polynomial in this polynomial function is exactly one, or by saying that the line which is the graph of a linear function is a ''slanted'' line (neither vertical nor horizontal). This requirement will not be imposed in this article, thus constant functions, <math>f(x) = b</math>, will be considered to be linear functions (their graphs are horizontal lines).

A linear function is a [[polynomial function]] in which the [[variable (mathematics)|variable]] {{mvar|x}} has degree at most one:{{sfn|Stewart|2012|p=24}}

:<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.{{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 [[Domain_of_a_function|___domain]] or set of allowed values for {{mvar|x}} of a linear function is the entire set of [[real number]]s {{math|'''R'''}}. This means that any real number can be substituted for {{mvar|x}}.

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.

Because two different points determine a line, it is enough to substitute two different values for {{mvar|x}} in the linear function and determine {{mvar|f(x)}} for each of these values. This will give the coordinates of two different points that lie on the line. Because {{mvar|f}} is a function, this line will not be vertical.

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>

Since the graph of a linear function is a nonvertical line, this line has exactly one intersection point with the {{mvar|y}}-axis. This point is {{math|(0, ''b'')}}.

==Slope==

A the graph of a nonconstant linear function has exactly one intersection point with the {{mvar|x}}-axis. This point is {{math|({{sfrac|−''b''|''a''}}, 0)}}. From this, it follows that a nonconstant linear function has exactly one [[Zero_of_a_function|zero]] or root. That is, there is exactly one solution to the equation {{math|1=''ax'' + ''b'' = 0}}. The zero is {{math|1=''x'' =}}&nbsp;{{sfrac|−''b''|''a''}}.

[[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). If the line is the graph of the linear function <math>f(x) = ax + b</math>, this slope is given by the constant {{mvar|a}}.

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

The points on a line have coordinates which can also be thought of as the solutions of [[linear equation]]s in two variables (the equation of the line). These solution sets define functions which are linear functions. This connection between linear equations and linear functions provides the most common way to produce linear functions.

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>.

There are many standard forms of linear equations, but only three will be examined to see how to obtain linear functions:

*[[#General form|General form]]

*[[#Slope-intercept form|Slope-intercept form]]

*[[#Parametric form|Parametric form]]

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 form ==

[[Image: wiki_linearna_funkcija_eks1.png|thumb||right]]<!-- are PNG and a translit from Ukrainian necessary? -->

==Slope-intercept, point-slope, and two-point forms==

The slope-intercept form of a linear function is an equation of the form

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> y(x)=ax+b </math>.

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

The slope-intercept form has two variables {{mvar|x}} and {{mvar|y}} and two coefficients {{mvar|a}} and {{mvar|b}}. The slope-intercept form is also called the ''explicit form'' because it defines {{math|''y''(''x'')}} explicitly (directly) in terms of {{mvar|x}}.

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=f(x)</math>.

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

The slope-intercept form of a linear function is unique. That is, if the value of either or both of the coefficient letters {{mvar|a}} and {{mvar|b}} are changed, a different function is obtained.

:<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_1-y_0}{x_1-x_0}</math> and inserts this into the point-slope form:

The constant {{mvar|b}} is the so-called {{mvar|y}}-intercept. It is the {{mvar|y}}-value at which the line intersects the {{mvar|y}}-axis.

:<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:

:<math>\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}</math>.

==Relationship with linear equations==

The coefficient {{mvar|a}} is the [[slope]] of the line, which measures of the rate of change of the linear function. Since {{mvar|a}} is a constant, this rate of change is constant. Moving from any point on the line to the right by one unit (that is, increasing {{mvar|x}} by 1), the {{mvar|y}}-value of the function changes by {{mvar|a}}.

[[Image:wiki linearna funkcija eks1.png|thumb|right]]<!-- are PNG and a translit from a foreign language necessary? -->

Linear functions commonly arise from practical problems involving variables <math>x,y</math> with a linear relationship, that is, obeying a [[linear equation]] <math>Ax+By=C</math>. If <math>B\neq 0</math>, one can solve this equation for ''y'', obtaining

:<math>y = -\tfrac{A}{B}x +\tfrac{C}{B}=ax+b,</math>

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 initial 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 slope-intercept form <math> y(x)=-2x+4 </math> has {{math|1=''a'' = −2}} and {{math|1=''b'' = 4}}. The point {{math|1=(0, ''b'') = (0, 4)}} is the intersection of the line and the {{mvar|y}}-axis, the point {{math|1=({{sfrac|−''b''|''a''}}, 0)}} = ({{sfrac|−4|−2}}, 0) = (2, 0)}} is the intersection of the line and the {{mvar|x}}-axis, and {{math|1=''a'' = −2}} is the slope of the line. For every step to the right ({{mvar|x}} increases by 1), the value of {{mvar|y}} changes by −2 (goes down).

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>.

== General form ==

[[Image: wiki_linearna_funkcija_stand1.png|180px|right]]

===Example===

The general form for the equation of a line is:

Suppose salami and sausage cost €6 and €3 per kilogram, and we wish to buy €12 worth. How much of each can we purchase? If ''x'' kilograms of salami and ''y'' kilograms of sausage costs a total of €12 then, €6×''x'' + €3×''y'' = €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 4&nbsp;kg of sausage; while the ''x''-intercept point <math>(x,y)=(2,0)</math> corresponds to buying only 2&nbsp;kg of salami.

: <math> Ax+By=C. </math>

When <math> B \ne 0 </math> this equation may be solved for the variable {{mvar|y}} and thus used to define a linear function (namely, <math> y = - \tfrac{A}{B} x + \tfrac{C}{B} = f(x)</math>). While all lines have equations in the general form, only the non-vertical lines have equations which can give rise to linear functions.

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>.

The general form has 2 variables {{mvar|x}} and {{mvar|y}} and 3 coefficients {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}.

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>.

This form is not unique. If one multiplies {{mvar|A}}, {{mvar|B}} and {{mvar|C}} by a constant factor {{mvar|k}}, the coefficients change, but the line remains the same. The linear function obtained from this form is unique since it depends only on the coordinates of the points on the line. For example, {{math|1=3''x'' − 2''y'' = 1}} and {{math|1=9''x'' − 6''y'' = 3}} are general forms of the equation of the same line which is associated with the linear function <math> f(x) = \tfrac{3}{2} x - \tfrac{1}{2} </math>.

== Relationship with other classes of functions ==

This general form is used mainly in geometry and in systems of two linear equations in two unknowns.

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.

==Parametric form==

[[Image: wiki_linearna_funkcija_par1.png|300px|right]]

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

The [[parametric form]] of a line consists of two equations:

:<math>x(t) = {b_1}+{a_1}t </math>

:<math>y(t) = {b_2}+{a_2}t </math>

where <math> a_1 \ne 0 </math>.

For example, it may represent an [[exponential growth|exponential function]] when its [[codomain|values]] are expressed in the [[logarithmic scale]]. It means that when {{math|[[logarithm|log]](''g''(''x''))}} is a linear function of {{mvar|x}}, the function {{mvar|g}} is exponential. With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function. With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function.

The parametric form has one [[parameter]] {{mvar|t}}, two variables {{mvar|x}} and {{mvar|y}}, and four coefficients {{math|''a''₁}}, {{math|''a''₂}}, {{math|''b''₁}}, and {{math|''b''₂}}. The coefficients are not unique, but they are related.

If ''both'' [[___domain of a function|arguments]] and values of a function are in the logarithmic scale (i.e., when {{math|[[logarithm|log]](''y'')}} is a linear function of {{math|[[logarithm|log]](''x'')}}), then the straight line represents a [[power law]]:

The line passes through the points {{math|(''b''₁, ''b''₂)}} and {{math|(''b''₁ + ''a''₁, ''b''₂ + ''a''₂)}}.

:<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]]

The vector-parametric form is used in engineering (it is simple to model the path from one point to another point with {{mvar|t}}=time). Engineers tend to use parametric notation and the letter {{mvar|t}} for the parameter; mathematicians use vector notation and the letter &lambda;.

On the other hand, the graph of a linear function in terms of [[polar coordinates]]:

:<math>r =f(\theta ) = a\theta + b</math>

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

== See also ==

This form of a linear function can easily be extended to lines in higher dimensions, which is not true of the other forms.

* [[Affine map]], a generalization

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

; Example: <math>{{X}} = ({-1},{1}) + t({2},{3})</math>

* Here: {{math|1=''a''₁ = 2}} and {{math|1=''a''₂ = 3}} and {{math|1=''b''₁ = −1}} and {{math|1=''b''₂ = 1}}

* The line passes through the points> {{math|1=(''b''₁, ''b''₂) = (−1, 1)}} and {{math|(''b''₁ + ''a''₁, ''b''₂ + ''a''₂)}}

* The parametric form of this line is:

*:<math>x(t) = {-1}+{2}t </math>

*:<math>y(t) = {1}+{3}t </math>

* The slope-intercept form of this line is: {{math|1=''y''(''x'') = 1.5''x'' + 2.5}} (solve the first parametric equation for {{mvar|t}} and substitute in$

* One general form of this line is: {{math|1=−3''x'' + 2''y'' = 5}}.

== Notes ==

{{Reflist}}

<references/>

== 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 &amp;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.columbiamath.okstate.edu/itc~noell/sipa/mathebsm/linear.html

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

* http://www.math.okstate.edu/~noell/ebsm/linear.html

* http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

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