In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
) is a non-vertical
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
in the plane.
The characteristic property of linear functions is that when the input variable is changed, the change in the output is
proportional to the change in the input.
Linear functions are related to
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
s.
Properties
A linear function is a
polynomial function
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
in which the
variable has degree at most one:
:
.
Such a function is called ''linear'' because its
graph, the set of all points
in the
Cartesian plane
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, is a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
. The coefficient ''a'' is called the ''slope'' of the function and of the line (see below).
If the slope is
, this is a ''constant function''
defining a horizontal line, which some authors exclude from the class of linear functions.
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,
is required, so constant functions will be considered linear.
If
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
. In advanced mathematics texts, the term ''linear function'' often denotes specifically homogeneous linear functions, while the term
affine function
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
is used for the general case, which includes
.
The natural
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
of a linear function
, the set of allowed input values for , is the entire set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s,
One can also consider such functions with in an arbitrary
field, taking the coefficients in that field.
The graph
is a non-vertical line having exactly one intersection with the -axis, its -intercept point
The -intercept value
is also called the ''initial value'' of
If
the graph is a non-horizontal line having exactly one intersection with the -axis, the -intercept point
The -intercept value
the solution of the equation
is also called the ''root'' or
''zero'' of
Slope
The
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
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
, this slope is given by the constant .
The slope measures the constant rate of change of
per unit change in ''x'': whenever the input is increased by one unit, the output changes by units:
, and more generally
for any number
. If the slope is positive,
, then the function
is increasing; if
, then
is decreasing
In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the derivative of a general function measures its rate of change. A linear function
has a constant rate of change equal to its slope , so its derivative is the constant function
.
The fundamental idea of differential calculus is that any
smooth function
(not necessarily linear) can be closely
approximated near a given point
by a unique linear function. The
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is the slope of this linear function, and the approximation is:
for
. The graph of the linear approximation is the
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
of the graph
at the point
. The derivative slope
generally varies with the point ''c''. Linear functions can be characterized as the only real functions whose derivative is constant: if
for all ''x'', then
for
.
Slope-intercept, point-slope, and two-point forms
A given linear function
can be written in several standard formulas displaying its various properties. The simplest is the ''slope-intercept form'':
:
,
from which one can immediately see the slope ''a'' and the initial value
, which is the ''y''-intercept of the graph
.
Given a slope ''a'' and one known value
, we write the ''point-slope form'':
:
.
In graphical terms, this gives the line
with slope ''a'' passing through the point
.
The ''two-point form'' starts with two known values
and
. One computes the slope
and inserts this into the point-slope form:
:
.
Its graph
is the unique line passing through the points
. The equation
may also be written to emphasize the constant slope:
:
.
Relationship with linear equations
Linear functions commonly arise from practical problems involving variables
with a linear relationship, that is, obeying a
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
. If
, one can solve this equation for ''y'', obtaining
:
where we denote
and
. That is, one may consider ''y'' as a dependent variable (output) obtained from the independent variable (input) ''x'' via a linear function:
. In the ''xy''-coordinate plane, the possible values of
form a line, the graph of the function
. If
in the original equation, the resulting line
is vertical, and cannot be written as
.
The features of the graph
can be interpreted in terms of the variables ''x'' and ''y''. The ''y''-intercept is the initial value
at
. 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,
. Negative slope ''a'' indicates a decrease in ''y'' for each increase in ''x''.
For example, the linear function
has slope
, ''y''-intercept point
, and ''x''-intercept point
.
Example
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
, as above. That is, if we first choose the amount of salami ''x'', the amount of sausage can be computed as a function
. Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos:
, and the slope is −2. The ''y''-intercept point
corresponds to buying only 4 kg of sausage; while the ''x''-intercept point
corresponds to buying only 2 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
to the domain
.
Also, we could choose ''y'' as the independent variable, and compute ''x'' by the
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
linear function:
over the domain
.
Relationship with other classes of functions
If the coefficient of the variable is not zero (), then a linear function is represented by a
degree 1
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
(also called a ''linear polynomial''), otherwise it is a
constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic propertie ...
– also a polynomial function, but of zero degree.
A straight line, when drawn in a different kind of coordinate system may represent other functions.
For example, it may represent an
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
when its
values are expressed in the
logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
. It means that when is a linear function of , the function 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.
If ''both''
arguments and values of a function are in the logarithmic scale (i.e., when is a linear function of ), then the straight line represents a
power law
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
:
:
On the other hand, the graph of a linear function in terms of
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
:
:
is an
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a ...
if
and a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
otherwise.
See also
*
Affine map
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
, a generalization
*
Arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
, a linear function of integer argument
Notes
References
* James Stewart (2012), ''Calculus: Early Transcendentals'', edition 7E, Brooks/Cole.
*
External links
* https://web.archive.org/web/20130524101825/http://www.math.okstate.edu/~noell/ebsm/linear.html
* http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
{{Authority control
Calculus
Polynomial functions