Linear function

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As a linear function is often (especially in the school math) is a function of the form

that is, a polynomial function of at most first degree.

However, this is not a linear mapping in the sense of linear algebra , but an affine mapping , since the linearity condition is generally not fulfilled. This is why one speaks of an affine-linear function. A linear mapping or linear function in the sense of linear algebra is only involved in a special case , i.e. such functions are also referred to as homogeneous linear functions or proportionality . Based on this designation, the function for the case is also called a general linear function or linear-inhomogeneous function . The term linear function is used throughout this article .

Linear functions are one of the relatively simple functions in mathematics. They are continuous and differentiable . Many problems can be easily solved for linear functions; therefore one often tries to approximate complicated problems by means of linear relationships .


Slope triangles on the graph of the linear function

The graph of a linear function is a straight line. The following applies in Cartesian coordinates

with real numbers and where (the abscissa ) is an independent variable and (the ordinate ) is the dependent variable .

There are numerous other naming conventions for the functional term, e.g. B. or in Austria is frequently used in Switzerland, however, in Belgium you can also find or

This representation is also known as the normal form of a linear function. Its two parameters can be interpreted as follows:

The graph of a linear function never runs parallel to the y-axis, as more than one would be assigned to one, which would contradict the (legal) uniqueness of a function, which is required by definition.

Determination of the functional term from two points

Slope of a linear function through two given points

It is assumed that the points and lie on the graph of the linear function and are different from one another.

The slope can be calculated with

The y-axis intercept results from


The function term we are looking for is therefore given by

or easier through


Function equation

A function with is called a linear function. In the case , “fully rational function of the first degree” or “polynomial of the first degree” is used as a designation.
The graphical representation of the function graph is a straight line.

Axis intersections

Intersection with the axis:
Intersection with the axis:


Zlinfkt 01.gif

The slope of the graph of a linear function can be read off as a coefficient from the function equation.

It is calculated from the coordinates of two points on the straight line as follows:

Establish functional equation

  • The slope and a point on the straight line are known.
  • The coordinates of two points and those on the straight line are known.
First the slope factor is calculated, then with it :

Intersection of two straight lines

The solution to this equation is the coordinate of the intersection of the two straight lines.
is then the coordinate of this intersection

Orthogonal straight lines

For the slopes and two perpendicular straight lines , the following applies:

Derivative and antiderivative

The derivative of is is always a constant function , since the derivative of a function gives the slope of its tangent at the point .

Antiderivatives of have the form This can be shown as follows:

Limit values

If the coefficient of a function is positive, then and The graph develops from "bottom left" to "top right". If, however, is negative, then and The graph runs from “top left” to “bottom right”. In the special case there is a constant function, so in this case the graph runs parallel to the axis.

Web links

Commons : Linear Equations  - collection of images, videos and audio files
Commons : Linear functions  - collection of images, videos and audio files


  • Manfred Leppig: Learning Levels in Mathematics. Girardet 1981, ISBN 3-7736-2005-5 , pp. 61-74.