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