Linear function (calculus)

The term linear function is often (especially in the school math ) an image of the form

So at most a polynomial of first degree refers.

In the strict mathematical sense, these are not a linear map, but to an affine transformation, since the linearity condition A., is not satisfied. It is only for the special case of it is a linear function in the strict sense, also referred to as a homogeneous linear function or proportionality. Only on the basis of this description, the function for the case also general linear function or linear inhomogeneous function is called (see also line equation ), but this can sometimes lead to confusion. Nevertheless, in this article, the term often used is maintained linear function.

Linear functions are among the relatively simple functions in mathematics. You are continuous and differentiable. Many problems can easily be solved for linear functions; Therefore, one often tries to approximate complex problems by linear correlations.

Graph

The graph of a linear function is always a straight line. In Cartesian coordinates, such straight lines thus satisfy the equation

Wherein ( the abscissa) and an independent ( the ordinate ) is the dependent variable.

There are numerous other naming conventions for the function term, for example, or in Austria is commonly used in Switzerland, however, is also found in Belgium or

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

  • The number m is the slope of the straight line.
  • The number n is the y -axis or ordinate, the inhomogeneity or the shift constant.

The graph of a linear function never runs parallel to the y axis, since this would have an associated x greater than y. This would contradict the definition of a function as a unique assignment.

Determination of the functional term from two points

It is assumed that the points are on the graph the linear function, and are different from each other.

The slope can be calculated

The y -intercept is given by

The desired function term is given by

Or more simply by

Summary

Functional equation

Axis intersections

Slope

Establish functional equation

  • The slope and a point located on the straight line, are known.
  • The coordinates of two points, which lie on the straight line, are known.

Intersection of two lines

Orthogonal lines

Derivative and antiderivative

The derivation of is thus always a constant function ( a linear function can be also as a function of constant dissipation define ) as the derivative of a function gives the slope of its tangent at the point.

An antiderivative of this is can be shown as follows:

Homogeneous polynomial
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