Linear function

This article covers the functions in elementary calculus. For linear functions in linear algebra see linear mapping .

As a linear function is often (especially in the school math) is a function of the form ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle f(x)=m\cdot x+n;\quad m,n\in \mathbb {R} ,}$

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 . ${\displaystyle n=0}$ ${\displaystyle f(x)=mx.}$ ${\displaystyle n\neq 0}$

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 .

graph

Slope triangles on the graph of the linear function ${\displaystyle x\mapsto {\tfrac {1}{2}}x+2}$

The graph of a linear function is a straight line. The following applies in Cartesian coordinates ${\displaystyle (x|y)}$

${\displaystyle y=m\cdot x+n}$

with real numbers and where (the abscissa ) is an independent variable and (the ordinate ) is the dependent variable . ${\displaystyle m}$ ${\displaystyle n,}$ ${\displaystyle x}$ ${\displaystyle y}$

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 ${\displaystyle ax+b,}$ ${\displaystyle mx+c,}$ ${\displaystyle mx+b}$ ${\displaystyle mx+t.}$ ${\displaystyle y=kx+d}$ ${\displaystyle y=mx+q.}$ ${\displaystyle y=mx+p}$ ${\displaystyle y=kx+t.}$

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

• The number indicates the slope of the straight line. ${\displaystyle m}$
• The number is the y-axis or ordinate segment , the inhomogeneity or the displacement constant. ${\displaystyle n}$

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. ${\displaystyle x}$ ${\displaystyle y}$

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. ${\displaystyle (x_{1}|y_{1})}$ ${\displaystyle (x_{2}|y_{2})}$ ${\displaystyle f}$

The slope can be calculated with ${\displaystyle m}$

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}$

The y-axis intercept results from ${\displaystyle n}$

${\displaystyle n=y_{1}-m\cdot x_{1}}$ or ${\displaystyle n=y_{2}-m\cdot x_{2}.}$

The function term we are looking for is therefore given by ${\displaystyle f(x)}$

${\displaystyle f(x)={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot x+\left(y_{1}-{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot x_{1}\right)}$

or easier through

${\displaystyle f(x)={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot (x-x_{1})+y_{1}.}$

Summary

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. ${\displaystyle f}$ ${\displaystyle f(x)=mx+n}$ ${\displaystyle m\neq 0}$

The graphical representation of the function graph is a straight line.

Axis intersections

Intersection with the axis: ${\displaystyle P}$ ${\displaystyle x}$ ${\displaystyle P(x_{P}|0)\Rightarrow f(x_{P})=0}$

Intersection with the axis: ${\displaystyle Q}$ ${\displaystyle y}$ ${\displaystyle Q(0|y_{Q})\Rightarrow y_{Q}=f(0)}$

pitch

The slope of the graph of a linear function can be read off as a coefficient from the function equation. ${\displaystyle \tan \alpha }$ ${\displaystyle f}$ ${\displaystyle m}$ ${\displaystyle f(x)=mx+n}$

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

${\displaystyle \tan \alpha ={\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {\Delta y}{\Delta x}}}$

Establish functional equation

• The slope and a point on the straight line are known. ${\displaystyle m}$ ${\displaystyle P_{1}(x_{1}|y_{1}),}$

Approach: ${\displaystyle f(x)=mx+n}$

${\displaystyle P_{1}(x_{1}|y_{1})\quad \Rightarrow \quad f(x_{1})=y_{1}\quad \Rightarrow \quad mx_{1}+n=y_{1}\quad \Rightarrow \quad n=y_{1}-mx_{1}}$

• The coordinates of two points and those on the straight line are known. ${\displaystyle P_{1}(x_{1}|y_{1})}$ ${\displaystyle P_{2}(x_{2}|y_{2}),}$

First the slope factor is calculated, then with it : ${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$ ${\displaystyle n}$

${\displaystyle P_{1}(x_{1}|y_{1})\quad \Rightarrow \quad f(x_{1})=y_{1}\quad \Rightarrow \quad mx_{1}+n=y_{1}\quad \Rightarrow \quad n=y_{1}-mx_{1}}$

or

${\displaystyle P_{2}(x_{2}|y_{2})\quad \Rightarrow \quad f(x_{2})=y_{2}\quad \Rightarrow \quad mx_{2}+n=y_{2}\quad \Rightarrow \quad n=y_{2}-mx_{2}}$

Intersection of two straight lines

Approach: ${\displaystyle f(x)=g(x)}$

The solution to this equation is the coordinate of the intersection of the two straight lines. ${\displaystyle x_{S}}$ ${\displaystyle x}$

${\displaystyle y_{S}=f(x_{S})=g(x_{S})}$is then the coordinate of this intersection ${\displaystyle y}$ ${\displaystyle S(x_{S}|y_{S}).}$

Orthogonal straight lines

For the slopes and two perpendicular straight lines , the following applies: ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$ ${\displaystyle g_{1}}$ ${\displaystyle g_{2}}$

${\displaystyle m_{1}\cdot m_{2}=-1}$

${\displaystyle m_{1}=-{\frac {1}{m_{2}}}}$

${\displaystyle m_{2}=-{\frac {1}{m_{1}}}}$

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 . ${\displaystyle f\left(x\right)=mx+n}$ ${\displaystyle f'\left(x\right)=m.}$ ${\displaystyle f'}$ ${\displaystyle P\left(x|f(x)\right)}$

Antiderivatives of have the form This can be shown as follows: ${\displaystyle f}$ ${\displaystyle F(x)={\frac {m}{2}}x^{2}+nx+c.}$

${\displaystyle F'(x)=\left({\frac {m}{2}}x^{2}+nx+c\right)'={\frac {m}{2}}\cdot \left(x^{2}\right)'+n\cdot (x)'+0={\frac {m}{2}}\cdot 2x+n=mx+n=f(x)}$

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. ${\displaystyle f(x)=mx+n}$ ${\displaystyle m}$ ${\displaystyle \lim _{x\to -\infty }f(x)=-\infty }$ ${\displaystyle \lim _{x\to \infty }f(x)=\infty .}$ ${\displaystyle m}$ ${\displaystyle \lim _{x\to -\infty }f(x)=\infty }$ ${\displaystyle \lim _{x\to \infty }f(x)=-\infty .}$ ${\displaystyle m=0}$ ${\displaystyle \lim _{x\to -\infty }f(x)=\lim _{x\to \infty }f(x)=n,}$ ${\displaystyle x}$

Web links

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

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

Wikibooks:: Mathematics for School ${\displaystyle {\begin{smallmatrix}{\mathbf {MATHE} \mu \alpha T\mathbb {R} ix}\end{smallmatrix}}}$

• Calculator and theory of linear function
• Linear Functions Introduction for Students (Video)

literature

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