Parameters (math)

A parameter ( Greek παρά para , German 'besides' and μέτρον metron 'measure'), also form variable, is a variable in mathematics that occurs together with other variables, but is of a different quality. It is also said that a parameter is arbitrary, but fixed. It differs from a constant in that the parameter is only constant for one case under consideration, but can be varied for the next case.

In the equation are both and variables. Depending on whether or as a parameter is considered, a function of the other variables is then described by, each with a different character: ${\displaystyle y=b\cdot x^{2}}$ ${\displaystyle b}$ ${\displaystyle x}$ ${\displaystyle b}$ ${\displaystyle x}$ ${\displaystyle y=b\cdot x^{2}}$

• If you hold firm, then a resulting quadratic function with whose graph a parabola with the opening is. This opening depends on the particular choice of the parameter . ${\displaystyle b}$ ${\displaystyle y=f\left(x\right)=b\cdot x^{2}}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle b}$
• If you hold on, you get a linear function whose graph represents a straight line with the slope through the origin of the yb-plane. The slope depends on the particular choice of parameter . ${\displaystyle x}$ ${\displaystyle y=f\left(b\right)=x^{2}\cdot b}$ ${\displaystyle m=x^{2}}$ ${\displaystyle m}$ ${\displaystyle x}$

If different values ​​are used one after the other for a parameter, a family of curves is obtained .

For example, a parameter can influence the graph of a function with in several ways: ${\displaystyle p}$ ${\displaystyle y=f(x)}$

• ${\displaystyle y=f\left(x\right)+p}$: A change in the parameter opposite leads to a shift of the graph in the direction of the y-axis by units. ${\displaystyle p}$ ${\displaystyle p=0}$ ${\displaystyle p}$
• ${\displaystyle y=f\left(x+p\right)}$: A change in the parameter opposite leads to a shift of the graph in the direction of the x-axis by units. ${\displaystyle p}$ ${\displaystyle p=0}$ ${\displaystyle -p}$
• ${\displaystyle y=p\cdot f(x)}$: A change in the parameter opposite leads to stretching or compression in the direction of the y-axis. If the amount is less than 1, then there is compression. If negative, the graph is also mirrored on the x-axis. ${\displaystyle p}$ ${\displaystyle p=1}$ ${\displaystyle p}$ ${\displaystyle p}$
• ${\displaystyle y=f(p\cdot x)}$: A change in the parameter opposite leads to stretching or compression in the direction of the x-axis. If the amount is less than 1, then there is a stretch. If negative, the graph is also mirrored on the y-axis. ${\displaystyle p}$ ${\displaystyle p=1}$ ${\displaystyle p}$ ${\displaystyle p}$

For a further use of the term parameter in mathematics, see parametric representation .

By convention , parameters are usually designated with letters from the beginning of the Latin or Greek alphabet ( or with indices or etc.), while variables are designated with letters from the end of the alphabet ( ). ${\displaystyle a,b,c,\dotsc }$ ${\displaystyle a_{1},a_{2},a_{3},\dotsc }$ ${\displaystyle \alpha ,\beta ,\gamma ,\dotsc }$ ${\displaystyle x,y,z;x_{1},x_{2},\dotsc ;\xi _{1},\xi _{2},\dotsc }$

literature

• Stefan Harald Kaufmann: The meaning of the term parameter for mathematics lessons . In: Michael Neubrand (Ed.) (2009): Contributions to mathematics teaching, 2009 annual conference of the Society for Didactics of Mathematics , pp. 657–660. [1]

Web links

• The parameters of the vertex shape of a parabola . Online math; accessed October 1, 2015