Zoning

The zoning is a method of curve discussion in mathematics . It is used to mark the areas of a coordinate system in which the curve of a rational function does not run. With the help of a few curve points, the curve shape can then be sketched relatively easily in the areas that remain free.

The three forms

The function is brought into two new forms for the territorial divisions.

The first step is to convert it into a real broken function.

example

The function is given . Since the numerator contains larger or equal exponents of x, it is an improperly broken term. ${\ displaystyle y = f (x) = {\ frac {4x ^ {2} + 8x-12} {x ^ {2} -1}}}$

By polynomial division of the numerator by the denominator you get the real broken Term: . ${\ displaystyle 4 + {\ frac {8x-8} {x ^ {2} -1}}}$

The third form (product form) is obtained by factoring the polynomials into numerators and denominators. For our example it would look like this: . ${\ displaystyle {\ frac {4 (x + 3) (x-1)} {(x-1) (x + 1)}}}$

Curve discussion without derivations

This is followed by a curve discussion : The zeros , the poles , the symmetry and the behavior in infinity are determined. The actual territorial division takes place in the next step.

The first zoning

First you multiply the equation ( , polynomials) by the denominator so that you no longer have a fraction, so . Now you set every single (linear) factor of the equation equal to zero. This gives you several new equations, mostly vertical, horizontal or bisecting lines, which you draw as limits in a coordinate system. It is recommended to work with two colors in order to distinguish boundaries that come from the left side of the equation from those of the right side. Points of intersection of different colored borders are curve points. When the limits are drawn in, the value of each limit is written. For there is a double vertical limit at x = 2. ${\ displaystyle y = {\ tfrac {p (x)} {q (x)}}}$ ${\ displaystyle p}$ ${\ displaystyle q}$ ${\ displaystyle y \ cdot q (x) = p (x)}$ ${\ displaystyle x ^ {2} = 0}$

A sample point is then taken. However, this point must not lie on one of the boundaries of the first division. The point is inserted into the equation, but it is only of interest whether the result on the respective side is negative or positive. Then you compare the result. If one has different signs , there are no function values in the area from which the sample point was taken. Now you can go over a simple boundary to get into an area where there are curve points.

If it applies to an area that the curve runs in it, then a curve course through another area that is separated from this by a single, triple, five-fold, etc. border is excluded.

Individual evidence

^ Karl-Heinz Pfeffer: Analysis for technical high schools . Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-1024-3 , pp. 99, 178 .

[1] Karl-Heinz Pfeffer: Analysis for technical high schools . Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-1024-3 , pp. 99, 178 .