# Reconstruction of functions

The **reconstruction of functions** deals with the establishment of functional equations. Some reconstruction tasks require differential calculus.

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### Remember

In the

To do this, you set up equations and solve them using systems of equations.

**reconstruction of functions**you look for a special function that satisfies given properties (e.g. type, points, slope, ...).To do this, you set up equations and solve them using systems of equations.

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### Method

- Function and derivation
- Set up equations
- Solve equations
- Specify function equation

### Example

We are looking for a second degree function that has an intersection with the y axis at $(0|-3)$ and a maximum point at $H(3|2)$.

#### Function and derivation

*A second degree function is a quadratic function. This looks like this:*

$f(x)=ax^2+bx+c$*The derivative is also needed:*

$f'(x)=2ax+b$*The goal is now to find the variables $a$, $b$ and $c$ with the given points.*-
#### Set up equations

*The other information is now used to build equations.**The intersection with the y-axis $S_y(0|-3)$ is inserted into the function $f(x)=ax^2+bx+c$:*

$f(0)=-3$

$a\cdot0^2+b\cdot0+c=-3$

$c=-3$*The same happens with the maximum point at $H(3|2)$*

$f(3)=2$

$a\cdot3^2+b\cdot3+c=2$

$9a+3b+c=2$*The derivative is zero at maximum points.*

$f'(3)=0$

$2a\cdot3+b=0$

$6a+b=0$ -
#### Solve equations

*The equations can be solved with a system of linear equations.*- $c=-3$
- $9a+3b+c=2$
- $6a+b=0$

*First of all, the insertion method lends itself by inserting the I. equation into II.*- $9a+3b-3=2$
- $6a+b=0$

*There are now several possibilities, but the insertion process makes sense here. First change and then use.*- $9a+3b-3=2$
- $6a+b=0\quad|-6a$

$b=-6a$

**II in I**

$9a-18a-3=2\quad|+3$

$-9a=5\quad|:(-9)$

$a=-\frac59$ -
#### Specify function equation

*The following variables are already known:*

$a=-\frac59$ and $c=-3$*$b$ can be calculated with one of the equations:*

$b=-6a$ $=-6\cdot(-\frac59)$ $=\frac{10}3$*The variables are used and we get the function we are looking for.*

$f(x)=ax^2+bx+c$

$f(x)=-\frac59x^2+\frac{10}3x-3$