symmetry - Symmetrie
symmetry to the y-axis - Achsensymmetrie zur y-Achse
Following has to be true: $${f(-x) = f(x)}$$
rotational symmetry about the origin - Punktsymmetrie zum Ursprung
Following has to be true: $${f(-x) = - f(x)}$$ or in another form $${- f(-x) = f(x)}$$
roots (zeroes) - Nullstellen
set the function $${f(x) = 0}$$ and attempt to find solutions.
Numerical method - see Newton-Raphson method - Das Newtonverfahren
y-intercept - x = 0
Substitute x = 0 and calculate f(0) This is quite useful for sketching the graph of the function.
stationary points - Kritische Punkte - 1. Ableitung
Notwendiges Kriterium für lokale Extrema $${f’(x) = 0}$$ Die Funktion sei an der Stelle
$x_E$differenzierbar. Dann gilt: Wenn bei$x_E$ein lokales Extremum liegt, dann ist$f'(x_E) = 0$.
Hinreichendes Kriterium für lokale Extrema Die Funktion f sei in einer Umgebung von x zweimal differenzierbar. Dann gilt:
local maximum - Hochpunkt (Extrema)
| $${x < x_E}$$ | $${x_E}$$ | $${x_E < x}$$ |
|---|---|---|
| $${f’(x) > 0}$$ | $${f’(x_E) = 0}$$ | $${f^\prime (x) < 0}$$ |
| $${f^{\prime \prime}(x) < 0}$$ |
local minimum - Tiefpunkt (Extrema)
| $${x < x_E}$$ | $${x_E}$$ | $${x_E < x}$$ |
|---|---|---|
| $${f’(x) < 0}$$ | $${f’(x_E) = 0}$$ | $${f^\prime (x) > 0}$$ |
| $${f^{\prime \prime}(x) > 0}$$ |
point of inflexion (inflection) - Sattelpunkt
Note: this is not an extrema - Der Sattelpunkt ist kein Extrempunkt
| $${x < x_E}$$ | $${x_E}$$ | $${x_E < x}$$ |
|---|---|---|
| $${f’(x) < 0}$$ | $${f’(x_E) = 0}$$ | $${f^\prime (x) < 0}$$ |
| or | ||
| $${f’(x) > 0}$$ | $${f’(x_E) = 0}$$ | $${f^\prime (x) > 0}$$ |
Curvature - Krümmung
A parabola is either curved in clockwise direction (when it opens downwards) or it is curved in anti-clockwise direction (when it opens upwards). For polynomials of higher degree, the curvature may change direction.
The point where this change takes place is called the point of inflexion - Wendepunkt (in German)
point of inflexion - Wendepunkte - 2. Ableitung
To look at where a function has a possible point of inflexion (Wendepunkt), we need to have a closer look at the 2nd derivative and also the 3rd derivative to be sure of the nature of the point of inflexion.
$${f^{\prime\prime}(x) = 0}$$
Notwendiges Kriterium für Wendepunkte
Let the function f be differentiatable twice at
$x_W$, then following holds. If$x_w$is a point of inflexion of, then${f^{\prime\prime}(x_W) = 0}$.
3rd derivative - 3. Ableitung (f’’’ -Kriterium)
The third derivative is used to determine the nature of the point of inflection.
If ${f^{\prime\prime\prime}(x) = 0}$ it may not be a point of inflexion but a saddle point.
To be sure that it is a point of inflexion,
$${f^{\prime\prime\prime}(x) \neq 0}$$
Hinreichendes Kriterium für Wendpunkte
Let the function f be differentiatable three times in the region of
$x$. If${f^{\prime\prime}(x_W) = 0}$and${f^{\prime\prime\prime}(x_W) \neq 0}$, then f has a point of inflexion at$x_W$.
Following criteria (Vorzeichenwechsel-Kriterium) is useful.
Let the function be twice differentiable at
$x_W$and${f^{\prime\prime}(x_W) = 0}$. If the second derivative has a change of sign (Vorzeichenwechsel), then there is a point of inflexion (ein Wendepunkt) at$x_W$
vertical asymptotes - Polstellen
This can happen for x values where the function is not defined because the denominator would be 0 which is not possible.
Example
$${f(x) = \frac{1}{x}}$$ $${f(x) = \frac{1}{x-3}}$$ $${f(x) = \frac{2x + 1}{x^2 -4x -5}}$$
non-vertical asymptotes - waagrechte und schräge Asymptoten
polynomial division - Polynomdivision
see here
Sketching the curve - Der Graph
types of functions you will need to know
linear function
see content from previous years here and in following GeoGebra book I created for 8th grade.
quadratics
see content from previous years - quadratic equation and parabolas