quadratic function - quadratische Funktion
A quadratic function has the form $${y = ax^2 + bx + c}$$, where ${a, b, c \in \mathbb{R}}$ and ${a \ne 0}$.
The graph of a quadratic function is called parabola.
source: jsxgraph.org/wiki/index.php/Slider_and_function_plot
task
- change the slider a. What do you notice?
- change the slider c. What do you notice?
- change the slider b. What kind of path does the parabola move in?
basic parabola - Normalparabel
quadratic of the form $${y = x^2}$$
stretching and compressing the basic parabola
$${y = ax^2}$$, where $${|a| > 1}$$ means that the graph appears to be compressed in x-direction or stretched in y-direction. If ${|a| < 1}$ means that the graph appears to be “pulled apart” in x-direction.
translating a basic parabola
(along the x and y axes in a coordinate system)
Here is a short video on translating a parabola up or down, left or right, or both.
Enter your own parabolas here - function plotter
an animation (new)
Thanks to manim with documentation here
vertex form (a = 1)
$${f(x) = (x - d)^2 + e }$$
vertex form with variable a
$${f(x) = a \cdot (x - d)^2 + e }$$
another example of graphing a parabola
reflecting and translating the parabola
reflecting and stretching and compressing a parabola
exercise - state the coordinates of the vertex
Given a quadratic equation in vertex form, you should be able to state the coordinates of the parabola just from looking at the equation.
converting a quadratic function from general form to vertex form
(von der Normalform in die Scheitelpunktform)
This uses completing the square (quadratische Ergänzung).
example with video
Convert ${f(x) = x^2 - 6x + 8}$ or ${y = x^2 - 6x + 8}$ into vertex form.
axis intercepts
A y-intercept is a value of y where the graph meets the x-axis.
An x-intercept is a value of x where the graph meets the y-axis.
investigation with function plotter
Use Geogebra, Desmos or other function plotter to look at the following quadratic functions.
intercept with the y-axis - Schnittpunkt mit der y-Achse
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Set ${x = 0}$ in the expression of the function.
intercept(s) with the x-axis (if there are any) - Schnittpunkt(e) mit der x-Achse (falls vorhanden)
`${\underline
Set ${y = 0}$ or it the function is given as f(x): ${f(x) = 0}$.
Note the x-intercepts are generally known as zeroes or roots of a function.
$${f(x) = ax^2 + bx + c }$$
Here, you are looking for the solution of
$${f(x) = 0}$$
So, for the quadratic funtion it means
$${f(x) = ax^2 + bx + c = 0}$$
methods - see quadratic equations
- factorising (Vieta) if the equation has integer solutions
- quadratic formula - always works, even for surd solutions.
- pq-formula - if in Normalform
- completing the square - works well if the function is in vertex form.
video examples
finding the root (x-intercept) when in vertex form
Kurzes Video um die Nullstellen mit der Scheitelpunktform zu bestimmen.
1 - already in vertex form
2 - in the general form using the quadratic formula
3 - factorising the general form to find the zeros and reading off the y-intercept
axis of symmetry - Symmetrieachse
The line of symmetry passes through the vertex.
A fast way to find the line of sysmmetry if the parabola is given in
- general form
- general form where a = 1 (Normalform)
- vertex form
- factorised form
applications
word problems
Have a look at questions 15-26 on Aufgabenfuchs.de
links to other stuff
Links to pages on graphs of quadratic functions (parabolas)
- water fountain I
- water fountain II
- adjusting b and c
- [quadratische Funktion - Aufgabenfuchs] (https://mathe.aufgabenfuchs.de/funktion/quadratische-funktion.shtml)