## 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 ErgĂ¤nzung*).

#### 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*

`${\underline{x = 0}}$`

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)*

`${\underline{y = 0}}$`

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)