# Parabolas

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.

• 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

$${f(x) = a \cdot (x - d)^2 + e }$$

### 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

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

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.

## 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