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The standard parabola

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

A thrown ball flies along a curve. This curve is called a parabola. You learn the simplest parabola, \(y = x^2\), with a table of values, the vertex and symmetry.
📄 Open worksheet (PDF, German)

The video

In the video the parabola grows from a table of values. Then you see the vertex and the mirror points.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:24 Table of values
  3. 0:50 The graph
  4. 1:04 Vertex and symmetry
  5. 1:28 Careful, mistake!
  6. 1:53 Try it yourself

The most important points

The equation \(y = x^2\)

For every \(x\) you calculate \(x \cdot x\). Example \(x = -3\): \(y = (-3)^2 = 9\). The \(y\)-values are never negative.

Table of values for y = x^2.
Table of values for \(y = x^2\).

You plot the points and connect them with a smooth curve, not with straight lines. This is the standard parabola (die Normalparabel).

The points lie on the standard parabola.
The points lie on the standard parabola.

Vertex and symmetry

The lowest point is called the vertex (der Scheitelpunkt): \(S(0|0)\). The parabola is symmetric about the \(y\)-axis: the point \((-2|4)\) has the mirror point \((2|4)\).

Vertex and mirror points.
Vertex and mirror points.

Careful: \((-3)^2 = 9\), not \(-9\). Minus times minus gives plus.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
Page 1/3
The worksheet – Page 2
Page 2/3
Show the worksheet solutions
The worksheet – Page 3
Page 3/3

Try it yourself

Task

a) Does \(P(4|16)\) lie on the standard parabola? b) Does \(Q(3|6)\) lie on it?

Show solution

a) \(4^2 = 16\): yes. b) \(3^2 = 9 \neq 6\): no.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: die Normalparabel y = x² (Wertetabelle, Scheitelpunkt, Symmetrie). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.