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Stretching, compressing and reflecting parabolas

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

You multiply \(x^2\) by a number \(a\). You learn how \(a\) makes the parabola narrower, wider or open downwards.
📄 Open worksheet (PDF, German)

The video

In the video three parabolas lie over the standard parabola: times 2, times 0.5 and times \(-1\).

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

Contents of the video

  1. 0:00 Stretching: times two
  2. 0:29 Compressing: times one half
  3. 0:51 Reflecting: negative factor
  4. 1:17 The factor a
  5. 1:43 Careful, mistake!
  6. 2:13 Try it yourself

The most important points

Times 2: narrower

\(y = 2 \cdot x^2\): All \(y\)-values become twice as big (\(x = 2\): 8 instead of 4). The parabola is narrower; it is stretched (gestreckt).

y = 2x^2 compared with the standard parabola.
\(y = 2x^2\) compared with the standard parabola.

Times 0.5: wider

\(y = 0{,}5 \cdot x^2\): All \(y\)-values become half as big (\(x = 4\): 8 instead of 16). The parabola is wider; it is compressed (gestaucht).

y = 0{,}5x^2 compared with the standard parabola.
\(y = 0{,}5x^2\) compared with the standard parabola.

Negative factor

\(y = -x^2\): The \(y\)-values get the opposite sign. The parabola is reflected in the \(x\)-axis and open downwards. The vertex is the highest point.

y = -x^2: reflected in the x-axis.
\(y = -x^2\): reflected in the \(x\)-axis.

Summary for \(y = a \cdot x^2\): \(a > 1\) narrower, \(0 < a < 1\) wider, \(a < 0\) open downwards. The vertex stays \((0|0)\). Careful: \(y = -x^2\) at \(x = 3\): first \(3^2 = 9\), then the minus. So \(y = -9\).

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

The parabola \(y = -0{,}5 \cdot x^2\): open upwards or downwards? Wider or narrower than the standard parabola? How big is \(y\) for \(x = 2\)?

Show solution

Open downwards (\(a < 0\)), wider (\(0{,}5 < 1\)), \(y = -0{,}5 \cdot 4 = -2\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Parabeln y = a·x² (strecken, stauchen, spiegeln). 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.