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Vertex form with a factor a

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

In \(y = a\cdot(x - d)^2 + e\), \(a\) can be any number except \(0\). You read off the vertex, the opening and the shape, draw from the vertex and set up the equation from a graph.
📄 Open worksheet (PDF, German)

The video

In the video we compare narrow and wide parabolas and calculate \(a\) with a point.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:52 Drawing from the vertex
  3. 1:21 Setting up the equation
  4. 1:58 Careful, mistake!
  5. 2:29 Try it yourself

The most important points

Reading off

\(y = 2\cdot(x-1)^2 + 3\): vertex (der Scheitelpunkt) \(S(1\,|\,3)\), \(a = 2 > 0\): opens upwards, narrower. \(y = -0{,}5\cdot(x+2)^2 + 4\): \(S(-2\,|\,4)\), opens downwards, wider.

Two parabolas with a factor: narrower (blue) and wider (green).
Two parabolas with a factor: narrower (blue) and wider (green).

From the vertex: 1 to the right, \(a\) up; 2 to the right, \(4\cdot a\) up.

y = 2·(x-1)^2 + 3: 1 right, 2 up; 2 right, 8 up.
\(y = 2\cdot(x-1)^2 + 3\): 1 right, 2 up; 2 right, 8 up.

Setting up the equation

Vertex \(S(2\,|\,-1)\): \(y = a\cdot(x-2)^2 - 1\). Insert the point \(P(4\,|\,1)\): \(1 = a\cdot 2^2 - 1\), so \(4a = 2\) and \(a = 0{,}5\).

Vertex and one point give the equation.
Vertex and one point give the equation.

Careful: \(2\cdot(x-1)^2\) is not \((2x-2)^2\). First the bracket, then the square, then times \(a\), then plus \(e\).

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
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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
The worksheet – Page 5
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Try it yourself

Task

\(y = 3\cdot(x+1)^2 - 2\): vertex? Up or down? Narrower or wider? Which \(y\)-value at \(x = 0\)?

Show solution

\(S(-1\,|\,-2)\), opens upwards, narrower; \(x = 0\): \(3\cdot 1^2 - 2 = 1\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Scheitelpunktform y = a·(x − d)² + e mit Faktor a (ablesen, zeichnen, aufstellen). 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.