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Extreme values: minimum and maximum

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

The vertex is the lowest or highest point of a parabola. You read off the minimum (das Minimum) and the maximum (das Maximum).
📄 Open worksheet (PDF, German)

The video

In the video we read extreme values from the vertex form and from the standard form.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:23 From the vertex form
  3. 0:45 From the standard form
  4. 1:07 The range
  5. 1:35 Careful, mistake!
  6. 1:59 Try it yourself

The most important points

Minimum and maximum

\(a > 0\): opens upwards, the vertex is the lowest point, the minimum. \(a < 0\): opens downwards, the vertex is the highest point, the maximum. The extreme value is the \(y\)-value of the vertex.

A parabola with minimum -3 and one with maximum 4.
A parabola with minimum \(-3\) and one with maximum \(4\).

Examples

\(y = -(x-2)^2 + 3\): \(S(2\,|\,3)\), \(3\) is the maximum, at \(x = 2\). \(y = x^2 - 4x + 1 = (x-2)^2 - 3\): \(S(2\,|\,-3)\), \(-3\) is the minimum, at \(x = 2\).

y = (x-2)^2 - 3 has the minimum -3.
\(y = (x-2)^2 - 3\) has the minimum \(-3\).

Range

Minimum \(-3\): \(y \geq -3\). Maximum \(3\): \(y \leq 3\). Careful: The extreme value is the \(y\)-value, not the \(x\)-coordinate.

The dashed lines at y = -3 and y = 3 are the limits.
The dashed lines at \(y = -3\) and \(y = 3\) are the limits.

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 1/4
The worksheet – Page 2
Page 2/4
The worksheet – Page 3
Page 3/4
Show the worksheet solutions
The worksheet – Page 4
Page 4/4

Try it yourself

Task

Does \(y = x^2 + 6x + 4\) have a minimum or a maximum? How big is the extreme value and at which \(x\) is it reached?

Show solution

\(y = (x+3)^2 - 5\), \(a = 1 > 0\): minimum \(-5\) at \(x = -3\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Extremwerte quadratischer Funktionen (Minimum, Maximum, Wertebereich). 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.