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Starting value and growth factor

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

You multiply the exponential function by a number \(a\). You learn starting value and growth factor, growth and decay, and how to find the equation from two points.
📄 Open worksheet (PDF, German)

The video

In the video you see how \(a\) lets the curve start on the \(y\)-axis and how \(b\) comes from two \(y\)-values.

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

Contents of the video

  1. 0:00 The starting value a
  2. 0:27 The growth factor b
  3. 0:52 Growth and decay
  4. 1:14 Setting up the equation
  5. 1:44 Careful, mistake!
  6. 2:08 Try it yourself

The most important points

The starting value \(a\)

\(y = a \cdot b^x\). For \(x = 0\), \(b^0 = 1\), so \(y = a\). The number \(a\) is the starting value (der Startwert) and the \(y\)-intercept. Example: \(y = 2 \cdot 2^x\) starts at \(2\).

y = 2 · 2^x starts at 2.
\(y = 2 \cdot 2^x\) starts at \(2\).

The growth factor \(b\)

Divide two \(y\)-values in a row: for \(y = 2 \cdot 3^x\), \(y\) is \(2, 6, 18, 54\). \(6 : 2 = 3\). The number \(b\) is the growth factor (der Wachstumsfaktor).

Always times 3.
Always times \(3\).

\(b > 1\): growth. \(b = 1{,}05\) means \(+5\,\%\). \(0 < b < 1\): decay (der Zerfall). \(b = 0{,}9\) means \(-10\,\%\).

Equation from two points

\((0|3)\) and \((2|12)\): \(a = 3\). \(12 = 3 \cdot b^2\), so \(b^2 = 4\) and \(b = 2\). \(y = 3 \cdot 2^x\).

The curve through (0|3) and (2|12).
The curve through \((0|3)\) and \((2|12)\).

Careful: \(y = 2 \cdot 3^x\) at \(x = 2\): first the power \(3^2 = 9\), then times \(2\). So \(18\), not \(6^2 = 36\).

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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Show the worksheet solutions
The worksheet – Page 3
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Try it yourself

Task

An exponential function goes through \((0|4)\) and \((1|6)\). What is the equation?

Show solution

\(a = 4\). \(6 = 4 \cdot b\), so \(b = 1{,}5\). \(y = 4 \cdot 1{,}5^x\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Exponentialfunktionen y = a·b^x (Startwert, Wachstumsfaktor, Gleichung aus zwei Punkten). 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.