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The graph of the exponential function

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

Bacteria divide into two every hour: always times 2. This is exponential growth. You draw the graph of \(y = 2^x\) and compare different bases.
📄 Open worksheet (PDF, German)

The video

In the video the graph grows from a table of values. Then you see the bases \(2\), \(3\) and \(0{,}5\) compared.

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

Contents of the video

  1. 0:00 Introduction: doubling
  2. 0:23 Negative x
  3. 0:43 The graph
  4. 1:02 Other bases
  5. 1:28 Careful, mistake!
  6. 1:49 Try it yourself

The most important points

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

\(2^0 = 1\), \(2^{-1} = \frac{1}{2}\), \(2^{-2} = \frac{1}{4}\), \(2^{-3} = \frac{1}{8}\). For positive \(x\), \(y\) grows faster and faster. For negative \(x\), \(y\) gets small, but never zero.

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

The graph

The graph always goes through \((0|1)\), because \(b^0 = 1\). To the right it rises ever more steeply. To the left it approaches the \(x\)-axis but never touches it: the \(x\)-axis is the asymptote.

The graph of y = 2^x.
The graph of \(y = 2^x\).

Other bases

\(b > 1\): the graph rises. \(0 < b < 1\): the graph falls. \(y = 0{,}5^x\) is the mirror image of \(y = 2^x\) in the \(y\)-axis. All graphs go through \((0|1)\).

2^x, 3^x and 0{,}5^x compared.
\(2^x\), \(3^x\) and \(0{,}5^x\) compared.

Careful: \(2^3 = 2 \cdot 2 \cdot 2 = 8\), not \(2 \cdot 3 = 6\). \(y = 2x\) is a straight line, \(y = 2^x\) is a curve.

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

Try it yourself

Task

Calculate \(y\) for \(y = 3^x\) with \(x = -1\), \(0\) and \(2\). Does the graph rise or fall?

Show solution

\(\frac{1}{3}\), \(1\), \(9\). The base \(3\) is greater than \(1\): the graph rises.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: der Graph der Exponentialfunktion y = b^x (Wertetabelle, Basis, steigen und fallen). 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.