Starting from a graph of the original function, it's easy to graph a linear coordinate transformation of it. The key principles are these:

- A coordinate transformation outside the function acts vertically, while a coordinate transformation inside the function acts horizontally;
- Adding and subtracting shift the graph, while multiplying and dividing change the scale;
- Anything inside (horizontal) acts
*backwards*.

More concretely, consider these examples:

Coordinate transformation of f: |
Effect on the graph: |
---|---|

f(x) + 1, | Shift 1 unit upwards; |

f(x) − 1, |
Shift 1 unit downwards; |

2f(x), |
Stretch vertically by a factor of 2; |

f(x)/2, |
Compress vertically by a factor of 2; |

−f(x), |
Flip vertically across the horizontal axis; |

−2f(x), |
Flip and stretch vertically; |

2f(x) + 1, |
Stretch vertically and then shift upwards; |

f(x + 1), |
Shift 1 unit to the left; |

f(x − 1), |
Shift 1 unit to the right; |

f(2x), |
Compress horizontally by a factor of 2; |

f(x/2), |
Stretch horizontally by a factor of 2; |

f(−x), |
Flip horizontally across the vertical axis; |

f(−2x), |
Flip and compress horizontally; |

f(2x + 1), |
Shift to the left and then compress horizontally; |

2f(x + 1), |
Stretch vertically and shift to the left, in either order. |

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This web page was written between 2010 and 2015 by Toby Bartels, last edited on 2015 August 15. Toby reserves no legal rights to it.

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