For example, if *f*(*x*) = *x*^{2} for all *x*
(that is, *f* is the squaring function)
and *g*(*x*) = *x* + 1
(that is, *g* is the linear function
whose rate of change is 1 and whose initial value is also 1),
then both *f* ∘ *g* and *g* ∘ *f*
are linear coordinate transformations of *f*.
In particular, (*f* ∘ *g*)(*x*) =
(*x* + 1)^{2};
this is called a *passive* or *inside* coordinate transformation.
On the other hand,
(*g* ∘ *f*)(*x*) =
*x*^{2} + 1;
this is called
an *active* or *outside* coordinate transformation.

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 (following the order of operations); |

1 − f(x), |
Flip vertically and then shift upwards
(same as −f(x) + 1); |

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

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 (forwards and backwards are the same here); |

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

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

f(1 − x), |
Shift to the left and then flip horizontally
(same as f(−x + 1)); |

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

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This web page was written by Toby Bartels, last edited on 2021 August 30. Toby reserves no legal rights to it.

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