# Linear coordinate transformations (§3.5)

A **linear coordinate transformation** of a function
is a composite of that function with one or more non-constant linear functions.
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
with rate of change and initial value both 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.

More generally, instead of a non-constant linear function,
we could use any invertible function with a sufficiently large domain or range.
(A non-constant linear function is always invertible,
and its domain and range always consist of all real numbers.)

## Graphs of transformed functions

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; |

2*f*(*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; |

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

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

*f*(2*x* + 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)); |

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

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