Linear coordinate transformations

A linear coordinate transformation of a function is a composite of that function with one or more non-constant linear functions. (More generally, a coordinate transformation is a composite with an invertible function with a sufficiently large domain or range. Notice that a non-constant linear function is always invertible, with a domain and a range both as large as possible.)

For example, if f(x) = x² 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) = x² + 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 (same as −1f(x));
−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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