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:

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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This web page was written by Toby Bartels, last edited on 2020 September 22. Toby reserves no legal rights to it.

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