Linear coordinate transformations (§2.5)

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;
−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).

Go back to the course homepage.

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

The permanent URI of this web page is https://tobybartels.name/MATH-1300/2024FA/transformations/.