Linear coordinate transformations

A linear coordinate transformation of a function is a composite of that function with one or more non-constant linear 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) + 3,	Shift 3 units upwards;
f(x) − 3,	Shift 3 units 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) + 3,	Stretch vertically and then shift upwards;
f(x + 3),	Shift 3 units to the left;
f(x − 3),	Shift 3 units 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;
f(−2x),	Flip and compress horizontally;
f(2x + 3),	Shift to the left and then compress horizontally;
2f(x + 3),	Stretch vertically and shift to the left, in either order.

This web page was written between 2010 and 2012 by Toby Bartels, last edited on 2012 April 20. Toby reserves no legal rights to it.