# Linear transformations

A linear transformation of a function is a composite of that function with one or more linear functions.

Linear transformations are easy to graph. The key principles are these:

• A transformation outside the function acts vertically, while a transformation inside the function acts horizontally;
• Adding and subtract shift the graph, while multiplying and dividing change the scale;
• Anything inside (horizontal) acts backwards.

More concretely, consider these examples:

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 vertically 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.

Go back to the course homepage.
This web page was written in 2010 by Toby Bartels, last edited on 2010 November 28. Toby reserves no legal rights to it.

The permanent URI of this web page is `http://tobybartels.name/MATH-1150/2010FA/transformations/`.