Linear coordinate transformations

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

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

• A coordinate transformation outside the function acts vertically, while a coordinate 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:

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.

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