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; |
−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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