What is y=f(2x) transformation?

The transformation y = f(2x) represents a horizontal compression of the original function y = f(x).

• Compression Factor: The function is compressed horizontally by a factor of 1/2. This means that every x-coordinate on the graph of y = f(x) is halved to obtain the corresponding x-coordinate on the graph of y = f(2x).

• Effect on the Graph: If the original graph of f(x) stretches from x = a to x = b, the graph of f(2x) will stretch from x = a/2 to x = b/2. The y-values remain the same. For example, if f(x) has a root at x = r, then f(2x) will have a root at x = r/2.

• Important Notes:

  • If you have a point (x, y) on the graph of y = f(x), then the corresponding point on the graph of y = f(2x) is (x/2, y).
  • Consider the implications on key features of the graph such as its period (if it's a periodic function), its domain, and range. The domain will change, but the range remains the same.
  • For trigonometric functions, this transformation affects the period. For example, if f(x) = sin(x), then f(2x) = sin(2x), which has a period of π (π = 2π/2) instead of 2π.

• Example: If f(x) is a curve that crosses the x-axis at x = 4, then f(2x) will cross the x-axis at x = 2. If f(x) reaches a maximum at x = 6, then f(2x) will reach a maximum at x = 3, and the maximum y-value will be unchanged.