Transforming sine & cosine graphs

Four parameters shape an entire wave.

The idea

For nonzero A and B, |A| is the amplitude, 2π/|B| is the period, h is the horizontal shift, and k is the vertical shift. Cosine follows the same rules. The midline is y = k and the output ranges from k − |A| to k + |A|.

Work through an example

For y = 2 sin(3(x − π/6)) + 1, the amplitude is 2, period is 2π/3, phase shift is π/6 to the right, and midline is y = 1. Its range is [−1, 3].

What to watch for

In y = A sin(Bx − C) + k, the phase shift is C/B, not C. If A = 0, the output is constant and there is no unique fundamental period.