Changing the Period

Sine and cosine graphs repeat because the point tracing them goes around a circle again and again. The plain function y = sin(x) finishes one full lap every 2pi units of x. Multiplying the input by b, as in y = sin(bx), changes how quickly x reaches a full lap's worth of angle. If b is 3, the input bx hits 2pi when x is only 2pi/3, so the wave finishes a cycle three times faster. If b is 1/2, the input needs x to go all the way to 4pi before bx reaches 2pi, so the wave stretches out and takes twice as long to repeat. That is why a bigger b means a faster, more compressed wave, not a wider one: b scales the x-axis itself, squeezing more or less of the original pattern into the same horizontal space.

The formula period = 2pi / |b| comes straight from this reasoning: set bx equal to 2pi and solve for x. The absolute value matters because a negative b, like in sin(-2x), flips the graph left to right across the y-axis but does not change how often it repeats. sin(-2x) and sin(2x) have the same period, 2pi/2 = π, just mirrored.

This sits next to amplitude changes and phase shifts in the same family of transformations. Together, a, b, h, and k in y = a sin(b(x - h)) + k control height, speed, horizontal slide, and vertical slide, and period changes are the piece that stretches or compresses time itself along the curve.