Phase Shift

A sinusoid is built from a parent wave, sin(x) or cos(x), by stretching it vertically, stretching it horizontally, sliding it sideways, and sliding it up or down. Phase shift is the sideways slide. It moves every point on the graph left or right by the same amount, so the shape, the height, and the width of the wave stay exactly the same. Nothing about the wave changes except where it starts along the x-axis — the max, min, and zero crossings all shift together by one fixed distance.

The catch is in how that slide is written. In y = sin(b(x - c)), c is the phase shift, plain and simple: the graph of sin(bx) moves c units right. But if the equation comes as sin(bx + k), that k is not the phase shift. You have to factor b out of the whole argument first: bx + k = b(x + k/b). Written that way, the shift is -k/b. Skipping that factoring step is the single most common error with this topic.

This matters because phase shift is one of four transformations working together on the same graph — amplitude and vertical shift change the output values, while period (from b) and phase shift (from c) change how the input is read. Once you can pull c and b apart cleanly, graphing any sinusoid becomes a matter of finding one key point, like a maximum, and sliding the whole wave to match. That skill carries straight into later work rewriting sine and cosine in terms of each other using shifts of π/2.