The sine function \sin(x) traces a smooth, repeating wave. Three numbers control its shape:

• Amplitude A — how tall the wave is. It stretches the curve vertically, so the peaks reach A and the troughs reach -A.
• Frequency f — how often it repeats. A larger f squeezes the wave horizontally, so more cycles fit across the same width.
• Phase \varphi — how far it is shifted sideways. A positive \varphi slides the whole wave to the left.

Play with the wave

Drag a slider (or type a value) and watch the curve respond. The angle x runs along the bottom in degrees. The faint line is the plain \sin(x) for comparison; the bold curve is y = A\,\sin(f x + \varphi).

Test yourself

Pick the graph that matches each description.