Maths: From Sine Graph to Equation, Step by Step

Sinusoidal functions model countless real-world patterns, from sound waves to seasonal temperatures, and their graphs are governed by three parameters: a, b, and c in f(x) = a sin(bx) + c. Each parameter shapes the curve in a distinct way, and reading a graph backwards to recover these values is a foundational skill in Maths AA HL. The amplitude a controls the vertical stretch, found from half the distance between the maximum and minimum values, while c gives the vertical shift, the midpoint between those extremes. The parameter b governs horizontal compression through the period, since b = 2π/T. Crucially, the horizontal gap between a maximum and the next minimum equals half a period, linking graph features directly to b. Transformations such as f(x + π/4) then shift the curve horizontally, changing the phase of the sine function. Understanding how amplitude, period, and phase interact lets you move fluently between a graph, its equation, and any transformed version of it.