Representative curves and transformations

Sine and Cosine Curves

The representative curves of the sine and cosine functions are called sine and cosine curves. They have exactly the same shape, but are shifted relative to one another:

cos(x) = sin(x + π/2)

The cosine curve starts at its maximum (1) at x = 0; the sine curve starts at 0 at x = 0 and then increases. Both oscillate between -1 and 1, with a period of 2π.

Modifying a sinusoid

By modifying the expression of the function, we alter its graph:

Function	Effect on the graph
sin(x) + k	vertical translation by k
sin(x + a)	horizontal translation (phase shift)
A * sin(x), A > 0	amplitude multiplied by A: oscillates between -A and A
sin(w * x), w > 0	period divided by w: new period = 2π/w

Example 1

Let f(x) = 3cos(x) - 1. This function oscillates between 3(-1) – 1 = –4 and 3 × 1 – 1 = 2. Its period remains 2π (since w = 1); its graph is that of the cosine function, vertically stretched and then shifted downwards by 1.

Example 2

Let g(x) = sin(2x). Here w = 2, so the period becomes 2pi/2 = pi: the curve ‘narrows’; it repeats its pattern twice as fast as a standard sine function.

Common pitfall

Do not confuse the effect of A (amplitude, vertical) with that of w (period, horizontal). Multiplying x by w before applying sin compresses or stretches the curve horizontally; multiplying sin(x) by A stretches it vertically, without affecting the period.