Graphs of Sine & Cosine: Amplitude, Period, Phase

You’ll be able to

Read amplitude, period, midline and phase shift from an equation in the form a·sin(b(x − h)) + k
Compute the period as 2π/|b| and explain the reciprocal relationship
Write an equation for a sinusoid from its graph or from described features

Four numbers determine a sinusoid

Every sine or cosine graph is fixed by four quantities. The midline y = k is the horizontal line the curve oscillates about. The amplitude |a| is the distance from midline to peak — always positive. The period is how long one full cycle takes. The phase shift h says how far the cycle has been slid horizontally. Extract those four and you have the graph; read them off a graph and you have the equation.

The standard sinusoid

y = a·sin(b(x − h)) + k · amplitude = |a| · period = 2π/|b| · midline y = k · phase shift h

The equation must be factored as b(x − h) before h can be read. In y = sin(2x − π), the shift is not π; factoring gives sin(2(x − π/2)), so h = π/2.

Why b and the period are reciprocal

The parent sine completes a cycle as its argument runs from 0 to 2π. In sin(bx) the argument is bx, which reaches 2π when x = 2π/b. So a larger b compresses the graph horizontally and shortens the period. This inverse relationship trips people up because it feels backward — b = 4 makes the graph faster, with period π/2 rather than 8π.

Worked example

For y = −3·cos(2(x − π/4)) + 5, state amplitude, period, midline, phase shift, and the maximum and minimum values.

1.Amplitude = |−3| = 3.
2.Period = 2π/2 = π.
3.Midline: y = 5.
4.Phase shift: π/4 to the right (the expression is already factored).
5.Maximum = midline + amplitude = 5 + 3 = 8; minimum = 5 − 3 = 2.
6.The negative sign reflects the cosine, so where the parent cosine peaks the graph now has its minimum: at x = π/4 the value is 2, not 8.

Answer: Amplitude 3, period π, midline y = 5, shifted π/4 right, maximum 8, minimum 2. The reflection changes where the extremes occur, but not their values — those come from the midline and amplitude alone.

Watch out

Amplitude is never negative. For y = −4 sin x the amplitude is 4; the minus sign is a reflection across the midline, not a negative distance. Writing "amplitude = −4" loses a point.

Checkpoint

What is the period of y = 2 sin(πx/3)?

Building the equation from a graph

Work in a fixed order and the ambiguity disappears. Midline: average the maximum and minimum, k = (max + min)/2. Amplitude: half their difference, |a| = (max − min)/2. Period: measure peak to peak, then b = 2π/period. Phase: pick a landmark and match it — for a cosine model, the easiest landmark is a maximum, since the unshifted cosine peaks at its argument zero.

Worked example

A sinusoid has maximum 14 at x = 2, minimum 4 at x = 8, and these are consecutive extremes. Write a cosine equation.

1.Midline: k = (14 + 4)/2 = 9.
2.Amplitude: a = (14 − 4)/2 = 5.
3.Consecutive maximum and minimum are half a period apart: 8 − 2 = 6 is half a period, so the period is 12.
4.b = 2π/12 = π/6.
5.Cosine peaks when its argument is 0, and the maximum is at x = 2, so h = 2.
6.Equation: y = 5·cos((π/6)(x − 2)) + 9.

Answer: y = 5·cos((π/6)(x − 2)) + 9. Using cosine and anchoring to the maximum avoids the extra reasoning a sine model needs, since sine's landmark is a midline crossing on the way up rather than a peak.

Tip

Consecutive max and min are half a period apart, not a full period. Getting this wrong doubles or halves b — the most common error in writing an equation from described features.

Checkpoint

What is the phase shift of y = sin(3x + π)?

Answer the 2 checkpoints as you read.