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Sinusoidal Models

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The four parameters

A sinusoidal model has the form y = A sin(Bx − C) + D. Each letter controls one feature. A is the amplitude — half the distance between the highest and lowest values, always taken as |A|. D is the midline, the horizontal center the wave oscillates around. B controls the period (how fast it cycles), and C produces a horizontal phase shift.

Reading the period from B

A basic sine wave repeats every 2π. The coefficient B compresses or stretches that: the period equals 2π/B. A larger B means a shorter period (more cycles packed in). For y = sin(2x), B = 2, so the period is 2π/2 = π — the wave completes a full cycle twice as fast as the parent sine.

Sinusoid parameters
y = A sin(Bx) + D · amplitude = |A| · period = 2π/B · midline y = D
Amplitude is a distance (never negative). The period comes only from B; the amplitude A and midline D do not affect it.
Worked example

For y = 3 sin(2x) + 1, find the amplitude, period, and midline.

  1. 1.The coefficient in front of sine is A = 3, so the amplitude is |3| = 3.
  2. 2.The coefficient of x inside is B = 2, so the period is 2π/B = 2π/2 = π.
  3. 3.The constant added is D = 1, so the midline is y = 1.
  4. 4.The wave therefore oscillates between y = 1 − 3 = −2 and y = 1 + 3 = 4.
Answer: Amplitude 3, period π, midline y = 1. The graph swings between −2 and 4 and completes a cycle every π units.
Checkpoint

In y = 3 sin(2x) + 1, what is the amplitude?

Watch out

Amplitude is half the total vertical swing, not the whole thing. If a wave runs from −2 to 4, the swing is 6 but the amplitude is 6/2 = 3. Divide the peak-to-trough distance by two.

Checkpoint

What is the period of y = sin(2x)?

On the exam

From a graph, read the midline and amplitude off the max and min: midline D = (max + min)/2 and amplitude A = (max − min)/2. Then find the period by measuring one full cycle and solve B = 2π/period.

Answer the 2 checkpoints as you read.

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