notesonly.in

One notebook for every subject — open it anywhere.

Log in

Amplitude Period and Phase Shift: Explained with Examples

MathematicsTry the interactive 3D model →

Every sine or cosine wave can be written as y = A sin(Bx + C) + D, and the four numbers A, B, C and D control exactly four visible features of the graph: its height, its speed of repetition, its left-right position and its up-down position.

This matters because real waves are never the plain y = sin x. A sound wave has a loudness and a pitch; an alternating current has a voltage and a frequency. Amplitude, period and phase shift are the vocabulary engineers use to describe all of them.

Key Concepts

The amplitude is |A|, the height of each peak above the middle line — always a positive number. The period is 2π/|B|, the horizontal length of one complete wave: a larger B squeezes the wave so it repeats faster. The phase shift is −C/B, the horizontal slide of the graph (positive means right, negative means left). The number D is the vertical shift: the whole wave is lifted so it oscillates around the midline y = D instead of y = 0.

Formulas

For y = A sin(Bx + C) + D, with x in radians:

  • Amplitude = |A|; the midline is y = D, peaks at y = D + |A|, valleys at y = D − |A|.
  • Period = 2π/|B|.
  • Phase shift = −C/B (right if positive, left if negative).
  • Vertical shift = D.

The phase shift is easiest to see by factoring: Bx + C = B(x + C/B), so the ordinary sine wave slides by −C/B. The same four rules work for y = A cos(Bx + C) + D.

Worked Example

Find the amplitude, period, phase shift and vertical shift of y = 3 sin(2x − π/2) + 1.

  1. Read the numbers: A = 3, B = 2, C = −π/2, D = 1.
  2. Amplitude = |3| = 3.
  3. Period = 2π/2 = π.
  4. Phase shift = −(−π/2)/2 = π/4, so the graph slides π/4 to the right.
  5. Vertical shift = 1: the midline is y = 1, with peaks at y = 4 and valleys at y = −2.

Check: at x = π/4 the angle is 2(π/4) − π/2 = 0, so y = 3 sin 0 + 1 = 1 — the graph crosses its midline rising, as a right-shifted sine wave should. At x = π/2 the angle is π/2, so y = 3(1) + 1 = 4, the peak. The numbers match the picture.

Common Misconceptions

  • "The phase shift is just −C." Wrong — it is the full fraction −C/B. Here the shift is π/4, not π/2. Forgetting to divide by B is the most common error.
  • "The amplitude is A." Wrong when A is negative: amplitude is always |A|.
  • "A bigger B makes a longer wave." Exactly the opposite: since the period is 2π/|B|, doubling B halves the period and the wave repeats twice as fast.

Real-World Applications

Amplitude, period and phase shift describe every real wave. In audio engineering, amplitude is loudness, period (or its reciprocal, frequency) is pitch, and phase shift aligns speakers so waves add instead of cancelling. In electrical engineering, the 50 Hz mains supply is a sine wave with period 1/50 of a second, and three-phase power uses three waves shifted by 120 degrees. Tides, seasons and daylight hours are all modelled with this same formula.

In short: in y = A sin(Bx + C) + D, |A| sets the height, 2π/|B| sets the repeat length, −C/B slides the wave sideways and D lifts it up or down.

Related blogs