Amplitude, Period, and Phase Shift
Simple Explanation
A transformed sine wave, y = A sin(B(xβC)) + D, has amplitude |A| (how tall the wave is), period 2Ο/B (how often it repeats), phase shift C (how far it is shifted horizontally), and vertical shift D.
Why Do We Need It?
Real periodic phenomena rarely match the basic y=sinx exactly β this general form describes any stretched, compressed, or shifted wave.
See It
Two overlapping wave curves: the basic sine curve, and a taller, horizontally shifted version of it
Formula
The General Sine Transformation
y = A sin(B(xβC)) + D
A general transformed sine wave, with amplitude |A|, period 2Ο/B, phase shift C, and vertical shift D.
- A
- β controls the amplitude (height) of the wave
- B
- β controls the period, via period = 2Ο/B
- C
- β the horizontal (phase) shift
- D
- β the vertical shift
When to use it: Whenever a sine (or cosine) graph has been stretched, compressed, or shifted from its basic form.
Worked Example
Identify amplitude, period, and phase shift
Find the amplitude, period, and phase shift of y = 3sin(2(xβΟ/4)).
Why Does This Work?
Multiplying sin(x) by A stretches its output (height) by a factor of A; replacing x with B(xβC) compresses the input by a factor of B (shrinking the period by that same factor) and shifts the whole graph right by C, since the wave now reaches its "starting point" C units later.
Real-Life Example
Modeling seasonal temperature variation
Average monthly temperature rises and falls once per year, peaking in summer rather than at the start of the calendar year.
A phase-shifted sine curve models this real cycle β the amplitude captures the temperature swing's size, the period is fixed at 12 months, and the phase shift aligns the peak with the actual warmest month.
Practice
Find the amplitude of y = 5sin(x).
MediumCommon mistake
Forgetting to divide 2Ο by B to find the period β using B itself as the period instead.
Quick Review
- y = A sin(B(xβC)) + D.
- Amplitude = |A|; Period = 2Ο/B; Phase shift = C; Vertical shift = D.
- Larger B compresses the graph (shorter period); larger A stretches it taller.