Amplitude and period
What you'll learn
Control the height and speed of any sine wave using A and B in y = A sin(Bx + C) + D.
The basic wave y = sin(x) has a height of 1 and completes one cycle every 2π. Four parameters let you reshape it into any wave you need.
The general form
y = A sin(Bx + C) + D
Each parameter controls exactly one dimension of the wave.
Amplitude and period, side by side
Amplitude = |A| — how tall the wave is (max value above the midline). In y = 2 sin(x), every output is doubled: the wave reaches +2 and −2 instead of +1 and −1.
Period = 2π / |B| — the width of one complete cycle. In y = sin(2x), B = 2 so the period is 2π/2 = π — the wave completes twice as fast, fitting two cycles where one used to be.
Phase shift and vertical shift
| Parameter | Effect | Formula |
|---|---|---|
| A | Amplitude (vertical scale) | stretch/compress by |A|; flip if A < 0 |
| B | Frequency (horizontal scale) | period = 2π / |B| |
| C | Phase shift (horizontal slide) | shift = −C / B |
| D | Vertical shift (move up/down) | midline moves to y = D |
Working it out
For y = 3 sin(2x − π) + 1:
- Amplitude: |3| = 3
- Period: 2π / 2 = π
- Phase shift: −(−π) / 2 = π/2 to the right
- Vertical shift: +1 (midline at y = 1)
Start by reading A and B — they control the wave's size and speed. Add C and D to position it.
Check your understanding
Question 1 of 2
In y = 3 sin(x), what is the amplitude?