Three dials — amplitude, frequency, phase — describe every vibration.
Pluck a guitar string, push a child on a swing, watch the tide rise and fall — and the same shape keeps coming back. It is the sine curve from 25.4, but now we put it to work. A plain y = sin x has a fixed height of 1, a fixed repeat-length of 2π, and starts dead-on at the origin. Real vibrations are taller or shorter, faster or slower, and rarely start at the perfect moment. So we hang three dials on the wave — A for height, ω for how tightly it packs, and φ for how far it has slid — and write every one of them as y = A·sin(ωx + φ). Turn the three dials and you can match a heartbeat, a sound wave, or the swing of a pendulum. Let us learn what each dial does.
Start with the cleanest change. Leave the inside of the sine alone and just multiply the whole thing by a number: y = A·sin x. Every output of sin x already lives between −1 and 1, so multiplying by A stretches those outputs to live between −A and A (when A > 0). The shape is untouched — same crests, same troughs, same crossings — only the height changes. We call |A| the amplitude: the distance from the calm middle line up to a crest.
In y = A·sin(ωx + φ), the number A out front is the amplitude. The wave swings between −|A| and +|A|; its amplitude is |A|, the distance from the midline to a peak. A is a vertical stretch — it never changes how often the wave repeats.
Amplitude is |A|, not A. If A = −3, the amplitude is still 3 — the minus sign just flips the wave upside down (a crest becomes a trough), it does not give a "negative height."
Drag A and watch the wave grow and shrink against the ghost of y = sin x. The range updates live.
Now reach inside the sine and scale the input: y = sin(ωx). The plain wave finishes one full cycle when its angle has run through 2π. Here the angle is ωx, so one cycle finishes when ωx = 2π, that is when x = 2π/ω. That length is the period — how far along the x-axis the wave travels before it repeats. The bigger ω is, the sooner ωx reaches 2π, so a bigger ω makes a shorter period — the wave packs more cycles into the same stretch.
Read it as a one-line conclusion: period = 2π/ω. If ω = 2, the period is 2π/2 = π — the wave repeats twice as fast as y = sin x. If ω = ½, the period is 2π/(½) = 4π — it crawls, taking twice as long to come around.
The number ω inside the sine is the angular frequency. The period — the length of one full cycle — is T = 2π/ω. A larger ω squeezes more cycles into each unit of x, so the wave looks tighter.
ω is not the period. They are reciprocals (up to the factor 2π): a bigger ω makes a shorter period, because period = 2π/ω. Confusing the two is the classic slip — if ω doubles, the wave gets faster, not slower.
The last dial lives inside too, but as an added constant: y = sin(ωx + φ). The wave normally begins its rising cycle where its angle is 0. Here the angle is ωx + φ, which equals 0 when x = −φ/ω. So the whole wave has slid: the cycle that used to start at x = 0 now starts at x = −φ/ω. The number φ is the phase; the amount −φ/ω is the phase shift.
Sign matters and it surprises people. A positive φ slides the wave left (because −φ/ω is negative), and a negative φ slides it right. For example y = sin(x + π/2) starts its cycle at x = −π/2 — shifted left by π/2 — which is exactly the cosine curve. That is the cleanest way to see cos x = sin(x + π/2), the shift we met in 25.4.
The added constant φ is the phase. The wave's cycle starts at x = −φ/ω. A positive φ shifts the graph left, a negative φ shifts it right — the opposite of what the plus sign first suggests.
Set A, ω, and φ and watch y = A·sin(ωx + φ) (blue) ride over the ghost y = sin x (slate). The readout reports the amplitude, the period 2π/ω, and the phase.
Run it backward. If someone hands you a sinusoidal graph, you can recover all three dials by measuring three things:
Height ⇒ A. Measure from the midline up to a crest. That distance is the amplitude |A|. Period ⇒ ω. Measure how far the wave travels before it repeats — that is the period T — then turn it into ω by ω = 2π/T. Shift ⇒ φ. Find where a rising cycle crosses the midline going up; that x is the phase shift −φ/ω, so φ = −ω·(that x).
A wave rises to +3 and falls to −3, repeats every π units, and its upward midline crossing sits at x = −π/4. Then A = 3; ω = 2π/π = 2; and the start is −φ/ω = −π/4, so φ = −2·(−π/4) = π/2. The equation is y = 3 sin(2x + π/2). Check: at x = −π/4 the angle is 2(−π/4) + π/2 = 0, so y = 0 and rising — exactly as drawn.
A target wave is drawn in green. Dial your A, ω, φ until your blue wave lands on top of it. When all three match, the readout turns green.
Here is the payoff. A mass bouncing on a spring, a pendulum swinging through a small angle, the air pressure of a pure musical note, the height of the tide — all of them, when nothing drains their energy, trace a sine curve in time. We call this simple harmonic motion (SHM), and its template is exactly our wave with x replaced by time t:
Displacement y = A·sin(ωt + φ), with t in seconds. Then amplitude = |A| (the maximum swing, in metres), period T = 2π/ω (seconds per cycle), and frequency f = 1/T = ω/2π (cycles per second, in hertz). Same three dials — now wearing real units.
A spring obeys y = 0.05 sin(4πt) metres. Read off the dials: amplitude A = 0.05 m (it swings 5 cm each way); ω = 4π, so the period is T = 2π/(4π) = 0.5 s; and the frequency is f = 1/0.5 = 2 Hz — two full bounces every second. The phase φ = 0 means it starts at rest, moving up.
The same template describes a tuning fork at 440 Hz (ω = 2π·440), a tide cresting every 12.4 hours, or a pendulum clock ticking once a second. Learn the three dials once and you can model them all. In Stage 26 · Sequences, we will see periodic patterns return in a different costume — as terms that repeat — but that is a story for the next stage.
Three dials reshape the sine curve, each in its own way. Keep this table and the SHM template at your fingertips.
| Dial | In y = A·sin(ωx + φ) | What it does to the graph |
|---|---|---|
| Amplitude A | multiplies the whole sine | vertical stretch to height |A|; range [−|A|, |A|]; A < 0 flips it |
| Frequency ω | scales the input x | sets the period = 2π/ω; bigger ω ⇒ shorter period (tighter) |
| Phase φ | adds inside the sine | slides sideways; cycle starts at x = −φ/ω; φ > 0 shifts left |
SHM template: y = A·sin(ωt + φ) · amplitude |A| · period T = 2π/ω · frequency f = ω/2π.
State the amplitude and the range of y = 4 sin x.
The amplitude is |4| = 4. The wave swings between −4 and 4, so the range is [−4, 4].
Find the period of y = sin(3x) and of y = sin(½x).
Period = 2π/ω. For ω = 3: 2π/3. For ω = ½: 2π/(½) = 4π. The bigger ω gives the shorter period.
By how much, and in which direction, does the graph of y = sin(x − π/3) shift compared with y = sin x?
Here φ = −π/3 and ω = 1, so the cycle starts at x = −φ/ω = π/3. A negative φ shifts the wave right by π/3.
A wave reaches a maximum of 2 and a minimum of −2, repeats every 4π units, and crosses the midline going up at x = 0. Write its equation y = A sin(ωx + φ).
Amplitude A = 2. Period 4π ⇒ ω = 2π/4π = ½. The upward crossing at x = 0 means −φ/ω = 0, so φ = 0. Equation: y = 2 sin(½x).
A spring's displacement is y = 0.1 sin(πt) metres, t in seconds. Give its amplitude, period, and frequency in real units.
Amplitude 0.1 m (10 cm each way). ω = π, so period T = 2π/π = 2 s. Frequency f = 1/T = 0.5 Hz — one full bounce every two seconds.
True or false: in y = A sin(ωx + φ), doubling ω doubles the period. Explain.
False. Period = 2π/ω, so doubling ω halves the period — the wave gets faster, not slower. ω and the period are inversely related.
Six questions to lock it in. Tap the answer you think is right.
This lesson covers the sinusoidal model y = A·sin(ωx + φ) and simple harmonic motion, aligned with CCSS HSF-TF.B.5 (model periodic phenomena with specified amplitude, frequency, and midline) and HSF-IF.C.7e (graph trigonometric functions, showing period, midline, and amplitude). It builds directly on graphing sine and cosine (Stage 25.4) and on transformations of functions (Stage 20). Encourage learners to drive the interactive dials and read the amplitude, period = 2π/ω, and phase aloud — the most common error is treating ω as the period rather than its reciprocal partner.