Amplitude, period & phase shift, without the rule-memorizing

study guide · graphing trig functions

Every sinusoidal graph problem — in precalculus, physics, or a signals course years later — is the same four questions in a trench coat: how tall, how fast, slid how far sideways, and slid how far up. Those four answers live in four letters:

y = A · sin(B(x − C)) + D

Learn to read this form fluently and you can sketch any sine or cosine curve in under a minute, and — just as important — write the equation of a curve someone hands you.

A — amplitude: how tall

Plain sin x swings between −1 and 1. Multiplying by A stretches that swing: y = 3 sin x runs from −3 to 3. The amplitude is |A|, the distance from the midline to a peak — not from bottom to top. If A is negative the graph flips upside down; the amplitude is still the absolute value. A quick check that catches many errors: amplitude = (max − min) / 2.

B — angular frequency: how fast

B compresses or stretches the graph horizontally. The standard sine wave completes one full cycle in 2π; with the factor B, cycles come B times as fast, so:

period = 2π / B

y = sin 2x repeats every π. y = sin(x/2) takes 4π. Note the inversion — bigger B means a shorter period — which is exactly the relationship between frequency and period everywhere in science. If a problem gives you the period and asks for the equation, run the formula backwards: B = 2π / period.

C — phase shift: slid sideways (the trap)

Written in the form sin(B(x − C)), the graph shifts right by C when C is positive. The minus sign trips nearly everyone: sin(x − π/4) moves the curve to the right, not the left. Why? The curve does whatever plain sine does, but it needs x to be π/4 bigger before the inside reaches the same value — so every feature of the graph happens π/4 later. "Later" on the x-axis means rightward.

D — vertical shift: the midline

Adding D lifts the whole curve so it oscillates around the horizontal line y = D, called the midline. Real-world sinusoids almost always have one: average tide height, resting blood pressure, the 12-hour mark in a daylight model. From a graph, D = (max + min) / 2 — the midline is always halfway between the extremes.

The five-point method

One clean cycle of a sine curve is defined by five landmarks: start on the midline going up, peak at a quarter period, midline again at the half, trough at three quarters, and midline at the full period. To sketch y = A sin(B(x − C)) + D:

1. Draw the midline y = D, and guide lines at D + |A| and D − |A|.
2. Mark the cycle start at x = C and the cycle end at x = C + 2π/B.
3. Split that interval into quarters and place the five landmark points.
4. Connect them with a smooth wave — no straight segments, no sharp corners.

For cosine, the same five landmarks apply with one change: cosine starts at the peak (peak, midline, trough, midline, peak). In fact cosine is just sine shifted left by a quarter period — cos x = sin(x + π/2) — which is why every "sine or cosine?" modeling question has more than one correct answer.

Worked example

y = 2 sin(3(x − π/6)) + 1

• Amplitude |A| = 2 → the curve spans from −1 to 3.
• Period 2π/3 → three full waves fit in the usual 2π window.
• Phase shift π/6 right → the cycle starts at x = π/6.
• Midline y = 1 → landmarks: (π/6, 1), peak (π/6 + π/6, 3) = (π/3, 3), (π/2, 1), trough (2π/3, −1), (5π/6, 1).

Five points, one smooth curve through them, done. Reading the transformations took longer to type than to do.

What about tangent?

Tangent plays by different rules: its natural period is π (not 2π), it has no amplitude — it runs off to ±∞ — and it owns vertical asymptotes wherever cosine is zero, at x = π/2 + kπ. Under y = A tan(B(x − C)) + D, the period becomes π/B and the asymptotes shift along with everything else. When sketching, draw the asymptotes first and let the curve climb between them; treating tangent like a bounded wave is a classic way to lose easy marks.