Sine and cosine graphs: amplitude, period and shift

TL;DR:

• In y = a sin(bx − c) + d, |a| is the amplitude, 2π/b is the period, c/b is the shift and d moves the midline.

• Sketch one period from five key points: start, quarter, half, three quarters and end.

• Cosine is sine shifted by a quarter period.

A sound wave, a spinning wheel seen from the side, and the tide at a harbour can all be modelled by a sine wave. A small equation tells you how tall the wave is, how long it takes to repeat and where it starts. Learn to read those three things, and sketching becomes a quick routine.

The basic wave

The graph of y = sin x starts at 0, rises to 1 at x = π/2, returns to 0 at π, falls to −1 at 3π/2 and is back to 0 at 2π. Then it repeats. The distance for one full cycle is the period, 2π. The height above the midline is the amplitude, 1.

The cosine graph has the same shape but starts at its maximum: cos 0 = 1. Shifting the sine wave left by π/2 gives cosine, because sin(x + π/2) = cos x.

What each number in the equation does

• a stretches vertically. The amplitude is |a|. A negative a flips the wave upside down.

• b squeezes horizontally. The period is 2π / b. A bigger b means more cycles in the same space.

• c slides the wave sideways. The phase shift is c / b to the right when the equation is sin(bx − c).

• d lifts the whole wave. The midline is y = d.

Worked example: y = 3 sin(2x)

1. Amplitude: |a| = 3, so the graph goes between −3 and 3.

2. Period: 2π / 2 = π, so one full cycle fits in a stretch of length π.

3. Phase shift: there is none, so the wave starts at the origin, rising.

4. Divide one period into four equal steps of π/4: key x values are 0, π/4, π/2, 3π/4, π.

5. Evaluate: y = 3 sin(0) = 0, 3 sin(π/2) = 3, 3 sin(π) = 0, 3 sin(3π/2) = −3, 3 sin(2π) = 0.

Plot those five points and join them with a smooth wave: up to 3, down through 0 to −3, and back up to 0. Then repeat the pattern on both sides. Check one more value from the equation: at x = π/8, 2x = π/4, so y = 3 × 0.707 = 2.12, which lies between the points (0, 0) and (π/4, 3), as it should.

Adding a shift

Now take y = 3 sin(2x − π/3). Write it as 3 sin(2(x − π/6)), so the shift is (π/3) / 2 = π/6 to the right. Check that the wave starts rising at x = π/6: then 2x − π/3 = 2(π/6) − π/3 = 0, and y = 0. A mistake many people make is to say the shift is π/3. You must divide by b first.

With a vertical shift, y = 3 sin(2x) + 1 has the same shape with midline y = 1. It goes between 1 − 3 = −2 and 1 + 3 = 4. Check at x = π/4: 3 × 1 + 1 = 4, the maximum.

Try one yourself: y = −2 cos(x/2) + 1

Read the numbers. Amplitude: |−2| = 2. Period: 2π / (1/2) = 4π. Midline: y = 1, so the range is 1 − 2 = −1 to 1 + 2 = 3. The minus sign flips the cosine, so the graph starts at a minimum rather than a maximum. Check: at x = 0, y = −2(1) + 1 = −1. At x = 2π, y = −2 cos π + 1 = −2(−1) + 1 = 3, the maximum, half a period later. Quarter-period points at x = π and x = 3π give −2 cos(π/2) + 1 = 1, on the midline.

Frequency and the real world

The reciprocal of the period is the number of cycles per unit of x. If the horizontal axis is time in seconds and the period is 0.5 s, the frequency is 1 / 0.5 = 2 cycles per second. For y = 3 sin(2x) with x in seconds, the period is π ≈ 3.14 s and the frequency is 1 / π ≈ 0.32 cycles per second.

How to check your answer

• Range. The highest and lowest points should be d ± |a|.

• Period test. Evaluate at x and at x + period. The y values should match.

• Starting point. For sine, the wave crosses the midline going up at the start of a cycle.

• Count the cycles. For b = 2, you should see two full waves between 0 and 2π.

Common mistakes

• Using b as the period instead of 2π / b.

• Forgetting to factor b out before reading the shift.

• Reading the shift direction backwards: x − c shifts right, x + c shifts left.

• Drawing the wave in degrees when the axis is in radians.

• Calling the full height the amplitude. The amplitude is half the distance from top to bottom.

Where you will use it next

Radians are the natural axis units, as explained in radians: why 180 degrees equals π. The idea of a pattern that repeats links to arithmetic and geometric sequences, and the slope of the wave at any point is the subject of what a derivative measures.

Frequently asked questions

How do I tell a sine graph from a cosine graph?

Look at x = 0. Sine passes through the midline, cosine is at a peak or trough. Either can describe the same wave with the right shift.

Can the period be negative?

No. A negative b flips the wave horizontally, but the period is 2π / |b|, always positive.