Period, Amplitude, Midline and Transformations of Trig Functions

A general sinusoidal function y = A sin(B(x − C)) + D has amplitude |A|, period 2π/|B|, horizontal shift (phase shift) C and midline y = D. In brief: for y = 3 sin(2x − π) + 1, the amplitude is 3, the period is π, the phase shift is π/2 to the right and the midline is y = 1, so the graph oscillates between −2 and 4.

Almost every real periodic quantity can be modelled by a transformed sine or cosine: tides, daylight, temperature, sound and alternating current. The four parameters have simple geometric meanings and can be read directly from a graph. This guide explains each parameter, shows how to graph and how to find a formula from a graph or from data, with a table of effects.

What is the amplitude?

The amplitude is the distance from the midline to a maximum (or to a minimum): amplitude = (max − min)/2. In y = A sin x it is |A|. If A < 0, the graph is also reflected in the x-axis. For y = −2 cos x, the amplitude is 2 and the graph starts at its minimum −2. The amplitude stretches the graph vertically. For tangent, there is no amplitude, because it is unbounded.

How do you find the period?

The period is the length of one full cycle. For y = sin(Bx) or cos(Bx), the period is 2π/|B|. For tangent, the period is π/|B|. Increasing B compresses the graph horizontally, and the frequency B/(2π) is the number of cycles per unit of x.

Examples: y = sin 3x has period 2π/3. y = cos(x/2) has period 4π. y = tan 2x has period π/2. In degrees, y = sin(4x) has a period of 90°.

What do the phase shift and midline do?

In y = A sin(B(x − C)) + D, the number C is the horizontal shift to the right (to the left if negative), and D is the vertical shift of the midline. If the function is given as y = A sin(Bx − E) + D, then the phase shift is E/B. For y = 3 sin(2x − π) + 1: B = 2, E = π, so the phase shift is π/2.

The maximum is D + |A| and the minimum is D − |A|. For the example, the range is [1 − 3, 1 + 3] = [−2, 4].

How do you graph a transformed function step by step?

1. Draw the midline y = D and mark the maximum and minimum lines.
2. Compute the period P = 2π/|B| and divide one period into four equal parts.
3. Find the starting point of a cycle: for sine, at x = C on the midline, going up (if A > 0).
4. Plot the five key points: midline, maximum, midline, minimum, midline (for sine), spaced by P/4.
5. Draw a smooth curve and repeat it in both directions.

Example 1: y = 2 sin(x) + 1

Midline y = 1, amplitude 2, period 2π. Key points: (0, 1), (π/2, 3), (π, 1), (3π/2, −1), (2π, 1). The range is [−1, 3].

Example 2: y = 3 sin(2x − π) + 1

Period π, so the quarter period is π/4. A cycle starts at x = π/2 on the midline. Key points: (π/2, 1), (3π/4, 4), (π, 1), (5π/4, −2), (3π/2, 1). Check the value at x = 3π/4: 3 sin(3π/2 − π) + 1 = 3 sin(π/2) + 1 = 4 ✓.

Example 3: cosine with a negative sign

y = −4 cos(πx/3) − 1. Amplitude 4, period 2π/(π/3) = 6, midline y = −1, range [−5, 3]. At x = 0, y = −4 − 1 = −5, the minimum. At x = 3: −4 cos π − 1 = 4 − 1 = 3, the maximum.

Example 4: tangent

y = tan(2x) has period π/2, vertical asymptotes at x = π/4 + kπ/2, and zeros at x = kπ/2.

How do you write a formula from a description?

1. Compute D = (max + min)/2 and A = (max − min)/2.
2. Compute B = 2π/period.
3. Choose sine or cosine: cosine starts at a maximum, so it is simpler if the data begin at a maximum. Find the shift C.
4. Check the formula with two known points.

Example 5: a Ferris wheel

A Ferris wheel has a lowest point 2 m above the ground and a highest point 42 m above the ground, and one rotation takes 8 minutes. A rider starts at the lowest point. Then D = (42 + 2)/2 = 22, A = 20 and B = 2π/8 = π/4. Starting at the minimum suggests h(t) = −20 cos(πt/4) + 22. Check: h(0) = −20 + 22 = 2 ✓ and h(4) = 20 + 22 = 42 ✓.

Example 6: daylight

In a city, the daylight is at most 15.5 hours and at least 8.5 hours, with the maximum at day 172 and a period of 365 days. D = 12, A = 3.5, B = 2π/365. Using cosine with a maximum at 172: L(d) = 3.5 cos(2π(d − 172)/365) + 12. For d = 172 + 91 ≈ 263, L ≈ 12, since a quarter period brings it to the midline. The daylight pattern is an example of a seasonal cycle.

Table of parameter effects

Where do these ideas come from?

Using sinusoids to describe periodic phenomena goes back to the astronomers’ models of planetary motion with circles, and to the study of waves in the seventeenth and eighteenth centuries. The notions of amplitude, frequency and phase were developed in the nineteenth century with the theory of sound and light. Joseph Fourier’s work showed that general periodic signals are sums of such sinusoids.

Where are they used today?

Sound is described by the amplitude (loudness) and the frequency (pitch): the standard concert pitch A has a frequency of 440 Hz, so B = 2π · 440 and the period is 1/440 s ≈ 2.27 ms. Alternating current in a household is V(t) = 325 sin(2π · 50t) volts for a 230 V (rms) supply at 50 Hz. Tides, seasonal temperature and economic cycles are modelled with similar functions.

Which mistakes are common?

Confusing B with the period

B = 2 means the period is π, not 2. The period is 2π/B.

Reading the phase shift from the unfactored form

In sin(2x − π), the shift is π/2, not π. Factor out B first.

Mixing up the amplitude with the range

The range of y = 3 sin x + 1 is [−2, 4], and the amplitude is 3, not 6.

Forgetting that a negative A reflects the graph

The amplitude is |A|, but the sign of A decides whether the graph starts going up or down.