Periodic Phenomena

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

A periodic function repeats its values at regular intervals. The period is the length of one full cycle; the amplitude is half the vertical range; the midline is the average of the maximum and minimum. These three numbers describe any sinusoidal or tide-like phenomenon. Topic 3.1 sets up the language that the next topics (sin, cos, sinusoidal modeling) will use.

Key quantities

Amplitude

A = \frac{\text{max} - \text{min}}{2}

Midline

y = \frac{\text{max} + \text{min}}{2}

Period

P = \text{length of one full cycle}

Caution: Amplitude is half the vertical range, not the full range. If a tide varies from 2 ft to 8 ft, amplitude is 3 ft, not 6 ft.

Type 1

Identify period and amplitude

Find one full cycle of the graph and measure its horizontal length (period). Take half of (max − min) for amplitude.

Example 1

A periodic graph completes one full cycle from

x = 0

x = 4

. Its max value is

7

and min is

-3

. State period and amplitude. (A) Period 4, amplitude 10 (B) Period 4, amplitude 5 (C) Period 2, amplitude 5 (D) Period 8, amplitude 10

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Type 2

Find midline, max, min

Average max and min to find the midline. Combine with amplitude to recover max and min from any sinusoidal description.

Example 2

A periodic function has max value

11

and min value

3

. What is the midline? (A)

y = 4

(B)

y = 7

(C)

y = 8

(D)

y = 14

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Type 3

Periodic real-world context

Translate the verbal description of a periodic system into period, amplitude, and midline.

Example 3

A tide varies between

2

ft and

8

ft, completing one full cycle every

12

hours. State the period, amplitude, and midline. (A) Period 12 h, amplitude 6 ft, midline 5 ft (B) Period 12 h, amplitude 3 ft, midline 5 ft (C) Period 6 h, amplitude 3 ft, midline 5 ft (D) Period 12 h, amplitude 8 ft, midline 4 ft

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