AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.1 — Periodic Phenomena
Notes — Describing Cyclic Behavior
💡 Learning Objectives (3.1.A)
By the end of this lesson you will be able to:
- Recognize periodic behavior in real-world contexts and in graphs
- Identify the period and the amplitude of a periodic function from a graph
- Locate the midline and identify maxima and minima
- Predict future values of a periodic function based on its period
1. What Makes Something Periodic?
A function f is periodic if its output pattern repeats at regular intervals of the input. Formally: there exists a positive number k such that f(x + k) = f(x) for every x in the domain. The smallest such k is called the period of the function.
Periodic behavior is everywhere once you start looking. Daylight hours over a year, tides, heartbeat voltages, Ferris wheel heights, vibrating strings — all of these repeat the same pattern over and over, with the same time to complete one full cycle.
2. Period, Amplitude, and Midline
💡 Key Features of a Periodic Graph
- Period: the horizontal length of one complete cycle.
- Midline: the horizontal line halfway between the maximum and minimum. Its equation is y = (max + min)/2.
- Amplitude: half the vertical distance between the maximum and minimum. Amplitude = (max − min)/2. It is always positive.
- Maximum / Minimum: the greatest and least output values.
⚠️ Common mistake
Amplitude is NOT the full swing from max to min — it is HALF of that swing. If a wave goes from y = 2 up to y = 10, the amplitude is 4, not 8.
3. Reading a Periodic Graph
To find the period from a graph, pick any feature (a peak, a zero, an upward crossing of the midline) and measure the horizontal distance to the next occurrence of that same feature. Don't measure to a different feature — crest-to-trough gives you half a period, not a full one.
📘 Example — Extracting features
A temperature sensor reads a repeating pattern that peaks at 26 °C, bottoms out at 14 °C, and takes 24 hours to repeat.
- Period: 24 hours
- Midline: y = (26 + 14)/2 = 20 °C
- Amplitude: (26 − 14)/2 = 6 °C
- Maximum value: 26 °C; Minimum value: 14 °C
4. Periodicity and Prediction
Because f(x + k) = f(x) whenever k is a multiple of the period, you can predict far-future values without plotting the whole graph. Take the input, subtract the period as many times as needed, and read off the resulting value.
📘 Example — Future prediction
Suppose a tide height T(t) has period 12 hours and T(2) = 5 ft.
- T(14) = T(2 + 12) = T(2) = 5 ft
- T(50) = T(2 + 48) = T(2 + 4·12) = T(2) = 5 ft
- T(17) = T(5 + 12) = T(5). To find T(5), you'd need more information about the graph between 2 and 12.
5. A Numerical Glance
The table below records the height (m) of a point on a rotating Ferris wheel every 2 seconds. Look for the period: where does the pattern first repeat?
t (s) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 |
|---|---|---|---|---|---|---|---|---|
h (m) | 3 | 8 | 13 | 8 | 3 | 8 | 13 | 8 |
The pattern repeats every 8 seconds: h(0) = h(8) = 3. The period is 8 s, the max is 13 m, the min is 3 m, so amplitude = 5 m and midline h = 8 m.
6. Not Every Repeating Shape Is Periodic
Periodicity requires EXACT repetition with the same spacing. A graph whose peaks get taller each cycle, or whose spacing drifts, is not truly periodic — even if the overall shape looks wavy. Be careful to check both the shape AND the spacing.
7. Summary
- Periodic: f(x + k) = f(x) for some smallest positive k (the period)
- Midline sits halfway between max and min; amplitude is half the swing
- Measure the period between matching features (peak-to-peak, not peak-to-trough)
- Use f(x + nk) = f(x) to make far-future predictions
- Periodicity requires exact repetition in both shape and spacing