Unit 5 · Topic 3.1
Periodic Phenomena
Recognize repeating relationships, determine the smallest interval that produces a full cycle, and use one cycle to describe or construct the entire graph.
Learning Goals
- Decide whether a verbal description, table, or graph represents periodic behavior.
- Distinguish a period from a half-cycle or from isolated equal outputs.
- Construct an entire graph from one complete cycle.
- Describe extrema, range, direction, and rates of change over repeated cycles.
- Estimate a period from contextual data and state the model's limitations.
1. What Makes a Relationship Periodic?
A function is periodic when its complete output pattern repeats after a fixed positive input interval. If the smallest such interval is \(P\), then
for every applicable input \(x\). The number \(P\) is the period, and one repetition is a cycle.
2. Identify Periodicity in Context
| Context | Input | Output | Reason |
|---|---|---|---|
| Observation wheel | Elapsed time | Rider height | Each revolution returns to the same position |
| Automatic beacon | Elapsed time | Light intensity | The programmed sequence restarts |
| Seasonal temperature | Day of year | Average temperature | The yearly pattern is approximately recurring |
| Cooling cup | Elapsed time | Temperature | No cycle; it approaches room temperature |
Words such as oscillates, rotates, and repeats every can signal periodicity, but a response must identify both quantities and the repeat interval.
3. Find the Smallest Positive Repeat
If \(f(x+12)=f(x)\), then shifts of 24 and 36 also reproduce the outputs. Those are repeats, but 12 is the fundamental period if no smaller positive shift works.
A maximum followed by a minimum usually spans half a cycle. Measure from one feature to the next matching feature moving in the same direction.
4. Read a Periodic Graph
A horizontal translation by one period places the graph on itself. Check that maxima, minima, crossings, direction, and shape all align.
5. Build the Whole Graph from One Cycle
Suppose one complete cycle is given on \([2,8]\). Its width is \(P=6\). Copy the same shape to neighboring intervals by translating every point:
- Locate one complete pattern.
- Compute its width.
- Translate by integer multiples of that width.
- Preserve the order, height, direction, and shape of all features.
6. Determine a Period from a Table
A rider's height \(h(t)\), in meters, is sampled every two seconds.
| \(t\) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \(h(t)\) | 2 | 8 | 14 | 8 | 2 | 8 | 14 |
The block \(2,8,14,8\) starts again at \(t=8\), so the data support a period of 8 seconds:
A 4-second shift fails because \(h(0)=2\ne14=h(4)\).
7. Describe Extrema, Range, and Center
The rider's maximum is 14 meters, minimum is 2 meters, and observed range is \([2,14]\). Its balanced center and distance to either extreme are
These features repeat in every cycle, but a periodic function need not be smooth or sinusoidal.
8. Repeat Increasing and Decreasing Intervals
Height increases from \(t=0\) to \(t=4\) and decreases from \(t=4\) to \(t=8\). With period 8,
Turning points, concavity, and other characteristics in one period recur at corresponding locations in every period.
9. Rates of Change Also Repeat
The rider's average rate from 0 to 2 seconds is \((8-2)/(2-0)=3\) meters per second. From 8 to 10 seconds it is also 3. This does not mean the rate is constant throughout a cycle.
10. Locate an Input within a Cycle
Inputs separated by whole periods occupy the same location in the cycle. If \(P=8\), then
Processes can share a period and range but reach their maxima at different times, so their starting positions within the cycle differ.
11. Periodic Does Not Mean Sinusoidal
A four-second beacon can be described on one cycle by
It ramps up, remains bright, ramps down, and remains dark. Its corners and flat sections make it periodic but not sinusoidal.
12. Estimate a Period from Noisy Data
| Hour | 0 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 |
|---|---|---|---|---|---|---|---|---|---|
| Sensor | 10.1 | 14.8 | 10.3 | 5.2 | 9.9 | 15.1 | 10.0 | 4.9 | 10.2 |
Highs near hours 3 and 15 and lows near hours 9 and 21 support an estimated period of about 12 hours.
13. Construct a Graph from Words
A pump has a six-minute cycle: flow rises from 0 to 12 liters per minute during minute 1, remains at 12 through minute 3, falls to 0 during minute 4, and remains off through minute 6.
- Draw the rise on \([0,1]\).
- Draw the high plateau on \([1,3]\).
- Draw the fall on \([3,4]\).
- Draw the zero plateau on \([4,6]\).
- Translate the shape by \(6k\).
The period is 6 minutes and range is \([0,12]\). Smoothing the corners into a sine curve would contradict the description.
14. AP Reasoning Workflow
- Name quantities and units.
- Compare successive equal-length intervals.
- Test the smallest candidate period against several features.
- Describe one cycle: extrema, range, direction, and rates.
- Extend by horizontal translations and interpret.
15. Common Errors
- Calling maximum-to-minimum distance the period.
- Using two equal outputs as the only evidence.
- Reporting \(2P\) without checking for a smaller repeat.
- Changing feature order or direction when copying a cycle.
- Assuming every periodic relationship is sinusoidal.
- Claiming exact permanent periodicity from noisy data.
- Giving a period without input units.
Key Takeaways
- Identify repeating behavior, period, cycles, maxima, minima, and phase in contextual data.
- Core relationship: \(f(t+P)=f(t)\)
- Error check: Do not confuse half a cycle with the full period.
- Consecutive maxima occur at \(t=5\) and \(t=17\). State the candidate period and additional evidence needed.
- A function has period 6 and \(f(2)=9\). Find \(f(20)\).
- Use the rider table to state its period, range, maximum times, and increasing intervals.
- Explain why the beacon is periodic but not sinusoidal.