Periodic phenomena

A function f is periodic when there is a fixed length k with $f(x + k) = f(x)$ for every x in the domain: slide the graph k units and it lands exactly on itself. The smallest such k is the period. Once you know one full cycle, you know the function everywhere; that is the entire power of periodicity.

Be strict about the definition. A graph that wiggles but whose bumps grow, shrink, or drift is not periodic, and repeating values are not enough: the whole local pattern must recur. A function can hit the value 3 many times without any two stretches of graph matching.

Repeating behavior gets summarized by three measurements, and each has a precise home on the graph:

1. Period: the horizontal length of one full cycle. Measure between matching features: peak to next peak, trough to next trough, or any point to the next point where the pattern truly restarts. Peak to the NEXT TROUGH is only half a cycle.
2. Midline: the horizontal center line the oscillation straddles, at the average of the maximum and minimum values: $k = (\max + \min)/2$.
3. Amplitude: the vertical distance from the midline up to a peak (equivalently, down to a trough): $(\max - \min)/2$. It is half the total swing, never the full peak-to-trough distance, and never the peak's height above zero unless the midline happens to be zero.

Example: water depth at a pier oscillates between 2 m and 8 m. Midline: (8 + 2)/2 = 5 m. Amplitude: (8 − 2)/2 = 3 m. If high tides come 12 hours apart, the period is 12 hours; the high-to-low gap is only 6.

Periodicity is a prediction machine. If the period is k, then the function's value at any time t equals its value at $t + k$, $t + 2k$, at $t$ plus any whole number of periods. To find a value far away, subtract whole periods until you land inside a cycle you know.

A ferris wheel completes a revolution every 8 minutes and a rider is at the top at t = 3. When are they next at the top? At t = 11, 19, 27, ..., every 8 minutes, full stop. The classic error is answering with half the period ("they come back around at t = 7") by confusing the top-to-bottom trip with the full loop. The second classic is counting the number of cycles wrong over a long interval: from t = 3 to t = 43 is 40 minutes, which is exactly 5 revolutions, so the rider is at the top again, not somewhere else.

Frequency counts cycles per unit of input; the period measures input per cycle. They are reciprocals: a wheel turning once every 4 seconds has frequency 1/4 revolution per second; a sound wave with frequency 440 cycles per second has period 1/440 seconds. Doubling the frequency halves the period.

Keep the units attached and the two cannot swap on you: period carries input units (seconds, hours, meters), frequency carries "per" units (per second, per hour). When Unit 3 later writes sinusoids as $\sin(bx)$, the b will be neither the period nor the frequency but a rate tied to both; the habit of asking "is this number an amount of input, or a count per input?" starts here.