Periodic Phenomena

A Ferris wheel car's height traces the same curve again and again because the motion that produces it, steady rotation, never changes character. Every time the wheel turns through a full revolution, the car returns to the exact same position, so the height as a function of angle or time must retrace its steps exactly. That repeating chunk is called a cycle, and the angle or time it takes to complete one is the period. Anything driven by constant rotation, a pedal on a bike, a point on a record, the tip of a clock hand, behaves this way.

To build the whole graph you only need one cycle. Find the pattern over one period, from start to the moment it first repeats, then copy that piece end to end as many times as you like, left and right. The key features to nail down on that single cycle are where it starts, where it peaks, where it crosses the midline, and where it bottoms out. Once those checkpoints are placed, the rest of the graph is just that shape marching on forever in both directions.

Speed changes the period without touching the shape's height. A wheel spinning twice as fast completes each revolution in half the time, so the graph's cycle compresses horizontally, repeating twice as often, while the amplitude and midline, which depend only on the wheel's size and center height, stay exactly the same. This separation of how tall from how fast is the idea that the next lessons turn into precise formulas, amplitude, period, and midline each get their own knob once sine and cosine functions take over as the formal model.