3.1 Periodic Phenomena

A relationship is periodic when its outputs run through the same pattern again and again, and every repeat takes up the same amount of input. Think of the height of a pedal on a bike moving at a steady speed: up, down, up, down, with the same timing every turn.

The period is the smallest positive number k such that f(x + k) = f(x) for every x in the domain. In words: shift the input by one period and you get exactly the same output.

One complete repetition of the pattern is called a cycle. A cycle can start anywhere. Max to the next max, min to the next min, or any point to the next matching point all span exactly one period.

From a graph, measure the horizontal distance between two consecutive maximum points, or two consecutive minimum points. Using consecutive peaks avoids the most common mistake, which is measuring from a peak to a valley (that's often only half a period).

From a table, look for where the outputs start to repeat over equal-length steps. If the outputs go 3, 5, 8, 5, 3, 5, 8, 5, 3 at inputs 0 through 8, the pattern 3, 5, 8, 5 repeats every 4 units, so the period is 4.

The period must be the smallest repeat length. A function that repeats every 4 units also repeats every 8 and every 12, but its period is 4.

A periodic function can have all the features you studied in unit 1: intervals where it increases or decreases, intervals where it is concave up or concave down, zeros, maximums and minimums, and different rates of change. The difference is that whatever happens in one period happens in every period.

So if f increases on 1 < x < 3 and has period 10, it also increases on 11 < x < 13, on 21 < x < 23, and on −9 < x < −7. The average rate of change over [1, 3] equals the average rate of change over [11, 13].

This is also how you evaluate far-off inputs: subtract (or add) whole periods until the input lands in a cycle you know. With period k, f(x + nk) = f(x) for every integer n.

To graph a periodic situation, sketch one cycle carefully from the description, then copy it to the left and right.

For one cycle, ask: where does it start, when does it reach its highest and lowest values, how long does each part take, and does it change quickly or slowly in each part? For example, a pendulum clock's pendulum moves fastest at the bottom of its swing and slows near each end, so its horizontal position changes quickly in the middle of each swing and slowly near the turnarounds.