Topic 3.1 Periodic Phenomena: identify a periodic relationship, and describe its period, amplitude and key features from a graph, table or context.

What this topic is asking

The College Board (Topic 3.1) wants you to recognize a periodic relationship and describe its key features. A relationship is periodic if its output values repeat over successive equal-length intervals of the input. You must identify the period (how long one cycle takes), the amplitude (half the gap between the maximum and minimum), and the midline (the horizontal line halfway between them), reading these from a graph, a table or a worded context.

What makes a relationship periodic

Periodic behavior appears whenever something cycles: the height of a point on a rotating wheel, the tide, daylight hours through the year, or a vibrating string. The defining test is repetition over equal input intervals, not the particular shape of one cycle.

Period, amplitude and midline

Amplitude and midline come from the output values ($M$ and $m$); the period comes from the input axis. Keeping these two readings separate, one vertical and one horizontal, prevents most confusion in this topic.

Reading features from a graph

The period is the horizontal distance between two matching points on consecutive cycles, most easily measured peak to peak or trough to trough. The maximum and minimum heights give the amplitude and midline directly.

Concavity within a cycle

Within one cycle a smooth periodic function changes concavity. From a trough up to the midline it is increasing and concave up; from the midline up to a peak it is increasing and concave down; coming down from the peak it is decreasing and concave down until the next midline crossing, then decreasing and concave up into the trough. This is the change-in-tandem language of Topic 1.1 applied to a repeating curve, and it explains the smooth S-shape of one sinusoidal cycle.

A point worth stating once is that amplitude is always reported as a positive distance. Even if a function dips far below zero, the amplitude is half the total vertical swing, never a negative number, and the midline need not be the $x$-axis. Reading the midline as the average of the extremes, rather than assuming it is $y = 0$, is the habit that makes the transformation work in Topics 3.5 and 3.6 straightforward.

Try this

Q1. A periodic function has maximum $20$ and minimum $4$. What is its amplitude and midline? [1 point]

• Cue. Amplitude $\frac{20 - 4}{2} = 8$; midline $y = \frac{20 + 4}{2} = 12$.

Q2. Consecutive troughs of a periodic graph occur at $x = 3$ and $x = 10$. What is the period? [1 point]

• Cue. $10 - 3 = 7$, so the period is $7$.