Find the least positive input interval over which the whole pattern repeats. Matching height and direction are useful checkpoints, but alone do not prove that an arbitrary pattern repeats. Measure a period in the input variable’s units; the wave drawing is conceptual, not a scaled graph.
What a strong response shows
Match the same state and direction, not just two equal output values.
Measure the horizontal distance between adjacent matching cycles; the period uses input units.
Reduce a later input by an integer number of periods before reading the output.
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What You Need to Know
A periodic relationship repeats a recognizable pattern after a fixed amount of input.
Find the repeat interval first; then mark the high point, low point, midline, and direction of change.
Translate the context into an input such as time, identify one complete repeat, and place the corresponding output values on a graph.
Calling a pattern periodic just because it increases or decreases once.
What input interval brings the output back to the same stage of the pattern?
Learning Objectives
3.1.A — Construct graphs of periodic relationships based on verbal representations. 3.1.B — Describe key characteristics of a periodic function based on a verbal representation.
If the prompt says construct, sketch the cycle from the context; if it says describe, name repeat length and key values.
. Label input and output. . Locate two matching stages. . Use their horizontal separation as the period. . Sketch one cycle and repeat it.
Using the vertical distance between a peak and trough as the period.
Which feature is measured horizontally: amplitude or period?
Key Concepts
Period is the horizontal length of one repeat. Amplitude measures distance from the midline to a peak or trough. The midline is the average level around which the pattern moves.
Horizontal repeat means period; vertical distance from midline means amplitude.
For a tide, mark high tide and the next high tide. Their time difference is the period; halfway between high and low is the midline.
Measuring peak-to-trough as amplitude instead of twice the amplitude.
If peak-to-trough is units, what is the amplitude?
Relevant Representations
A periodic context can be represented by stages in a table and a repeating graph. Use a sinusoidal model only when its smooth wave shape and assumptions fit the context.
| Stage | Context cue | Graph cue |
|---|---|---|
| start | midline, rising | crosses midline upward |
| high | maximum output | peak |
| middle | midline, falling | crosses midline downward |
| low | minimum output | trough |
Keep the same input and output units as you move between context, table, graph, and formula.
Write four stages of one cycle, plot them in order, connect smoothly, and repeat the horizontal pattern.
Connecting periodic data with straight corners when the context changes smoothly.
Which representation makes the repeat interval easiest to see?
Essential Relationships or Formulas
Period, amplitude, and midline are read from the graph before any formula is chosen.
Use horizontal differences for period and vertical averages for midline.
Read the highest and lowest outputs, compute their average for the midline, then compute half their difference for amplitude.
Using the first and last x-values of a displayed window as one period without checking that the endpoints match.
Why does the midline use an average of the maximum and minimum?
Worked Examples
A context can be graphed by anchoring one cycle before extending the pattern.
Start with a meaningful stage such as a peak or trough, then move by one period.
A machine vibration reaches a maximum every seconds. Mark maxima at 0.8-second intervals, place the midline halfway between extremes, and draw a smooth repeat.
Starting a new cycle at a different stage and shifting the graph accidentally.
What evidence would confirm that the sketch has the correct period?
Common Mistakes
Periodic graphs are judged by repeat length, vertical scale, and the order of stages.
Check repeat, height, and stage order before checking appearance.
Compare the proposed graph with the context: locate two matching events, compare extreme values, and verify rising or falling direction between them.
Changing units midway, such as mixing minutes and hours.
What three features should you verify before accepting a periodic sketch?
AP Exam Tips
Use context words to anchor the graph: every, cycle, high, low, average, and time between repeats.
Use the Questionbank set below to test this skill.
Write units on both axes and identify one complete cycle before describing the whole graph.
State the period and amplitude in context, then support the claim with two visible graph features.
Reporting a number without saying what it measures.
Can your answer name the feature, value, and unit?
Can you transfer the method?
A periodic signal has period minutes and . Explain how to obtain without extending its graph.
Compare your reasoning
, so five complete periods give . State the repeat in input units; do not add to the output.
Work independently, then compare the mathematical evidence in your answer.