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AP Exam Kit

UNIT 3 · PERIODIC MODELING

How do you find amplitude, period and the starting point?

Turn a familiar trigonometric function into a contextual model: identify the extremes, find a complete cycle, then use the starting event to check the phase.

AP Exam Kit original guide6 min read
In this guide · 6 sections

A note from your study buddyCheck the starting value and the value half a period later. A familiar equation still needs to fit the situation.

01 / DIRECT ANSWER

Use four quantities to describe one cycle

Amplitude is half the difference between the maximum and minimum outputs; the midline is their average. A period is the time between consecutive matching positions, such as two successive minima. The starting event determines the phase, which you must check by substitution.

For a model written in radians as y = A sin(B(t − C)) + D, the amplitude is |A| and the midline is y = D. For B ≠ 0, the period is T = 2π/|B|. C controls the horizontal shift. Different but equivalent equations can describe the same motion.

From familiar mathematics to AP reasoning

Once you can sketch sine and cosine, practice explaining what each number represents, which units it uses and why the starting point makes sense. Aim for a model you can justify rather than a single preferred equation form.

02 / ORIGINAL WORKED EXAMPLE

An observation wheel starting at its lowest point

An observation wheel has a radius of 12 meters, a center 18 meters above the ground and a rotation time of 30 seconds. Let t be the number of seconds since a cabin left the lowest point, and h(t) its height above the ground in meters. Assume constant speed.

Observation-wheel height over timeHeight starts at 6 meters, reaches 18 meters at 7.5 seconds and 30 meters at 15 seconds, returns to 18 meters at 22.5 seconds and 6 meters at 30 seconds. The midline is 18 meters.6183007.51522.530Height h (meters)Time t (seconds)
The same model as a graph. Key values also appear in the accessible table below.
  1. Find the extremes. The minimum height is 18 − 12 = 6 meters; the maximum is 18 + 12 = 30 meters. The amplitude is 12 meters and the midline is 18 meters.
  2. Translate the period into a coefficient. The motion repeats every 30 seconds, so B = 2π/30 = π/15 radians per second.
  3. Match the minimum at the start. Since cos(0) = 1, a negative cosine term starts the cabin at its lowest height.
h(t)=18−12cos⁡(πt15)h(t)=18-12\cos\left(\frac{\pi t}{15}\right)
Five key times during the first rotation
t (seconds)h(t) (meters)Position
06Lowest point
7.518Rising through the midline
1530Highest point
22.518Falling through the midline
306Back at the lowest point

Checking h(0) = 6, h(15) = 30 and h(30) = 6 confirms that the model fits these events. Here, 0 ≤ t ≤ 30 describes the first rotation. Extending the model to a longer interval requires the constant-speed assumption to remain reasonable.

03 / COMMON MISTAKES

Three distinctions that matter

Calling the maximum-to-minimum time a period

A maximum followed by the next minimum spans only half a period. Compare consecutive maxima or consecutive minima to find a complete cycle.

Using the maximum height as the amplitude

The maximum height includes the height of the center. Amplitude measures the greatest distance from the midline: 12 meters here.

Forgetting radian mode

The coefficient π/15 comes from the radian measure 2π. Keep the same angle units when evaluating predictions, and round only at the end.

04 / YOUR TURN

Use a different model to make a prediction

This existing original question uses different parameters and measures time in minutes. Read the units before substituting.

ORIGINAL MCQ · NO ACCOUNT NEEDED

Using H(t)=10+8sin⁡(πt/6)H(t)=10+8\sin(\pi t/6), what height is predicted at t=4.7t=4.7 minutes, to the nearest tenth?

Choose one answer

Answers and explanations are returned by the practice server after submission. This article does not save guest practice history.

Start with this idea

Set the calculator to radian mode.

05 / KEEP GOING

Move from one calculation to an explanation

The practice link opens Original, Unit 3 and topic 3.7. Before each answer, check the starting point, half a period and the units. Then explain what the model’s prediction means in context.

Open math practice →Review the Unit 3 guide

SOURCES & SCOPE

Sources and scope

Course scope was checked against these primary sources on 2026-10-17. Worked examples, explanations and study advice are by AP Exam Kit, which is not affiliated with or endorsed by College Board.

AP Exam Kit authored this supplemental guide and checked its cited course sources. Independent teacher review has not been completed; this is not official College Board material.