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Modeling Periodic
Phenomena

Learn to choose trigonometric functions that model real-world cycles: specify amplitude, frequency, and midline to match any repeating pattern.

Precalculus

What Does Periodic Mean?

A phenomenon is periodic if it repeats the same pattern at regular intervals. Think of ocean tides, the swing of a pendulum, or daily temperature cycles: each follows a predictable rhythm that trigonometric functions can capture precisely.

One Period
\(t\)
\(y\)

Three Key Parameters

Amplitude \(A\)
The height from the midline to a peak (or trough). It measures how far the function swings.
Period \(T\)
The length of one complete cycle. Related to frequency by \( T = \frac{2\pi}{B} \).
Midline \(D\)
The horizontal axis the wave oscillates around: the average of the max and min values.

Anatomy of a Sinusoidal Wave

Midline \(D\)
 Max Min
\(A\)
Period \(T\)

The General Sinusoidal Model

\(A\)
Amplitude: \(\frac{\text{max} - \text{min}}{2}\)
\(B\)
Frequency factor: \(B = \frac{2\pi}{T}\)
\(C\)
Phase shift: horizontal slide left or right
\(D\)
Midline: \(\frac{\text{max} + \text{min}}{2}\)

The same formula works with \(\cos\) instead of \(\sin\). Choose whichever aligns more naturally with your data's starting point.

Period and Frequency Factor \(B\)

\(B = 1\)
\(B = 2\)
\(x\)

Sine vs. Cosine: Which to Use?

\(\sin\) \(\cos\)
Starts at Midline (going up) Maximum
Best when data begins at An average value A peak or trough
Relationship \(\cos(x) = \sin\!\left(x + \frac{\pi}{2}\right)\)
Phase shift Use \(C\) to shift start Use \(C\) to shift start

How to Build a Trig Model

1
Find the midline \(D\): average the maximum and minimum values. \( D = \frac{\text{max} + \text{min}}{2} \).
2
Find the amplitude \(A\): half the distance from max to min. \( A = \frac{\text{max} - \text{min}}{2} \).
3
Find \(B\): identify the period \(T\), then \( B = \frac{2\pi}{T} \).
4
Determine phase shift \(C\): if the data starts at midline rising, no shift for \(\sin\). If it starts at a peak, use \(\cos\) (or shift \(\sin\)).
5
Write the equation: \( y = A\sin\!(B(x - C)) + D \) and verify against the data.

Example: Ferris Wheel

A Ferris wheel has a diameter of 40 m and its center is 25 m above the ground. It completes one revolution every 3 minutes. A rider starts at the bottom. Model the rider's height \(h(t)\) in meters.

25 m 45 m 5 m
\(A = 20\)
 Start Ground (0 m)

Example: Daily Temperature

A city's temperature ranges from a high of \(88^{\circ}\mathrm{F}\) at 3:00 PM to a low of \(64^{\circ}\mathrm{F}\) at 3:00 AM. Model the temperature \(T(t)\) where \(t\) is hours after midnight.

1
Midline: \( D = \frac{88 + 64}{2} = 76^{\circ}\mathrm{F} \)
2
Amplitude: \( A = \frac{88 - 64}{2} = 12^{\circ}\mathrm{F} \)
3
Period: \( T = 24 \) hours, so \( B = \frac{2\pi}{24} = \frac{\pi}{12} \)
4
Phase shift: Peak at \(t = 15\) (3 PM). Using \(\cos\), shift right by 15: \( C = 15 \)
5
Model: \( T(t) = 12\cos\!\left(\frac{\pi}{12}(t - 15)\right) + 76 \)

Example: Ocean Tides

High tide reaches 11.2 ft at 6:00 AM and low tide is 2.8 ft at 12:15 PM. The cycle repeats every 12.5 hours.

7.0 ft 11.2 ft 2.8 ft 6 AM 12:15 6:30 PM

Understanding Phase Shift \(C\)

\(C\)
\(\sin(x)\)
\(\sin(x - C)\)

Reflections: Negative Amplitude

\(\cos(x)\)
\(-\cos(x)\)

Common Mistakes to Avoid

Wrong Amplitude
Using max - min instead of \(\frac{\text{max} - \text{min}}{2}\). The amplitude is half the total range, not the full range.
Forgetting \(B\)
Writing \(\sin(x)\) with period \(2\pi\) when the real period is different. Always compute \(B = \frac{2\pi}{T}\).
Phase vs. Midline
Confusing vertical shift \(D\) (midline) with horizontal shift \(C\) (phase). \(D\) moves the wave up/down; \(C\) moves it left/right.

Parameter Summary

Parameter Formula Effect
\(A\) \(\frac{\text{max} - \text{min}}{2}\) Vertical stretch (height)
\(B\) \(\frac{2\pi}{T}\) Horizontal compression (speed)
\(C\) Horizontal shift value Slides wave left/right
\(D\) \(\frac{\text{max} + \text{min}}{2}\) Vertical shift (midline)

Question 1 of 5

A sound wave oscillates between 3 decibels and 15 decibels. What is the amplitude?

Question 2 of 5

A phenomenon repeats every 8 seconds. What is the value of \(B\) in the model \( y = A\sin(Bx) + D \)?

Question 3 of 5

A Ferris wheel has max height 50 ft, min height 6 ft, and completes one revolution in 4 minutes. A rider starts at the top. Which model is correct?

Question 4 of 5

In the model \( y = 5\sin\!\left(\frac{2\pi}{12}(t - 3)\right) + 10 \), what is the midline?

Question 5 of 5

Daylight hours range from 9.5 to 14.5 with a period of 365 days. The maximum occurs at day 172. Which model fits?