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Modeling Periodic
Phenomena

Learn to choose and build trigonometric functions that capture real-world cycles: tides, temperatures, Ferris wheels, and more.

Precalculus

What Does Periodic Mean?

A function is periodic if its values repeat at regular intervals. Formally, \( f(x + P) = f(x) \) for all \( x \), where \( P \) is the period: the smallest positive number after which the pattern repeats.

Period P 0

Anatomy of a Sinusoidal Curve

midline
\(A\)
 max Period P max min
Amplitude \(A\)
Half the distance from max to min: \( A = \frac{\text{max} - \text{min}}{2} \)
Period \(P\)
The length of one complete cycle. Related to \(B\) by \( P = \frac{2\pi}{|B|} \)
Midline \(D\)
The horizontal center: \( D = \frac{\text{max} + \text{min}}{2} \)

Why Trigonometric Functions?

Many real-world quantities oscillate: they rise and fall in a smooth, repeating pattern. Sine and cosine are the simplest functions that produce this behavior, making them the natural choice for modeling.

1
Smooth and continuous: no sharp corners or jumps, just like real-world cycles.
2
Adjustable parameters: amplitude, period, phase shift, and midline let you fine-tune the model.
3
Predictive power: once built, the function forecasts future values beyond your data.

The General Sinusoidal Model

\(A\)
Amplitude: vertical stretch. \(|A|\) = half the range.
\(B\)
Frequency factor: \(P = \frac{2\pi}{|B|}\). Larger \(|B|\) = shorter period.
\(C\)
Phase shift: horizontal translation. Positive \(C\) shifts right.
\(D\)
Vertical shift: the midline of the curve.

Visualizing Each Parameter

\(D\)
\(A\)
\(C\)
\(D\)
\( P = \frac{2\pi}{B} \)
\(x\)
\(y\)

Sine vs Cosine: Which to Choose?

\(\sin\)
\(\cos\)
 0 1
\(\sin\) \(\cos\)
Starts at midline (going up) maximum
Best when data begins at the midline value a peak or trough
Relationship \(\cos(x) = \sin\!\left(x + \frac{\pi}{2}\right)\)

How to Build the 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 the period \(P\): the time (or distance) for one full cycle. Then \( B = \frac{2\pi}{P} \).
4
Find the phase shift \(C\): determine where the cycle starts relative to \(x = 0\). Choose \(\sin\) or \(\cos\) to minimize \(C\).
5
Write the equation: \( y = A\sin\!\bigl(B(x - C)\bigr) + D \) or \( y = A\cos\!\bigl(B(x - C)\bigr) + D \).

Example 1: Daily Temperature

A city's temperature ranges from a low of 50°F at 4 AM to a high of 90°F at 4 PM. The cycle repeats every 24 hours. Model \(T(t)\) where \(t\) is hours after midnight.

70°
 4AM
50°
 4PM
90°
 0 6 12 18 24 hours after midnight

Example 2: Ferris Wheel

A Ferris wheel has a diameter of 40 m, its center is 25 m above the ground, and it completes one revolution every 3 minutes. You board at the bottom. Model your height \(h(t)\) in meters.

Example 3: Ocean Tides

High tide of 10 ft occurs at 2 AM. Low tide of 2 ft occurs at 8 AM. The cycle repeats every 12 hours. Write \(W(t)\) for water depth at hour \(t\) after midnight.

6 ft 2AM: 10ft 8AM: 2ft 0 6 12 18 24

Frequency vs Period

These two terms describe the same idea from opposite perspectives. Period is how long one cycle takes; frequency is how many cycles fit in one unit.

Period \(P\) Frequency \(f\)
Definition Time for one cycle Cycles per unit time
Units hours, seconds, etc. cycles/hour, Hz, etc.
Relationship \( f = \frac{1}{P} \) and \( P = \frac{1}{f} \)
Link to \(B\) \( P = \frac{2\pi}{|B|} \) \( f = \frac{|B|}{2\pi} \)

Negative \(A\) and Reflections

When \(A\) is negative, the curve is reflected vertically across the midline. A negative-cosine starts at the minimum instead of the maximum. This is often the simplest way to model data that begins at a low point.

\( A > 0 \)
\( A < 0 \)
 midline

Finding the Phase Shift

The phase shift \(C\) aligns the model with the data. The strategy depends on whether you choose sine or cosine:

\(\cos\) approach
Set \(C\) equal to the \(x\)-value where the maximum occurs. (With \(-A\): where the minimum occurs.)
\(\sin\) approach
Set \(C\) equal to the \(x\)-value where the curve crosses the midline going up.

Common Mistakes

✕ Wrong \(B\) formula
Using \(B = P\) instead of \(B = \frac{2\pi}{P}\). Always divide \(2\pi\) by the period.
✕ Amplitude = range
Using max \(-\) min as \(A\). Remember: \(A\) is half the distance: \(\frac{\text{max}-\text{min}}{2}\).
✕ Forgetting the midline
Omitting \(D\). If the data oscillates around 70, you need \(+70\) in the equation.
✕ Phase shift sign
Writing \(\sin(B(x + C))\) when you mean shift right. The form is \(\sin(B(x - C))\): subtract \(C\).
✕ Degrees vs radians
Using \(B = \frac{360}{P}\) in radian mode. In radians, always use \(B = \frac{2\pi}{P}\).

Parameter Summary

Parameter Formula What It Controls
\(A\) \(\frac{\text{max} - \text{min}}{2}\) Height of oscillation
\(B\) \(\frac{2\pi}{P}\) Speed of repetition
\(C\) Horizontal shift Where the cycle starts
\(D\) \(\frac{\text{max} + \text{min}}{2}\) Center line of oscillation
\(P\) \(\frac{2\pi}{|B|}\) Length of one full cycle

Question 1 of 5

A pendulum swings to a max height of 14 cm and a min height of 2 cm. What is the amplitude of the sinusoidal model?

Question 2 of 5

A phenomenon repeats every 8 seconds. What is the value of \(B\) in the sinusoidal model?

Question 3 of 5

A Ferris wheel has a max height of 50 m and a min height of 2 m. It completes one revolution every 4 minutes. You board at the bottom. Which model is correct?

Question 4 of 5

The function \( y = 5\sin\!\left(3x - \frac{\pi}{2}\right) + 7 \) models a periodic phenomenon. What is the midline and period?

Question 5 of 5

High tide of 11 ft occurs at 6 AM and low tide of 3 ft occurs at 12 PM. The tidal cycle is 12 hours. Which function models the water depth \(W(t)\), where \(t\) is hours after midnight?