7.6Modeling Periodic Phenomena
Build sinusoidal models y = A·sin(Bx + C) + D from real-world data. Determine amplitude, period, phase shift, and vertical shift from context, and use the model to make predictions.
Sinusoidal models describe tides, temperature cycles, sound waves, and AC circuits. Building a trig model from data — finding amplitude, period, and phase shift from context — is a high-value applied math skill.
How do you translate real-world periodic data — a maximum, a minimum, and a period — into a precise mathematical model of the form y = A·sin(Bx + C) + D?
A sinusoidal model has the form y = A·sin(Bx + C) + D (or with cosine). Each parameter controls a specific feature of the graph:
Parameter Meanings
|A| = amplitude
Period = 2π/|B|
Phase shift = −C/B
D = vertical shift (midline)
From Data to Model
A = (max − min)/2
D = (max + min)/2
B = 2π/period
C: use a known point to solve
The average monthly temperature in a city has a maximum of 88°F in July (month 7) and a minimum of 28°F in January (month 1). Write a sinusoidal model T(t) where t is the month.
A = (88 − 28)/2 = 30.
D = (88 + 28)/2 = 58.
Period = 12 months → B = 2π/12 = π/6.
Use cosine: maximum occurs at t = 7. T(t) = 30cos(B(t − 7)) + 58.
T(t) = 30cos((π/6)(t − 7)) + 58.
A Ferris wheel has a diameter of 40 m. The center is 25 m above the ground. It completes one revolution every 60 seconds. A rider starts at the bottom. Write h(t) for the rider's height.
A = 40/2 = 20 (radius = amplitude).
D = 25 (center height = midline).
Period = 60 s → B = 2π/60 = π/30.
Rider starts at the bottom (minimum). Use −cosine: h(t) = −20cos(Bt) + 25.
Or equivalently: h(t) = 20sin(Bt − π/2) + 25.
A sinusoidal function has amplitude 4, period 2π/3, phase shift π/6 to the right, and midline y = 1. Write the equation.
A = 4, D = 1.
Period = 2π/3 → B = 2π/(2π/3) = 3.
Phase shift = −C/B = π/6 → C = −π/2.
y = 4sin(3x − π/2) + 1.
The function y = 3sin(2x + π/4) − 1 models a wave. Identify the amplitude, period, phase shift, and midline.
Amplitude: |A| = |3| = 3.
Period: 2π/|B| = 2π/2 = π.
Phase shift: −C/B = −(π/4)/2 = −π/8 (shift left π/8).
Midline: y = D = −1.
Ocean tides at a harbor have a high tide of 12 ft at 2:00 AM and a low tide of 2 ft at 8:00 AM. Write a model h(t) where t is hours after midnight.
A = (12 − 2)/2 = 5. D = (12 + 2)/2 = 7.
Half-period = 8 − 2 = 6 hours → period = 12 hours → B = 2π/12 = π/6.
Use cosine with maximum at t = 2: h(t) = 5cos((π/6)(t − 2)) + 7.
A sound wave has amplitude 0.3 cm and frequency 440 Hz (cycles per second). Write a model y(t) for the displacement.
Hint: Period = 1/frequency. B = 2π/period = 2π · frequency. Assume no phase shift and midline y = 0.
A sinusoidal function passes through a maximum of 10 at x = π/4 and a minimum of −2. Write the equation using cosine.
Hint: Find A and D from max and min. The maximum of cosine occurs at x = 0, so a phase shift of π/4 is needed.
Identify the amplitude, period, phase shift, and midline of y = −2cos(πx/3 + π/6) + 4.
Hint: Read off A = −2 (amplitude = 2), B = π/3, C = π/6, D = 4. Phase shift = −C/B.
A city's average high temperature follows a sinusoidal pattern with a maximum of 95°F in July (month 7) and a minimum of 35°F in January (month 1). Predict the average high in April (month 4).
Hint: Build the model first, then evaluate at t = 4.
A spring oscillates so that its displacement from rest is 8 cm at t = 0 and 0 cm at t = π/4 seconds. The amplitude is 8 cm. Write a model d(t).
Hint: The displacement starts at maximum (8 cm) and decreases, suggesting a cosine model with no phase shift.
Interactive Practice — 5 Questions
For y = 5sin(2x − π/3) + 1, the amplitude is:
For y = 3cos(πx/4) − 2, the period is:
A sinusoidal function has max = 14 and min = 2. The midline is:
For y = A·sin(Bx + C) + D, the phase shift is:
A Ferris wheel with radius 15 m and center 18 m high completes one revolution in 40 seconds. The amplitude of the height model is:
Common Mistakes
Using the maximum value as the amplitude instead of half the range.
Amplitude A = (max − min)/2. The maximum value is D + A, not A itself.
Setting B equal to the period instead of 2π/period.
B = 2π/period. If the period is 12, then B = 2π/12 = π/6.
Confusing phase shift direction: thinking a positive C shifts right.
In y = A·sin(Bx + C) + D, phase shift = −C/B. A positive C shifts the graph LEFT.
Forgetting to account for the midline when reading max/min from a graph.
The maximum of the function is D + A, and the minimum is D − A. Always identify D first.
Math Tips
When a problem gives you a maximum at a specific x-value, cosine is often the cleaner choice — the maximum of cosine occurs at x = 0, so the phase shift equals the x-value of the maximum.
When a problem says a quantity 'starts at the minimum' (like a Ferris wheel rider at the bottom), use −cos(Bx) + D, which starts at its minimum.
The four-step recipe: (1) find A and D from max/min, (2) find B from the period, (3) find C from a known point, (4) write the equation.
Always verify your model by checking that it gives the correct max and min values, and that it passes through at least one given data point.
Frequency and period are reciprocals: f = 1/T and T = 1/f. For sound waves, frequency is given in Hz (cycles per second).