Modeling Periodic Phenomena
Aligned to F-TF.B.5 — Common Core State Standards for Mathematics.
What this lesson teaches
A sinusoid y = A × sin(B(x − C)) + D models repeating data. A is the amplitude (half the peak-to-trough distance), D is the midline, and the period is 2 × pi / B.
Worked examples
Example 1
A Ferris wheel carries riders between 2 m and 20 m every 60 seconds. Midline D = (2 + 20)/2 = 11, amplitude A = (20 − 2)/2 = 9, and period 60 gives B = 2 × pi / 60 = pi/30. So height ≈ 9 × sin((pi/30) × t) + 11.
Example 2
An ocean tide runs from a low of 0.4 m to a high of 3.6 m, with about 12.4 hours between consecutive highs. Midline = (3.6 + 0.4)/2 = 2.0 m, amplitude = (3.6 − 0.4)/2 = 1.6 m, and B = 2π/12.4 which is about 0.507. With t measured from a high tide, h(t) = 1.6·cos(0.507t) + 2.0.
Example 3
Common mistake: choosing sine when the data starts at a peak. Sine starts at the midline and rises; cosine starts at the maximum. Modelling a Ferris wheel that begins at the bottom needs a NEGATIVE cosine, −A·cos(Bt) + D, or a sine with a phase shift. Look at where t = 0 falls on the cycle — bottom, middle or top — before picking the function.
Practice questions
- Find the amplitude and midline for values ranging from 4 to 10.
- What is B if the period is 12?
- Write a sine model with amplitude 5, midline 20, and period 8.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Modelar fenómenos periódicos
Una sinusoide y = A × sin(B(x − C)) + D modela datos que se repiten. A es la amplitud (la mitad de la distancia de cresta a valle), D es la línea media, y el periodo es 2 × pi / B.
Ejemplo 1: Una rueda de la fortuna lleva a los pasajeros entre 2 m y 20 m cada 60 segundos. Línea media D = (2 + 20)/2 = 11, amplitud A = (20 − 2)/2 = 9, y un periodo de 60 da B = 2 × pi / 60 = pi/30. Así, la altura ≈ 9 × sin((pi/30) × t) + 11.
Ejemplo 2: Una marea oceánica va de un mínimo de 0.4 m a un máximo de 3.6 m, con unas 12.4 horas entre pleamares consecutivas. Línea media = (3.6 + 0.4)/2 = 2.0 m, amplitud = (3.6 − 0.4)/2 = 1.6 m, y B = 2π/12.4, aproximadamente 0.507. Con t medido desde una pleamar, h(t) = 1.6·cos(0.507t) + 2.0.
Ejemplo 3: Error común: elegir seno cuando los datos empiezan en un máximo. El seno arranca en la línea media y sube; el coseno arranca en el máximo. Modelar una rueda de la fortuna que empieza abajo necesita un coseno NEGATIVO, −A·cos(Bt) + D, o un seno con desfase. Mira dónde cae t = 0 en el ciclo — abajo, en medio o arriba — antes de elegir la función.
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