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Modeling Periodic Phenomena

  • Grade 11
  • Mathematics
  • F-TF.B.5
  • Functions
  • Free lesson

Aligned to F-TF.B.5 — Common Core State Standards for Mathematics.

What this lesson teaches

Repeating patterns can be modeled with a sine or cosine function of the form y = A·sin(Bx) + D. Here A is the amplitude (half the height), D is the midline, and the period equals 2π/B.

Worked examples

Example 1

A Ferris wheel rises to 25 m and dips to 5 m, one turn every 20 seconds. The midline is (25 + 5)/2 = 15, the amplitude is (25 − 5)/2 = 10, and B = 2π/20 = π/10. A model is h(t) = 10·sin(π/10 · t) + 15.

Example 2

Daylight in a northern city runs from 8 hours in December to 16 hours in June, a full cycle each year. Midline = (16 + 8)/2 = 12 hours, amplitude = (16 − 8)/2 = 4 hours, period = 365 days so B = 2π/365. With day 0 at the spring equinox, a model is L(t) = 4·sin(2πt/365) + 12.

Example 3

Common mistake: confusing B with the period. In y = 3·sin(4t), the period is NOT 4 — it is 2π/4 = π/2, so the wave repeats four times as often. B counts cycles, the period measures one cycle's length, and they are reciprocally related by period = 2π/B. Writing B = 20 for a 20-second Ferris wheel makes it spin 20 times faster than intended.

Practice questions

  1. A wave has amplitude 4 and midline 0: write y = A·sin(x) with these values.
  2. Find the period of y = sin(2x).
  3. A tide varies from 2 ft to 8 ft: find its amplitude and midline.

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En español

Modelar Fenómenos Periódicos

Los patrones que se repiten se pueden modelar con una función seno o coseno de la forma y = A·sen(Bx) + D. Aquí A es la amplitud (la mitad de la altura), D es la línea media, y el período es igual a 2π/B.

Ejemplo 1: Una rueda de la fortuna sube hasta 25 m y baja hasta 5 m, una vuelta cada 20 segundos. La línea media es (25 + 5)/2 = 15, la amplitud es (25 − 5)/2 = 10, y B = 2π/20 = π/10. Un modelo es h(t) = 10·sen(π/10 · t) + 15.

Ejemplo 2: La luz del día en una ciudad del norte va de 8 horas en diciembre a 16 horas en junio, con un ciclo completo por año. Línea media = (16 + 8)/2 = 12 horas, amplitud = (16 − 8)/2 = 4 horas, periodo = 365 días, así que B = 2π/365. Con el día 0 en el equinoccio de primavera, un modelo es L(t) = 4·sen(2πt/365) + 12.

Ejemplo 3: Error común: confundir B con el periodo. En y = 3·sen(4t), el periodo NO es 4 — es 2π/4 = π/2, así que la onda se repite cuatro veces más seguido. B cuenta ciclos, el periodo mide la duración de un ciclo, y se relacionan de forma recíproca: periodo = 2π/B. Escribir B = 20 para una rueda de la fortuna de 20 segundos la hace girar 20 veces más rápido de lo previsto.

More Functions lessons

Logarithms and Exponential EquationsF-LE.A.4Graphing Key Features of FunctionsF-IF.C.7Equivalent Forms of a FunctionF-IF.C.8Building Functions from RelationshipsF-BF.A.1Transformations of GraphsF-BF.B.3Radian Measure of AnglesF-TF.A.1

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