3.5 Sinusoidal Function Context and Data Modeling

Building a sinusoidal model

To model periodic data, use f(x) = A·sin(B(x - C)) + D or cosine form. A is the amplitude (half the max-min range), D is the midline (average of max and min), B = 2π/period controls how fast it cycles, and C is the phase shift (horizontal shift). Identify max/min values or key points from context to solve for each parameter.

Reading context clues

Real-world scenarios like tides, temperature, or Ferris wheels give clues: amplitude = (max - min)/2, midline = (max + min)/2, and the period is the time to complete one full cycle. Choose sine if the data starts at the midline going up, or cosine if it starts at a max or min — this simplifies finding the phase shift C.

Interpreting and validating the model

Once built, use the model to predict values, find when a quantity reaches a certain level, or check reasonableness against given data points. Restricting the domain may be necessary since real contexts (like a Ferris wheel ride) only apply over a limited time. Always verify units and that the model matches known data at key points.

Key terms

Amplitude (A) in a sinusoidal model

Half the difference between the maximum and minimum values: A = (max - min)/2.

Midline (D) in a sinusoidal model

The average of the maximum and minimum values: D = (max + min)/2; it's the horizontal axis of symmetry.

Formula for B (angular frequency)

B = 2π / period, where period is the length of one complete cycle.

Phase shift (C)

The horizontal shift of the sine or cosine graph, chosen to align the model with the data's starting behavior.

Choosing sine vs. cosine for a model

Use sine if data starts at the midline increasing; use cosine if data starts at a maximum (or negative cosine for a minimum).

Try an AP-style question

A Ferris wheel has diameter 50 m, center 30 m above ground, and takes 10 min per turn. A rider starts at the bottom. Which models height h(t)?

1. h(t) = 30 + 25cos(πt/5)
2. h(t) = 30 − 25cos(πt/5)
3. h(t) = 25 − 30cos(πt/5)
4. h(t) = 30 − 25sin(πt/10)