2.3.5 Sinusoidal Models

Sinusoidal Models

A sinusoidal model is of the form
- $f (x) = a \sin (b (x - c)) + d$
- $f (x) = a \cos (b (x - c)) + d$
The a represents the amplitude of the function
- The bigger the value of a the bigger the range of values of the function
The b determines the period of the function
- The bigger the value of b the quicker the function repeats a cycle
- The period is $\frac{360 °}{b}$ (in degrees) or $\frac{2 π}{b}$ (in radians)
The c represents the phase shift
- This is a horizontal translation by c units
The d represents the principal axis
- This is the line that the function fluctuates around

Anything that oscillates (fluctuates periodically)
Examples include:
- D(t) is the depth of water at a shore t hours after midnight
- T(d) is the temperature of a city d days after the 1st January
- H(t) is vertical height above ground of a person t second after entering a Ferris wheel

The amplitude is the same for each cycle
- In real-life this might not be the case
- The function might get closer to the principal axis over time
The period is the same for each cycle
- In real-life this might not be the case
- The time to complete a cycle might change over time

Read and re-read the question carefully, try to get involved in the context of the question!
Sketch a graph of the function being used as the model, use your GDC to help you and focus on the given domain
Remember that if the model is adjusted, horizontal translations happen before horizontal stretches

The water depth, $D$ , in metres, at a port can be modelled by the function

$D (t) = 3 \sin (\frac{π}{12} (t - 2)) + 12, 0 \leq t < 24$

where $t$ is the elapsed time, in hours, since midnight.

Write down the depth of the water at midnight.

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Find the minimum water depth and the number of hours after midnight that this depth occurs.

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Calculate how long the water depth is at least 13.5 metres each day.

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