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, for a model of the form $y = a \sin (b (x - c)) + d$ , horizontal stretches happen before horizontal translations

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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Sinusoidal Models (DP IB Maths: AI HL)