Sinusoidal Models

A sinusoidal model is of the form
- $f (x) = a \sin (b x) + d$
- $f (x) = a \cos (b x) + d$
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 period = $\frac{360}{b}$
- The bigger the value of b the quicker the function repeats a cycle
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

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

$D (t) = 3 \sin (15 ° \times t) + 12, 0 \leq t < 24$

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

Write down the depth of the water at midnight.

2-3-5-ib-ai-sl-sinusoidal-models-a-we-solution

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

2-3-5-ib-ai-sl-sinusoidal-models-b-we-solution

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

2-3-5-ib-ai-sl-sinusoidal-models-c-we-solution

Sinusoidal Models (DP IB Maths: AI SL)