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IBMaths: AI HLDPRevision Notes2. Functions2.3 Modelling with Functions2.3.5 Sinusoidal Models

Sinusoidal Models (DP IB Maths: AI HL)

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Dan

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Dan

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Sinusoidal Models

What are the parameters of a sinusoidal model?

  • A sinusoidal model is of the form
    • space f open parentheses x close parentheses equals a sin left parenthesis b open parentheses x minus c close parentheses right parenthesis plus d
    • space f open parentheses x close parentheses equals a cos open parentheses b open parentheses x minus c close parentheses close parentheses plus 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 fraction numerator 360 degree over denominator b end fraction (in degrees) or fraction numerator 2 pi over denominator b end fraction (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

What can be modelled as a sinusoidal model?

  • 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

What are possible limitations of a sinusoidal model?

  • 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

Examiner Tip

  • 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 equals a sin open parentheses b open parentheses x minus c close parentheses close parentheses plus d, horizontal stretches happen before horizontal translations 

Worked example

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

 space D open parentheses t close parentheses equals 3 sin open parentheses pi over 12 open parentheses t minus 2 close parentheses close parentheses plus 12 comma blank 0 less or equal than t less than 24

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

a)
Write down the depth of the water at midnight.

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

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

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

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

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

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.