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IBMathsDPApplications & Interpretation (AI)HLRevision Notes2. FunctionsFurther Functions & GraphsSinusoidal Functions & Graphs

Sinusoidal Functions & Graphs (DP IB Applications & Interpretation (AI)): Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Updated on

DP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SL

Sinusoidal functions & graphs

What are the key features of sinusoidal graphs?

  • A sinusoidal graph is of the form

    • space y equals a sin left parenthesis b open parentheses x minus c close parentheses right parenthesis plus d

    • space y equals a cos left parenthesis b open parentheses x minus c close parentheses right parenthesis plus d

  • The y-intercept is at the point where x equals 0

    • left parenthesis 0 comma space minus a sin left parenthesis b c right parenthesis space plus space d right parenthesis for space y equals a sin left parenthesis b open parentheses x minus c close parentheses right parenthesis plus d

    • (0 comma space a cos left parenthesis b c right parenthesis space plus space d right parenthesis for space y equals a cos left parenthesis b open parentheses x minus c close parentheses right parenthesis plus d

  • The period of the graph is the length of the interval of a full cycle

    • This 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 maximum value is y space equals space a space plus space d

    • You can find this by letting sin open parentheses b open parentheses x minus c close parentheses close parentheses equals 1 or cos open parentheses b open parentheses x minus c close parentheses close parentheses equals 1

  • The minimum value isspace y space equals space minus a space plus space d

    • You can find this by letting sin open parentheses b open parentheses x minus c close parentheses close parentheses equals negative 1 or cos open parentheses b open parentheses x minus c close parentheses close parentheses equals negative 1

  • The principal axis is the horizontal line halfway between the maximum and minimum values

    • This is y space equals space d

  • The amplitude is the vertical distance from the principal axis to the maximum value

    • This is the value of a

  • The phase shift is the horizontal distance from its usual position

    • This is the value of c

Graph depicting sine and cosine functions. Amplitude and period are marked. Equations: y=asin(b(x-c))+d in red, y=acos(b(x-c))+d in blue.
Example of sinusoidal graphs and their key features

Examiner Tips and Tricks

Make sure your angle setting is in the correct mode (degrees or radians) at the start of a question involving sinusoidal functions.

Worked Example

a) Sketch the graph space y equals 3 sin left parenthesis 2 open parentheses x degree minus 15 degree close parentheses right parenthesis plus 1 for the values space 0 less or equal than x less or equal than 360.

2-2-3-ib-ai-hl-sinusoidal-graphs-a-we-solution

b) State the equation of the principal axis of the curve.

2-2-3-ib-ai-sl-sinusoidal-graphs-b-we-solution

c) State the period and amplitude.

2-2-3-ib-ai-sl-sinusoidal-graphs-c-we-solution

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    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.

    Lucy Kirkham

    Reviewer: Lucy Kirkham

    Expertise: Head of Content Creation

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.