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Periodic Functions (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

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DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

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Periodic Functions

What are periodic functions?

  • A function space f left parenthesis x right parenthesis is called periodic, with period k, if

    • space f left parenthesis x plus k right parenthesis equals f left parenthesis x right parenthesis for all values of x

  • Examples of periodic functions include:

    • sin x & cos x: The period is 2π or 360° 

    • tan x: The period is π or 180°

    • Linear combinations of periodic functions with the same period

      • For example: f open parentheses x close parentheses equals 2 sin invisible function application open parentheses 3 x close parentheses minus 5 cos invisible function application open parentheses 3 x plus 2 close parentheses

What are the symmetries of graphs of periodic functions?

  • The graph of a periodic function has translational symmetry

    • The graph is unchanged by translations that are integer multiples of stretchy left parenthesis table row k row 0 end table stretchy right parenthesis

    • The means that the graph appears to repeat the same section (cycle) infinitely

2-3-3-ib-aa-hl-periodic-functions

Examiner Tips and Tricks

  • There may be several intersections between the graph of a periodic function and another function

    • i.e.  Equations may have several solutions so only answers within a certain range of x-values may be required

      • e.g. Solve  tan space x equals square root of 3  for  0 degree space less or equal than space x less or equal than space 360 degree

      • x equals 60 degree comma space 240 degree

    • Alternatively you may have to write all solutions in a general form

      • e.g.  x equals 60 left parenthesis 3 k plus 1 right parenthesis degree comma space space space k equals 0 comma space plus-or-minus 1 comma space plus-or-minus 2 comma space...

Worked Example

The graph space y equals f left parenthesis x right parenthesis is shown below. Given that space f is periodic, write down the period.

2-3-3-ib-aa--ai-we-image-c
2-3-3-ib-aa-hl-periodic-functions-we-solution

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    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.