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

Periodic functions

What are periodic functions?

• A function $f\left(x\right)$ is called periodic, with period k, if

  • $f(x+k)=f\left(x\right)$ for all values of $x$

• Examples of periodic functions include

  • $\sin x$ and $\cos x$ with period $2\pi$ (360°)

  • $\tan x$ with period $\pi$ (180°)

  • Linear combinations of periodic functions with the same period

    • For example: $f\left(x\right)=2\sin \left(3x\right)-5\cos \left(3x+2\right)$

What do graphs of periodic functions look like?

• The graph of a periodic function has translational symmetry

  • The graph is unchanged by translations that are integer multiples of $\left(\begin{array}{c}k\\ 0\end{array}\right)$

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

• Periodic graphs can have infinitely many points of intersection with other graphs

  • e.g. the graph of $y=\tan x$ and the horizontal line $y=\sqrt{3}$ intersect

    • at $x=60°,240°,420°,...$ and at $x=-120°,-300°,...$

    • i.e. $x=60°+180k°$ where $k\in \mathrm{\mathbb{Z}}$