Practice Periodic Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A function that repeats its values at regular intervals: f(x+T)=f(x)f(x + T) = f(x) for all xx, where TT is the smallest positive period.

The same pattern over and over. Like a heartbeat or the seasons.

Showing a random 20 of 50 problems.

Example 1

challenge
Find the smallest positive period of f(x)=sinโกx+sinโก(2x)f(x)=\sin x + \sin(2x).

Example 2

hard
A function satisfies f(x+4)=f(x)f(x+4) = f(x) for all xx, and is defined on [0,4)[0,4) by f(x)=x2โˆ’4x+3f(x) = x^2 - 4x + 3. Find f(13.5)f(13.5).

Example 3

medium
Find the midline of f(x)=2sinโกx+4f(x)=2\sin x + 4.

Example 4

easy
Are all repeating patterns sinusoidal?

Example 5

easy
Does f(x+T)=f(x)f(x+T)=f(x) for all xx define a periodic function?

Example 6

medium
A function has period 44 and f(1)=7f(1) = 7. Find f(9)f(9).

Example 7

medium
A function repeats every 55 with f(x)=xf(x) = x on [0,5)[0, 5). Find f(12)f(12).

Example 8

medium
Find the period of f(x)=sinโกโ€‰โฃ(ฯ€x3)f(x) = \sin\!\left(\tfrac{\pi x}{3}\right).

Example 9

challenge
Show that f(x)=sinโก(x)f(x)=\sin(x) has ฯ€\pi as NOT a period but 2ฯ€2\pi as a period.

Example 10

medium
Find the amplitude, period, and midline of f(x)=3sinโก(2x)+1f(x) = 3\sin(2x) + 1.

Example 11

hard
Average daily temperature in a city is modeled by T(d)=60+20sinโกโ€‰โฃ(2ฯ€(dโˆ’80)365)T(d) = 60 + 20\sin\!\left(\tfrac{2\pi(d - 80)}{365}\right). Find max TT and the day it occurs.

Example 12

medium
Find the period of f(x)=sinโก(2x)f(x)=\sin(2x).

Example 13

hard
A pendulum's displacement is x(t)=5cosโกโ€‰โฃ(ฯ€t2)x(t) = 5\cos\!\left(\tfrac{\pi t}{2}\right) cm. How many complete oscillations in 2020 s?

Example 14

medium
Find the period of g(x)=cosโก(3x)g(x) = \cos(3x) and sketch one complete cycle.

Example 15

challenge
Write a sine function with amplitude 4, period ฯ€\pi, and midline y=1y=1.

Example 16

medium
Find the maximum value of f(x)=โˆ’3sinโกx+4f(x) = -3\sin x + 4 and the xx at which it occurs in [0,2ฯ€)[0, 2\pi).

Example 17

medium
Find the period of f(x)=cosโกโ€‰โฃ(13x)f(x)=\cos\!\left(\tfrac{1}{3}x\right).

Example 18

easy
Find the amplitude of f(x)=4cosโก(x)f(x) = 4\cos(x).

Example 19

medium
Solve 2sinโกx=12\sin x = 1 for xโˆˆ[0,2ฯ€)x \in [0, 2\pi).

Example 20

medium
Find the period of f(x)=cosโก(ฯ€x)f(x) = \cos(\pi x).
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