Periodic Functions Examples: 42 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Periodic Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

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

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A periodic function returns to the same value every fixed interval called the period.

Common stuck point: The procedure for periodic functions is the easy part; the trap is calling a drifting or growing pattern periodic. Asking "Does the function return to the exact same value after a fixed, repeating interval?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the function return to the exact same value after a fixed, repeating interval?

Worked Examples

Example 1

easy

Verify that

f(x) = \sin(x)

is periodic with period

2\pi

by checking the definition

f(x + p) = f(x)

Answer

f(x) = \sin(x)

has period

2\pi

First step

Recall the identity:

\sin(x + 2\pi) = \sin x \cos 2\pi + \cos x \sin 2\pi

Full solution

2
Substitute known values $\cos 2\pi = 1$ and $\sin 2\pi = 0$ : $\sin(x + 2\pi) = \sin x \cdot 1 + \cos x \cdot 0 = \sin x$ .
3
Since $f(x + 2\pi) = f(x)$ for all $x$ , and $2\pi$ is the smallest such positive number, the period is $p = 2\pi$ .

A function is periodic if it repeats its values at regular intervals. The sine function satisfies

f(x+2\pi)=f(x)

for all real

x

, making

2\pi

its fundamental period.

Example 2

medium

Find the period of

g(x) = \cos(3x)

and sketch one complete cycle.

Example 3

medium

Find the amplitude, period, and midline of

f(x) = 3\sin(2x) + 1

Example 4

medium

Find the period of

f(x) = \sin\!\left(\tfrac{\pi x}{3}\right)

Example 5

medium

Find the maximum value of

f(x) = -3\sin x + 4

and the

x

at which it occurs in

[0, 2\pi)

Example 6

medium

A function repeats every

5

with

f(x) = x

[0, 5)

. Find

f(12)

Example 7

hard

Find the period of

f(x) = \sin(3x) + \cos(4x)

Example 8

hard

Average daily temperature in a city is modeled by

T(d) = 60 + 20\sin\!\left(\tfrac{2\pi(d - 80)}{365}\right)

. Find max

T

and the day it occurs.

Example 9

hard

A pendulum's displacement is

x(t) = 5\cos\!\left(\tfrac{\pi t}{2}\right)

cm. How many complete oscillations in

20

Example 10

challenge

Show that

f(x) = \sin(x) + \sin(\pi x)

is NOT periodic.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

The function

h(x) = \tan(x)

has period

\pi

. What is

\tan\!\left(\frac{5\pi}{4}\right)

given that

\tan\!\left(\frac{\pi}{4}\right) = 1

Example 2

hard

A function satisfies

f(x+4) = f(x)

for all

x

, and is defined on

[0,4)

f(x) = x^2 - 4x + 3

. Find

f(13.5)

Example 3

easy

What is the period of

\sin x

Example 4

easy

What is the period of

\cos x

Example 5

easy

Does

f(x+T)=f(x)

for all

x

define a periodic function?

Example 6

easy

Is the period a horizontal or vertical measurement?

Example 7

easy

What is the amplitude of

f(x)=3\sin x

Example 8

easy

Are all repeating patterns sinusoidal?

Example 9

easy

The graph of

f

repeats every 4 units. What is its period?

Example 10

easy

What is the range of

f(x)=\sin x

Example 11

medium

Find the period of

f(x)=\sin(2x)

Example 12

medium

Find the period of

f(x)=\cos\!\left(\tfrac{1}{3}x\right)

Example 13

medium

State the amplitude and period of

f(x)=2\sin(3x)

Example 14

medium

A wave repeats every 6 seconds. What is its frequency in cycles per second?

Example 15

medium

Find the period of

f(x)=\tan x

Example 16

medium

f(x+5)=f(x)

for all

x

and

f(2)=7

, find

f(12)

Example 17

medium

The graph reaches a max of 5 and a min of

-1

. What is the amplitude?

Example 18

medium

Find the midline of

f(x)=2\sin x + 4

Example 19

medium

Find the period of

f(x)=3\sin(\pi x)

Example 20

challenge

Find the smallest positive period of

f(x)=\sin x + \sin(2x)

Example 21

challenge

Write a sine function with amplitude 4, period

\pi

, and midline

y=1

Example 22

challenge

Show that

f(x)=\sin(x)

has

\pi

as NOT a period but

2\pi

as a period.

Example 23

easy

Find the amplitude of

f(x) = 4\cos(x)

Example 24

easy

Compute

\sin(2\pi + \tfrac{\pi}{6})

using periodicity.

Example 25

easy

Find the period of

f(x) = \tan(2x)

Example 26

medium

A function has period

4

and

f(1) = 7

. Find

f(9)

Example 27

medium

Find the midline of

f(x) = -2\cos x + 5

Example 28

medium

Find the period of

f(x) = \cos(\pi x)

Example 29

medium

A Ferris wheel of radius

20

m completes one revolution in

40

s. Riders board at the bottom,

1

m above ground. Model height

h(t)

Example 30

medium

Solve

2\sin x = 1

for

x \in [0, 2\pi)

Example 31

hard

A function satisfies

f(x + 6) = f(x)

and on

[0, 6)

f(x) = (x - 3)^2

. Find

f(20)

Example 32

hard

Solve

\cos(2x) = \tfrac{1}{2}

for

x \in [0, 2\pi)

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Related Concepts

Trigonometric Functions Amplitude Frequency

Background Knowledge

These ideas may be useful before you work through the harder examples.

trigonometric functions