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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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.

Showing a random 20 of 50 problems.

Example 1

challenge
Find the smallest positive period of f(x)=sin⁡x+sin⁡(2x).

Example 2

hard
A function satisfies f(x+4)=f(x) for all x, and is defined on [0,4) by f(x)=x2−4x+3. Find f(13.5).

Example 3

medium
Find the midline of f(x)=2sin⁡x+4.

Example 4

easy
Are all repeating patterns sinusoidal?

Example 5

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

Example 6

medium
A function has period 4 and f(1)=7. Find f(9).

Example 7

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

Example 8

medium
Find the period of f(x)=sin⁡ ⁣(πx3).

Example 9

challenge
Show that f(x)=sin⁡(x) has π as NOT a period but 2π as a period.

Example 10

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

Example 11

hard
Average daily temperature in a city is modeled by T(d)=60+20sin⁡ ⁣(2π(d−80)365). Find max T and the day it occurs.

Example 12

medium
Find the period of f(x)=sin⁡(2x).

Example 13

hard
A pendulum's displacement is x(t)=5cos⁡ ⁣(πt2) cm. How many complete oscillations in 20 s?

Example 14

medium
Find the period of g(x)=cos⁡(3x) and sketch one complete cycle.

Example 15

challenge
Write a sine function with amplitude 4, period π, and midline y=1.

Example 16

medium
Find the maximum value of f(x)=−3sin⁡x+4 and the x at which it occurs in [0,2π).

Example 17

medium
Find the period of f(x)=cos⁡ ⁣(13x).

Example 18

easy
Find the amplitude of f(x)=4cos⁡(x).

Example 19

medium
Solve 2sin⁡x=1 for x∈[0,2π).

Example 20

medium
Find the period of f(x)=cos⁡(πx).
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