Periodic Function: Definition, Formula, Graph & Examples

TL;DR

A periodic function is a function whose values repeat at fixed, regular intervals. The length of one full repeat is the period, written P, and it satisfies f(x+P)=f(x) for every x.

Key terms: period (the length of one full cycle), amplitude (the height from the midline to the peak, a separate measurement), cycle (one complete repeat), midline (the horizontal line halfway between peak and trough).

What Is A Periodic Function?

A periodic function is a function that repeats its output values after a fixed horizontal distance called the period. If you slide the graph sideways by that distance, it lands exactly on top of itself.

The defining condition is short:

f(x+P)=f(x) for all x

The smallest positive value of P that makes this true is the fundamental period (usually just called the period). The sine function repeats every 2π, so sin(x+2π)=sin(x), and its period is 2π.

A function that never settles into a repeat, like f(x)=x or f(x)=x², is non-periodic.

The Period Formula For A Transformed Function

When a function is stretched or squeezed horizontally, its period changes by a predictable factor.

If f(x) has period P, then f(ax+b) has period:

P/|a|

Table 1: Variable key for the transformed-period formula.

The shift b slides the graph left or right but never changes how often it repeats. Only the coefficient a on the variable matters for the period, and it always enters as |a|.

Table 2: Periods of common functions.

Properties of Periodic Functions

A few properties hold for every periodic function and are worth keeping in view before working examples.

• Repetition by the period. If P is a period, so is every integer multiple: f(x+nP)=f(x) for any whole number n. The smallest positive P is the fundamental period.
• Sum and difference. If f and g share the period P, then f+g and f−g are also periodic with period P (or a divisor of it).
• Horizontal scaling changes the period. Replacing x by ax scales the period by 1/|a|.
• Domain and range. A periodic function is defined for all real inputs, and its range is fixed by one cycle — the outputs over a single period are the only outputs it ever takes.
• Shifting does not change the period. Adding a constant inside, f(x+b), slides the graph sideways but leaves the period untouched.

Examples of Periodic Functions

Example 1: Verify that cosine is periodic

Show that f(x)=cos(x) satisfies the periodic condition with P=2π.

Test the condition f(x+P)=f(x):

f(x+2π)=cos(x+2π)=cos(x)

Final answer: cosine is periodic with period 2π.

Example 2: Find the period of sin(3x)

Find the period of f(x)=sin(3x).

Here a=3:

2π/|3| = 2π/3

Final answer: the period is 2π/3.

Example 3: A wrong path first, then the fix

Find the period of f(x)=cos(4x+5).

The correct move uses only a=4:

2π/|4| = π/2

Final answer: the period is π/2.

Example 4: Find the period of tan(2x)

Find the period of f(x)=tan(2x).

Here a=2:

π/|2| = π/2

Final answer: the period is π/2.

Example 5: A negative coefficient

Find the period of f(x)=sin(−3x).

The coefficient is a=−3, so:

2π/|−3| = 2π/3

Final answer: the period is 2π/3.

Example 6: A function that is not periodic

Decide whether f(x)=x+sin(x) is periodic.

A period must be positive, so no positive P works.

Final answer: x+sin(x) is not periodic.

Why Periodic Functions Exist

Periodic functions exist because the world is full of motion that comes back to where it started. The Earth turns, the Moon cycles through its phases, and a heartbeat, a sound wave, a pendulum, and the tide all return to the same state after a fixed interval.

Where the idea earns its keep:

• Sound and music. A pure musical note is a sine wave; its period sets the pitch.
• Tides. Coastal tide tables are built from periodic functions.
• Electricity. The current from a wall socket is alternating current, a sine wave.
• Climate and biology. Daylight hours, body temperature, and migration cycles all follow periodic patterns over a day or a year.

The Mathematicians Behind Periodic Functions

• Joseph Fourier (1768–1830, France) showed that almost any periodic signal can be written as a sum of simple sine and cosine waves.
• Leonhard Euler (1707–1783, Switzerland) tied sine and cosine waves to circular motion.
• Brook Taylor (1685–1731, England) studied the vibrating string, an early periodic-motion problem.

Common Mistakes With Periodic Functions

Mistake 1: Confusing period with amplitude

Don't do this: Do not call the vertical height the period. The correct way: The period is horizontal, how wide one full cycle is.

Mistake 2: Measuring only half a cycle

Don't do this: Do not measure peak-to-trough and call it the period. The correct way: Measure peak-to-peak, a full cycle.

Mistake 3: Reporting a negative amplitude

Don't do this: Do not let the sign of the coefficient become the amplitude. The correct way: Take the absolute value.

Mistake 4: Adding the shift into the period formula

Don't do this: Do not divide by anything other than |a|. The correct way: Only the coefficient on the variable changes the period.

Practice Questions on Periodic Functions

Work through these:

1. Find the period of f(x)=sin(5x).
2. Find the period of f(x)=cos(x/3).
3. Find the period of f(x)=tan(3x).
4. Find the period of f(x)=sin(πx).
5. Is f(x)=x²+cos(x) periodic? Explain in one line.

Answers

1. 2π/|5| = 2π/5.
2. 2π/|1/3| = 6π.
3. π/|3| = π/3.
4. 2π/|π| = 2.
5. No, the rising x² term never repeats.