What Are Periodic Functions?

$\text{Periodic Functions}$

Here are some important definitions you'll need to know about periodic functions. $\\$ Periodic function: A function with a graph that repeats identically from left to right. It can be represented by $f(x)=f(x+p)$. $\\$ Period: The horizontal distance required for a complete cycle of the graph.$\\$ Frequency: $\frac{1}{\text{period}}$$\\$ Phase shift: A horizontal shift of a periodic function. $\\$ Vertical shift: A shift up or down of a periodic function.$\\$ Amplitude: One half of the distance between the minimum and maximum $y$ values. This is always positive. $\\$

Now, we will discuss an example of a periodic function. $f(x)=\sin(x)$. Look at the graph below.

This function is periodic because it repeats identically. We can see that the graph on the interval $\left(-\frac{3\pi}{2}, \frac{\pi}{2}\right)$ is identical in the interval $\left(\frac{\pi}{2}, \frac{5\pi}{2}\right)$. This means that $f(x)=f(x+p)$. We can see that below. The graph of $\color{Red}\sin{x}$ is in red, the graph of $\color{Blue}\sin(x+2\pi)$ is in blue and the graph of $\color{purple}\sin(x+4\pi)$ in in purple. The blue and purple graphs both of a shift of $p$ and are still periodic.

The period of this graph $(\sin{x})$ is $2\pi$ because that is the distance it takes for the curve to complete one cycle. $\\$ The frequency of this graph is $\frac{1}{2\pi}$.$\\$ The amplitude is $1$.

If you click on the continue button, we will look at a phase shift and a vertical shift of a periodic function $y=\cos(x)$.

The graph below shows a phase shift right $2$ units. The green dashed graph is $\color{green}y=\cos(x)$.

The black graph is a phase shift of the function $y=\cos(x)$. It is the function $\color{black}y=\cos{(x-2)}$. It still has a period of $2\pi$, a frequency of $\frac{1}{2\pi}$, and an amplitude of $1$. Now, we'll look at a vertical shift. The green dashed graph is $\color{green}y=\cos(x)$.

The black graph above is the function $y=\cos(x)+3$. It still has a period of $2\pi,$ a frequency of $\frac{1}{2\pi}$, and an amplitude of $1$.

Now let's do a practice problem.

Now you know some more about periodic functions! Click the practice button to get more practice with these functions!

Periodic Functions

When we have dealt with trigonometric functions, we were dealing with periodic functions, so we have seen these before!

A periodic function is a function that repeats itself over a regular interval. The sine and cosine functions are two very common periodic functions.

The period is the interval over which a periodic function repeats. The fundamental period is the smallest possible period of a particular function.

Common math questions may ask you to find the period, graph the periodic function given the period, or write an equation based on the graph of a periodic function. We've done these types of problems before with sine and cosine functions!

Intro to Periodic Functions

by Dina Yagodich

In this video by Dina Yagodich, you will learn about what periodic functions are and major characteristics.

Summary

A periodic function is a function whose graph repeats itself identically from left to right. Mathematically, this looks like:

$$f(x) = f(x+p) \text{ for all x }$$

The sine and cosine functions are examples of periodic functions.

The period of a function is the horizontal distance required for a complete cycle. On a standard function, the period goes from $0$ to $2\pi$ in radians, or $0^{\circ}$ to $360^{\circ}$. This won't be the case for all periodic functions, but they will be fairly standard for basic trigonometric (sine or cosine) functions.

The frequency of a function is the reciprocal of the period. Mathematically, that looks like:

$$\text{frequency }= \frac{1}{\text{period}}$$

A phase shift of a function is the horizontal shift of a periodic function. We can also think of this as a horizontal translation.

The amplitude of a function is half the distance between the maximum and the minimum values of a periodic function, or the distance to the midline. We can also think of this as the radius of a periodic function. Amplitudes will always be positive.

The vertical shift of a function is the vertical shift of a periodic function along the $y$-axis. We can also think of this as a vertical translation.