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The

**Triangle Wave Function**is a periodic function used in signal processing. It is an even function, which means it is symmetrical around the y-axis.

This function is sometimes also called the *continuous sawtooth function*, however, the actual “sawtooth” has a slightly different shape:

## Constructing a Triangle Wave Function

Graphing the triangle wave function is a **challenge**, because there isn’t a single definition, in terms of a graphable function, for the triangle wave.

One way to define the function is in terms of the floor function (⌊ x ⌋ ):

This may be a helpful definition, but with the inclusion of a floor function, it isn’t graphable as-is. An alternate definition is as the absolute value of the sawtooth wave, but that definition also includes the floor function.

Perhaps the **simplest way to approximate the shape **is with the sine function and inverse sine function. For example, the following graph uses a combination of sine and inverse sine to create the triangular waves:

## Creating a Triangle Wave with Piecewise Functions

One **relatively simple way to create a graph** of the triangle wave function is to construct a series of piecewise functions. In other words, each individual “tooth” can be built with one or two functions.

As an example, the following graph of one tooth (actually a sawtooth function) is represented by two different functions: one for the upslope and one for the downslope:

## References

Desmos.com

Gustafson, S. Math 257/316 Assignment 5 Solutions. Retrieved December 20, 2019 from: http://www.math.ubc.ca/~gustaf/M316/homework5sol.pdf

Orloff, J. ES.1803 Topic 22 Notes. Retrieved December 20, 2019 from: http://web.mit.edu/jorloff/www/18.03-esg/notes/topic22.pdf

Weisstein, Eric W. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TriangleWave.html

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