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How to Graph Trigonometric Functions

How to Graph Trigonometric Functions
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  • 0:04 Trigonometric Functions
  • 1:26 The Period & Midline
  • 2:59 The Phase Shift
  • 3:46 The Amplitude
  • 4:26 Brief Example
  • 4:53 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn how to read a trigonometric function so that you could recognize it on a graph. There isn't much math to it, you just have to know which numbers to look at.

Trigonometric Function

Graphing a trigonometric function is actually pretty easy if you know what numbers to look at. Recall that a trigonometric function ('trig function') is simply a mathematical function of an angle. There are three basic trig functions:

  • the sine function
  • the cosine function
  • the tangent function

There are also three inverse functions:

  • the cosecant function - the reciprocal of the sine function
  • the secant function - the reciprocal of the cosine function
  • the cotangent function - the reciprocal of the tangent function.

To graph these functions, it's helpful to know what their basic graphs look like without any additional numbers changing the graph:


The basic graphs of trigonometric functions
graph trigonometric functions


Now, you can use the properties of trigonometric functions to help you graph any one. This is because all trigonometric functions follow the same rules.

Take the standard trigonometric function:


graph trigonometric functions


  • trig stands for basic trigonometric function, which could be either cosine, sine, or tangent.
  • A is amplitude
  • The phase shift is given by C / B
  • The vertical shift is given by D

Let's take a look at how these values change the basic graphs.

The Period and Midline

The period of your graph is how often the graph repeats itself. This is found by dividing your regular period by the absolute value of B.

For sine and cosine functions, the regular period is 2pi. This means your graph will repeat itself every 2pi, so at 6.28, at 12.56, etc. If your function has a B value, then it will change that period. For example, say you have this function:


graph trigonometric functions


Because you now have a value of 2 for B, your period is no longer 2pi. It now is 2pi / 2 = pi. So now your graph will repeat every pi, so at 3.14, 6.28, 9.42, etc.


graph trigonometric functions


What do you think your period will be if your B value is 3? If you said 2pi / 3, then you are 100% right!

For tangent and cotangent functions, the regular period is pi. So if your B value here is 2, then your new period is pi / 2, or roughly 1.57.

The midline is the line in the middle of the graph. For plain functions without added values, the midline is the y = 0 line. This midline changes if the D value creates a vertical shift.

If D = 1, then your graph will be shifted up by 1. If D = -2, then your graph will be shifted down by 2.

For example, the graph of f(x) = tan(x) - 1 will be your tan(x)' graph shifted down by 1.


graph trigonometric functions


The Phase Shift

The phase shift tells you where your period begins. Basic functions all begin at x = 0. A phase shift moves this value either to the left or right.

A phase shift of 2 means you are shifting your period 2 spaces to the right. You get a phase shift of 2 whenever the ratio of your C to B is 2, so C / B = 2 (for example, if C = 2 and B = 1).

Remember that in the standard function, C is being subtracted, so if you see a minus, C is actually positive. But if you see a plus, then C is actually negative. A negative phase shift moves the graph to the left.

So, the function f(x) = sin(x - 2) moves the graph of sin(x) two spaces to the right.


graph trigonometric functions


Now your function begins at x = 2.

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