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4.6 Graphs of Other Trigonometric FUNctions. How can I sketch the graphs of all of the cool quadratic FUNctions?. Graph of the tangent FUNction. The tangent FUNction is odd and periodic with period π .

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4 6 graphs of other trigonometric functions

4.6 Graphs of Other Trigonometric FUNctions

How can I sketch the graphs of all of the cool quadratic FUNctions?


Graph of the tangent function
Graph of the tangent FUNction

  • The tangent FUNction is odd and periodic with period π.

  • As we saw in Section 2.6, FUNctions that are fractions can have vertical asymptotes where the denominator is zero and the numerator is not.

  • Therefore, since , the graph of will have vertical asymptotes at , where n is an integer.


Let s graph y tan x
Let’s graph y = tan x.

  • The tangent graph is so much easier to work with then the sine graph or the cosine graph.

    • We know the asymptotes.

    • We know the x-intercepts.


Y 2 tan 2x
y = 2 tan (2x)

  • Now, our period will be

  • Additionally, the graph will get larger twice as quickly.

  • The asymptotes will be at

  • The x-intercept will be (0,0)



Graph of a cotangent function
Graph of a Cotangent FUNction

  • Like the tangent FUNction, the cotangent FUNction is

    • odd.

    • periodic.

    • has a period of π.

  • Unlike the tangent FUNction, the cotangent FUNction has

    • asymptotes at period πn.


Y cot x
y = cot x

  • The asymptotes are at ±πn.

  • There is an x-intercept at


Y 2 cot 2x
y = -2 cot (2x)

  • The period is

  • There is an x-intercept at

  • There is an asymptote at


Graphs of the reciprocal functions
Graphs of the Reciprocal FUNctions

  • Just a reminder

    • the sine and cosecant FUNctions are reciprocal FUNctions

    • the cosine and secant FUNctions are reciprocal FUNctions

  • So….

    • where the sine FUNction is zero, the cosecant FUNction has a vertical asymptote

    • where the cosine FUNction is zero, the secant FUNction has a vertical asymptote


  • And…

    • where the sine FUNction has a relative minimum, the cosecant FUNction has a relative maximum

    • where the sine FUNction has a relative maximum, the cosecant FUNction has a relative minimum

    • the same is true for the cosine and secant FUNctions

  • Let’s graph y = csc x



Now you try your own
Now, you try your own….

  • Just graph the FUNction as if it were a sine or cosine FUNction, then make the changes we have already made.


Damped trigonometric graphs just for fun
Damped Trigonometric Graphs (Just for Fun!)

  • Some FUNctions, when multiplied by a sine or cosine FUNction, become damping factors.

  • We use the properties of both FUNctions to graph the new FUNction.

  • For more fun on damping FUNctions, please read p 339 in your textbook.



Writing about math
Writing About Math check out page 340.

  • Please turn to page 340 and complete the Writing About Math – Combining Trigonometric Functions.

  • You may work with your group.

  • This activity is due at the end of the class.


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