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4.6 Graphs of Other Trigonometric FUNctionsPowerPoint Presentation

4.6 Graphs of Other Trigonometric FUNctions

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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.
- 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.

- 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)

- 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)

- The period is 2π.
- The asymptotes are at ±π.
- The x-intercept is (0,0).

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

- The asymptotes are at ±πn.
- There is an x-intercept at

y = -2 cot (2x)

- The period is
- There is an x-intercept at
- There is an asymptote at

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….

- 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!)

- 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.

- For a nifty summary of the trigonometric FUNctions, please check out page 340.
- As a matter of fact, I would make sure I memorized all of the information on page 340.

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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