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

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4.6 Graphs of Other Trigonometric FUNctions

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  1. 4.6 Graphs of Other Trigonometric FUNctions How can I sketch the graphs of all of the cool quadratic FUNctions?

  2. 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.

  3. 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.

  4. 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)

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

  6. 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.

  7. y = cot x • The asymptotes are at ±πn. • There is an x-intercept at

  8. y = -2 cot (2x) • The period is • There is an x-intercept at • There is an asymptote at

  9. 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

  10. 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

  11. Now, let’s graph y = sec x

  12. 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.

  13. 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.

  14. 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.

  15. Writing About Math • 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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