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, let’s graph y = sec 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 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.