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. sin x tan x y tan x • Therefore, since cos x , the graph of will have vertical asymptotes at 2 n , where 2 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 x y = 2 tan (2x) • Now, our period will be b 2 • Additionally, the graph will get larger twice as quickly. • The asymptotes will be at 4 • The x-intercept will be (0,0) y x x y tan 2 • The period is 2π. • The asymptotes are at ±π. • The x-intercept is (0,0). y x 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. , 0 • There is an x-intercept at 2 y x y = -2 cot (2x) • The period is 2 • There is an x-intercept at • There is an asymptote at ,0 4 2 y x 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 y x Now, let’s graph y = sec x y 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. x y 2 csc 2 y sec x y x y x 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.