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.

xxx

cossintan xy tan

n22

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.

x

y

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)

2

b

4

x

y

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

2

tan xy

x

y

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

0,2

x

y

y = -2 cot (2x)

• The period is

• There is an x-intercept at

• There is an asymptote at

2

0,4

2

x

y

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

x

y

x

y

x

y

Now, let’s graph y = sec x

x

y

x

y

x

y

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.

xy

xy

sec

2csc2

x

y

x

y

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.