166 I CHAPTER 4 Graphs ofthe Circular Functions

lf@ I Exercises

20. y=tan(~+1T)

22. y = cot(2X _ 3;)

24. y = 1 - tan x

26. y = - 2 - cot x128. y = 3 + - tan x 2

30. y = -2 + 3 tan(4x + 1T)

32. y = -2 + ~tan(~x - 1T)

27. y = -1 + 2 tan x

21. y=cot(3X+~)

23. y = 1 + tan x

25. y = 1 - cot x

19. y=tan(2x-1T)

Graph each function over a one-period interval. See Examples 1-3.

17. y = tan 4x

8. y = tan - x2.: y = 2 tan x2

1

110. y = 2 cot x

11. y=2tan-x12. y = - cot x4

2x 1

113. y = cot 3x

14 y = -cot - x15. y = -2 tan-x. 2

- 4

1

1 116. y = 3 tan-x

17. y = -cot4x18. y = - - cot 2x2

-. 2 2

x Graph each function over a two-period interval. See Examples 4 and 5.

1

29. y = -1 + "2cot(2x - 31T)

31. y = 1 - 2 cot [ 2 (x + ~) ]

17. 18.y

y

~ 2"I

-2I

Y=~COI4X

y;;; ~cotx

13.

14.y

y

x

-1 y=cot3x15.

16.y

y

.1

1

1I1 x

1. C 2. A Concept Check In Exercises 1-6, match each function with its graphfrom choices A-F

3. B 4. D ( )5. F 6. E 1. y = -tan x 2. y = -cot x 1y = tan x - ~

7. 8. Y ,_. 1 ( 1T) ( 1T) ( 1T)) -IIan 2x 4. y = cot x - "4 ~ y = cot x + "4 6. y = tan x + "4

x I 1 X I 1 •• x A. y B. y C. y•

~l

I 1

1 I

9. 10. 01/1 . x I y' I • x O~

y y 7T' _7!. 31T _IE. 7!...• 2 2

I

x

D. y E. y F. Y

y=2colx ~ I I ~ It11 I I I I,

11. 12. I I I I I 'o x 0 x 10

y Y 7r 57r 37r 7r _7!.1 't37r4 4 4 4 4 41 I 1 I

x ~k1 .x

"r.

SECTION4.3 Graphs of the Tangent and Cotangent Functions I 167

34. y

2

x

0II-2

36.

y•II11II x

38.

y

'h I

I+-x7Tt \2-2

-4

x

x

40. The least positive number k for which x = k is an asymptote for the cotangent func­tion is ~.

Concept Check In Exercises 39-42, tell whether each statement is true or false. iffalse,tell why.

39. The least positive number k for which x = k is an asymptote for the tangent function• 1T

IS 2'

42. The graph of y = cot x in Figure 26 suggests that cot( -x) = -cot x for all x in thedomain of cot x.

41. The graph of y = tan x in Figure 23 suggests that tan( -x) = tan x for all x in thedomain of tan x.

44. Concept Check Consider the function defined by f(x) = -4 tan(2x + 7T). What isthe domain of f? What is its range?

45. Show that tan( -x) = -tan x by writing tan( -x) as ;~~i=:; and then using the rela­tionships for sin( -x) and cos( -x).

46. Show that cot( -x) = -cot x by writing cot( -x) as ~~:~=:j and then using the rela-tionships for cos( -x) and sin( -x).

//

/

Work each exercise.

43. Concept Check If c is any number, then how many solutions does the equationc = tan x have in the interval (-27T, 27TJ?

x

Connecting Graphs with Equations Determine the simplest form of an equation foreach graph. Choose b > 0, and include no phase shifts. (Midpoints and quarter-pointsare identified by dots.) See Example 6.

x 33. y

x

x

y

22.y

20.y

28.

y = -2 + 3 tan (4x + 7T)

30.

32.

35.y

t .'7T) ~y=cot 2x-Z

24.yt ~

Ix

I

x

x '" UJ. \~h~-1

y= I-tau x26.

37.yy

~ rfJr x

I

III

y = tan (2x - 7T)

19.y

II

)i )iIH:<!:

31T 51T

44" 4"

x

y

25.y

23.

y

y

x-t-­III

y=-I+"2tanx

y

21.

27.

29.y

31.

L.I-­II

T :j: \¥ \¥ I

y= 1-2cot[2(x+¥)] y= -2+~tan(~x-7T)