1. 4.3 Period Changes and Graphs other Trig Functions
Obj: Graph sine and cosine with period changes
Obj: Graph other Trig Functions

2. 2 EX:  Graph y = 3 – 2 cos x Ref, Amp Yes, -2 Per 2 ππ/
St.Pt
Vert. Shift 3

3. 2 EX:  Graph y = 3 – 2 cos x 0 π π 3π 2π 2 2 1 0 -1 0 1
0 π π 3π 2π
-2( )

4. 2 EX:  Graph y = 3 – 2 cos x 0 π π 3π 2π -2(1 0 -1 0 1) 2 2
0 π π 3π 2π
-2( )

5. 2 EX:  Graph y = 3 – 2 cos x 0 π π 3π 2π -2(1 0 -1 0 1) 2 2
0 π π 3π 2π
-2( )

6. 3 EX:  Graph y =  –  sinx 1
0 π π 3π 2π

7. 3 EX:  Graph y =  –  sinx 1
0 π π 3π 2π

8. 3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2 0 1 0 -1 0
0 π π 3π 2π
-2/3( )
0 -2/ /3 0

9. 3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2 0 -2/3 0 2/3 0
0 π π 3π 2π
0 -2/ /3 0
+½ +½ +½ +½ +½
½ -1/6 ½ 5/6 ½

10. 3 EX:  Graph y =  –  sinx 1
0 π π 3π 2π

11. 4 EX:  Graph y = 4 cos (x – π)1
07π 10π 13π 16π 19

12. 4 EX:  Graph y = 4 cos (x – π)1
π 10π 13π 16π 19

13. 4 EX:  Graph y = 4 cos (x – π)1
π 10π 13π 16π 19
4( )

14. 4 EX:  Graph y = 4 cos (x – π)1
π 10π 13π 16π 19

15. 5 EX:  Graph y = 1 +sin(x + π/6)
- 2π 5π 8π 11π
½( )
0 ½ ½ 0

16. 5 EX:  Graph y = 1 +sin(x + π/6)
- 2π 5π 8π 11π
0 ½ ½ 0
1 1½ 1 ½ 1

17. 5 EX:  Graph y = 1 +sin(x + π/6)
- 2π 5π 8π 11π
0 ½ ½ 0
1 1½ 1 ½ 1

18. 4.1 Period changes in graphs of Sine and Cosine
OBJ: Find the period for a sine and cosine graph

19. y = d + a(trig b (x + c)
a (amplitude) multiply a times (0 | )
Sin|Cos
-a Reflection
b (period) 2π
b can be factored out OR
c (starting point) Set (bx + __) = 0 instead of
completing factoring with b
d (vertical shift)

20. DEF:  Period of Sine and Cosine
The graph of y = sin b x will look like that
of sin x, but with a period of  2  .
 b 
Also the graph of y = cos b x looks like
that of y = cos x, but with a period of  2 

21. 8 EX: • Graph y = sin 2x Ref. no Amp. 1 Per. 2π/2 = π ¼ Per. π/4
St. Pt.
Vert. Sh.
none

22. 8 EX: • Graph y = sin 2x
 2 3 4

23. 8 EX: • Graph y = sin 2x
 2 3 4

24. 8 EX: • Graph y = sin 2x Ref. no Amp. 1 Per. 2π/2 = π ¼ Per. π/4
St. Pt.
Vert. Sh.
none 1
π/4 3π/4 4π/4

25. EX: • Graph y = -2cos 3x EX 9 • Graph y = 3 – 2cos 3x
 2 3 4

26. EX: • Graph y = -2cos 3x EX 9 • Graph y = 3 – 2cos 3x
 2 3 4

27. EX: • Graph y = -2cos 3x EX 9 • Graph y = 3 – 2cos 3x
 2 3 4
-2( )

28.  2 3 4

29. 10 EX: Graph y = –2cos3(x+π)

30. 10 EX: Graph y = –2cos3(x+π)
-2 -  2

31. 10 EX: Graph y = –2cos3(x+π)
-2 -  2

32. 10 EX: Graph y = –2cos3(x+π)
-2 -  2

33. 11 EX: • Graph y = cos(2x/3)

34. 11 EX: • Graph y = cos(2x/3) 1
π 6π 9π 12π

35. 11 EX: • Graph y = cos(2x/3) 1
π 6π 9π 12π

36. 11 EX: • Graph y = cos(2x/3) 1
π 6π 9π 12π

37. 12 EX: Graph y = –2 sin 3x

38. 12 EX: Graph y = –2 sin 3x 1
π 2π 3π 4π

39. 12 EX: Graph y = –2 sin 3x 1
π 2π 3π 4π

40. 12 EX: Graph y = –2 sin 3x 1
π 2π 3π 4π

41. 13 EX: Graph y = 3 cos ½ x

42. 13 EX: Graph y = 3 cos ½ x 1
0 π 2π 3π 4π -1

43. 13 EX: Graph y = 3 cos ½ x 1
0 π 2π 3π 4π -1

44. 13 EX: Graph y = 3 cos ½ x 1
0 π 2π 3π 4π -1

45. 4.2 Graphs of the Other Trigonometric Functions
OBJ: Graph Other Trigonometric Functions

46. y = d + a(trig b (x + c)
a (amplitude) multiply a times (0 | )
b (period) 2π b
c (starting point)
d (vertical shift)

47. Graph y = cos x
0 π π 3π 2π
2 2

48. 14 EX: Graph y = sec x
0 π π 3π 2π
2 2

49. 14 EX: Graph y = sec x
0 π π 3π 2π
2 2

50. 15 EX: Graph y = 2 + sec(2x–π)

51. 15 EX: Graph y = 2 + sec2(x–π)
0 2π 3π 4π 5π 6

52. 15 EX: Graph y = 2 + sec2(x–π)
0 2π 3π 4π 5π 6

53. Graph y = sin x 1
0 π π 3π 2π

54. 16 EX: Graph y = csc x 1
0 π π 3π 2π

55. 17 EX: Graph y = csc (x + π/3)

56. 17 EX: Graph y = csc (x + π/3)
-2π -π π 2

57. 17 EX: Graph y = csc (x + π/3)
-2π -π π 2

58. 17 EX: Graph y = csc (x + π/3)
-2π -π π 2

59. 18 y = tan x
Ref.
Amp.
Per.
¼ Per.
St. Pt.
Vert. Sh. No 1  4
none

60. 19 y = tan (2x + π/2) y = tan 2 (x + /4)
Ref.
Amp.
Per.
¼ Per.
St. Pt.
Vert. Sh. No 1  2 8 - 4
none

61. 20 y = 2 + ¼ tan (½x + π) y=2+¼ tan½(x + 2 π)
Ref.
Amp.
Per.
¼ Per.
St. Pt.
Vert. Sh. No 2  2
-2 2↑

62. 21 y = cot x
Ref.
Amp.
Per.
¼ Per.
St. Pt.
Vert. Sh. No 1  4
none

63. 22 y = 2 + cot x
Ref.
Amp.
Per.
¼ Per.
St. Pt.
Vert. Sh. No 1  4 2↑

64. 6 EX: Graph y =-3 – 2cos(x+5π/6)

65. 6 EX: Graph y =-3 – 2cos(x+5π/6)
-5 -2π π 4π 7π

66. 6 EX: Graph y =-3 – 2cos(x+5π/6)
-5 -2π π 4π 7π

67. 6 EX: Graph y =-3 – 2cos(x+5π/6)
-5 -2π π 4π 7π