1. 4.1 Graphs of Sine and Cosine OBJ: Graph sine and cosine

2. 1 DEF:  Sine Graph 1 0ππ3π 2π -12 2 0 1 0 -1 0

3. 1 DEF:  Sine Graph 1 0ππ3π 2π -12 2 0 1 0 -1 0

4. y = d + a (trig b ( x + c) ) a (amplitude) multiply a times (0 |1 0 -1 0 1) b (period) 2π b c (starting point) d (vertical shift)

5. y = sin x Ref. no Amp. 1 Per. 2π ¼ Per. π/2 St. Pt. 0 Vert. Sh. none 0 1 0 1 0 1 0 -1 π/2 3π/2 4π/2

6. 2 DEF:  Cosine Graph 0ππ3π 2π 2 1 0 -1 0 1

7. 2 DEF:  Cosine Graph - π 0ππ3π 2π 22 2 1 0 -1 0 1

8. DEF:  Periodic function A function f with the property f(x) = f(x+p) for every real number x in the domain of f and for some real positive number p. The smallest possible positive value of p is the period of the function f.

9. 3 EX:  Graph y = 2 sin x 0ππ3π 2π 2 0 1 0 -1 0 2(0 1 0 -1 0) 0 2 0 -2 0

10. 3 EX:  Graph y = 2 sin x 0ππ3π 2π 2 0 2 0 -2 0

11. DEF:  Amplitude of Sine and Cosine The graph of y = a sin x or y = a cos x will have the same shape as y = sin x or y cos x, respectively, except with range -  a   y   a . The number  a  is called the amplitude.

12. y = d + a(trig b ( x + c) ) a (amplitude) multiply a times (0 |1 0 -1 0 1) b (period) 2π b c (starting point) d (vertical shift)

13. 4 y = -2 cos x 1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2 2 1 0 -1 π/2 3π/2 4π/2 -2

14. 4 y = -2 cos x Ref. yes Amp. - 2 Per. 2π ¼ Per. π/2 St. Pt. 0 Vert. Sh. none 1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2 2 1 0 -1 π/2 3π/2 4π/2 -2

15. 4 y = -2 cos x 2 1 0 -1 π/2 3π/2 4π/2 -2

16. DEF:  Vertical Translation A function of the form y =d + a sin b x or of the form y = d + a cos b x is shifted vertically when compared with y = a sin b x or y =a cos b x.

17. y = d + a(trig b ( x + c) ) a (amplitude) multiply a times (0 |1 0 -1 0 1) b (period) 2π b c (starting point) d (vertical shift)

18. 5 EX:  Graph y = – 3 + 2 sin x 0ππ3π 2π 2

19. 1 DEF:  Sine Graph 1 0ππ3π 2π -12 2 0 1 0 -1 0

20. 3 EX:  Graph y = 2 sin x 0ππ3π 2π 2 2(0 1 0 -1 0) 0 2 0 -2 0

21. 5 EX:  Graph y = – 3 + 2 sin x 1 0ππ3π 2π -12 2 2(0 1 0 -1 0) 0 2 0 -2 0

22. 5 EX:  Graph y = – 3 + 2 sin x 1 0ππ3π 2π -12 2 2(0 1 0 -1 0) 0 2 0 -2 0 -3-3-3 -3 -3

23. 5 EX:  Graph y = – 3 + 2 sin x 1 0ππ3π 2π -12 2 -3 -1 -3 -5 -3

24. DEF:  Phase Shift The function y = sin (x + c) has the shape of the basic sine graph y = sin x, but with a translation  c  units: to the right if c < 0 and to the left if c > 0. The number c is the phase shift of the graph. The cosine graph has the same function traits.

25. y = d + a(trig b (x + c) a (amplitude) multiply a times (0 |1 0 -1 0 1) b (period) 2π b c (starting point) d (vertical shift)

26. EX:  Graph y = sin (x – π/3) 6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14  -1 6 6 6 6 6 0 1 0 -1 0

27. 6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14  -1 6 6 6 6 6 0 1 0 -1 0 -1(0 1 0 -1 0 0 -1 0 1 0

28. 6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14  -1 6 6 6 6 6 0 -1 0 1 0 +4 +4 +4 +4 +4 4 3 4 5 4

29. 6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14  -1 6 6 6 6 6 4 3 4 5 4

30. EX:  Graph y = 3cos (x + π/4) 7 EX: Graph y =-3 + 3cos(x+π/4)

31. -   3  5  7  4 4 4 4 4 1 0 -1 0 1

32. EX:  Graph y = 3cos (x + π/4) 7 EX: Graph y =-3 + 3cos(x+π/4) -   3  5  7  4 4 4 4 4 1 0 -1 0 1 3(1 0 -1 0 1) 3 0 -3 0 3

33. 7 EX: Graph y =-3 + 3cos(x+π/4) -   3  5  7  4 4 4 4 4 3 0 -3 0 3 -3 -3 -3 -3 -3 0 -3 -6 -3 0 __ __ __ __ __ __ __ __ __

34. 7 EX: Graph y =-3 + 3cos(x+π/4) -   3  5  7  4 4 4 4 4 __ __ __ __ __ __ __ __ __

35. 1 EX:  Graph y = -2 +sin x Ref, Amp No, 1 Per 2 π ¼ Per 0ππ3π 2π π/2 2 2 St.Pt. 0 Vert. Shift 2

36. 1 EX:  Graph y = -2 +sin x 0ππ3π 2π 2 0 1 0 -1 0 -2 -2 -2 -2 -2 -2 -1 -2 -3 -2

37. 1 EX:  Graph y = -2 +sin x 0ππ3π 2π 2 0 1 0 -1 0 -2 -2 -2 -2 -2 -2 -1 -2 -3 -2

38. 2 EX:  Graph y = 3 – 2 cos x Ref, Amp Yes, -2 Per 2 π ¼ Per 0ππ3π 2π π/2 2 2 St.Pt. 0 Vert. Shift 3

39. 2 EX:  Graph y = 3 – 2 cos x 0ππ3π 2π 2 1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2

40. 2 EX:  Graph y = 3 – 2 cos x 0ππ3π 2π -2(1 0 -1 0 1) 2 2 -2 0 2 0 -2 +3 +3 +3 +3 +3 1 3 5 3 1

41. 2 EX:  Graph y = 3 – 2 cos x 0ππ3π 2π -2(1 0 -1 0 1) 2 2 -2 0 2 0 -2 +3 +3 +3 +3 +3 1 3 5 3 1