Chapter 4: Graphing & Inverse Functions

1. Chapter 4:Graphing & Inverse Functions Sections 4.2, 4.3, & 4.5 Transformations

2. What happens when the function has a coefficient?

3. f (x) = sin x

4. f (x) = sin x f (x) = 2sin x

5. f (x) = sin x f (x) = 3sin x

6. f (x) = sin x f (x) = 4sin x

7. f (x) = sin x f (x) = ½sin x

8. f (x) = sin x f (x) = ¼sin x

9. f (x) = A sin x A indicates the amplitude A indicates the ??????????? the amplitude is A times larger than that of the basic sine curve (amp = 1)

10. What about cosine?

11. f (x) = cos x

12. f (x) = cosx f (x) = cos x

13. f (x) = cosx f (x) = 2cos x f (x) = 3cos x

14. f (x) = cosx f (x) = ½cos x f (x) = ¼cos x

15. Amplitude is the distance from the midline…so always positive. f (x) = A sin x or f (x) = A cosx A indicates the amplitude |A| indicates the amplitude the amplitude is A times larger than that of the basic sine or cosine curve the amplitude is |A| times larger than that of the basic sine or cosine curve the amplitude is A times larger than that of the basic sine curve

16. What about negative coefficients?

17. f (x) = sin x f (x) = -sin x

18. f (x) = cos x f (x) = -cos x

19. f (x) = A sin x or f (x) = A cosx If A is negative the graph is reflected across the x-axis.

20. f (x) = A sin x or f (x) = A cosx Domain: Range: Amplitude: Period:

21. f (x)=cosx f (x)=-3cosx

22. What happens when the ANGLE has a coefficient?

23. f (x) = sin x f (x) = sin 2x

24. f (x) = sin 2x f (x) = sin x

25. f (x) = sin x f (x) = sin 4x

26. f (x) = sin x f (x) = sin ½x

27. f (x) = sin x f (x) = sin ¼x

28. f (x) = sin Bx 2p ___ B indicates the period indicates the ????????? B indicates the ????????? the period of the function is the period of the basic curve divided by B (period = 2p)

29. f (x) = cosx f (x) = cos 2x f (x) = cos ½x

30. f (x) = sin Bx or f (x) = cos Bx f (x) = sin Bx Period for tangent and cotangent will be based on its period of π. 2p ___ B indicates the period the period of the function is the period of the basic curve divided by B (period = 2p)

31. What about negative coefficients of the angle?

32. f (x) = sin x SINE odd function f(-x) = - f(x) origin symmetry f (x) = sin -x Also graph of: f (x) = -sin x !

33. f (x) = cosx COSINE even function f(-x) = f(x) y-axis symmetry f (x) = cos -x Also graph of: f (x) = cosx !

34. f (x) = sin Bx or f (x) = cos Bx If B is negative the graph is reflected across the y-axis.

35. f(x)=sin Bxor f(x)=cosBx Domain: Range: Amplitude: Period: B < 0 means y-axis reflection

36. What happens when a constant is added to the function?

37. f(x)= sinx f(x)= sinx + 1

38. f(x)= sinx f(x)= sinx - 2

39. f(x)= cosx f(x)= cosx – 3

40. f (x) = sin x + D or f (x) = cosx + D D indicates displacement. The displacement is a vertical translation (shift) upward for D > 0 and downward for D < 0

41. What happens when a constant is added to the angle?

42. f(x)=sinx f(x)=sin(x+ )

43. f(x)=cosx f(x)=cos(x- )

44. f(x)=cosx Phase shift is NOT p! Phase shift is NOT p! Coefficients affect the phase shift! = cos[2(x- )] f(x)=cos(2x- )

45. f(x)=sinx Alternate method: (negative means left) = sin[ (x+ )] f(x)=sin( x+ )

46. f(x)=sinx = sin[ (x+ )] f(x)=sin( x+ )

47. = sin(Bx+C) = cos(Bx+C) = sin[B(x+C)] = cos[B(x+C)] f (x) f (x) or -C ___ B indicates phase shift. -C The phase shift is a horizontal translation left for C > 0 and right for C < 0

48. Amplitude: A A < 0 reflect x Period: B < 0 reflect y 2p ___ B f (x)= A sin[B(x + C )] + D f (x)= Acos[B(x + C )] + D Phase shift: Displacement: D -C D > 0 up D < 0 down C > 0 left C < 0 right

49. f(x)=3sin(2x)+1

50. f(x)=2sin(x-π)-2 Looks like sine reflected also…