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  1. Feb 11, 2011 The transformed trigonometric functions

  2. f(x) = a sin b(x – h) + k • Recall which is which in the rule:

  3. Match the parameters to the number: k h b a

  4. Match the parameters to the number: k h b a 5 7 4 1

  5. Which is affected by parameter a? a = 1 Amplitude Period Frequency l.o.o.

  6. Which is affected by parameter a? a = 2 Amplitude Period Frequency l.o.o.

  7. Which is affected by parameter a? a = 3 Amplitude Period Frequency l.o.o.

  8. Which is affected by parameter a? Amplitude Period Frequency l.o.o.

  9. In fact, parameter a = amplitude Amplitude Period Frequency l.o.o.

  10. y = 2 cos x y = 8 sin 2x y = -3 cos x y = 4 sin 9x - 2 What would be the amplitude:

  11. y = 2 cos x y = 8 sin 2x y = -3 cos x y = 2.4 sin 9x - 2 amplitude = 2 amplitude = 8 amplitude = 3 amplitude = 2.4 What would be the amplitude:

  12. What would be the value of a in the rule?

  13. What would be the value of a in the rule? a = 5

  14. What would be the value of a in the rule?

  15. What would be the value of a in the rule? a = 4

  16. What would be the value of a in the rule? a = 4

  17. Another way to find amplitude: Amplitude = half the distance between the Max and min values = (M – m)  2 = (2 - -6)  2 = 8  2 = 4

  18. Another way to find amplitude: Amplitude = half the distance between the Max and min values = (M – m)  2 = (2 - -6)  2 = 8  2 = 4 2 -6

  19. What would be the value of a in the rule?

  20. What would be the value of a in the rule? a = 1 Amplitude = half the distance between the Max and min values = (M – m)  2 = (2 - 0)  2 = 2  2 = 1

  21. In general then: • For f(x) = a sin b(x – h) + k OR: • f(x) = a cos b(x – h) + k Amplitude =

  22. In general then: • For f(x) = a sin b(x – h) + k OR: • f(x) = a cos b(x – h) + k Amplitude = |a|

  23. In general then: • For f(x) = a sin b(x – h) + k OR: • f(x) = a cos b(x – h) + k Amplitude = |a|

  24. Which is affected by parameter b? b = 1 Amplitude Period Frequency l.o.o.

  25. Which is affected by parameter b? b = 2 Amplitude Period Frequency l.o.o.

  26. Which is affected by parameter b? b = 4 Amplitude Period Frequency l.o.o.

  27. Which is affected by parameter b? Amplitude Period Frequency l.o.o.

  28. Which is affected by parameter b? 4 cycles Amplitude Period Frequency l.o.o.

  29. Which is affected by parameter b? Amplitude Period Frequency l.o.o.

  30. In fact, b = frequency y = sin 4x Amplitude Period Frequency = 4 = b l.o.o.

  31. y = cos 4x y = 8 sin 2x y = -3 cos (x + 1) -2 y = 2.4 sin (-9x) - 2 What would be the frequency:

  32. y = cos 4x y = 8 sin 2x y = -3 cos (x + 1) -2 y = 2.4 sin (-9x) - 2 frequency = 4 frequency = 2 frequency =  frequency = 9 What would be the frequency:

  33. What would be the value of b in the rule?

  34. What would be the value of b in the rule? b = 1

  35. What would be the value of b in the rule?

  36. What would be the value of b in the rule? b = 3

  37. What would be the value of b in the rule?

  38. What would be the value of b in the rule? b = 0.5

  39. In general then: • For f(x) = a sin b(x – h) + k OR: • f(x) = a cos b(x – h) + k Frequency =

  40. In general then: • For f(x) = a sin b(x – h) + k OR: • f(x) = a cos b(x – h) + k Frequency = |b|

  41. And if 4 cycles have a total width of 2.... ...then one of those cycles must have a width of... y = sin 4x Amplitude Period Frequency l.o.o.

  42. And if 4 cycles have a total width of 2.... ...then one of those cycles must have a width of... y = sin 4x Amplitude Period Frequency l.o.o. ?

  43. And if 4 cycles have a total width of 2.... ...then one of those cycles must have a width of... y = sin 4x Amplitude Period = Frequency l.o.o.

  44. And if 4 cycles have a total width of 2.... ...then one of those cycles must have a width of... y = sin 4x Amplitude Period = Frequency l.o.o.

  45. In fact, period = y = sin 4x Amplitude Period = Frequency l.o.o.

  46. In fact, period = y = sin 4x Amplitude Period = Frequency l.o.o.

  47. y = cos 4x y = 8 sin 2x y = -3 cos (x + 1) -2 y = 2.4 sin (-9x) - 2 period = period = period = period = What would be the period:

  48. y = cos 4x y = 8 sin 2x y = -3 cos (x + 1) -2 y = 2.4 sin (-9x) - 2 period = period = period = period = What would be the period:

  49. In general then: • For f(x) = a sin b(x – h) + k OR: • f(x) = a cos b(x – h) + k Frequency = |b| Period =

  50. Which is affected by parameter h? h = 0 Amplitude Period Frequency l.o.o.