2. Copyright © 2009 Pearson Education, Inc. CHAPTER 6: The Trigonometric Functions 6.1The Trigonometric Functions of Acute Angles 6.2Applications of Right Triangles 6.3Trigonometric Functions of Any Angle 6.4Radians, Arc Length, and Angular Speed 6.5Circular Functions: Graphs and Properties 6.6Graphs of Transformed Sine and Cosine Functions

3. Copyright © 2009 Pearson Education, Inc. 6.6 Graphs of Transformed Sine and Cosine Functions  Graph transformations of y = sin x and y = cos x in the form y = A sin (Bx – C) + D and y = A cos (Bx – C) + D and determine the amplitude, the period, and the phase shift.  Graph sums of functions.  Graph functions (damped oscillations) found by multiplying trigonometric functions by other functions.

4. Slide 6.6 - 4 Copyright © 2009 Pearson Education, Inc. Variations of the Basic Graphs We are interested in the graphs of functions in the form y = A sin (Bx – C) + D and y = A cos (Bx – C) + D where A, B, C, and D are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs.

5. Slide 6.6 - 5 Copyright © 2009 Pearson Education, Inc. The Constant D Let’s observe the effect of the constant D.

6. Slide 6.6 - 6 Copyright © 2009 Pearson Education, Inc. The Constant D

7. Slide 6.6 - 7 Copyright © 2009 Pearson Education, Inc. The Constant D The constant D in y = A sin (Bx – C) + D and y = A cos (Bx – C) + D translates the graphs up D units if D > 0 or down |D| units if D < 0.

8. Slide 6.6 - 8 Copyright © 2009 Pearson Education, Inc. The Constant A Let’s observe the effect of the constant A.

9. Slide 6.6 - 9 Copyright © 2009 Pearson Education, Inc. The Constant A

10. Slide 6.6 - 10 Copyright © 2009 Pearson Education, Inc. The Constant A If |A| > 1, then there will be a vertical stretching. If |A| < 1, then there will be a vertical shrinking. If A < 0, the graph is also reflected across the x- axis.

11. Slide 6.6 - 11 Copyright © 2009 Pearson Education, Inc. Amplitude The amplitude of the graphs of is |A|. y = A sin (Bx – C) + D and y = A cos (Bx – C) + D

12. Slide 6.6 - 12 Copyright © 2009 Pearson Education, Inc. The Constant B Let’s observe the effect of the constant B.

13. Slide 6.6 - 13 Copyright © 2009 Pearson Education, Inc. The Constant B

14. Slide 6.6 - 14 Copyright © 2009 Pearson Education, Inc. The Constant B

15. Slide 6.6 - 15 Copyright © 2009 Pearson Education, Inc. The Constant B

16. Slide 6.6 - 16 Copyright © 2009 Pearson Education, Inc. The Constant B If |B| < 1, then there will be a horizontal stretching. If |B| > 1, then there will be a horizontal shrinking. If B < 0, the graph is also reflected across the y-axis.

17. Slide 6.6 - 17 Copyright © 2009 Pearson Education, Inc. Period The period of the graphs of is y = A sin (Bx – C) + D and y = A cos (Bx – C) + D

18. Slide 6.6 - 18 Copyright © 2009 Pearson Education, Inc. Period The period of the graphs of is y = A csc (Bx – C) + D and y = A sec (Bx – C) + D

19. Slide 6.6 - 19 Copyright © 2009 Pearson Education, Inc. Period The period of the graphs of is y = A tan (Bx – C) + D and y = A cot (Bx – C) + D

20. Slide 6.6 - 20 Copyright © 2009 Pearson Education, Inc. The Constant C Let’s observe the effect of the constant C.

21. Slide 6.6 - 21 Copyright © 2009 Pearson Education, Inc. The Constant C

22. Slide 6.6 - 22 Copyright © 2009 Pearson Education, Inc. The Constant C

23. Slide 6.6 - 23 Copyright © 2009 Pearson Education, Inc. The Constant C

24. Slide 6.6 - 24 Copyright © 2009 Pearson Education, Inc. The Constant C if |C| < 0, then there will be a horizontal translation of |C| units to the right, and if |C| > 0, then there will be a horizontal translation of |C| units to the left. If B = 1, then

25. Slide 6.6 - 25 Copyright © 2009 Pearson Education, Inc. Combined Transformations It is helpful to rewrite as y = A sin (Bx – C) + D and y = A cos (Bx – C) + D and

26. Slide 6.6 - 26 Copyright © 2009 Pearson Education, Inc. Phase Shift The phase shift of the graphs is the quantity and

27. Slide 6.6 - 27 Copyright © 2009 Pearson Education, Inc. Phase Shift If C/B > 0, the graph is translated to the right |C/B| units. If C/B < 0, the graph is translated to the right |C/B| units.

28. Slide 6.6 - 28 Copyright © 2009 Pearson Education, Inc. Transformations of Sine and Cosine Functions To graph follow the steps listed below in the order in which they are listed. and

29. Slide 6.6 - 29 Copyright © 2009 Pearson Education, Inc. Transformations of Sine and Cosine Functions 1.Stretch or shrink the graph horizontally according to B. The period is |B| < 1 Stretch horizontally |B| > 1 Shrink horizontally B < 0 Reflect across the y-axis

30. Slide 6.6 - 30 Copyright © 2009 Pearson Education, Inc. Transformations of Sine and Cosine Functions 2.Stretch or shrink the graph vertically according to A. The amplitude is A. |A| < 1 Shrink vertically |A| > 1 Stretch vertically A < 0 Reflect across the x-axis

31. Slide 6.6 - 31 Copyright © 2009 Pearson Education, Inc. Transformations of Sine and Cosine Functions 3.Translate the graph horizontally according to C/B. The phase shift is

32. Slide 6.6 - 32 Copyright © 2009 Pearson Education, Inc. Transformations of Sine and Cosine Functions 4.Translate the graph vertically according to D. D < 0 |D| units down D > 0 D units up

33. Slide 6.6 - 33 Copyright © 2009 Pearson Education, Inc. Example Sketch the graph of Solution: Find the amplitude, the period, and the phase shift.

34. Slide 6.6 - 34 Copyright © 2009 Pearson Education, Inc. Example Solution continued Then we sketch graphs of each of the following equations in sequence. To create the final graph, we begin with the basic sine curve, y = sin x.

35. Slide 6.6 - 35 Copyright © 2009 Pearson Education, Inc. Example Solution continued

36. Slide 6.6 - 36 Copyright © 2009 Pearson Education, Inc. Example Solution continued

37. Slide 6.6 - 37 Copyright © 2009 Pearson Education, Inc. Example Solution continued

38. Slide 6.6 - 38 Copyright © 2009 Pearson Education, Inc. Example Solution continued

39. Slide 6.6 - 39 Copyright © 2009 Pearson Education, Inc. Example Solution continued

40. Slide 6.6 - 40 Copyright © 2009 Pearson Education, Inc. Example Graph: y = 2 sin x + sin 2x Solution: Graph: y = 2 sin x and y = sin 2x on the same axes.

41. Slide 6.6 - 41 Copyright © 2009 Pearson Education, Inc. Example Solution continued Graphically add some y-coordinates, or ordinates, to obtain points on the graph that we seek. At x = π/4, transfer h up to add it to 2 sin x, yielding P 1. At x = – π/4, transfer m down to add it to 2 sin x, yielding P 2. At x = – 5π/4, add the negative ordinate of sin 2x to the positive ordinate of 2 sin x, yielding P 3. This method is called addition of ordinates, because we add the y-values (ordinates) of y = sin 2x to the y- values (ordinates) of y = 2 sin x.

42. Slide 6.6 - 42 Copyright © 2009 Pearson Education, Inc. Example Solution continued The period of the sum 2 sin x + sin 2x is 2π, the least common multiple of 2π and π.

43. Slide 6.6 - 43 Copyright © 2009 Pearson Education, Inc. Example Sketch a graph of Solution f is the product of two functions g and h, where To find the function values, we can multiply ordinates. Start with The graph crosses the x-axis at values of x for which sin x = 0, kπ for integer values of k.

44. Slide 6.6 - 44 Copyright © 2009 Pearson Education, Inc. Example Solution continued f is constrained between the graphs of y = –e –x/2 and y = e –x/2. Start by graphing these functions using dashed lines. Since f(x) = 0 when x = kπ, k an integer, we mark those points on the graph. Use a calculator to compute other function values. The graph is on the next slide.

45. Slide 6.6 - 45 Copyright © 2009 Pearson Education, Inc. Example Solution continued