1. Copyright © 2005 Pearson Education, Inc. Chapter 4 Graphs of the Circular Functions

2. Copyright © 2005 Pearson Education, Inc. 4.1 Graphs of the Sine and Cosine Functions

3. Copyright © 2005 Pearson Education, Inc. Slide 4-3 Periodic Functions (Conceptual View) Periodic Functions are functions whose values repeat in a regular pattern for every member of their domain Example:

4. Copyright © 2005 Pearson Education, Inc. Slide 4-4 Periodic Functions (Definition) A “periodic function” is a function f such that, for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function.

5. Copyright © 2005 Pearson Education, Inc. Slide 4-5 Sine Function The periodic nature of the circular sine function can be seen by considering the unit circle and by graphing the (x, sin x) pairs:

6. Copyright © 2005 Pearson Education, Inc. Slide 4-6 Sine Function f(x) = sin x

7. Copyright © 2005 Pearson Education, Inc. Slide 4-7 Notes Concerning the Sine Function Since the period of the sine function is, all possible values of the sine function will occur in any interval, and in each adjoining interval the exact pattern of values will be repeated Any interval of values of the sine function is called “one period” of the sine function We will define the “primary period” of the sine function to be those values in the interval

8. Copyright © 2005 Pearson Education, Inc. Slide 4-8 Sketching the Graph of the Primary Period of the Sine Function In a rectangular coordinate system, mark the positions of and on the x-axis Divide this interval on the x-axis into four intervals and label the endpoints of each (quarter points) For each of these five numbers, the corresponding sine values are 0, 1, 0 -1, 0 (see unit circle) Plot the (x, sin x) pairs and connect them with a smooth curve Graph may be extended left and right

9. Copyright © 2005 Pearson Education, Inc. Slide 4-9 Cosine Function The periodic nature of the circular cosine function can be seen by considering the unit circle and by graphing the (x, cos x) pairs:

10. Copyright © 2005 Pearson Education, Inc. Slide 4-10 Cosine Function f(x) = cos x

11. Copyright © 2005 Pearson Education, Inc. Slide 4-11 Notes Concerning the Cosine Function Since the period of the cosine function is, all possible values of the cosine function will occur in any interval, and in each adjoining interval the exact pattern of values will be repeated Any interval of values of the cosine function is called “one period” of the cosine function We will define the “primary period” of the cosine function to be those values in the interval

12. Copyright © 2005 Pearson Education, Inc. Slide 4-12 Sketching the Graph of the Primary Period of the Cosine Function In a rectangular coordinate system, mark the positions of and on the x-axis Divide this interval on the x-axis into four intervals and label the endpoints of each (quarter points) For each of these five numbers, the corresponding cosine values are 1, 0, -1, 0, 1 (see unit circle) Plot the (x, cos x) pairs and connect them with a smooth curve Graph may be extended left and right

13. Copyright © 2005 Pearson Education, Inc. Slide 4-13 Amplitude of Sine and Cosine Functions The amplitude of both the sine and the cosine function is defined to be “one half of the difference between the maximum and minimum values of the function” Since the maximum value of both functions is “1” and the minimum value of both is “-1”, the amplitude of both is: ½[1- (-1)] = ½(2) = 1

14. Copyright © 2005 Pearson Education, Inc. Slide 4-14 Graph of y = a sin x compared with y = sin x Both graphs will have a period of All the “y” values on y = a sin x, will be multiplied by “a”, compared with the “y” values on y = sin x The amplitude of y = a sin x will be: | a | instead of “1” (the graph will be vertically stretched or squeezed depending on whether | a | > 1 or | a | < 1) If a < 0, the graph of y = a sin x will be inverted with respect to y = sin x Example: Compare the graphs of: y = 3 sin x and y = sin x

15. Copyright © 2005 Pearson Education, Inc. Slide 4-15 Example: Compare graph of y = 3 sin x with graph of y = sin x. Make a table of values. 0 33 0303sin x 0 11 010sin x 22 3  /2  /2 0x

16. Copyright © 2005 Pearson Education, Inc. Slide 4-16 Effects of Changing a Factor of the “Argument” of a Circular Function In the circular function “sin x”, “x” is called the argument of the function (The same terminology applies for each of the other circular functions) If the argument of a circular function is changed from “x” to “bx”, the effect is always that: The original period is changed from The original graph is horizontally squeezed if | b | > 1 and horizontally stretched if | b | < 1 These concepts will be discussed and verified on the following slides

17. Copyright © 2005 Pearson Education, Inc. Slide 4-17 Graph of y = sin x compared with y = sin bx y = sin x will have primary period for (has a period of ) y = sin bx will have a primary period for, but solving this inequality for x: tells us that it has period: The graph of y = sin bx will have exactly the same shape* as the graph of y = sin x, except that it will have the period described above. In effect, the original graph will be horizontally squeezed or stretched depending on whether | b | > 1 or | b | < 1 *The same shape means at the left end of the interval: it will have a value of 0, at its first quarter point a value of 1, at its half point a value of 0, at its three quarter point a value of -1 and at its right end a value of 0

18. Copyright © 2005 Pearson Education, Inc. Slide 4-18 Function values for: y = sin x and y = sin 2x (From Unit Circle or Calculator) xsin xsin 2x

19. Copyright © 2005 Pearson Education, Inc. Slide 4-19 Example: Graph y = sin 2x continued

20. Copyright © 2005 Pearson Education, Inc. Slide 4-20 Doing a Quick Sketch of y = sin 4x Primary period of the graph will occur for: Divide this interval into four quarters: Assign to each of these the sine pattern: Graph:

21. Copyright © 2005 Pearson Education, Inc. Slide 4-21 Periods of Modified Sine and Cosine Functions For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period. For b > 0, the graph of y = cos bx will resemble that of y = cos x, with period.

22. Copyright © 2005 Pearson Education, Inc. Slide 4-22 Doing a Quick Sketch of: Primary period of the graph will occur for: Find numbers to divide this interval into four quarters: Assign to each of these the cosine pattern: Graph:

23. Copyright © 2005 Pearson Education, Inc. Slide 4-23 Summary Comments about Graphs of y = a sin bx and y = a cos bx Each of these graphs will, respectively, look like the the graphs of the basic functions, y = sin x and y = cos x, except that: “a” alters the amplitude of the graph to | a | “a” inverts the graph, if “a” is negative “b”, when it is positive, changes the period to:

24. Copyright © 2005 Pearson Education, Inc. Slide 4-24 Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = a sin bx or y = a cos bx, with b > 0, follow these steps. Step 1To find the interval for one period, solve the inequality: (the number at the right end will be the new period) and lay off this interval on the x-axis Step 2Find numbers to divide the interval into four equal parts. Step 3Assign the appropriate sine or cosine patterns, multiplied by “a”, to each of these five x-values. (The points will be maximum points, minimum points, and x-intercepts.)

25. Copyright © 2005 Pearson Education, Inc. Slide 4-25 Guidelines for Sketching Graphs of Sine and Cosine Functions continued Step 4Plot the points found in Step 3, and join them with a smooth curve having amplitude |a|. Step 5Draw the graph over additional periods, to the right and to the left, as needed.

26. Copyright © 2005 Pearson Education, Inc. Slide 4-26 Graph y =  2 sin 4x Step 1Solve inequality: Step 2Divide the interval into four equal parts. Step 3Assign pattern of sine values (0, 1, 0, -1, 0) multiplied by “a” (-2) to each of these five numbers:

27. Copyright © 2005 Pearson Education, Inc. Slide 4-27 Graph y =  2 sin 4x continued Step 4Plot the points and join them with a smooth curve

28. Copyright © 2005 Pearson Education, Inc. Slide 4-28 Homework 4.1 Page 141 All: 1 – 32 MyMathLab Assignment 4.1 for practice MyMathLab Homework Quiz 4.1 will be due for a grade on the date of our next class meeting

29. Copyright © 2005 Pearson Education, Inc. 4.2 Translations of the Graphs of the Sine and Cosine Functions

30. Copyright © 2005 Pearson Education, Inc. Slide 4-30 Horizontal Translations The horizontal movement of a graph to the left or right of its original position is called a “horizontal translation” In circular functions, a horizontal translation is called a phase shift. A phase shift in a circular function occurs when a term is added or subtracted from an argument: When a positive number “d” is added to an argument the phase shift is “d” units to the left When a positive number “d” is subtracted from an argument, the phase shift is “d” units to the right

31. Copyright © 2005 Pearson Education, Inc. Slide 4-31 Example of a Phase Shift in the Sine Function Compared with, the graph of will be moved units to the left. The amplitude will still be “1” and the period will still be

32. Copyright © 2005 Pearson Education, Inc. Slide 4-32 Sketch the graph of The primary period of this function will occur when: Solving for x gives:

33. Copyright © 2005 Pearson Education, Inc. Slide 4-33 Sketch the graph of Divide this interval: into quarter points: Assign pattern of sine values (0, 1, 0, -1, 0) to each of these five numbers: Connect the points with a smooth curve:

34. Copyright © 2005 Pearson Education, Inc. Slide 4-34 Sketch the graph of Points to be connected with smooth curve:

35. Copyright © 2005 Pearson Education, Inc. Slide 4-35 Graph y = sin (x   /3) Find the interval for one primary period. Divide the interval into four equal parts. Assign pattern of sine values (0, 1, 0, -1, 0) to each of these five numbers:

36. Copyright © 2005 Pearson Education, Inc. Slide 4-36 Graph y = sin (x   /3) continued Plot:

37. Copyright © 2005 Pearson Education, Inc. Slide 4-37 Graph Note: There will be both an amplitude change and a phase shift. Find the interval of primary period: Divide into four equal parts. Assign pattern of cosine values, (1, 0, -1, 0, 1), multiplied by 3, to each of these five numbers (3, 0, -3, 0, 3):

38. Copyright © 2005 Pearson Education, Inc. Slide 4-38 Graph continued  Plot:

39. Copyright © 2005 Pearson Education, Inc. Slide 4-39 Vertical Translations The vertical movement of a graph up or down from its original position is called a “vertical translation” A vertical translation of a circular function occurs when a term is added to, or subtracted from, a basic function When a positive number “c” is added to a basic function the vertical translation is “c” units up When a positive number “c” is subtracted from a basic function the vertical translation is “c” units down In both these cases the effect is to move the “x-axis” from y = 0 to y = c

40. Copyright © 2005 Pearson Education, Inc. Slide 4-40 Example of a Vertical Translation in the Sine Function Compared with, the graph of will be moved unit up The amplitude will still be “1” and the period will still be There will be no phase shift

41. Copyright © 2005 Pearson Education, Inc. Slide 4-41 Sketch the graph of The primary period of this function will occur when: Divide this interval into quarter points: Assign pattern of sine values (0, 1, 0, -1, 0), with “1” added to each, to each of these five numbers: Connect the points with a smooth curve:

42. Copyright © 2005 Pearson Education, Inc. Slide 4-42 Sketch the graph of Points to be connected with smooth curve:

43. Copyright © 2005 Pearson Education, Inc. Slide 4-43 Analyzing Functions of the Form: Before analyzing transformations, these functions should be written in the exact form shown here, including having the argument factored as shown. Compared with the basic functions: “a” changes the amplitude from 1 to | a | if “a” is negative, the graph is inverted “b” changes the period from “c” causes a vertical translation of “c” units “d” causes a phase shift (horizontal translation) of “d” units (If “d” is positive, then “x – d” gives a phase shift “d” units to the right, and “x + d” gives a phase shift “d” units left)

44. Copyright © 2005 Pearson Education, Inc. Slide 4-44 Analyze y = 2  2 sin 3x compared with y = sin x The graph will have an amplitude of 2 and will be inverted Its period will be It will have a vertical translation of +2 (2 units up) It will have no phase shift

45. Copyright © 2005 Pearson Education, Inc. Slide 4-45 Further Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = c + a sin b(x - d) or y = c + a cos b(x – d), with b > 0, follow these steps. Step 1To find the interval for one period, solve the inequality: and lay off this interval on the x-axis (the number at the right end will be the new period) Step 2Find numbers to divide the interval into four equal parts. Step 3Assign the appropriate sine (0, 1, 0, -1, 0) or cosine (1, 0, -1, 0, 1) patterns, multiplied by “a”, and with “c” added, to each of these five x-values.

46. Copyright © 2005 Pearson Education, Inc. Slide 4-46 Guidelines for Sketching Graphs of Sine and Cosine Functions continued Step 4Plot the points found in Step 3, and join them with a smooth curve having amplitude |a|. Step 5Draw the graph over additional periods, to the right and to the left, as needed.

47. Copyright © 2005 Pearson Education, Inc. Slide 4-47 Sketch the graph of

48. Copyright © 2005 Pearson Education, Inc. Slide 4-48 Analyze y =  1 + 2 sin (4x +  ) compared with y = sin x First factor the coefficient of x from argument: Period: Phase Shift: Amplitude: Vertical Translation:

49. Copyright © 2005 Pearson Education, Inc. Slide 4-49 Graph y =  1 + 2 sin (4x +  )

50. Copyright © 2005 Pearson Education, Inc. Slide 4-50 Homework 4.2 Page 152 All: 1 – 12, 17 – 22, 27 – 28, 31 – 34, 39 – 46 MyMathLab Assignment 4.2 for practice MyMathLab Homework Quiz 4.2 will be due for a grade on the date of our next class meeting

51. Copyright © 2005 Pearson Education, Inc. 4.3 Graphs of Other Circular Functions

52. Copyright © 2005 Pearson Education, Inc. Slide 4-52 Graphs of Cosecant and Secant Functions Since these functions are reciprocals of the sine and cosine functions, they also have period The graph of one period can be done over any interval of the domain that has this length, but we will call the interval the primary period At the values of the domain where sine and cosine have values of 0, cosecant and secant will be undefined, these values of the domain establish vertical asymptotes, shown on the graphs as vertical dashed lines (asymptotes are not actually part of the graph, but they show where a function is undefined) Graphs will be continuous between these vertical asymptotes

53. Copyright © 2005 Pearson Education, Inc. Slide 4-53 Graphs of Cosecant and Secant Functions You will recall that values of the cosecant and secant functions (ranges) will be so no portion of their graphs will be in the interval Cosecant and secant functions will have value 1 and -1 when sine and cosine functions have these values Other values of cosecant and secant functions can be found by reciprocating sine and cosine values

54. Copyright © 2005 Pearson Education, Inc. Slide 4-54 Cosecant Function

55. Copyright © 2005 Pearson Education, Inc. Slide 4-55 Secant Function

56. Copyright © 2005 Pearson Education, Inc. Slide 4-56 Reference Graphs for Cosecant and Secant By reciprocal identities already learned To sketch the graphs of cosecant or secant with these arguments, we can first sketch the corresponding sine and cosine graphs with those same arguments as references Where the sine and cosine graphs have value of 0 draw vertical asymptotes to show places where the cosecant and secant functions are undefined Reciprocate the values of the sine and cosine functions and sketch the graphs of cosecant and secant

57. Copyright © 2005 Pearson Education, Inc. Slide 4-57 Graph Primary Period of

58. Copyright © 2005 Pearson Education, Inc. Slide 4-58 Graphing First graph the reference function: Note that functions boxed in red are reciprocals Also note that both functions have undergone the same vertical stretching or squeezing and, the same vertical translation Therefore, to complete the graph is it only necessary to attached the U-shaped cosecant curve to the reference curve

59. Copyright © 2005 Pearson Education, Inc. Slide 4-59 Graphing First graph the reference function: Note that functions boxed in red are reciprocals Also note that both functions have undergone the same vertical stretching or squeezing and, the same vertical translation Therefore, to complete the graph is it only necessary to attached the U-shaped secant curve to the reference curve

60. Copyright © 2005 Pearson Education, Inc. Slide 4-60 Sketch Graph:

61. Copyright © 2005 Pearson Education, Inc. Slide 4-61 Visualizing and Graphing the Tangent Function

62. Copyright © 2005 Pearson Education, Inc. Slide 4-62 Observations About the Tangent Function and Graph of y = tan x The tangent will be undefined at and every odd multiple of it (the graph will have vertical asymptotes at these values of the domain) The graph will be continuous and all possible values of tangent will be obtained as x varies between Based on this last observation, the period of the tangent function will be and will be called the primary period Since there are no maximum or minimum values for tangent, amplitude is not defined

63. Copyright © 2005 Pearson Education, Inc. Slide 4-63 Tangent Function Endpoints & Quarter Points: Tangent Values:

64. Copyright © 2005 Pearson Education, Inc. Slide 4-64 Cotangent Function This function is the reciprocal of the tangent function At places where the tangent has value of 0, cotangent will be undefined and the graph will have vertical asymptotes at those values At odd multiples of where the tangent is undefined, the value of cotangent is 0 Using these facts and taking reciprocals of tangent values the graph of cotangent can be established as follows:

65. Copyright © 2005 Pearson Education, Inc. Slide 4-65 Cotangent Function as a Reciprocal of Tangent The period is: Primary period: Endpoints and Quarter Points: Cotangent values:

66. Copyright © 2005 Pearson Education, Inc. Slide 4-66 Cotangent Function

67. Copyright © 2005 Pearson Education, Inc. Slide 4-67 Sketching Graphs of Transformed Tangent Functions To graph Determine the primary period by solving for x: Find endpoints and quarter points Assign pattern of tangent values multiplied by “a” with “c” added Note: Analysis of phase shift, period change, vertical stretching or squeezing, and vertical translation will be the same as for sine and cosine

68. Copyright © 2005 Pearson Education, Inc. Slide 4-68 Sketch

69. Copyright © 2005 Pearson Education, Inc. Slide 4-69 Sketching Graphs of Transformed Cotangent Functions To graph Determine the primary period by solving for x: Find endpoints and quarter points Assign pattern of tangent values multiplied by “a” with “c” added Note: Analysis of phase shift, period change, vertical stretching or squeezing, and vertical translation will be the same as for sine and cosine

70. Copyright © 2005 Pearson Education, Inc. Slide 4-70 Sketch

71. Copyright © 2005 Pearson Education, Inc. Slide 4-71 Homework 4.3 Page 165 All: 1 – 6, Even: 8 – 46 MyMathLab Assignment 4.3 for practice MyMathLab Homework Quiz 4.3 will be due for a grade on the date of our next class meeting

72. Copyright © 2005 Pearson Education, Inc. 4.4 Harmonic Motion

73. Copyright © 2005 Pearson Education, Inc. Slide 4-73 Simple Harmonic Motion The position of a point oscillating about an equilibrium position at time t is modeled by either where a and are constants, with The amplitude of the motion is |a|, the period is and the frequency is

74. Copyright © 2005 Pearson Education, Inc. Slide 4-74 Example Suppose that an object is attached to a coiled spring such the one shown (on the next slide). It is pulled down a distance of 5 in. from its equilibrium position, and then released. The time for one complete oscillation is 4 sec. a) Give an equation that models the position of the object at time t. b) Determine the position at t = 1.5 sec. c) Find the frequency.

75. Copyright © 2005 Pearson Education, Inc. Slide 4-75 Example continued When the object is released at t = 0, distance the object of the object from its equilibrium position 5 in. below equilibrium. We use  a =  5 

76. Copyright © 2005 Pearson Education, Inc. Slide 4-76 Example continued The motion is modeled by b) After 1.5 sec, the position is Since 3.54 > 0, the object is above the equilibrium position. c) The frequency is the reciprocal of the period, of ¼.