CHAPTER 8. Sinusoidal Functions

Transcription

1 CHAPTER 8 Sinusoidal Functions

2 8.1 UNDERSTANDING ANGLES Chapter 8

3 DEFINITIONS A central angle is an angle whose vertex is at the center of a circle. The unit circle is a circle whose radius is 1.

4 Arc length refers to part of the circumference of a circle subtended by a central angle.

5 MEASURING ANGLES Up to this point, we have only measured angles in degrees (i.e ). Here, we will learn about a different unit for measuring angles called radians.

6 INVESTIGATING RADIAN MEASURE Answer the following 1. What is the circumference of the unit circle? (Radius = 1) 2. How many degrees are in a full circle?

7 3. What conclusion/connection can we make between the circumference of a circle and its measure in degrees? 4. What is another way of writing 180 0?

8 WE WILL USE THIS STATEMENT TO HELP US CONVERT FROM DEGREE MEASURE TO RADIAN MEASURE AND VICE VERSA. Angles measured in radians can be written in decimal form, or they can be written in terms of. (i.e. 3, 2, etc ) 4 Note! Angles written in radians will not have a unit/symbol. Degrees Radians 2 4

9 CONVERSIONS FACTORS Converting from degrees to radians, we multiply by 180 Note! An easy way to remember this is, in radian measure, the pi symbol is on top, so it must be on top in the conversion factor.

10 CONVERSIONS FACTORS Converting from radians to degrees, we multiply by 180 Note! We want the pi symbol in the radian measure to cancel out for the degree value, so we must put in in the bottom of the conversion factor to ensure that it will cancel.

11

12 EXAMPLES. CONVERT THE FOLLOWING DEGREES TO RADIAN MEASURE. A B C

13 D E F

14 EXAMPLES. CONVERT THE FOLLOWING RADIANS TO DEGREES. ROUND YOUR ANSWER TO THE NEAREST DEGREE. A. 4 B. 5 6 C. 2 7

15 D. 5 E. 3.2

16 EXAMPLE. HOW MANY DEGREES ARE IN ONE RADIAN?

17 EXAMPLE. WHICH IS LARGER 8 OR 3?

18 P #1, 2, 4, 5, 8 NOTE! DO NOT SKETCH Independent Practice THE ANGLES.

19 JUST TO RECAP! Radian measure is an alternate way to express the size of an angle. Using radians allows you to express the measure of an angle as a real number without units. Think of radians as the length of a piece of string needed to cover the arc formed by the central angle on the unit circle. The central angle formed by one complete revolution in a circle is or 2π radians.

20 NEED TO KNOW! Use benchmarks to estimate the degree measure of an angle given in radians. π = π = radian 60 0

21 NEED TO KNOW! Degrees Radians (in terms of π) Radians (as a decimal) π / π π / π 6.28

22 8.2 EXPLORING GRAPHS OF PERIODIC Chapter 8 FUNCTIONS

23 DEFINITIONS A periodic function is a function whose graph repeats in regular intervals or cycles.

24 THESE ARE NOT PERIODIC FUNCTIONS

25 DEFINITIONS The midline is the horizontal line halfway between the maximum and minimum values of a periodic function.

26 DEFINITIONS The amplitude is the distance from the midline to either the maximum or minimum value of a periodic function.

27 DEFINITIONS The period is the length of the interval to complete one cycle.

28 CONSIDER Y = SIN(X) Complete the following table of values: x 0 o 30 o 60 o 90 o 120 o 150 o 180 o 210 o 240 o y x 270 o 300 o 330 o 360 o 390 o 420 o 450 o 480 o 510 o y

29 CONSIDER Y = SIN(X)

30 Y = SIN(X) 1. What is the y-intercept (starting point) of the graph? 2. What are the x-intercepts?

31 Y = SIN(X) 3. What is the maximum value? 4. For what value of x does the maximum value occur?

32 Y = SIN(X) 5. What is the minimum value? 6. For what value(s) of x does the minimum value occur?

33 Y = SIN(X) 7. Is the graph periodic? 8. What is the period of the graph?

34 Y = SIN(X) 9. What is the amplitude of the graph? 10. What is the equation of the midline?

35 Y = SIN(X) 11. What is the domain of y=sin(x)? 12. What is the range of y=sin(x)?

36 CONSIDER Y = COS(X) Complete the following table of values: x 0 o 30 o 60 o 90 o 120 o 150 o 180 o 210 o 240 o y x 270 o 300 o 330 o 360 o 390 o 420 o 450 o 480 o 510 o y

37 CONSIDER Y = COS(X)

38 Y = COS(X) 1. What is the y-intercept (starting point) of the graph? 2. What are the x-intercepts?

39 Y = COS(X) 3. What is the maximum value? 4. For what value of x does the maximum value occur?

40 Y = COS(X) 5. What is the minimum value? 6. For what value(s) of x does the minimum value occur?

41 Y = COS(X) 7. Is the graph periodic? 8. What is the period of the graph?

42 Y = COS(X) 9. What is the amplitude of the graph? 10. What is the equation of the midline?

43 Y = COS(X) 11. What is the domain of y=cos(x)? 12. What is the range of y=cos(x)?

44 COMPARING Y = SIN(X) AND Y = COS(X) 1. How are the graphs of the sine and cosine functions similar?

45 COMPARING Y = SIN(X) AND Y = COS(X) 2. How are the graphs different?

46 COMPARING Y = SIN(X) AND Y = COS(X) 3.Without having to rely on our calculators and complete a table of values each time, which 5 points could we use to get an accurate sketch of either the sine or cosine graph?

47

48 NOW WITH RADIANS y = sin(x) Complete the following table of values: x 0 π / 2 π 3π / 2 2π 5π / 2 3π 7π / 2 4π y

49 NOW WITH RADIANS y = sinx π 2π 3π 4π

50 NOW WITH RADIANS y = cos(x) Complete the following table of values: x 0 π / 2 π 3π / 2 2π 5π / 2 3π 7π / 2 4π y

51 NOW WITH RADIANS y = cosx

52 IN SUMMARY Degrees: Radians:

53 THE GRAPHS OF Y = SIN(X) AND Y = COS(X) HAVE THE FOLLOWING COMMON CHARACTERISTICS: Multiple x-intercepts One y-intercept A domain of {x / x ε R} A range of {y / -1 y 1, y ε R} An amplitude of 1 A period of or 2π A midline defined by the equation y = 0

54 P #1, 2, 4, 5, 6 Independent Practice

55 8.3 THE GRAPHS OF SINUSOIDAL FUNCTIONS Chapter 8

56 PROPERTIES OF SINUSOIDAL FUNCTIONS A Sinusoidal Function is one whose graph has the same shape as the graphs of y=sin(x) and/or y=cos(x). Are the following graphs sinusoidal? Are they periodic? y x Sinusoidal And Periodic Sinusoidal And Periodic Neither Periodic NOR Sinusoidal

57 PROPERTIES OF SINUSOIDAL FUNCTIONS 1. Using the graphs above as reference, are all sinusoidal functions periodic? 2. Are all periodic functions sinusoidal?

58 PROPERTIES OF SINUSOIDAL FUNCTIONS Recall the terms Midline, Amplitude, Period, Minimum Value and Maximum Value

59 PROPERTIES OF SINUSOIDAL FUNCTIONS The maximum and minimum points can be easily read directly from a graph. The equation of the Midline, Amplitude, Period and Range require a little more work to determine.

60 EQUATION OF THE MIDLINE The equation of the midline is the average of the maximum and minimum values: y maximum value 2 minimum value

61 AMPLITUDE The amplitude is the positive vertical distance between the midline and either a maximum or minimum value. It is also half of the vertical distance between a maximum value and a minimum value. Amplitude maximum value 2 minimum value

62 PERIOD The period of a sinusoidal function can be calculated in three ways. It represents the horizontal distance between : 2 consecutive Maximum Values 2 consecutive Minimum Values 3 consecutive points on the Midline

63 RANGE The range is the difference in the maximum and minimum values. Range maximum value minimum value

64 EXAMPLE 1. The graphs of a sinusoidal function is shown. Describe this graph by determining its range, the equation of its midline, its amplitude, and its period. Range: Equation of Midline:

65 EXAMPLE 1. The graphs of a sinusoidal function is shown. Describe this graph by determining its range, the equation of its midline, its amplitude, and its period. Amplitude: Period:

66 EXAMPLE 2. The graph of a sinusoidal function is shown. Describe this graph by determining its range, the equation of its midline, its amplitude, and its period. Range: Equation of Midline:

67 EXAMPLE 2. The graph of a sinusoidal function is shown. Describe this graph by determining its range, the equation of its midline, its amplitude, and its period. Amplitude: Period:

68 3. Alexis and Colin own a car and a pickup. They noticed that the odometers of the two vehicles gave different values for the same distance. As part of their investigation into the cause, they put a chalk mark on the outer edge of a tire on each vehicle. The following graphs show the height of the tires as they rotated while the vehicles were driven at the same slow, constant speed. What can you determine about the characteristics of the tires from these graphs?

69 COMPLETE THE FOLLOWING TABLE Car Pickup Truck Range What does the range represent? Amplitude What does the amplitude represent? Period What does the period represent?

70 4. WHILE RIDING ON A FERRIS WHEEL, MASON'S HEIGHT ABOVE THE GROUND IN TERMS OF TIME CAN BE REPRESENTED BY THE FOLLOWING GRAPH:

71 1. Identify the period of the graph. What does this represent? 2. Identify the range of the data. What does this represent?

72 3. Identify the amplitude. What does this represent? 4. State the x- and y-intercepts. What do these values represent?

73 5. Determine the equation of the midline. What does this represent? 6. What was Mason s height at 10 seconds?

74 7. What was the first time at which Mason reached a height of 7 metres? 8. What was the first time at which Mason reached a height of 3 m while going downward on the Ferris Wheel?

75 PG , #1, 2, 3, 4, 5, 8, 9, 13, 14 Independent Practice

76 8.4 THE EQUATIONS OF SINUSOIDAL Chapter 8 FUNCTIONS

77 In Math 2201, we examined the quadratic function y=a(x - h) 2 + k, and learned how to read properties such as vertex, direction of opening, etc from the equation. Here we will examine transformations of the sine and cosine functions, and will learn how to read various properties from the equation.

78 TRANSFORMED SINUSOIDAL EQUATIONS ARE WRITTEN IN THE FOLLOWING FORMS: y = asinb(x-c) + d And y = acosb(x-c) + d

79 EXAMINING THE IMPACT OF THE VALUE OF A ON THE GRAPH OF A SINUSOIDAL FUNCTION. Consider the graphs shown for each equation (A) y = sinx (B) y = 2sinx y y x x

80 (C) y = 5sinx (D) y = 0.5sinx y y x x

81 QUESTIONS 1. What does the value of a represent? 2. How is the shape of the graph affected by a?

82 QUESTIONS 3. What other properties besides amplitude are affected by the value of a? 4. Will the value of a affect the cosine graph in the same way that it affects the sine graph?

83 SUMMARY OF THE A VALUE The a value stretches or compresses a graph vertically. It equals the amplitude of the function. Amplitude = a

84 EXAMPLES: DETERMINE THE AMPLITUDE OF EACH SINUSOIDAL FUNCTION. (A) y = 2sin3(x - 90 ) + 1 (B) y = 0.75cos2(x - 45 ) - 3

85 EXAMINING THE IMPACT OF THE VALUE OF D ON THE GRAPH OF A SINUSOIDAL FUNCTION. Consider the graphs shown for each equation (A) y=sinx (B) y=sinx + 2 y y x x

86 (C) y=sinx - 3 y x

87 QUESTIONS 1. What does the value of d represent? 2. Is the shape of the graph affected by d?

88 QUESTIONS 3. What other properties besides the equation of the midline are affected by the value of d? 4. Will the value of d affect the cosine graph in the same way that it affects the sine graph?

89 SUMMARY OF THE D VALUE The d value gives us the equation of the midline of a sinusoidal function Equation of Midline: y = d Equation of Midline: y = d

90 EXAMPLES: WRITE THE EQUATION OF THE MIDLINE OF EACH FUNCTION. (A) y = 2sin3(x - 90 ) + 1 (B) y = 0.75cos2(x + 45 ) - 3

91 EXAMINING THE IMPACT OF THE VALUE OF B ON THE GRAPH OF A SINUSOIDAL FUNCTION. Consider the graphs shown for each equation (A) y = sinx (B) y = sin2x y y x x

92 (C) y = sin0.5x y x

93 QUESTIONS 1. What impact does the value of b have on each graph? 2. Write an equation that relates period to the value of b?

94 QUESTIONS 3. What other properties besides period are affected by the value of b? 4. Will the value of b affect the cosine graph in the same way that it affects the sine graph?

95 SUMMARY OF THE B VALUE The d value gives us the period of a sinusoidal function. In degrees, Period = 360 b In radians, Period = 2 b

96 EXAMPLES: DETERMINE THE PERIOD OF EACH FUNCTION. (A) y = 2sin3(x - 90 ) + 1 (B) y = 0.75cos2(x + 45 ) 3

97 (C) y 4cos3 x π 4 5

98 EXAMINING THE IMPACT OF THE VALUE OF C ON THE GRAPH OF A SINUSOIDAL FUNCTION. Consider the graphs shown for each equation (A) y = sinx (B) y x y y = sin(x-1.57) y sin x π 2 x

99 (C) y sinx

100 QUESTIONS 1. What impact does the value of c have on each graph? 2. How can we determine the phase shift from the equation of the sinusoidal function?

101 SUMMARY OF THE C VALUE The c value shifts a graph horizontally. The shift is obtained by taking the opposite sign of the number after x in the equation. Horizontal Shift / Translation = c

102 EXAMPLES: DETERMINE THE PERIOD OF EACH FUNCTION. (A) y = 2sin3(x - 90 ) + 1 (B) y = 0.75cos2(x + 45 ) 3

103 (C) y 4cos3 x π 4 5

104 MAXIMUM/MINIMUM POINTS The midline can be read from a graph, or it can be calculated if we are given an equation. Maximum Value = midline + amplitude Minimum Value = midline amplitude

105 Recall Amplitude = a Equation of midline: y = d Thus, Maximum Value = midline + amplitude = d + a Minimum Value = midline - amplitude = d - a

106 Maximum Value = d + a Minimum Value = d - a

107 DOMAIN AND RANGE OF A SINUSOIDAL FUNCTION For a sinusoidal function the domain is: {x / x ε R} However! This may change in application problems where the x-values are restricted. i.e. time-related word problems where x > 0. For a sinusoidal function the range is: {y minimum < y < maximum, y ε R}

108 EXAMPLE. STATE THE DOMAIN AND RANGE FOR EACH SINUSOIDAL FUNCTION: (A) y = 2sin3(x - 90 ) + 1 Minimum Value: Maximum Value: Domain: Range:

109 EXAMPLE. STATE THE DOMAIN AND RANGE FOR EACH SINUSOIDAL FUNCTION: (B) y = 4sin2(x + 60 ) - 3 Minimum Value: Maximum Value: Domain: Range:

110 EXAMPLE. STATE THE DOMAIN AND RANGE FOR EACH SINUSOIDAL FUNCTION: (C) y = 5cos3(x - π) + 4 Minimum Value: Maximum Value: Domain: Range:

111 EXAMPLE. STATE THE DOMAIN AND RANGE FOR EACH SINUSOIDAL FUNCTION: (D) y = 0.5cos4(x + 2π) - 6 Minimum Value: Maximum Value: Domain: Range:

112 IN SUMMARY The standard form for a sinusoidal function is: y = a sin b(x c) + d y = a cos b(x c) + d a the amplitude c the horizontal translation of the graph d the midline of the graph b the number of cycles in 360º (or 2π) You can find the period using: or

113 FROM A GRAPH a amplitude maximum value 2 minimumvalue Period: Max to Max Min to Min 3 consecutive points on the midline c = the starting point d maximum value 2 minimum value

114 Amplitude: EXAMPLES 1. Consider the function y = 2 cos4x + 1 for {x 0º x 360º, x R} Describe the graph of the function by stating the amplitude, equation of the midline, range, and period, as well as the relevant horizontal translation of y = cosx. (You can always verify your description by drawing a graph of this function using graphing technology). Equation of midline: Range: Period: Horizontal Translation:

115 EXAMPLE 2. Consider the function y = 3 sin2(x 45º) a) Describe the graph of the function by stating the amplitude, equation of the midline, range, and period, as well as the relevant horizontal translation of y = sinx. Amplitude: Equation of midline: Range: Period: Horizontal Translation:

116 How are they the same? EXAMPLE 2. Consider the function y = 3 sin2(x 45º) b) How would the graph of y = 3 cos2(x + 45º) be the same? How would it be different? How are they different?

117 EXAMPLE 3. The number of hours of daylight in Iqaluit, Nunavut, can be represented by the function y = sin (x ) where x is the day number in the year. a) How many hours of daylight occur in Iqaluit on the following days? i) the shortest day of the year? ii) the longest day of the year?

118 EXAMPLE The number of hours of daylight in Iqaluit, Nunavut, can be represented by the function y = sin (x ) where x is the day number in the year. b) In some years, June 21 is the longest day. Suppose that the Sun were to set in Iqaluit at 11:01 pm on June 21. At what time did the sun rise?

119 EXAMPLE The number of hours of daylight in Iqaluit, Nunavut, can be represented by the function y = sin (x ) where x is the day number in the year. c) What is the period of this sinusoidal function? Explain how the period relates ot the context of the problem?

120 EXAMPLE The number of hours of daylight in Iqaluit, Nunavut, can be represented by the function y = sin (x ) where x is the day number in the year. d) What does the value of c, , represent in the context of the problem?

121 PG , #1, 3, 5, 7, 8, 10, 12, 13, 15, 17 Independent Practice

122 8.5 MODELLING DATA WITH SINUSOIDAL Chapter 8 FUNCTIONS

123 EXAMPLE 1. Celeste lives in Red Deer, Alberta. The predicted hours of daylight for two consecutive years are shown in the tables below. In the second year, the spring equinox will occur on March 20 and the fall equinox will occur on September 23. Compare the hours of daylight on these two days. Hours of Daylight in Red Deer Alberta This Year Date Day # Length of Day (h) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Predicted Hours of Daylight in Red Deer Alberta Next Year Date Day # Length of Day (h) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan

124 y x 1. What type of function is modelled above? Sinusoidal

125 y x 2. What is the period of the function? First maximum point occurs at about day 180 The second maximum point occurs at about day 545. Therefore the period is = 365

126 y 3. On what days will there be approximately 12 hours of daylight? x Day 77 (March 18), Day 266 (Sept. 23), Day 444 (March 20), and Day 633 (Sept. 25)

127 y x 4. Suppose that the summer solstice will be on June 21 and the winter solstice Dec. 21 this year. Determine the approx. hours of daylight on each day. June 21 (Day 172) 16.5 hours Dec. 21 (Day 355) 7.8 hours

128 y x 1. What type of function is modelled above? Sinusoidal

129 EXAMPLE 2. Tara s high school class is planning a trip to British Columbia to study ecotourism next year. The students want to finish the tour, on May 21, by watching the Sun set over the Pacific Ocean. Tara found the following data for the current year and the following January. All the times are in standard time, in hours after noon, as decimal values. (Table on page 537 of text book) y 1. What is the period of the function? approximately 365 days 2. At what time should the class be at a vantage point on the cliff? Day 141 is May 21 about 8 hours after noon. Therefore around 8 pm x

130 EXAMPLE 3. In 2011, the Singapore Flyer was the largest Ferris wheel in the world. The table below gives the height of a rider from the ground at different times.

131 EXAMPLE Plot the data on the grid below and estimate when Jordy was level with Yale. y x

132 4. METEOROLOGIST BRYAN NODDON RECORDED THE AVERAGE PRECIPITATION IN GRAND FALLS WINDSOR FOR 2012 AND CREATED A SINUSOIDAL REGRESSION FOR THE DATA. (A) Use the graph to predict the amount of precipitation in August 2013.

133 (B) Write the equation of the sinusoidal regression SinReg y = a*sin(bx + c) + d a = b = c = d = Note! The equation on the calculator is in a different format than we use. You will NOT be required to know this form other than to read the equation from a snapshot as shown above and/or use it to interpolate or extrapolate data as in part C.

134 (C) Using the equation, predict the amount of precipitation in March 2017.