Applications - Part I - Textbook 1 [Q&A]

CHECK Your Understanding 1. Determine an equation for each sinusoidal function at the right. 2. Deer he the function at models the data in the table and does nor 0° 45° 90° 135° 180° 225° 270° 7 5 7 9 7 5 3. A sinusoidal function has an amplitude of 4 units, a period of 120°, and a maximum at (0, 9). Determine the equation of the function. 4. Determine the equation for each sinusoidal function. 5. For each table of data, determine the equation of the function that is the simplest model. a) x 0° 30° 3 2 60° 1/ 90° 2 120° 3 150° 2 180° 1/ b) c) d) x - 180° 17 0° 13 180° 17 x 360° 21 60" 540° 17 720° 13 900° 17 x - 120° -4 -20° 2 - 60° -7 0° 4 10° 5 40° 2 70° -1 120° -4 100° 2 180° -7 240° -4 130° 5 160° 2 6. Determine the equation of the cosine function whose graph has each of the K following features: a) b) d) 6. Determine the equation of the cosine function whose graph has each of the I following features: Amplitude Period 3 4 2 0.5 360° 180° 40° 720° Equation of the Axis y = 11 y = 15 y = 0 y = -3 Horizontal Translation 0° 30° 7° -56° 7. A sinusoidal function has an amplitude of 6 units, a period of 45°, and a minimum at (0, 1). Determine an equation of the function. 8. The table shows the average monthly high temperature for one year in A Kapuskasing, Ontario. a) Draw a scatter plot of the data and the curve of best fit. Let January be month 0. b) What type of model describes the graph? Why did you select that model? c) Write an equation for your model. Describe the constants and the variables in the context of this problem. d) What is the average monthly temperature for month 20? 9. The table shows the velocity of air of Nicole's breathing while she is at rest. Time (s) Velocity (L/s) 0 0.25 0.5 0.22 0.45 0.75 0.61 1.0 0.75 1.25 0.82 1.5 0.85 1.75 2 0.83 0.74 2.25 0.61 2.5 0.43 2.75 0.23 a) Explain why breathing is an example of a periodic function. b) Graph the data, and determine an equation chat models the situation. c) Using a graphing calculator, graph the data as a scatter plot. Enter your equation into the equation editor and graph. Comment on the closeness of fit between the scatter plot and the graph. d) What is the velocity of Nicole's breathing at 6 s? Justify. e) How many seconds have passed when the velocity is 0.5 L/s? 10. The table shows the average monthly temperature for three cities: Athens, Lisbon, and Moscow. a) Graph the data to show that temperature is a function of time for each city. b) Write the equations that model each function. c) Explain the differences in the amplitude and the vertical translation for each city. d) What does this tell you about the cities? 11. The relationship between the stress on the shaft of an electric motor and time can be modelled with a sinusoidal function. (The units of stress are megapascals (MiPa).) a) Determine an equation of the function that describes the equivalent stress in terms of time. b) What do the peaks of the function represent in this situation? c) How much stress was the motor undergoing at 0.143 s? 12. Describe a procedure for writing the equation of a sinusoidal function based c on a given graph. Extending 13. The diameter of a car's tire is 60 cm. While the car is being driven, the tire picks I up a nail. How high above the ground is the nail after the car has travelled 1 km? 14. Matthew is riding a Ferris wheel at a constant speed of 10 km/h. The boarding height for the wheel is 1 m, and the wheel has a radius of 7 m. What is the equation of the function that describes Matthew's height in terms of time, assuming Matthew starts at the highest point on the wheel? 6.6 Investigating Models of Sinusoidal Functions, pp. 391-393 a) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0° and x = 90°, for example, the period is 90 - 0 or 90°. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the mini- mum. Since the minimum is 4 and the maximum 8 + 4 is 8, the equation of the axis is y = y - 6. The amplitude is half the distance between the maximum and minimum values. Since the minimum is 4 and the maximum is 8, the amplitude is or 2. Also, the function crosses the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore. the equation for the sinusoidal 360 function is y = 2 cos 90 -x)° + 6 or y = 2 cos (4x)° + 6. This can also be expressed as y = 2 sin (4x + 90)° + 6. b) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0° and x = 180%, for exam- ple, the period is 180 - 0 or 180°. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is 1 and the maximum is 3, the equation of the axis is y -3+1 or y - 2. The amplitude is half the distance between the maximum and minimum values. Since the minimum is 1 and the maxi- mum is 3, the amplitude is or 1. Also. if the function had not been shifted 90° to the right, then it would have crossed the y-axis at its maximum valve, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal function 360 is y * COs (x - 90)° + 2 or 180 y = cos (2x - 180)º + 2. This can also be expressed as y = sin (2x - 90)* + 2. c) The period is the change in x that occurs as the function goes through one complete cycic. Since the graph goes through one complete cycle between x = 0° and x = 120°, for exam- pic, the period is 120 - 0 or 120°. The equation of the axis is the equation of the horizontal line that is halfway berween the maximum and the minimum. Since the minimum is - 4 and the maximum is 0, the equation of the axis is y = 0+ (-4) or y = - 2. The amplitude is half the distance between the maximam and minimum values. Since the minimum is - 4 and 0 - (-4) the maximum is 0, the amplitude is 2. Also, the function crosses the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal function is y = 2 cos/ 360 - 2 or y = 2 cos (3x)° - 2. 120 This can also be expressed as y = 2 sin (3x + 90)° - - 2 2. The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0° and x = 180°, for exam- ple, the period is 180 - 0 or 180°. The equation of the axis is the equation of the horizontal line that is halfway berween the maximum and the minimum. Since the minimum is 5 and the maximum is 9, the equation of the axis is 9 + 5 y = or y = 7. The amplitude is half the distance between the maximum and minimum values. Since the minimum is 5 and the maxi- mum is 9, the amplitude is - - or 2. Also, the function crosses the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, the equation for the 360 sinusoidal function is y = 2 cos 180" + 7 or y = 2 cos(2x)° + 7. This can also be expressed as y = 2 sin (2x + 90)° + 7. 3. Since the sinusoidal function has an amplitude of 4 units and a maximum at (0, 9), the equation of the axis must be y = 9 - 4 or y = 5. Also, since the maximum is at (0, 9), the graph crosses the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, since the period of 120° is already known, the equation for the sinusoidal function is y = 4 cos 300, 120* + 5 or y = 4 cos (3x)° + 5. This can also be expressed as y = 4 sin (3x + 90)º + 5. i) The period is the change in x that occurs as the function gocs through one complete cycle. Since the graph for i goes through one complete cycle between x = 4 and x = 10. for example, the period is 10 - 4 or 6. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is 2 and the maximum is 8, the 8 + 2 equation of the axis is y = • or y = 5. The amplitude is half the distance berween the max- imum and minimum values. Since the minimum 8-2 is 2 and the maximum is 8, the amplitude is or 3. Also, if the function had not been shifted 4 units to the right, it would have crossed the y-axis at its maximum, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal function is 360 y = 3 cos 6 (x - 4) ° + 5 05 y = 3 cos [60(* - 4)J° + 5. This can also be expressed as y = 3 sin (60(x - 2.5)]° + 5. ii) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph for ii goes through one com- plete cycle between x = 0 and x = 3, for exam- ple, the period is 3 - 0 or 3. The equation of the axis is the equation of the horizontal lime that is halfway between the maximum and the mintmum. Since the mummum is 0.5 and the maximum is 1.5, the equation of the axis is y = 15 + 05 or y = 1. The amplitude is half the distance between the maximum and mini- mum values. Since the minimum is 0.5 and the 1.5 - 0.5 maximum is 1.5, the amplitude is or 0.5. Also, if the function had not been shifted 1.5 units to the right, then it would have crossed the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal function is 360 y = 0.5 cos (*- 15) + 1 or y = 0.5 cos (120x - 180)* + 1. This equation can also be expressed as y = 1-0.5 cos (120x)]° + 1 or y = [-0.5 sin (120x + 90)]° + 1. iii) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph for iii goes through one com- plete cycle between x = 0 and x = 4, for exam- ple, the period is 4 - 0 or 4. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is - 3 and the maximum is - 1, the equation of the axis is - 1 + (-3) y= - - or y = - 2. The amplitude is half the distance berween the maximum and minimum values. Since the minimum is - 3 and the maximum is - 1, the amplitude is - - (-3) or 1. Also, if the function had not been shifted 3 units to the right, then it would have crossed the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, the equation for the sinu- 360| soidal function is y = cos (х - 3) - 2 ог y = cos (90x - 270)° - 2. This equation can also be expressed as y = cos [90(x - 3)]º - 2 or as y = sin [90(x- 2)j° - 2. i) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph for i goes through one complete cycle between x = 0 and x = 2, for example, the period is 2 - 0 or 2. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the mini- mum. Since the minimum is 20 and the maxi- mum is 30, the equation of the axis is y = 30 + 20 or y = 25. The amplitude is half the distance between the maximum and mini- mum values. Since the minimum is 20 and the 30 - 20 maximum is 30. the amplitude is or 5. Also, the function crosses the y-axis at its midpoint and is reflected in the y-axis. so the equation for it should use the sine function and the value of & should be negative. Therefore, the equation for the sinusoidal - 360 function is y = 5 sin + 25 or y = 5 sin (-180x)° + 25. This equation can also be expressed as y = 5 cos (180(*-1.5)J° + 25. il) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph for ii goes through one com- plete cycle between x = 0.5 and x - 3.5, for example, the period is 3.5 - 0.5 or 3. The equa- tion of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is 5 and the maximum is 15, the equation of the axis is y = 15 + 5 or y = 10. The amplitude is half the distance between the maximum and mini- mum values. Since the minimum is 5 and the 15 - maximum is 15, the amplitude is or 5. Also, if the function had not been shifted 2 units to the right, then it would have crossed the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal 360 function is y = 5 cos (x - 2)] + 10 ог y = 5 cos (120x- 240)° + 10. This equation can also be expressed as y = 5 cos (120(x - 2)j* + 10 or y = Ssin (120(x - 1.25)J° + 10. ili) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph for ill goes through one com- piete cycle between x = 0 and x » 1, for exam- ple, the period is 1-0 or 1. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is - 15 and the maximum is 5, the equation of the axis is 5 + 15) or y = - 5. The amplitude is half the distance between the maximum and minimum values. Since the minimum is - 15 and the maximum is 5, the amplitude is 5 - 15) or 10. Also, the function crosses the y-axis at its maximum value, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal function is y = 10 cos - 5 ог y = 10cos (360x)° - 5. This can also be expressed as y = 10 sin (360x ÷ 90)° - 5. 5. a) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0° and x ~ 120°. for example, the period is 120 - 0 or 120°. The equation of the axis is the equation of the horizontal line that is halfway between the max- imum and the minimum. Since the minimum is 1 and the maximum is 3, the equation of the 3 + 1 axis is y = or y = 2. The amplitude is half the distance between the maximum and minimum values. Since the minimum is 1 and the maximum is 3, the amplitude is or 1. Also, the function crosses the y-axis at its maxi- mum value, so the equation for it should use the cosine function. Therefore, the equation for the 360 sinusoidal function is y = cos 120* + 2 or y = cos (3x)° + 2. This can also be expressed as y = sin(3r + 90)º + 2. b) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0º and x = 720°, for exam- ple, the period is 720 - 0 or 720°. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is 13 and the maximum is 21, the equation of the axis is y = 13 + 21 - or y = 17. The amplitude is half the distance between the maximum and mini- mum values. Since the minimum is 13 and the 21 13 maximum is 21, the amplitude is - or 4. Also, the function crosses the y-axis at its mini- mum value, so the equation for it should use the cosine function and be shifted 180° to the right. Therefore, the equation for the sinusoidal func- 360 tion is y = 4 cos (x - 180)° + 17 or 720 y = 4cos 2(* - 180)° + 17. This can also be expressed as or y = 4 sin 2* ° + 17. c) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = -120° and x = 120°, for example, the period is 120 - (- 120) or 240°. The equa- tion of the axis is the cquation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is -7 and the maximum is - 1. the equation of the axis is y = - 4. The amplitude is half the distance between the max- imum and minimum values. Since the minimum is - 7 and the maximum is - 1, the amplitude is (-1) - (-7) or 3. Also, the function crosses the y-axis at its midpoint, so the equation for it should use the sine function. Therefore, the equation for the sinusoidal function is 360 y = sin 240* -4 ory = 3sin (Ex) -4. This function can also be expressed as y = 3 cos 2(*- 60) - 4. d) The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = - 20° and x ~ 100", for example, the period is 100 - (-20) or 120°. The equation of the axis is the equation of the horizontal line that is halfway berween the max- imum and the minimum. Since the minimum is - 1 and the maximum is 5. the equation of the 5 + (1 axis is y or y = 2-The amplitude is half the distance berween the maximum and minimum values. Since the minimum is - 1 and 51 the maximum is 5, the amplitude is or 3. Also, the function would cross the y-axis at its maximum value, except that the function is shifted 10° to the right, so the equation for it should use the cosine function. Therefore, the equation for the sinusoidal function is y = 3 cos 360 120 (* - 10) * + 2 0г y = 3 cos (3(x - 10)J* + 2. This can also be expressed as y = 3 sin [3(x + 20)}° + 2. 6. a) Since the amplitude is 3, a in the function y = a cos (k(x - d)) + cis 3. Since the period is 360°, k in the function