40s applied review questions for periodic functions
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Choose the best estimate for 120° in radians.
A.
B.
C.
D.
____
2. Choose the best estimate for 55° in radians.
A.
B.
C.
D.
____
2.1
0.7
2.8
3.1
0.5
1.0
1.5
2.0
3. Choose the best estimate for 136° in radians.
A.
B.
C.
D.
____
4. Choose the best estimate for 280° in radians.
A.
B.
C.
D.
____
5. Choose the best estimate for 0.1 radians in degrees.
A.
B.
C.
D.
____
3
4
5
6
0.5°
1°
3°
6°
6. Choose the best estimate for 0.8 radians in degrees.
A. 8°
B. 15°
C. 30°
D. 45°
____
7. Choose the best estimate for 7 radians in degrees.
A.
B.
C.
D.
____
8. Choose the best estimate for 3.1 radians in degrees.
A.
B.
C.
D.
____
263°
273°
283°
293°
10. Choose the best estimate for the central angle in degrees.
A.
B.
C.
D.
____
31°
85°
135°
175°
9. Choose the best estimate for the central angle in degrees.
A.
B.
C.
D.
____
400°
460°
520°
580°
84°
276°
444°
636°
11. Choose the best estimate for the central angle in radians.
A.
B.
C.
D.
____
12. Choose the best estimate for the central angle in radians.
A.
B.
C.
D.
____
13. Imagine that it is now 2 p.m. What time will it be when the minute hand has rotated through 300°?
A.
B.
C.
D.
____
4.2
4.8
5.2
5.8
2:40
2:50
3:00
3:10
14. Imagine that it is now 2 p.m. What time will it be when the minute hand has rotated through
radians?
A.
B.
C.
D.
2:20
2:52
3:15
3:45
____
15. Imagine that it is now 2 p.m. What time will it be when the minute hand has rotated through
1260°?
A.
B.
C.
D.
____
16. How many turning points does the graph of y = sin x have from 0° to 360°?
A.
B.
C.
D.
____
y = cos x
y=x
y = –sin x
y = –2 cos x
20. Identify the range of the graph of y = 1 + sin x.
A.
B.
C.
D.
____
90°
–90°
180°
270°
19. Which of the following is not a periodic function?
A.
B.
C.
D.
____
0°
360°
–180°
270°
18. Which of the following is not an x-intercept of the graph of y = cos x?
A.
B.
C.
D.
____
0
1
2
3
17. Which of the following is not an x-intercept of the graph of y = sin x?
A.
B.
C.
D.
____
5:30
4:50
6:00
4:10
{y | –1  y  1, y  R}
{y | 0  y  2, y  R}
{y | –1  y  2, y  R}
{y | –2  y  2, y  R}
21. Determine the midline of the following graph.
A.
B.
C.
D.
____
22. Determine the midline of the following graph.
A.
B.
C.
D.
____
y=2
y=3
y=4
y=5
23. Determine the midline of the following graph.
A.
B.
C.
D.
____
y=2
y=3
y=4
y=5
y = –4
y=0
y=4
y=8
24. Determine the amplitude of the following graph.
A.
B.
C.
D.
____
25. Determine the amplitude of the following graph.
A.
B.
C.
D.
____
2
3
4
5
26. Determine the amplitude of the following graph.
A.
B.
C.
D.
____
2
3
4
5
2
3
4
5
27. Determine the period of the following graph.
A.
B.
C.
D.
____
28. Determine the period of the following graph.
A.
B.
C.
D.
____
5
6
7
8
29. Determine the period of the following graph.
A.
B.
C.
D.
____
120°
240°
300°
360°
2
2.5
5
1.25
30. Determine the range of the following graph.
A.
B.
C.
D.
____
31. Determine the range of the following graph.
A.
B.
C.
D.
____
{y | 1  y  5, y  R}
{y | –2  y  2, y  R}
{y | 0  y  4, y  R}
{y | y  R}
32. Determine the range of the following graph.
A.
B.
C.
D.
____
{y | 0  y  8, y  R}
{y | –2  y  6, y  R}
{y | –4  y  8, y  R}
{y | y  R}
{y | –8  y  8, y  R}
{y | –4  y  4, y  R}
{y | 0  y  15, y  R}
{y | y  R}
33. A sinusoidal graph has an amplitude of 10 and a maximum at the point (18, 5). Determine the
midline of the graph.
A.
B.
C.
D.
____
34. A sinusoidal graph has a maximum at the point (5, 12) and a minimum at the point (12, 5).
Determine the midline of the graph.
A.
B.
C.
D.
____
y=0
y=5
y = 12
y = 8.5
35. A sinusoidal graph has a maximum at the point (4, –8) and the next minimum is at the point (7, –
10). Determine the period of the graph.
A.
B.
C.
D.
____
y=0
y = –5
y = 13
y=8
2
3
4
6
36. Select the function with the greatest amplitude.
A. y = 2 sin 3(x + 90°) + 5
B. y = 3 sin 2(x – 90°) – 3
C.
y = sin (x + 90°) – 1
D. y = sin 0.5(x – 90°)
____
37. Select the function with the greatest period.
A. y = 2 sin 3(x + 90°) + 5
B. y = 3 sin 2(x – 90°) – 3
C.
y = sin (x + 90°) – 1
D. y = sin 0.5(x – 90°)
____
38. Select the function with the greatest maximum value.
A. y = 2 sin 3(x + 90°) + 5
B. y = 3 sin 2(x – 90°) – 3
C.
y = sin (x + 90°) – 1
D. y = sin 0.5(x – 90°)
____
39. Determine the amplitude of the following function.
y = 3 sin 2(x + 90°) – 1
A.
B.
C.
D.
____
2
3
4
5
40. Determine the amplitude of the following function.
y = cos
x + 12
A.
B. 1
C. 2
D. 12
____
41. Determine the amplitude of the following function.
y = 0.5 sin (x – 2)
A.
B.
C.
D.
____
42. Determine the period of the following function.
y = 3 sin 2(x + 90°) – 1
A.
B.
C.
D.
____
0.5
1
2
0
180°
360°
720°
1080°
43. Determine the period of the following function.
y = cos
A.
B.
C.
D.
____
x + 12
180°
360°
720°
1080°
44. Determine the period of the following function.
y = 0.5 sin (x – 2)
A.
B.
C.
D.
180°
360°
720°
1080°
____
45. Determine the midline of the following function.
y = 3 sin 2(x + 90°) – 1
A.
B.
C.
D.
____
y=2
y=3
y=0
y = –1
46. Determine the midline of the following function.
y = cos
A.
B.
C.
D.
____
y = –2
y = 0.5
y=0
y=2
48. Determine the range of the following function.
y = 3 sin 2(x + 90°) – 1
A.
B.
C.
D.
____
y = 12
y=3
y=4
y=0
47. Determine the midline of the following function.
y = 0.5 sin (x – 2)
A.
B.
C.
D.
____
{y | –3  y  3, y  R}
{y | –2  y  4, y  R}
{y | –4  y  2, y  R}
{y | y  R}
49. Determine the range of the following function.
y = cos
A.
B.
C.
D.
____
x + 12
x + 12
{y | 11  y  13, y  R}
{y | –4  y  4, y  R}
{y | 9  y  15, y  R}
{y | y  R}
50. Determine the range of the following function.
y = 0.5 sin (x – 2)
A. {y | –3  y  –1, y  R}
B. {y | –0.5  y  0.5, y  R}
C. {y | –2  y  2, y  R}
D. {y | y  R}
____
51. The following data set is sinusoidal. Determine the missing value from the table.
x
0
1
2
3
4
5
6
y
1.0
2.5
4.0
2.5
1.0
2.5
A.
B.
C.
D.
____
52. The following data set is sinusoidal. Determine the missing value from the table.
x
2
4
6
8
10
12
14
16
y
2.6
2.0
2.6
4.0
5.4
6.0
4.0
A.
B.
C.
D.
____
0.4
0.5
0.6
0.7
54. The following data set is sinusoidal. Determine the missing value from the table.
x
1
2
3
4
5
6
7
y
–5
–8
–5
–2
–5
–8
A.
B.
C.
D.
____
2.0
2.6
4.7
5.4
53. The following data set is sinusoidal. Determine the missing value from the table.
x
–3
–2
–1
0
1
2
3
4
y
0.6
0.5
0.5
0.6
0.5
0.4
0.5
A.
B.
C.
D.
____
–0.5
1.0
2.5
4.0
–2
–5
–8
–11
55. The following data set is sinusoidal. Determine the missing value from the table.
x
0
2
4
6
8
10
18
y
5.8
6.8
5.8
4.8
5.8
6.8
A.
B.
C.
D.
4.8
5.8
6.8
7.8
____
56. The following data set is sinusoidal. Determine the missing value from the table.
x
3
4
5
6
7
8
30
y
21
17
13
17
21
17
A.
B.
C.
D.
____
57. The following data set is sinusoidal. Determine the missing value from the table.
x
–3
–2
–1
0
1
2
3
6
y
1.0
1.7
2.0
1.7
1.0
0.3
0.0
A.
B.
C.
D.
____
2
0.1
y = 1.0 sin 0.8(x – 2.3) + 0.8
y = 1.0 sin 0.8(x + 2.3) + 1.0
y = 0.8 sin 1.0(x – 3.2) + 1.0
y = 0.8 sin 1.0(x + 3.2) + 0.8
60. Determine the equation of the sinusoidal regression function for the data.
x
0
5
10
15
20
25
30
y
120
138
122
105
121
140
125
A.
B.
C.
D.
____
0.0
0.3
1.7
2.0
59. Determine the equation of the sinusoidal regression function for the data.
x
–5
–4
–3
–2
–1
0
1
y
0.8
1.5
1.8
1.4
0.8
0.0
–0.2
A.
B.
C.
D.
____
0.0
0.3
1.7
2.0
58. The following data set is sinusoidal. Determine the missing value from the table.
x
–3
–2
–1
0
1
2
3
12
y
1.0
1.7
2.0
1.7
1.0
0.3
0.0
A.
B.
C.
D.
____
13
17
21
25
y = 16.5 sin (0.3x – 0.1) + 123
y = 17.1 sin (0.31x – 0.05) + 122
y = 17.6 sin (0.34x + 0.1) + 121
y = 18 sin (0.25x + 0.05) + 120
61. Determine the equation of the sinusoidal regression function for the data.
x
0
1
2
3
4
5
6
7
y
A.
B.
C.
D.
____
12.4
12.0
12.1
12.6
13.5
y = 4.35 sin (0.63x + 3.13) + 15.44
y = 4.35 sin (0.36x – 3.13) + 15.44
y = 3.45 sin (0.63x + 3.13) + 15.44
y = 3.45 sin (0.36x – 3.13) + 15.44
7
8.0
y = 7.4 sin (1.2x – 2.0) + 9.1
y = 7.4 sin (1.2x – 2.0) – 9.1
y = 9.1 sin (1.2x – 2.0) + 7.4
y = 9.1 sin (1.2x – 2.0) – 7.4
14.74 h
14.89 h
15.04 h
15.19 h
64. The amount of daylight in a town can be modelled by the sinusoidal function
d(t) = 4.37 cos 0.017t + 12.52
where d(t) represents the hours of daylight and t represents the number of days since June 20,
2012.
How many hours of daylight should be expected on June 20, 2013?
A.
B.
C.
D.
____
13.1
63. The amount of daylight in a town can be modelled by the sinusoidal function
d(t) = 4.37 cos 0.017t + 12.52
where d(t) represents the hours of daylight and t represents the number of days since June 20,
2012.
How many hours of daylight should be expected on August 20, 2012?
A.
B.
C.
D.
____
14.2
62. Determine the equation of the sinusoidal regression function for the data.
x
0
1
2
3
4
5
6
y
–1.0
1.1
11.1
16.5
10.5
0.6
–0.8
A.
B.
C.
D.
____
15.4
16.80 h
16.84 h
16.88 h
16.92 h
65. The height of a mass attached to a spring can be modelled by the sinusoidal function
h(t) = 84 – 6.7 cos 24.8t
where h(t) represents the height in centimetres and t represents the time in seconds.
What is the height of the mass after 10 s?
A.
B.
C.
D.
77.4 cm
84.0 cm
86.9 cm
90.6 cm
Short Answer
1. Estimate the value of 270° in radians, to the nearest tenth.
2. Estimate the value of 172° in radians, to the nearest tenth.
3. Estimate the value of 540° in radians, to the nearest tenth.
4. Estimate the value of 5 radians in degrees, to the nearest ten degrees.
5. Estimate the value of 1.2 radians in degrees, to the nearest ten degrees.
6. Estimate the value of 9.4 radians in degrees, to the nearest ten degrees.
7. Estimate, to the nearest degree, the measure of the central angle. Check your estimate with a
protractor.
8. Estimate, to the nearest degree, the measure of the central angle. Check your estimate with a
protractor.
9. Estimate, to the nearest radian, the measure of the central angle. Check your estimate with a
protractor.
10. Estimate, to the nearest radian, the measure of the central angle. Check your estimate with a
protractor.
11. Eddie is facing west. What direction will he be facing if he rotates 235° to his right?
12. Eddie is facing northwest. What direction will he be facing if he rotates
radians to his left?
13. For the following pair of angle measures, determine which measure is greater.
235°, 4.5
14. For the following pair of angle measures, determine which measure is greater.
75°,
15. For the following pair of angle measures, determine which measure is greater.
450°, 7.5
16. What is the equation of the midline of y = cos x?
17. Identify the domain of y = sin x.
18. How does the vertical distance from the maximum to the minimum of a periodic function relate to
the amplitude?
19. How many turning points does the graph of y = cos x have from –1.5 to 1.5?
20. What is the first x-intercept of the graph of y = cos x to the left of the y-axis?
21. Determine the midline of the following graph.
22. Determine the midline of the following graph.
23. Determine the midline of the following graph.
24. Determine the amplitude of the following graph.
25. Determine the amplitude of the following graph.
26. Determine the amplitude of the following graph.
27. Determine the period of the following graph.
28. Determine the period of the following graph.
29. Determine the period of the following graph.
30. Determine the range of the following graph.
31. Determine the range of the following graph.
32. Determine the range of the following graph.
33. A sinusoidal graph has a maximum at the point (5, 12) and a minimum at the point (–12, –5).
Determine the range of the graph.
34. A sinusoidal graph has a maximum at the point (–40, 3) and a midline of y = –12. Determine the
amplitude of the graph.
35. A sinusoidal graph has an amplitude of 9 and a midline of y = –2. Determine the range of the
graph.
36. Determine the amplitude of the following function.
y=
cos (x – )
37. Determine the amplitude of the following function.
y = 5 sin 1.5(x + 60°) – 5
38. Determine the amplitude of the following function.
y = 10 cos 4(x – 180°) + 2
39. Determine the midline of the following function.
y=
cos (x – )
40. Determine the midline of the following function.
y = 5 sin 1.5(x + 60°) – 5
41. Determine the midline of the following function.
y = 10 cos 4(x – 180°) + 2
42. Determine the period of the following function.
y=
cos (x – )
43. Determine the period of the following function.
y = 5 sin 1.5(x + 60°) – 5
44. Determine the period of the following function.
y = 10 cos 4(x – 180°) + 2
45. Determine the range of the following function.
y=
cos (x – )
46. Determine the range of the following function.
y = 5 sin 1.5(x + 60°) – 5
47. Determine the range of the following function.
y = 10 cos 4(x – 180°) + 2
48. Determine the horizontal translation applied to y = cos x to obtain the following function.
y=
cos (x – )
49. Determine the horizontal and vertical translations applied to y = sin x to obtain the following
function.
y = sin (x + 60°) – 5
50. A seat’s position on a Ferris wheel can be modelled by the function
y = 18 cos 2.8(x + 1.2) + 21,
where y represents the height in feet and x represents the time in minutes.
Determine the diameter of the Ferris wheel.
51. The following data set is sinusoidal. Determine the missing value from the table.
x
0
1
2
3
4
5
6
y
5.5
3.0
0.5
3.0
5.5
3.0
52. The following data set is sinusoidal. Determine the missing value from the table.
x
0
2
4
6
8
10
12
y
–1
3
–1
–1
3
–1
53. The following data set is sinusoidal. Determine the missing value from the table.
x
–13
–12
–11
–10
–9
–8
–7
y
17
20
17
3
0
3
54. The following data set is sinusoidal. Determine the missing value from the table.
x
0
1
2
3
4
5
6
35
y
0.4
1.0
2.4
3.8
4.4
3.8
2.4
55. The following data set is sinusoidal. Determine the missing value from the table.
x
4
8
12
16
20
24
28
y
124
135
124
113
124
135
56. The following data set is sinusoidal. Determine the missing value from the table.
x
4
8
12
16
20
24
40
y
124
135
124
113
124
135
57. Determine the equation of the sinusoidal regression function for the data. Round values to the
nearest tenth.
x
–7
–6
–5
–4
–3
–2
–1
0
y
6.8
8.6
4.8
–1.7
–6.6
–6.0
–0.5
5.8
58. Determine the equation of the sinusoidal regression function for the data. Round values to the
nearest tenth.
x
3
4
5
6
7
8
9
10
y
1.2
2.1
4.3
6.1
6.2
4.7
2.5
1.3
59. Determine the equation of the sinusoidal regression function for the data. Round values to the
nearest tenth.
x
–5
–4
–3
–2
–1
0
1
2
y
–1.5
–22.5 –41.0 –53.0 –56.5 –51.0 –38.5 –19.5
60. Use sinusoidal regression to determine the missing value, to the nearest tenth.
x
0
1
2
3
4
5
6
7
y
15.5
13.6
20.0
27.7
28.3
21.2
14.0
61. Use sinusoidal regression to determine the missing value, to the nearest tenth.
x
1
2
3
4
5
6
7
8
y
12.4
11.0
5.6
11.7
12.0
6.5
5.6
62. Use sinusoidal regression to determine the missing value, to the nearest tenth.
x
–4
–3
–2
–1
0
1
2
3
y
7.0
9.4
10.9
11.1
9.8
5.1
3.3
63. The height of a mass attached to a spring can be modelled by the sinusoidal function
h(t) = 53.5 – 4.2 cos 23.5t
where h(t) represents the height in centimetres and t represents the time in seconds.
What is the height of the mass, to the nearest tenth of a centimetre, after the first minute?
64. A seat’s position on a Ferris wheel can be modelled by the function
h(t) = 14 sin (2.1t + 0.8) + 15.5
where h(t) represents the height in metres and t represents the time in minutes.
What is the height of the seat, to the nearest tenth of a metre, after 1.5 min?
65. Brianna’s position on a Ferris wheel can be modelled by the function
h(t) = 15.4 sin (2.3t – 1.4) + 17.2
where h(t) represents her height in metres and t represents the time in minutes.
How much higher is she after 30 s than at the start of the ride? Round your answer to the nearest
tenth of a metre.
Problem
1. For the following pair of angle measures, determine which measure is greater. Explain your
reasoning.
800°, 4.5
2. For the following pair of angle measures, determine which measure is greater. Explain your
reasoning.
20°, 0.4
3. For the following pair of angle measures, determine which measure is greater. Explain your
reasoning.
370°, 6.2
4. Before trying to hit a piñata, Shen is blindfolded and spun 2475°. Estimate the angle he must turn
to face the piñata to the nearest tenth of a radian. Show your work.
5. Before trying to hit a piñata, Emma is blindfolded and spun 2095°. Estimate the angle she must
turn to face the piñata to the nearest tenth of a radian. Show your work.
6. Imagine that it is now 11:30 a.m.
a) How many degrees does the minute hand rotate through in five minutes? What is the equivalent
angle in radians? Show your work.
b) What time will it be when the minute hand has rotated through 870°? Show your work.
c) Estimate, to the nearest tenth, this angle measure in radian measure. Show your work.
7. Imagine that it is now 4:45 p.m.
a) How many degrees will the minute hand rotate through before midnight? What is the equivalent
angle in radians? Show your work.
b) What time will it be when the minute hand has rotated through
radians? Show your work.
c) Determine how many degrees are in this rotation. Show your work.
8. An airplane takes off from Edmonton International Airport, facing southeast. As it gains altitude,
the airplane rotates 1.5 counterclockwise.
a) Express this angle in degrees.
b) In what direction is the airplane now heading? Show your work.
9. A fishing boat is on a heading of S 64° W. The captain checks his GPS and notices that the
harbour is almost directly north of them.
a) How many degrees must the boat turn to be facing the harbour? Assume the captain chooses the
shorter turn. Show your work.
b) Estimate this angle in radians. Show your work.
10. A sailboat changes headings from S 25° E to N 72° W by turning clockwise.
a) How many degrees did the boat turn? Show your work.
b) Estimate this angle in radians. Show your work.
11. Sketch a possible graph of a sinusoidal function with the following set of characteristics. Explain
your decision.
Domain: {x | 0  x  1080°, x  R}
Maximum value: 5
Minimum value: –5
Period: 720°
y-intercept: 0
12. Sketch a possible graph of a sinusoidal function with the following set of characteristics. Explain
your decision.
Domain: {x | 0  x  360°, x  R}
Range: {y | –2  y  6, y  R}
Period: 120°
y-intercept: 0
13. Sketch a possible graph of a sinusoidal function with the following set of characteristics. Explain
your decision.
Domain: {x | 0  x  4, x  R}
Range: {y | 1  y  5, y  R}
Period: 8
y-intercept: 1
14. 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. Show your work.
15. 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. Show your work.
16. The graph of a sinusoidal function is shown. Describe this graph by determining its range, the
equation of its midline, and its amplitude. Show your work.
17. The graph of a sinusoidal function is shown.
a) Determine the period of this graph. Show your work.
b) Determine the y-value of this graph when x = 3. Explain your answer.
c) Determine the y-value of this graph when x = 1.75. Explain your answer.
18. Jeremy’s gymnastics coach graphs one particular series of jumps. Describe Jeremy’s jumps using
the graph. Show your work.
19. Kira is sitting in an inner tube in the wave pool. The depth of the water below her, in terms of time,
during a series of waves can be represented by the graph shown.
a) What is the depth of the water below Kira when no waves are being generated? Explain how
you know.
b) How high is each wave? Show your work.
20. Kira is sitting in an inner tube in the wave pool. The depth of the water below her, in terms of time,
during a series of waves can be represented by the graph shown.
a) How long does it take for one complete wave to pass? Show your work.
b) What is the approximate depth of the water below Kira after 4 s?
c) What is the depth of the water below Kira at 9 s? Assume that the waves continue at the same
rate. Explain your answer.
21. Match each graph with the corresponding equation below. Explain your answers.
i) y = 2 cos (x – 120°) + 4
ii) y = 2 cos (x – 60°) + 4
iii) y = 2 cos (x + 60°) + 4
iv) y = 2 cos (x – 120°)
v) y = 2 cos (x – 60°)
vi) y = 2 cos (x + 60°)
22. Match each graph with the corresponding equation below. Explain your answers.
i) y = 2 cos 2.5(x – 0.8) – 1
ii) y = 3 cos 2(x – 0.8) + 1
iii) y = 3 cos 2(x – 0.8) – 1
iv) y = 3 cos 2.5(x – 0.6) – 1
v) y = 3 cos 2.5(x – 0.6) + 1
vi) y = 2 cos 2(x – 0.6) – 1
23. Describe the graph of the following function by stating the amplitude, equation of its midline,
range, and period. Show your work.
y=
sin (2x) + 3.5
24. Describe the graph of the following function by stating the amplitude, equation of its midline,
range, and period. Show your work.
y = 6 cos 8(x – 1.4) – 4
25. Describe the graph of the following function by stating the amplitude, equation of its midline,
range, and period. Show your work.
y = cos
(x + 180°) + 3
26. Determine the equation of the function whose amplitude and period are both triple the amplitude
and the period of this function:
y = 2 sin 6x + 1
but the midline is 5 units below the midline of the original function. Show your work.
27. The graph of a sinusoidal function has a maximum at (4, 3) followed by a minimum at (8, 1).
a) Describe the graph of the function by stating the amplitude, equation of its midline, range, and
period. Show your work.
b) Determine the y-value of the function when x = 10. Show your work.
c) Determine the y-value of the function when x = 100. Show your work.
28. The height of a chair on a Ferris wheel is described by the function
h(t) = –14 cos 3.2t + 16
where h(t) represents the height of the chair in metres and t represents the time in minutes.
a) What are the maximum and minimum heights you can reach if you are riding the Ferris wheel?
b) What is the period of the function? What does the period tell you about the Ferris wheel in this
context?
29. A competitive gymnast’s coach analyzes one particular series of jumps. These jumps can be
modelled by the sinusoidal function
h(t) = 11.5 sin
+ 9.8
where h(t) represents the height of the gymnast in feet and t represents the time in seconds.
a) What is the maximum height of the gymnast jumps? Explain how you know.
b) How often does the gymnast jump? Explain how you know.
30. The height of a point on a bicycle wheel is described by the function
h(t) = 34 cos 4.6(t + 1.7) + 34
where h(t) represents the height of the point in centimetres and t represents the time in seconds.
a) What is the diameter of the bicycle wheel? Show your work.
b) How fast is the bicycle moving, to the nearest metre per second? Show your work.
31. The average depth of the water at an ocean port can be modelled by the function
h(t) = 0.76 cos (0.25t) + 3.82
where h(t) represents the depth in metres and t represents the time in hours after 5:00 p.m. on April
19, 2012.
a) What is the minimum depth of water, to the nearest centimetre? Show your work.
b) Estimate the depth of the water at 10:30 a.m. on April 20, 2012. Show your work.
32. Zack’s position on a Ferris wheel can be modelled by the function
h(t) = 25.4 sin (1.8t + 4.0) + 24.2
where h(t) represents his height in metres and t represents the time in minutes.
a) What is the maximum height of the Ferris wheel? Show your work.
b) What is the period of rotation of the Ferris wheel, to the nearest tenth of a minute? Show your
work.
c) What do the x-intercepts mean in the context of this question? Give a possible reason for these
values.
33. Meena is sitting in an inner tube in the wave pool at West Edmonton Mall. The following table
gives the depth of the water below her.
0
1
2
3
4
5
6
7
Time (s)
3.1
3.8
4.0
3.5
2.6
2.1
1.9
2.4
Depth (m)
a) Create a scatter plot, and draw a curve of best fit for the data using sinusoidal regression.
b) Determine the depth of the water after 9 s, to the nearest tenth of a metre. Show your work.
34. The following table gives the average depth of the water on an ocean port measured every 3 h for a
day.
3
6
9
12
15
18
21
24
Time (h)
2.0
2.5
2.9
3.1
2.7
2.2
1.8
1.7
Depth (m)
a) Create a scatter plot, and draw a curve of best fit for the data using sinusoidal regression.
b) Determine the locations of two maximums and calculate the period of the graph, to the nearest
minute. Show your work.
35. The following table gives the population of a predator animal over a decade.
1
2
3
4
5
Year
165
158
174
198
212
Population
6
7
8
9
10
Year
216
205
184
162
161
Population
a) Create a scatter plot, and draw a curve of best fit for the data using sinusoidal regression.
b) Use your graph to estimate the population in the 12th year.
36. The following table gives the time of sunrise recorded on the first of the month in a Saskatchewan
town. All times are Central Standard Time.
Jan. 1
Feb. 1
Mar. 1
Apr. 1
May 1
Jun. 1
Month
07:32
07:07
06:36
06:09
05:53
Sunrise
Jul. 1
Aug. 1
Sep. 1
Oct. 1
Nov. 1
Dec. 1
Month
05:48
06:02
06:28
06:55
07:18
07:37
Sunrise
a) Create a scatter plot, and draw a curve of best fit for the data using sinusoidal regression.
b) Determine the missing value from the table. Show your work.
37. The following table gives the average temperature in an Alberta town for the first nine months of
the year.
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep.
Month
Average
1.6
8.8
15.2 20.3 23.0 22.7 17.8
Temperature (°C) –10.8 –3.1
Use sinusoidal regression to estimate the average temperatures for October, November, and
December, to the nearest tenth of a degree. Show your work.
38. For a physics project, Miro and Alex had to graph and analyze an example of simple harmonic
motion. Alex swung on a swing, and Miro used a motion detector to measure Alex’s height above
the ground every second, as she swung back and forth. The following table gives the height of the
swing over time.
1
2
3
4
5
6
7
8
Time (s)
Height of
189
87
135
173
74
168
142
83
swing (cm)
Use sinusoidal regression to estimate Alex’s minimum and maximum heights, to the nearest
centimetre. Show your work.
39. The following table gives the population of a predator animal over eight years.
1
2
3
4
5
6
7
8
Year
304
294 290
292
297
305
316
326
Population
a) Use sinusoidal regression to estimate the time between minimum populations, to the nearest
tenth of a year. Show your work.
b) Estimate the population in the 10th year.
c) Estimate the population in the 20th year.
40. The following table gives the time of sunrise recorded on the first of the month in a British
Columbia town. All times are Pacific Standard Time.
Jan. 1
Feb. 1
Mar. 1
Apr. 1
May 1
Jun. 1
Month
07:28
07:01
06:34
06:10
05:47
05:38
Sunrise
Jul. 1
Aug. 1
Sep. 1
Oct. 1
Nov. 1
Dec. 1
Month
05:44
05:58
06:27
06:58
07:26
07:45
Sunrise
Use sinusoidal regression to determine the earliest and latest sunrises possible in this town. Round
values to the nearest minute. Show your work.
40s applied review questions for periodic functions
Answer Section
MULTIPLE CHOICE
1. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
2. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
3. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
4. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
5. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
6. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
7. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
8. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
9. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
10. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
11. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
12. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
13. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
14. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
15. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
16. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function | turning point
17. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
18. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
19. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
20. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
21. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
22. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
23. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
24. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
25. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
26. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
27.
28.
29.
30.
31.
32.
33.
34.
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
35.
36.
37.
38.
39.
40.
41.
42.
KEY: sinusoidal function | midline
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
43.
44.
45.
46.
47.
48.
49.
50.
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
51.
52.
53.
54.
55.
56.
57.
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | interpolate
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | interpolate
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
58.
59.
60.
61.
62.
63.
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
64. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
65. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
SHORT ANSWER
1. ANS:
4.7 radians
PTS: 1
KEY: radian
2. ANS:
3.0 radians
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
3. ANS:
9.4 radians
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
4. ANS:
290°
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
5. ANS:
70°
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
6. ANS:
540°
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
7. ANS:
150°
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
8. ANS:
150°
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
9. ANS:
5
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
10. ANS:
8
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
11. ANS:
southeast
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
12. ANS:
south
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
13. ANS:
4.5
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
14. ANS:
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
15. ANS:
450°
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
16. ANS:
y=0
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function | midline
17. ANS:
{x | x  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
18. ANS:
The vertical distance is twice the amplitude.
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function | amplitude
19. ANS:
3
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function | turning point
20. ANS:
–90° or
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
21. ANS:
y=3
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
22. ANS:
y = –1
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
23. ANS:
y=4
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
24. ANS:
5
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
25. ANS:
5
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
26. ANS:
8
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
27. ANS:
300°
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
28. ANS:
6
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
29. ANS:
6
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period
30. ANS:
{y | –2  y  8, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
31. ANS:
{y | –6  y  4, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
32. ANS:
{y | –4  y  12, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
33. ANS:
{y | –5  y  12, y R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function
34. ANS:
15
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline
35. ANS:
{y | –11  y  7, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline
36. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
37. ANS:
5
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
38. ANS:
10
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude
39. ANS:
y=0
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
40. ANS:
y = –5
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
41. ANS:
y=2
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
42. ANS:
360° or 2 radians
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
43. ANS:
240°
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
44. ANS:
90°
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | period
45. ANS:
{y | –
y
, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
46. ANS:
{y | –10  y  0, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
47. ANS:
{y | –8  y  12, y  R}
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
48. ANS:
y = cos x was translated  radians to the right
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
49. ANS:
y = sin x was translated 60° to the left and 5 units down
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
50. ANS:
36 m
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
51. ANS:
0.5
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
52. ANS:
–5
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | interpolate
53. ANS:
10
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | interpolate
54. ANS:
3.8
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
55. ANS:
124
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
56. ANS:
135
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | extrapolate
57. ANS:
y = 7.9 sin (0.9x + 0.7) + 0.8
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
58. ANS:
y = 2.6 sin (0.9x + 2.1) + 3.8
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
59. ANS:
y = 52.6 sin (0.4x – 1.1) – 4.0
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
60. ANS:
14.7
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
61. ANS:
6.3
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | interpolate
62. ANS:
7.6
PTS:
1
DIF:
Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function
63. ANS:
57.0 cm
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
64. ANS:
5.4 m
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
65. ANS:
11.4 m
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
PROBLEM
1. ANS:
I know that a full rotation in radians is 2. So 4.5 is two full rotations plus an additional quarter
rotation.
In degrees, 360° is a full rotation.
2.25(360°) = 810°
4.5 is equivalent to 810°.
4.5 is the greater angle.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
2. ANS:
I know that 1 radian is equivalent to about 60°.
Since 20° is one third of 60°, it is equivalent to about 0.33 radians, which is less than 0.4.
0.4 is the greater angle.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
3. ANS:
I know that 370° is 10° more than a full rotation.
A full rotation in radians is 2, or 6.28 (to the nearest hundredth), so 6.2 radians is less than a full
rotation.
370° is the greater angle.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
4. ANS:
2475° is nearly seven full rotations since 7(360°) = 2520°.
2520° – 2475° = 45°
Shen must turn 45° to face the piñata.
I know that  radians, which is a little less than 3.2, is equivalent to 180°.
Since 45° is one quarter of 180°, I estimate that Shen must turn 0.8 radians to face the piñata.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
5. ANS:
2100° is nearly six full rotations since 6(360°) = 2160°.
2160° – 2095° = 65°
Emma must turn 65° to face the piñata.
I know that 1 radian is equivalent to about 60°.
Since 65° is almost eleven tenths of 60°, I estimate that Emma must turn 1.1 radians to face the
piñata.
PTS: 1
KEY: radian
6. ANS:
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
a) I know that 5 min is one twelfth of an hour:
= 30°
The minute hand rotates through 30° or
radians, which is about 0.5 radians.
b) 870° = 360° + 360° + 150°
An 870° rotation is two full turns, plus another five twelfths.
That represents two more hours plus another 25 minutes.
The time is now 1:55 p.m.
c) I know that 2 complete rotations are about 2(6.3), or 12.6 radians.
I know that 1 radian is equivalent to about 60°.
Since 150° is 2.5 times 60°, 150° is about 2.5 radians.
12.6 + 2.5 = 15.1
The angle is about 15.1 radians.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
7. ANS:
a) The time from 4:45 to midnight is 7.25 hours. In that time, the minute hand will make 7.25 full
rotations.
7.25(360°) = 2610°
7.25(2) = 14.5
The minute hand will rotate through 2610° or 14.5 radians before midnight.
b)
A rotation of
radians is one full turn, plus a twelfth.
That represents one more hour plus another 5 minutes.
The time is now 5:50 p.m.
c) I know that a complete rotation is 360°. Add this to one twelfth of a complete rotation:
360° +
= 390°
The angle is 390°.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
8. ANS:
a) I know that 2 is one full rotation, or 360°, so 1.5 is the equivalent of three quarters of a
rotation, or 270°.
b) 270° counterclockwise puts the airplane facing the same direction as a 90° rotation clockwise,
which is a quarter turn. A quarter turn clockwise from southeast is southwest.
The airplane is now facing southwest.
PTS: 1
KEY: radian
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
9. ANS:
a) The boat is facing 64° west of south.
90° – 64° = 26°
The boat is facing 26° south of west.
26° + 90° = 116°
The captain needs to turn the boat 116° clockwise to face the harbour.
b) 116° is very close to 120°.
120° can be thought of as the difference between 180° and 60°.
I know that 180° is slightly more than 3.1 radians.
I know that 60° is slightly more than 1 radian.
I estimate 120° is equivalent to 2.1 radians.
Therefore, I estimate 116° is equivalent to 2.0 radians.
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
KEY: radian
10. ANS:
a) The boat is now facing 72° west of north.
90° – 72° = 18°
The boat is facing 18° north of west.
18° + 90° + 25° = 133°
The sailboat turned 133° clockwise.
b) 133° is very close to 135°.
135° can be thought of as three 45° angles.
I know that 45° is slightly less than 0.8 radians.
3(0.8) = 2.4
Therefore, I estimate 133° is equivalent to 2.3 or 2.4 radians.
TOP: Understanding angles
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
KEY: radian
11. ANS:
Answers may vary. Sample answer:
The period is two thirds the domain, so 1.5 cycles are included.
The graph has a maximum of 5 and a minimum of –5.
The graph starts at the point (0, 0):
TOP: Understanding angles
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
12. ANS:
Answers may vary. Sample answer:
The period is one third the domain, so three complete cycles are included.
The graph has a maximum of 6 and a minimum of –2.
The graph starts at the point (0, 0):
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
13. ANS:
The period is twice the domain, so only half a cycle is included.
The y-intercept is also the minimum, so the graph will increase from x = 0 to x = 4:
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
14. ANS:
Range:
Minimum value = –2.5
Maximum value = 0.5
The range of the graph is {y | –2.5  y  0.5, y  R}.
Equation of the midline (halfway between the maximum and minimum values):
y=
y=
y = –1
Amplitude (the vertical distance between the maximum value and the midline):
Amplitude = 0.5 – (–1)
Amplitude = 1.5
Period:
There is a maximum value at 90° and a maximum value at 330°.
Period = 330° – 90°
Period = 240°
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
15. ANS:
Range:
Minimum value = –1
Maximum value = 5
The range of the graph is {y | –1  y  5, y  R}.
Equation of the midline (halfway between the maximum and minimum values):
y=
y=
y=2
Amplitude (the vertical distance between the maximum value and the midline):
Amplitude = 5 – 2
Amplitude = 3
Period:
There is a maximum value at 3 and a maximum value at 15.
Period = 15 – 3
Period = 12
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
16. ANS:
Range:
Minimum value = –8
Maximum value = 4
The range of the graph is {y | –8  y  4, y  R}.
Equation of the midline (halfway between the maximum and minimum values):
y=
y=
y = –2
Amplitude (the vertical distance between the maximum value and the midline):
Amplitude = 4 – (–2)
Amplitude = 6
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
17. ANS:
a) There is a maximum value at 0.075 and a maximum value at 0.375.
Period = 0.375 – 0.075
Period = 0.3
b) I know that the period is 0.3, so the y-value at x = 3 is the same as the y-value at x = 0.
From the graph I can see that at x = 0, the y-value is –2.
The y-value of this graph when x = 3 is –2.
c) I know that the period is 0.3, so the y-value at x = 1.75 is the same as the y-value at x = 1.45, x =
1.15, and x = 0.85.
From the graph I can see that at x = 0.85, the y-value is –7.
The y-value of this graph when x = 1.75 is –7.
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period | extrapolate
18. ANS:
Range:
Minimum value = 1
Maximum value = 7
The range of the graph is {y | 1  y  7, y  R}.
Equation of the midline (halfway between the maximum and minimum values):
y=
y=
y=4
Amplitude (the vertical distance between the maximum value and the midline):
Amplitude = 7 – 4
Amplitude = 3
Period:
There is a maximum value at 13 and a maximum value at 17.
Period = 17 – 13
Period = 4
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
19. ANS:
a) The depth of the water below Kira when no waves are being generated would be the midline of
the sinusoidal function.
Equation of the midline (halfway between the maximum and minimum values):
y=
y=
y = 2.0
The depth of the water would be 2.0 m.
b) The height of a wave is the amplitude.
Amplitude (the vertical distance between the maximum value and the midline):
Amplitude = 2.8 – (2.0)
Amplitude = 0.8
The height of each wave is 0.8 m.
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
20. ANS:
a) The time for one wave to pass is the period of the sinusoidal function.
From the graph I can see that the period is 2 s.
b) From the graph I can see that at t = 4, the depth of water is 2.4 m.
c) I know that the period is 2 s, so the depth of water at t = 9 is the same as the depth of water at t =
7 and at t = 5.
From the graph I can see that at t = 5, the depth of water is 1.6 m.
The depth of water below Kira at 9 s is 1.6 m.
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
21. ANS:
The amplitude of both graphs is 2.
The period of both graphs is 360°.
The midline of graph A is y = 4, so equations iv), v), and vi) are eliminated.
If the graph represents a cosine function, then it has been translated by 120° to the right. Equation
i) represents a cosine function that has been translated by 120° to the right. Equation i) must be
correct for graph A.
The midline of graph B is y = 0, so equations i), ii), and iii) are eliminated.
If the graph represents a cosine function, then it has been translated by 60° to the right. Equation v)
represents a cosine function that has been translated by 60° to the right. Equation v) must be
correct for graph B.
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
22. ANS:
The amplitude of both graphs is 3, so equations i) and vi) are eliminated.
The midline of graph A is y = 1, so equations iii) and iv) are eliminated.
The period of graph A is about 2.5.
Equation v) is the only remaining equation with b = 2.5.
Also, equation v) represents a cosine function that has been translated by 0.6 to the right, just like
graph A. Equation v) must be correct for graph A.
The midline of graph B is y = –1, so equations ii) and v) are eliminated.
The period of graph B is about 3.1.
Equation iii) is the only remaining equation with b = 2.
Also, equation iii) represents a cosine function that has been translated by 0.8 to the right, just like
graph B. Equation iii) must be correct for graph B.
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
23. ANS:
y = a sin b(x – c) + d
y=
sin (2x) + 3.5
a=
, b = 2, c = 0, d = 3.5
The amplitude of the graph is a, which is
.
The equation of the midline is y = d, or y = 3.5.
Minimum value = d – a
Maximum value = d + a
Minimum value = 3.5 –
Maximum value = 3.5 +
Minimum value = 3.4
Maximum value = 3.6
The range of the graph is {y | 3.4  y  3.6, y  R}.
Since b = 2, the graph completes 2 cycles in 360° or 2 radians.
Period =
Period =
Period = 180°
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
24. ANS:
y = a cos b(x – c) + d
y = 6 cos 8(x – 1.4) – 4
a = 6, b = 8, c = 1.4, d = –4
The amplitude of the graph is a, which is 6.
The equation of the midline is y = d, or y = –4.
Minimum value = d – a
Maximum value = d + a
Minimum value = –4 – 6
Maximum value = –4 + 6
Minimum value = –10
Maximum value = 2
The range of the graph is {y | –10  y  2, y  R}.
Since b = 8, the graph completes 8 cycles in 360° or 2 radians.
Period =
Period =
Period = 45°
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
25. ANS:
y = a cos b(x – c) + d
y = cos
(x + 180°) + 3
a = 1, b =
, c = –180°, d = 3
The amplitude of the graph is a, which is 1.
The equation of the midline is y = d, or y = 3.
Minimum value = d – a
Maximum value = d + a
Minimum value = 3 – 1
Maximum value = 3 + 1
Minimum value = 2
Maximum value = 4
The range of the graph is {y | 2  y  4, y  R}.
Since b =
, the graph completes 0.75 cycles in 360° or 2 radians.
Period =
Period =
Period = 480°
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
26. ANS:
y = a sin b(x – c) + d
y = 2 sin 6x + 1
a = 2, b = 6, c = 0, d = 1
The amplitude of the graph is a, which is 2.
Triple 2 is 6, so the new graph has a = 6.
The equation of the midline is y = d, or y = 1.
The new graph has a midline 5 units below y = 1, or at y = –4.
The new graph has d = –4.
Since b = 6, the graph completes 6 cycles in 2 radians.
If the period triples, then the graph completes only 2 cycles in 2 radians.
The new graph has b = 2.
The function of the new graph is:
y = 6 sin 2x – 4
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
27. ANS:
a) Minimum value = 1
Maximum value = 3
The amplitude of the graph is 1.
The equation of the midline is y = 2.
Minimum value = 1
Maximum value = 3
The range of the graph is {y | 1  y  3, y  R}.
Since the distance from a maximum to a minimum is half the period, the period of the function
must be double 8 – 4, or 8.
b) I know that the period is 8, so (12, 3) must also be a point on the graph since (4, 3) is a point on
the graph.
The point at x = 10 is halfway between a minimum at (8, 1) and a maximum at (12, 3).
Therefore, the point is on the midline.
The equation of the midline is y = 2.
The y-value of this graph when x = 10 is 2.
c) I know that the period is 8, so the y-value at x = 100 is the same as the y-value at x = 4 because
100 – 8(12) = 4.
I know that the graph passes through the point (4, 3).
The y-value of this graph when x = 100 is 3.
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
28. ANS:
a) y = a cos b(x – c) + d
h(t) = –14 cos 3.2t + 16
a = –14, b = 3.2, c = 0, d = 16
The amplitude of the graph is the absolute value of a, which is 14.
The equation of the midline is y = d, or y = 16.
Minimum value = d – a
Maximum value = d + a
Minimum value = 16 – 14
Maximum value = 16 + 14
Minimum value = 2
Maximum value = 30
The range of the Ferris wheel is 2 m to 30 m.
b) Since b = 3.2, the graph completes 3.2 cycles in 2 minutes.
Period =
Period = 1.963...
The Ferris wheel completes one revolution every 1.96 min.
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
29. ANS:
a) y = a sin b(x – c) + d
h(t) = 11.5 sin
a = 11.5, b =
+ 9.8
, c = 0, d = 9.8
The amplitude of the graph is a, which is 11.5.
Maximum value = d + a
Maximum value = 9.8 + 11.5
Maximum value = 21.3
The maximum height of the gymnast jumps is 21.3 ft.
b) The time between consecutive jumps is the period of the graph.
Since b =
, the graph completes
cycles in 2 seconds.
Period =
Period =
Period = 4
The gymnast jumps once every 4 s.
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
30. ANS:
a) y = a cos b(x – c) + d
h(t) = 34 cos 4.6(t + 1.7) + 34
a = 34, b = 4.6, c = –1.7, d = 34
The diameter is double the amplitude, a, which is 68 cm.
b) Since b = 4.6, the graph completes 4.6 cycles in 2 seconds.
Period =
Period = 1.365...
The circumference of the wheel is 2 (34) centimetres, or about 213.6 cm.
(213.628...)(1.365...) = 291.796...
The bicycle is moving at 292 cm/s or about 3 m/s.
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 3.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 3.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 3.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | amplitude | period
31. ANS:
a) y = a cos b(x – c) + d
h(t) = 0.76 cos (0.25t) + 3.82
a = 0.76, b = 0.25, c = 0, d = 3.82
Minimum value = d – a
Minimum value = 3.82 – 0.76
Minimum value = 3.06
The minimum depth is 3.06 m or 306 cm.
b) 10:30 a.m. the next day is 17.5 h after 5:00 p.m. on April 19, 2012.
h(17.5) = 0.76 cos (0.25)(17.5) + 3.82
h(17.5) = 0.76(–0.3310...) + 3.82
h(17.5) = 3.568...
The depth of the water at 10:30 a.m. on April 20, 2012 is about 3.57 m.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | period
32. ANS:
a) y = a sin b(x – c) + d
h(t) = 25.4 sin (1.8t + 4.0) + 24.2
a = 25.4, b = 1.8, c =
= –2.222..., d = 24.2
25.4 + 24.2 = 49.6
The maximum height of the Ferris wheel is 49.6 m.
b) Since b = 1.8, the graph completes 1.8 cycles in 2 minutes.
Period =
Period = 3.490...
The period of rotation is 3.5 min.
c) The graph of this function has x-intercepts because a is greater than d. At these points, Zack is at
ground level. In the context of this question, the Ferris wheel is partially underground.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | period
33. ANS:
a)
b)
According to the curve of best fit, after 9 s, the depth of the water is 3.9 m.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate | curve of best fit
34. ANS:
a)
b)
According to the curve of best fit, there are maximums at (11.18, 3.05) and (34.87, 3.05).
34.87 – 11.18 = 23.69
0.69(60) = 41.4
The period of the graph is 23 h 41 min.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | period | curve of best fit
35. ANS:
a)
b)
According to the curve of best fit, the population in the 12th year is 195.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate | curve of best fit
36. ANS:
a)
b)
According to the curve of best fit, the missing value is (6, 5.728).
0.728(60) = 43.68
The missing value for June 1 is 05:44.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | interpolate | curve of best fit
37. ANS:
I used a spreadsheet to determine the equation of the sinusoidal regression function:
y = 18.063 sin (0.401x – 1.347) + 4.615
October is the 10th month:
y = 18.063 sin (0.401(10) – 1.347) + 4.615
y = 12.933...
The average temperature in October is 12.9 °C.
November is the 11th month:
y = 18.063 sin (0.401(11) – 1.347) + 4.615
y = 6.015...
The average temperature in November is 6.0 °C.
December is the 12th month:
y = 18.063 sin (0.401(12) – 1.347) + 4.615
y = –1.125...
The average temperature in December is –1.1 °C.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | extrapolate
38. ANS:
I used a spreadsheet to determine the equation of the sinusoidal regression function:
y = 57.87 sin (2.30x – 0.57) + 132.00
The amplitude of the graph is the value of a, which is 57.87.
The equation of the midline is y = d, or y = 132.00.
Minimum value = d – a
Maximum value = d + a
Minimum value = 132.00 – 57.87
Maximum value = 132.00 + 57.87
Minimum value = 74.13
Maximum value = 189.87
Alex’s maximum height is 190 cm and her minimum height is 74 cm.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | amplitude | midline
39. ANS:
a) I used a spreadsheet to determine the equation of the sinusoidal regression function:
y = 22.562 sin (0.476x + 3.101) + 312.757
The time between minimum populations is the period of the regression function.
Since b = 0.476, the graph completes 0.476 cycles in 2 years.
Period =
Period = 13.199...
The time between minimum populations is 13.2 years.
b) For the 10th year, x = 10:
y = 22.562 sin (0.476(10) + 3.101) + 312.757
y = 335.318...
The population in the 10th year should be about 335.
c) For the 20th year, x = 20:
y = 22.562 sin (0.476(20) + 3.101) + 312.757
y = 313.988...
The population in the 20th year should be about 314.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | period | extrapolate
40. ANS:
I used a spreadsheet to determine the equation of the sinusoidal regression function:
y = 1.123 sin (0.445x + 1.991) + 6.766
The amplitude of the graph is the value of a, which is 1.123.
The equation of the midline is y = d, or y = 6.766.
Minimum value = d – a
Maximum value = d + a
Minimum value = 6.766 – 1.123
Maximum value = 6.766 + 1.123
Minimum value = 5.643
Maximum value = 7.889
0.643(60) = 38.58
0.889(60) = 53.34
The earliest sunrise is at 05:39 and the latest sunrise is at 07:53.
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 3.4 Graph data and determine the sinusoidal function that best approximates the data. | 3.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 3.6
Solve, using technology, a contextual problem that involves data that is best represented by graphs
of sinusoidal functions, and explain the reasoning.
TOP: Modelling data with sinusoidal functions
KEY: sinusoidal function | regression function | amplitude | midline