Review for Unit 8 Sinusoidal Functions Name:_______________
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.
____
2.1
0.7
2.8
3.1
2. Choose the best estimate for 136° in radians.
A.
B.
C.
D.
____
3. Choose the best estimate for 0.8 radians in degrees.
A.
B.
C.
D.
____
4. Choose the best estimate for 3.1 radians in degrees.
A.
B.
C.
D.
____
8°
15°
30°
45°
31°
85°
135°
175°
5. Choose the best estimate for the central angle in degrees.
A. 263°
B. 273°
C. 283°
D. 293°
____
6. Choose the best estimate for the central angle in degrees.
A.
B.
C.
D.
____
84°
276°
444°
636°
7. Choose the best estimate for the central angle in radians.
A.
B.
C.
D.
____
8. How many turning points does the graph of y = sin x have from 0° to 360°?
A.
B.
C.
D.
____
0
1
2
3
9. Which of the following is not an x-intercept of the graph of y = sin x?
A. 0°
B. 360°
C. –180°
D. 270°
____
10. Which of the following is not an x-intercept of the graph of y = cos x?
A.
B.
C.
D.
____
11. Which of the following is not a periodic function?
A.
B.
C.
D.
____
y = cos x
y=x
y = –sin x
y = –2 cos x
12. Determine the midline of the following graph.
A.
B.
C.
D.
____
90°
–90°
180°
270°
y=2
y=3
y=4
y=5
13. Determine the amplitude of the following graph.
A. 2
B. 3
C. 4
D. 5
____
14. Determine the period of the following graph.
A.
B.
C.
D.
____
15. 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}
16. 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.
____
5
6
7
8
y=0
y = –5
y = 13
y=8
17. 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°)
____
18. 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°)
____
19. Determine the amplitude of the following function.
y = 0.5 sin (x – 2)
A.
B.
C.
D.
____
20. Determine the period of the following function.
y = 3 sin 2(x + 90°) – 1
A.
B.
C.
D.
____
y=2
y=3
y=0
y = –1
22. Determine the range of the following function.
y = 0.5 sin (x – 2)
A.
B.
C.
D.
____
180°
360°
720°
1080°
21. Determine the midline of the following function.
y = 3 sin 2(x + 90°) – 1
A.
B.
C.
D.
____
0.5
1
2
0
{y | –3  y  –1, y  R}
{y | –0.5  y  0.5, y  R}
{y | –2  y  2, y  R}
{y | y  R}
23. 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. –0.5
B. 1.0
C. 2.5
D. 4.0
____
24. 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.
____
25. 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.
____
–2
–5
–8
–11
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
26. Determine the equation of the sinusoidal regression function for the data.
x
0
1
2
3
4
5
6
7
y
15.4
14.2
13.1
12.4
12.0
12.1
12.6
13.5
A.
B.
C.
D.
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
Short Answer
1. Estimate the value of 540° in radians, to the nearest tenth.
2. Estimate, to the nearest radian, the measure of the central angle. Check your estimate with a
protractor.
3. For the following pair of angle measures, determine which measure is greater.
75°,
4. Identify the domain of y = sin x.
5. How does the vertical distance from the maximum to the minimum of a periodic function relate to
the amplitude?
6. How many turning points does the graph of y = cos x have from –1.5 to 1.5?
7. 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.
8. 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
9. 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
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.
370°, 6.2
3. 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
4. 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.
5. 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.
sss
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: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
4. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
5. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
6. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
7. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.1
TOP: Understanding angles
KEY: radian
8. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 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
9. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
10. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
11. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
12. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | midline
13. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
14.
15.
16.
17.
18.
19.
20.
21.
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.3
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
22.
23.
24.
25.
26.
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function | midline
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 8.4 Graph data and determine the sinusoidal function that best approximates the data. | 8.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 8.6
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: 8.4 Graph data and determine the sinusoidal function that best approximates the data. | 8.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 8.6
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: 8.4 Graph data and determine the sinusoidal function that best approximates the data. | 8.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 8.6
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: 8.4 Graph data and determine the sinusoidal function that best approximates the data. | 8.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 8.6
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
SHORT ANSWER
1. ANS:
9.4 radians
PTS: 1
KEY: radian
2. ANS:
5
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
3. ANS:
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
KEY: radian
4. ANS:
{x | x  R}
DIF:
Grade 12
REF: Lesson 8.1
TOP: Understanding angles
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs.
TOP: Exploring graphs of periodic functions
KEY: periodic function
5. ANS:
The vertical distance is twice the amplitude.
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 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
6. ANS:
3
PTS: 1
DIF: Grade 12
REF: Lesson 8.2
OBJ: 8.1 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
7. ANS:
36 m
PTS: 1
DIF: Grade 12
REF: Lesson 8.4
OBJ: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The equations of sinusoidal functions
KEY: sinusoidal function
8. ANS:
14.7
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 8.4 Graph data and determine the sinusoidal function that best approximates the data. | 8.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 8.6
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
9. ANS:
6.3
PTS: 1
DIF: Grade 12
REF: Lesson 8.5
OBJ: 8.4 Graph data and determine the sinusoidal function that best approximates the data. | 8.5
Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning. | 8.6
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
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 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
KEY: radian
3. 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
4. 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | amplitude | midline | period
5. 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: 8.1 Describe, orally and in written form, the characteristics of sinusoidal functions by
analyzing their graphs. | 8.2 Describe, orally and in written form, the characteristics of sinusoidal
functions by analyzing their equations. | 8.3 Match equations in a given set to their corresponding
graphs. TOP:
The graphs of sinusoidal functions
KEY: sinusoidal function | period | extrapolate