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Math 12 Sinusoidal functions 2 P
age1
Name:
____ 1. Imagine that it is now 2 p.m. What time will it be when the minute hand has rotated through 300°?
A. 2:40
B. 2:50
C. 3:00
D. 3:10
____ 2. How many turning points does the graph of y = sin x have from 0° to 360°?
A. 0
B. 1
C. 2
D. 3
____ 3. Determine the midline of the following graph.
A. y = 2
B. y = 3
C. y = 4
D. y = 5
____ 4. Determine the amplitude of the following graph.
A. 2
B. 3
C. 4
D. 5
Math 12 Sinusoidal functions 2 P
age2
____ 5. Determine the range of the following graph.
A. {y | 0 y 8, y R}
B. {y | –2 y 6, y R}
C. {y | –4 y 8, y R}
D. {y | y R}
____ 6. A sinusoidal graph has an amplitude of 10 and a maximum at the point (18, 5). Determine the
midline of the graph.
A. y = 0
B. y = –5
C. y = 13
D. y = 8
____ 7. 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°)
____ 8. Determine the amplitude of the following function.
y = cos x + 12
A.
B. 1
C. 2
D. 12
Math 12 Sinusoidal functions 2 P
age3
Note: You must show all work to receive full marks.
1.What is the equation of the midline of y = cos x? 2.
2. Sketch, identify the domain and range of y = sin x.
3. How does the vertical distance from the maximum to the minimum of a periodic function relate to
the amplitude?
4. How many turning points does the graph of y = cos x have from –1.5 to 1.5?
Math 12 Sinusoidal functions 2 P
age4
5. What is the first x-intercept of the graph of y = cos x to the left of the y-axis?
6. Determine the midline of the following graph.
7. Determine the amplitude of the following graph.
8. Determine the period of the following graph.
Math 12 Sinusoidal functions 2 P
age5
9. Determine the range of the following graph.
10. Determine the range of the following graph.
11. 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.
12. A sinusoidal graph has a maximum at the point (–40, 3) and a midline of y = –12. Determine the
amplitude of the graph.
Math 12 Sinusoidal functions 2 P
age6
13. Determine the amplitude of the following function.
y = 5 sin 1.5(x + 60°) – 5
14. Determine the period of the following function.
y = cos (x – )
15. Determine the range of the following function.
y = 5 sin 1.5(x + 60°) – 5
16. Determine the horizontal translation applied to y = cos x to obtain the following function.
y = cos (x – )
Math 12 Sinusoidal functions 2 P
age7
17. 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.
18. 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.
19. 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.
Math 12 Sinusoidal functions 2 P
age8
math 12
Answer Section
MULTIPLE CHOICE
1. ANS: B PTS: 1 DIF: Grade 12 REF: Lesson 8.1
TOP: Understanding angles KEY: radian
2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
Math 12 Sinusoidal functions 2 P
age9
KEY: sinusoidal function | amplitude
SHORT ANSWER
1. ANS:
y = 0
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
2. 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
3. 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
4. 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
5. 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
6. 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
Math 12 Sinusoidal functions 2 P
age1
0
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
7. 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
8. 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
9. 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
10. 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
11. 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
Math 12 Sinusoidal functions 2 P
age1
1
12. 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
13. 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
14. 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
15. 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
16. 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
17. ANS:
36 m
Math 12 Sinusoidal functions 2 P
age1
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
PROBLEM
1. 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
2. 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.
Math 12 Sinusoidal functions 2 P
age1
3
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