Math 12 Sinusoidal functions 2Math 12 Sinusoidal functions 2 Page 7 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

Math 12 Sinusoidal functions 2 P

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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


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____ 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


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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?


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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.


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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.


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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 – )


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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.


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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


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





KEY: sinusoidal function | amplitude





graphs. TOP: The graphs of sinusoidal functions KEY: sinusoidal function






KEY: sinusoidal function | amplitude | midline

7. ANS: D PTS: 1 DIF: Grade 12 REF: Lesson 8.4




graphs. TOP: The equations of sinusoidal functions

KEY: sinusoidal function | period







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SHORT ANSWER

1. ANS:

y = 0

PTS: 1 DIF: Grade 12 REF: Lesson 8.2



KEY: periodic function | midline

2. ANS:

{x | x R}




KEY: periodic function

3. ANS:

The vertical distance is twice the amplitude.




KEY: periodic function | amplitude

4. ANS:

3




KEY: periodic function | turning point

5. ANS:

–90° or




KEY: periodic function

6. ANS:

y = 3




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0




KEY: sinusoidal function | midline

7. ANS:

5







8. ANS:

6







9. ANS:

{y | –2 y 8, y R}






10. ANS:

{y | –4 y 12, y R}






11. ANS:

{y | –5 y 12, y R}







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1

12. ANS:

15






KEY: sinusoidal function | amplitude | midline

13. ANS:

5







14. ANS:

360° or 2 radians







15. ANS:

{y | –10 y 0, y R}





graphs. TOP: The equations of sinusoidal functions KEY: sinusoidal function

16. ANS:

y = cos x was translated radians to the right






17. ANS:

36 m


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2






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






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.


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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.






KEY: sinusoidal function | period | extrapolate

Documents

Math 12 Sinusoidal functions 2Math 12 Sinusoidal functions 2 Page 7 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