Multiple Choice [ /10]Identify the choice that best completes the statement or answers the question.

____ 1. For the function y = sin x, the range is

a. y c. –1 y 1

b. y 0 d. –360° y 360°

____ 2. The minimum value of y = sin x is

a. –1 c.

b. 0 d. 1

____ 3. If the graph of y = cos x is translated 3 units upward, the new function is defined by the equation

a. y = cos (x – 3) c. y = cos x – 3

b. y = cos (x + 3) d. y = cos x + 3

____ 4. If the graph of y = sin x is translated 60° to the left, the new function is defined by the equation

a. y = sin (x + 60°) c. y = sin x + 60°

b. y = sin (x – 60°) d. y = sin x – 60°

____ 5. What is the amplitude of the function y = cos (x + 180°) – 3?

a. 180° c. –3

b. 1 d. –60°

____ 6. What is the maximum value of the function y = sin (x + 45°) – 4?

a. –5 c. –3

b. –4 d. –1

____ 7. The graph of y = sin x is stretched vertically by a factor of 2 and reflected in the y-axis. For this transformation, determine the values of a and k in the equation y = a sin kx.

a. 2, –1 c. –1, –2

b. 1, 2 d. –2, 1

____ 8. The period of

is

a. 180° c. 720°

b. 360° d. 1440°

____ 9. A cosine function has an amplitude of 3, a period of 720°, and a maximum of (0°, 4). What is the equation of this function?

a. y = 3 cos 2x + 1 c. y = 4 cos 2x

b. y = 3 cos(1/2x )+ 1 d. y = 4 cos (1/2x)

____ 10. If the graph of y = cos x is translated 1 unit upward and 45° to the left, the new function is defined by the equation

a. y = cos (x + 45°) + 1 c. y = cos (x – 45°) + 1

b. y = cos (x + 45°) – 1 d. y = cos (x – 45°) – 1

Similarities Differences

11. State four similarities and two differences for the functions y = sin x and y = cos x. [ /3]

Communication [ /8]

Chapter #2 test sinusoidal functionSunday, October 07, 201211:23 AM

Chapter 2 test sinusoidal function Page 1

12. Determine the domain and range of the function y = sin (x + 45°) + 3. [ /2]

|k| > 1

0 < |k| < 1

k < 0

13. Describe what happens to the graph of y = 5 cos [k(x + 60°)] – 2 as k varies. [ /3]

Application [ /8]

14. Write one cosine and one sine equation that can be represented by this graph?

16. Sketch a graph of y = 2 sin [2(x – 30°)] + 1 for

15. Graph the function y = sin (x + 90°). What do you notice about this graph?

.

Determine the following:

Chapter 2 test sinusoidal function Page 2

Determine the following:

Phase shift:

Period:

Amplitude:

Vertical translation:

Domain:

Range:

17. A sine function has an amplitude of 4 units, a period of 120°, and a maximum at (60°, 4). What is the equation of thissine function?

y = 4 sin [3(x – 30°)]

a) What is the amplitude of the disturbance?

b) What is the period of the function?

c) Rewrite this equation as a cosine function.

18. An echo is the sound reflected from an object, disturbing the particles of the medium (air, for example) through which the sound travels. The disturbance in a certain medium can be represented by the equation y = 68 sin

( + 180°).

19. The Bay of Fundy is located on the east coast of Canada. Tides in one area of the bay affect the water level by raising it to 8 m above sea level and lowering it to 8 m below sea level. Approximately every 12 h, the tide completes one cycle. Write a sinusoidal function to represent the height, h, in metres, of the water after t hours, for this section of the Bay of Fundy. Identify and explain the restrictions on the domain of this function.

Thinking

sinusoidal functionAnswer Section

OBJ: Section 2.1 LOC: C2.1 | C2.2 TOP: Graphs of Sinusoidal FunctionsKEY: range

1. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 LOC: C2.1 | C2.2 TOP: Graphs of Sinusoidal FunctionsKEY: minimum value

2. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C2.3 TOP: Translations of Sinusoidal FunctionsKEY: vertical translation

3. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C2.3 TOP: Translations of Sinusoidal FunctionsKEY: horizontal translation

4. ANS: A PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 | Section 2.2 LOC: C2.1 | C2.2 | C2.3 TOP: Graphs of Sinusoidal Functions | Translations of Sinusoidal FunctionsKEY: amplitude | vertical translation | horizontal translation

5. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.1 | Section 2.2 LOC: C2.1 | C2.2 | C2.3 TOP: Graphs of Sinusoidal Functions | Translations of Sinusoidal FunctionsKEY: maximum value | vertical translation | horizontal translation

6. ANS: C PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.3 LOC: C2.4 TOP: Stretches and Compressions of Sinusoidal FunctionsKEY: vertical stretch | reflection

7. ANS: A PTS: 1 DIF: 3 REF: Knowledge and Understanding

MULTIPLE CHOICE

Chapter 2 test sinusoidal function Page 3

OBJ: Section 2.1 | Section 2.3 LOC: C2.1 | C2.2 | C2.4 TOP: Graphs of Sinusoidal Functions | Stretches and Compressions of Sinusoidal FunctionsKEY: period

8. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 2.5 LOC: C2.6 TOP: Representing Sinusoidal FunctionsKEY: amplitude | period | maximum value | equation

9. ANS: B PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 2.2 LOC: C2.3 TOP: Translations of Sinusoidal FunctionsKEY: vertical translation | horizontal translation

10. ANS: A PTS: 1 DIF: 3 REF: Knowledge and Understanding

Example:Similarities: four of the following: period, domain, range, amplitude, shape, maximum value, minimum valueDifferences: x-intercepts and y-intercepts, intervals of increase and decrease

PTS: 1 DIF: 3 REF: Communication OBJ: Section 2.1 LOC: C2.1 | C2.2 TOP: Graphs of Sinusoidal FunctionsKEY: period | domain | range | amplitude | shape | maximum value | minimum value | x-intercept | y-intercept

11. ANS:

domain: x

range: 2 x 4

PTS: 1 DIF: 2 REF: Knowledge and UnderstandingOBJ: Section 2.2 LOC: C2.3 TOP: Translations of Sinusoidal FunctionsKEY: domain | range | vertical translation | horizontal translation

12. ANS:

When |k| > 1, there is a horizontal compression. When 0 < |k| < 1, there is a horizontal expansion. When k < 0, there is a reflection in the y-axis.

PTS: 1 DIF: 3 REF: Communication OBJ: Section 2.3 | Section 2.4 LOC: C2.4 | C2.5 TOP: Stretches and Compressions of Sinusoidal Functions | Combining Transformations of Sinusoidal Functions KEY: horizontal stretch | horizontal compression | reflection in the y-axis

13. ANS:

y = 3 cos 2x – 2

PTS: 1 DIF: 3 REF: Knowledge and Understanding OBJ: Section 2.5 LOC: C2.6 TOP: Representing Sinusoidal FunctionsKEY: graph | equation | representing sinusoidal functions

14. ANS:

It is the same as the graph of y = cos x.

PTS: 1 DIF: 3 REF: Communication OBJ: Section 2.1 | Section 2.2 LOC: C2.2 | C2.3 TOP: Graphs of Sinusoidal Functions | Translations of Sinusoidal FunctionsKEY: graphing sinusoidal functions | translation

15. ANS:

Phase shift 30° to the right, period 180°, amplitude 2, vertical translation 1 unit up,

domain –360° x 360°, range –1 y 3

PTS: 1 DIF: 3 REF: Knowledge and UnderstandingOBJ: Section 2.1 | Section 2.4 LOC: C2.2 | C2.5

16. ANS:

SHORT ANSWER

Chapter 2 test sinusoidal function Page 4

OBJ: Section 2.1 | Section 2.4 LOC: C2.2 | C2.5 TOP: Graphs of Sinusoidal Functions | Combining Transformations of Sinusoidal FunctionsKEY: graph | equation | phase shift | period | amplitude | vertical translation | domain | range

Amplitude 3, vertical translation 3 units up, period 120°, equation y = 3 cos 3x + 3

PTS: 1 DIF: 3 REF: Knowledge and UnderstandingOBJ: Section 2.5 LOC: C2.6 TOP: Representing Sinusoidal FunctionsKEY: amplitude | vertical translation | period | equation

17. ANS:

a) The amplitude is 68.b) The period is 360°.

c) The equation as a cosine function is y = 68 cos ( + 90°).

PTS: 1 DIF: 3 REF: Application | CommunicationOBJ: Section 2.6 LOC: C3.3 TOP: Solving Problems Involving Sinusoidal FunctionsKEY: sinusoidal function | amplitude | cosine function | reflection

18. ANS:

The amplitude is 8.h(t) = 8 sin 30tSince the number of hours has to be positive, the domain is

.

PTS: 1 DIF: 3 REF: Application OBJ: Section 2.5 LOC: C2.6 TOP: Representing Sinusoidal Functions KEY: equation | sinusoidal function | restriction

19. ANS:

PROBLEM

Chapter 2 test sinusoidal function Page 5