chapter #2 test sinusoidal function - math for...
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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
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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
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