Chapter #2 test sinusoidal function

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1 Chapter #2 test sinusoidal function Sunday, October 07, :23 AM 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 The minimum value of y = sin x is a. 1 c. b. 0 d 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 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 What is the amplitude of the function y = cos (x ) 3? a. 180 c. 3 b. 1 d What is the maximum value of the function y = sin (x + 45 ) 4? a. 5 c. 3 b. 4 d 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 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 Communication [ /8] 11. State four similarities and two differences for the functions y = sin x and y = cos x. [ /3] Similarities Differences Chapter 2 test sinusoidal function Page 1

[ /3] k > 1 0 < k < 1 k < 0 Application [ /8] 14.

2 12. Determine the domain and range of the function y = sin (x + 45 ) + 3. [ /2] 13. Describe what happens to the graph of y = 5 cos [k(x + 60 )] 2 as k varies. [ /3] k > 1 0 < k < 1 k < 0 Application [ /8] 14. Write one cosine and one sine equation that can be represented by this graph? 15. Graph the function y = sin (x + 90 ). What do you notice about this graph? 16. Sketch a graph of y = 2 sin [2(x 30 )] + 1 for. Determine the following: Chapter 2 test sinusoidal function Page 2

3 Determine the following: Phase shift: Period: Amplitude: Vertical translation: Domain: Thinking 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 this sine function? y = 4 sin [3(x 30 )] 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 ( ). a) What is the amplitude of the disturbance? b) What is the period of the function? c) Rewrite this equation as a cosine function. 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. sinusoidal function Answer Section MULTIPLE CHOICE 1. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding KEY: range 2. ANS: A PTS: 1 DIF: 1 REF: Knowledge and Understanding KEY: minimum value 3. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding KEY: vertical translation 4. ANS: A PTS: 1 DIF: 2 REF: Knowledge and Understanding KEY: 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 KEY: amplitude vertical translation horizontal translation 6. ANS: C OBJ: Section 2.1 Section 2.2 LOC: C2.1 C2.2 C2.3 KEY: maximum value vertical translation horizontal translation 7. ANS: A OBJ: Section 2.3 LOC: C2.4 TOP: Stretches and Compressions of Sinusoidal Functions KEY: vertical stretch reflection Chapter 2 test sinusoidal function Page 3

ANS: A KEY: vertical translation horizontal translation SHORT ANSWER 11.

4 8. ANS: C PTS: 1 DIF: 2 REF: Knowledge and Understanding OBJ: Section 2.1 Section 2.3 LOC: C2.1 C2.2 C2.4 TOP: Graphs of Sinusoidal Functions Stretches and Compressions of Sinusoidal Functions KEY: period 9. ANS: B KEY: amplitude period maximum value equation 10. ANS: A KEY: vertical translation horizontal translation SHORT ANSWER 11. ANS: Example: Similarities: four of the following: period, domain, range, amplitude, shape, maximum value, minimum value Differences: x-intercepts and y-intercepts, intervals of increase and decrease KEY: period domain range amplitude shape maximum value minimum value x-intercept y- intercept 12. ANS: domain: x range: 2 x 4 PTS: 1 DIF: 2 REF: Knowledge and Understanding KEY: domain range vertical translation horizontal translation 13. 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. 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 14. ANS: y = 3 cos 2x 2 KEY: graph equation representing sinusoidal functions 15. ANS: It is the same as the graph of y = cos x. OBJ: Section 2.1 Section 2.2 LOC: C2.2 C2.3 KEY: graphing sinusoidal functions translation 16. ANS: Phase shift 30 to the right, period 180, amplitude 2, vertical translation 1 unit up, domain 360 x 360, range 1 y 3 OBJ: Section 2.1 Section 2.4 LOC: C2.2 C2.5 Chapter 2 test sinusoidal function Page 4

5 OBJ: Section 2.1 Section 2.4 LOC: C2.2 C2.5 TOP: Graphs of Sinusoidal Functions Combining Transformations of Sinusoidal Functions KEY: graph equation phase shift period amplitude vertical translation domain range PROBLEM 17. ANS: Amplitude 3, vertical translation 3 units up, period 120, equation y = 3 cos 3x + 3 KEY: amplitude vertical translation period equation 18. 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 Communication OBJ: Section 2.6 LOC: C3.3 TOP: Solving Problems Involving Sinusoidal Functions KEY: sinusoidal function amplitude cosine function reflection 19. ANS: The amplitude is 8. h(t) = 8 sin 30t Since 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 Chapter 2 test sinusoidal function Page 5

Section 8.4 Equations of Sinusoidal Functions soln.notebook. May 17, Section 8.4: The Equations of Sinusoidal Functions.

Section 8.4 Equations of Sinusoidal Functions soln.notebook. May 17, Section 8.4: The Equations of Sinusoidal Functions. Section 8.4: The Equations of Sinusoidal Functions Stop Sine 1 In this section, we will examine transformations of the sine and cosine function and learn how to read various properties from the equation.