Honors Precalculus A. Semester Exam Review

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1 Semester Eam Review Honors Precalculus A Semester Eam Review MCPS

2 Semester Eam Review The semester A eamination for Honors Precalculus consists of two parts. Part 1 is selected response on which a calculator will NOT be allowed. Part is short answer on which a calculator will be allowed. Pages with the symbol indicate that a student should be prepared to complete items like these with or without a calculator. The formulas below are provided in the eamination booklet. Trigonometric Identities sin cos 1 sec 1 tan csc 1 cot sin sincos cossin cos coscos sinsin cos cos sin cos 1 1 sin sin sin cos tan tan tan 1 tan tan tan tan 1 tan 1 cos sin 1 cos cos 1 cos sin tan sin 1 cos Triangle Formulas Law of Sines: sin A sin B sin C Law of Cosines: a b c c a b abcosc Area of a Triangle: 1 sin ab C Arc Length s r ( in radians) s r ( in degrees) 360 Linear Speed v r MCPS

3 PART 1 NO CALCULATOR SECTION 1. Sketch the graph of the piece-wise function f. Look at the graph of the piece-wise function below., if 0 1, if 0 Which of the following functions is represented by the graph? A B f f,if 0,if 0,if 0,if 0 f C, if 0, if 0 f D,if 0,if 0 MCPS

4 3. Look at the graph of the piece-wise function below. y O Which type of discontinuity does the graph have at the following -values? a. 3 b. 1 c Sketch the graph of the following piece-wise function. Be sure to indicate arrows, closed circle endpoints and open circle endpoints. 4, if f 3, if 3 5, if Suppose that lim f 8, and that f 4 4 is undefined. a. Write a limit statement so that there is a removable discontinuity at 4. b. Write a limit statement so that there is a jump discontinuity at 4. MCPS

5 6. Which of the following is true about the function f 4? 3 A The function is continuous for all real numbers. B The function is discontinuous at 3only. C The function is discontinuous at 4only. D The function is discontinuous at 3and Let f c 7,if 5, if 5 What is the value of c that will make f continuous at 5? 8. Let g 4 c, if 11 5, if 11 What is the value of c that will make f continuous at 11? MCPS

6 9. Look at the graph of a function below. Does the graph represent an odd function, an even function, or a function that is neither odd nor even? MCPS

7 10. Look at the graph of a function below. Does the graph represent an odd function, an even function, or a function that is neither odd nor even? 11. Determine whether each function below is odd, even, or neither odd nor even. 3 a. f sin b. g 4 c. rcos 4 d. f cos sin 4 MCPS

8 1. If f 3, which of the following statements is NOT true? A B C The graph of f is symmetric with respect to the y-ais. f is an even function. The range of f is all real numbers. D As, f. 13. Look at the graph of the function below. y O a. What is the domain of this function? b. What is the range of this function? MCPS

9 1 14. For each function below, find a formula for f domain of f 1. and state any restrictions on the a. f b. 3 c. f f Look at the graph of f below. y O a. Sketch the graph of y f. b. Sketch the graph of y f 16. True or False.. a. The function g f 5 represents a vertical stretch of the graph of f by a factor of 5, followed by a vertical translation down units. b. The function graph of f. 1 g f represents a vertical and horizontal shrinking of the 4 MCPS

10 17. Match the transformations that would create the graph of g from the graph of f. g 3 f g f 3 A Stretch the graph of f horizontally B Stretch the graph of f vertically 1 g f 3 C Shrink the graph of f horizontally g f 1 D Shrink the graph of f vertically 3 For items 18 and 19, use the graphs of f and g below. f g y y 18. Which of the following represents the relationship between f and g? A gf 3 1 B g f 3 C g f 3 D 1 g f Sketch the graph of y f. MCPS

11 0. Look at the graph of f below. The domain of f is 6 5. y O a. Describe how to transform the graph of f to g f b. Sketch the graph of g. c. What is the domain of g? d. What is the range of g? 1 3. MCPS

12 3? 1. Which of the following describes the right-end behavior of the function f A B C D lim f lim f 0 lim f 3 lim f. Let and y be related by the equation y 17. a. Determine the values of y if = 0, 1, and 8. b. Is this relation a function? Justify your answer. c. Determine two functions defined implicitly by this relation. 3. Write the definitions of the si circular functions of an angle in standard position, passing through the point, y, with r y. 4. If 4 sin with cos 0, what are the values of other five trigonometric functions? 5 5. For each of the following, state the quadrant in which the terminal side of lies. a. sin 0, tan 0 b. cos 0, tan 0 c. sec 0,csc 0 6. Convert to radian measure. Leave your answer in terms of. a. 40 o b. 165 o MCPS

13 7. On the unit circle below the coordinates of point A are 1, 0 and the coordinates of point B are 0.8,0.6. Find the value of the following. y a. sin B b. cos c. tan A 8. Determine the eact value of the following. a. sin 6 b. 5 cos 4 c. 5 tan 3 d. 3 sin cos f. tan e. g. 7 tan 4 h. 4 cos 3 i. 11 sin 6 j. sin 4 k. 5 cos 6 l. 7 tan 6 m. 5 sec 4 n. 5 cot 6 o. 4 csc 3 p. sec 9. Sketch the graphs of the si circular functions on the interval. 30. What are the periods of the si circular functions? 31. What is the period of y tan 8? 3. What is the value of b such that y cosb 33. What is the value of c such that y cscc has a period of 3? has a period of 10? MCPS

14 34. Complete the table for the inverse circular functions. Domain 1 Sin Cos Tan 1 1 Range 35. Identify the functions represented by the graphs below. y a. 3 1 O 1 3 b. y c. y MCPS

15 36. Determine the eact (not a decimal) value of the following. a. Sin 1 1 b. Cos c. Tan 3 d. 1 3 Sin e. Cos f. Tan g. Sin 1 h. Cos 0 1 i. Tan 0 j. cos Sin k. sin Tan 1 l. tan Cos 1 1 m. sin Csc n. 1 1 tan Sin 13 o Cos cos Write a sinusoidal equation for each of the following graphs. y a b. y c MCPS

16 38. Determine the equation that best describes a sine curve with amplitude 3, period of 6, and a phase shift of to the right. 39. For each equation below, state the amplitude, period, phase shift, vertical translation, and any reflections of the sinusoid relative to the basic function f sin or g cos Sketch the graph, marking the - and y-aes appropriately. 6 a. hsin 3 5 b. h5cos 1 c. h sin Assume that angle A and angle B are supplementary. Which of the following is equal to cos A? A B C D sin B sin B cos B cos B 41. Simplify each epression below as a single function of a single angle. Do not evaluate. o o a. sin17 cos17 b. 3 3 cos cos sin sin c. sin 7 cos3 cos 7sin 3 d. o o tan11 tan 5 o 1 tan11 tan5 o e. 1 sin 9 f. cos8cos5 sin 8sin 5 g. 1 cos 3 o h sin cos cos sin MCPS

17 4. If 1 cot A and 0 A, determine the eact value of the following. 5 a. sin A b. A cos A c. sin A d. tan A e. cos 43. Solve the following equations over the interval o. a. sin 0 b. 3cos 4 5cos Solve the following equations on the interval 0. a. tan 3 0 b. sin 3sin 1 0 c. sin Complete the following chart. Radius Angle (radians) Arc length 6 inches feet 6 10 meters 30 meters 46. How many triangles are possible if a40, b80, and A 30 o? MCPS

18 47. Prove the following identities. a. sin cot cos b. sin cos 1 sin csc c. sin 1 cot d. sin cos sec csc cos sin e. sin y sin y sin cos y f. sin sin tan tan g. tan cot csc h. cot sec sec csc cos cot MCPS

19 PART CALCULATOR SECTION Semester Eam Review A calculator may be used on items 48 through 65. Make sure that your calculator is in the appropriate mode (radian or degree) for each item. Unless otherwise specified, answers should be correct to three places after the decimal point. 48. A wheel of radius 1 cm turns at 7 revolutions per second. Determine the linear speed of a point on the circumference of the wheel in meters per second. 49. A moon makes a circular revolution around its planet in 80 hours. The radius of the circular path is 0,000 miles. a. What is the angular speed of the planet in radians per hour? b. What is the linear speed of the planet in feet per second? 50. A ball on a string is swinging back and forth from a ceiling, as shown in the figure below. d Let d represent the distance that the center of the ball is from the wall at time t. Assume that the distance varies sinusoidally with time. When t 0 seconds, the ball is farthest from the wall, d 160 cm. When t 3 seconds, the ball is closest from the wall, d 0 cm. When t 6 seconds, the ball is farthest from the wall, d 160 cm. a. Sketch a graph of the distance as a function of time. b. Write a trigonometric equation for the distance as a function of time. c. What is the distance of the ball from the wall at t 5 seconds? d. What is the value of t the first time the ball is 40 cm from the wall? Your answer should be correct to three places after the decimal point. MCPS

20 Semester Eam Review 51. Sara is riding a Ferris wheel. Her sister Kari starts a stopwatch and records some data. Let h represent Sara s height above the ground at time t. Kari notices that Sara is at the highest point, 80 feet above the ground, when t 3 seconds. When t 7 seconds, Sara is at the lowest point, 0 feet above the ground. Assume that the height varies sinusoidally with time. a. Write a trigonometric equation for the height of Sara above the ground as a function of time. b. What will Sara s height be at t 11.5 seconds? Your answer should be correct to three places after the decimal point. c. Determine the first two times, t 0, when Sara s height is 70 feet. Your answer should be correct to three places after the decimal point. 5. At Ocean Tide Dock, the first low tide of the day occurs at midnight, when the depth of the water is meters, and the first high tide occurs at 6:30 a.m., with a depth of 8 meters. Assume that the depth of the water varies sinusoidally with time. a. Sketch and label a graph showing the depth of the water as a function of the number of hours after midnight. b. Determine a trigonometric function that models the graph. c. Suppose a ship requires at least three meters of water depth is planning to dock after midnight. Determine the earliest possible time that the ship can dock. o o 53. Solve the following equations for, where a. 3cos 9 7 b. 3sin 7sin 0 o 54. How many triangles ABC are possible if A0, b40, and a 10? o o 55. Given ABC, where A41, B58, and c 19.7cm, determine the measure of side b. Your answer should be correct to three places after the decimal point. 56. In ABC, a 9, b 1, c 16. What is the measure of B? Your answer should be correct to the nearest tenth of a degree. o 57. Determine the remaining sides of a triangle with A58, a11.4, b 1.8. Your answers (side and angles) should be correct to the nearest tenth. MCPS

21 Semester Eam Review 58. From a point 00 feet from its base, the angle of elevation from the ground to the top of a lighthouse is 55 degrees. How tall is the lighthouse? Your answer should be correct to three places after the decimal point. 59. A truck is travelling down a mountain. A sign says that the degree of incline is 7 degrees. After the truck has travelled 1 mile (580 feet), how many feet in elevation has the truck fallen? Your answer should be correct to three places after the decimal point. 60. The owner of a garage shown below plans to install a trim along the roof. The lengths required are in bold. How many feet of trim should be purchased? Your answer should be correct to three places after the decimal point. 50 o 0 feet 50 o 61. An airplane needs to take a detour around a group of thunderstorms, as shown in the figure below. How much farther does the plane have to travel due to the detour? Your answer should be correct to three places after the decimal point. 34 o 0 o 50 miles 6. Determine the area of triangle ABC if a4, b10, and mc 30 o. MCPS

22 Semester Eam Review 63. A real estate appraiser wishes to find the value of the lot below. 160 feet 6 o 50 feet a. How long is the third side of the lot? Your answer should be correct to three decimal places. b. Find the area of the lot. Your answer should be correct to three places after the decimal point. c. An acre is 43,560 square feet. If land is valued at $56,000 per acre, what is the value of the lot? Your answer should be correct to the nearest cent. 64. Find the area of the quadrilateral below. Your answer should be correct to three places after the decimal point o 38 Figure NOT drawn to scale o 65. Triangle ABC has an area of,400 with AB 80, AC 100. Determine the two possible measures of angle A. Your answers should be correct to the nearest tenth of a degree. MCPS

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