2013-2014 Precalculus A Reviewmontgomeryschoolsmd.org/uploadedFiles/curriculum/math...PRECALCULUS A...

PRECALCULUS A Semester Exam Review

MCPS © 2013–2014 1

The semester A examination for Precalculus consists of two parts. Part 1 is selected response on which a calculator will NOT be allowed. Part 2 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 examination booklet.

Trigonometric Identities

2 2sin cos 1 2 2sec 1 tan 2 2csc 1 cot

sin sin cos cos sin cos cos cos sin sin

sin 2 2sin cos 2 2 2 2cos 2 cos sin 2cos 1 1 2sin

tan tantan1 tan tan

22 tantan 2

1 tan

1 cossin2 2

1 coscos2 2

1 cos sintan2 sin 1 cos

Triangle Formulas

Law of Sines: sin sin sinA B Ca b c

Law of Cosines: 2 2 2 2 cosc a b ab C

Area of a Triangle: 1 sin2

ab C

Arc Length

s r ( in radians) 2360

s r ( in degrees)


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PART 1 NO CALCULATOR SECTION

1. Sketch the graph of the piece-wise function 2 , if 0

1 , if 0

x xf x

x x

2. Look at the graph of the piece-wise function below.

Which of the following functions is represented by the graph?

A 2 , if 0, if 0

x xf x

x x

B 2 , if 0, if 0

x xf x

x x

C 2, if 0

, if 0x x

f xx x

D 2, if 0, if 0

x xf x

x x

x

y

O


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3. Look at the graph of the piece-wise function below.

Which type of discontinuity does the graph have at the following x-values?

a. 3x

b. 1x

c. 4x

4. Let 27 , if 5

, if 5cx x

f xx x

What is the value of c that will make f x continuous at 5x ?

5. Let 4 , if 115 2 , if 11

x c xg x

x x

What is the value of c that will make f x continuous at 11x ?

x

y

O


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6. Which of the following is true about the function 43

xf xx

?

A The function is continuous for all real numbers.

B The function is discontinuous at 3x only.

C The function is discontinuous at 4x only.

D The function is discontinuous at 3x and 4x .

7. 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?

8. 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?

x

y

x

y


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9. Determine whether each function below is odd, even, or neither odd nor even.

a. 3sinf x x x

b. 2 4g x x

c. cos 4r x x x

10. If 23f x x , which of the following statements is NOT true?

A The graph of f x is symmetric with respect to the y-axis.

B f x is an even function.

C The range of f x is all real numbers.

D As ,x f x .

11. Look at the graph of the function below.

a. What is the domain of this function?

b. What is the range of this function?

x

y

O


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12. For each function below, find a formula for 1f x and state any restrictions on the domain of 1f x .

a. 2f x x

b. 3 4f x x

13. True or False.

a. The function 5 2g x f x represents a vertical stretch of the graph of f x by a factor of 5, followed by a vertical translation down 2 units.

b. The function 12 4

xg x f

represents a vertical and horizontal shrinking of the

graph of f x .

14. Match the transformations that would create the graph of g x from the graph of f x .

_______ 3g x f x A Stretch the graph of f x horizontally

_______ 3g x f x B Stretch the graph of f x vertically

_______ 13

g x f x

C Shrink the graph of f x horizontally

_______ 13

g x f x D Shrink the graph of f x vertically


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For items 15 and 16, use the graphs of f x and g x below.

f x g x

15. Which of the following represents the relationship between f x and g x ?

A 2 3g x f x

B 1 32

g x f x

C 2 3g x f x

D 1 32

g x f x

16. Sketch the graph of y f x .

17. Write the definitions of the six circular functions of an angle in standard position,

passing through the point ,x y , with 2 2r x y .

18. If 4sin5

with cos 0 , what are the values of other five trigonometric functions?

x

y

x

y


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19. 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

20. Convert to radian measure. Leave your answer in terms of .

a. 40o

b. 165o

21. 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.

a. sin

b. cos

c. tan

22. Sketch the graphs of the six circular functions on the interval 2 2x .

23. What are the periods of the six circular functions?

24. What is the period of tan 8y x ?

25. What is the value of b such that cosy bx has a period of 3 ?

26. What is the value of c such that cscy cx has a period of 10?

x

y

A

B


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27. Determine the exact value of the following.

a. sin6

b. 5cos4

c. 5tan3

d. 3sin2

e. cos f. tan2

g. 7tan4

h. 4cos

3

i. 11sin6

j. sin

4

k. 5cos6

l. 7tan6

m. 5sec4

n. 5cot

6

o. 4csc3

p. sec

28. Complete the table for the inverse circular functions.

1Sin x 1Cos x 1Tan x Domain

Range

29. Identify the functions represented by the graphs below.

a.

b. c.

x

y

1 2 3

O–3 –2 –1

1 0 –1 x

y

1 0 –1 x

y


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30. Determine the exact value of the following.

a. 1 1Sin2

b. 1 1Cos2

c. 1Tan 3

d. 1 3Sin2

e. 1 1Cos2

f. 1Tan 1

g. 1Sin 1 h. 1Cos 0 i. 1Tan 0

j. 1 3cos Sin2

k. 1sin Tan 1 l. 1 1tan Cos2

m. 1 8sin Csc5

n. 1 12tan Sin13

o. 1 11Cos cos6

31. Write a sinusoidal equation for each of the following graphs.

a.

b. c.

1

3

5

0

y

x

4 –4

–3

3

2 x

y

2

–6


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32. Determine the equation that best describes a sine curve with amplitude 3, period of 6, and

a phase shift of 2 to the right.

33. For each equation below, state the amplitude, period, phase shift, vertical translation, and any reflections of the sinusoid relative to the basic function sinf x x or cosg x x . Sketch the graph, marking the x- and y-axes appropriately.

a. 2sin 3 56

h x x

b. 5cos 1h x x

c. sin 4 2h x x

34. Simplify each expression below as a single function of a single angle. Do not evaluate.

a. 2sin17 cos17o o b. 3 3cos cos sin sin7 7 7 7

c. sin 7 cos3 cos 7sin 3 d. tan11 tan 251 tan11 tan 25

o o

o o

e. 21 2sin9

f. cos8cos5 sin 8sin 5

g. 1 cos 232

o h. 7 5 7 5sin cos cos sin13 13 13 13

35. If 12cot and 05 2

A A , determine the following.

a. sin 2A b. cos 2A c. sin2A

d. cos2A

36. Solve the following equations over the interval 0 360o .

a. 2sin 2 0 b. 3cos 4 5cos 5

37. Solve the following equations on the interval 0 2x .

a. tan 1 0x b. 22sin 3sin 1 0x x


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38. Complete the following chart.

Radius Angle (radians) Arc length 6 inches

4

56 15 feet

10 meters

30 meters

39. Prove the following identities.

a. sin cot cos

b. 2sin cos 1 sin 2x x x

c. 2csc sin

1 cotx x

x

d. sin cos sec csccos sin

e. sin sin 2sin cosx y x y x y

f. 2 2 2 2sin sin tan tan


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PART 2 CALCULATOR SECTION

A calculator may be used on items 40 through 54. 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.

40. A ball on a string is swinging back and forth from a ceiling, as shown in the figure below.

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 0t seconds, the ball is farthest from the wall, 160d 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 5t 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.

d


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41. 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 3t seconds. When 7t seconds, Sara is at the lowest point, 20 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 11.5t seconds? Your answer should be correct to three places after the decimal point.

c. Determine the first two times, 0t , when Sara’s height is 70 feet. Your answer should be correct to three places after the decimal point.

42. At Ocean Tide Dock, the first low tide of the day occurs at midnight, when the depth of the water is 2 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.

43. Solve the following equations for , where 0 360o o .

a. 3cos 9 7

b. 23sin 7sin 2 0

44. How many triangles ABC are possible if 20 , 40, and 10oA b a ?

45. Given ABC , where 41 , 58 , and 19.7cmo oA B c , determine the measure of side b. Your answer should be correct to three places after the decimal point.

46. In , 9, 12, 16ABC a b c . What is the measure of B ? Your answer should be correct to the nearest tenth of a degree.

47. Determine the remaining sides of a triangle with 58 , 11.4, 12.8oA a b . Your answers (sides and angles) should be correct to the nearest tenth.


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48. From a point 200 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.

49. 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 (5280 feet), how many feet in elevation has the truck fallen? Your answer should be correct to three places after the decimal point.

50. 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.

51. 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 top three places after the decimal point.

52. Determine the area of triangle ABC if 4, 10, and 30oa b m C .

20 feet 50o 50o

50 miles 34o 20o


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53. A real estate appraiser wishes to find the value of the lot below.

a. Determine the length of the third side of the lot.

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.

54. Find the area of the quadrilateral below. Your answer should be correct to three places after the decimal point.

250 feet

160 feet

62o

35 38

50 58 72o

126o

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