PreCalculus Honors: Functions and Their Graphs€¦ · PreCalculus Honors: Functions and Their...

PreCalculus Honors: Functions and Their Graphs Semester 1, Unit 3: Activity 14

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SpringBoard-

PreCalculus

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PreCalculus

Springboard Text

Unit 3

Vocabulary:

Initial Side Terminal Side Standard Position Coterminal Angles Subtend Radians Angular Velocity Linear Velocity Reference Triangle Periodic Function Period Amplitude Phase Shift Trigonometric

Functions Sine Unit Circle Tangent Cosecant Secant Tangent Concentric Circles One-to-One Functions Inverse Trigonometric Function Reference Angle

Unit Overview

In this unit, students study trigonometric functions. They graph these functions and analyze their behaviors. They explore inverse trigonometric functions. Students also solve trigonometric equations.

Student Focus

Main Ideas for success in lessons 14-1 and 14-2.

Draw angles in standard position.

Find the initial side and terminal side of an angle.

Identify conterminal angles.

Measure angles in degrees and radians, and convert from one to the other.

Example

Lesson 14-1:

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Lesson 14-2:

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Resources:

SpringBoard-

PreCalculus

Online

Resources:

PreCalculus

Springboard Text

Unit 3

Vocabulary:


Functions Sine Unit Circle Tangent Cosecant Secant Tangent Concentric Circles One-to-One Functions Inverse Trigonometric Function Reference Angle

Unit Overview

In this unit, students study trigonometric functions. They graph these functions and analyze their behaviors. They explore inverse trigonometric functions. Students also solve trigonometric equations.

Student Focus

Main Ideas for success in lessons 16-1 and 16-2.

Label angles and coordinates on the unit circle.

Define reciprocal trigonometric functions using the unit circle.

Evaluate all six trigonometric functions of angles in standard position.

Example

Lesson 16-1:

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Lesson 16-2:

The reciprocal functions of sine, cosine, and tangent are cosecant, secant, and

cotangent, respectively.

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Resources:

SpringBoard-

PreCalculus

Online

Resources:

PreCalculus

Springboard Text

Unit 3

Vocabulary:


Functions Sine Unit Circle Tangent Cosecant Secant Tangent Concentric Circles One-to-One Functions Inverse Trigonometric

Function Reference Angle

Unit Overview

In this unit, students will build on their understanding of right triangle trigonometry as they study angles in radian measure, trigonometric functions, and circular functions. Students will investigate in depth the graphs of the sine and cosine functions and extend their knowledge of trigonometry to include tangent, cotangent, secant, and cosecant, as well as solving trigonometric equations.

Student Focus

Main Ideas for success in lessons 17-1 and 17-2:

Graph trigonometric functions over a given interval.

Describe how changes in the parameters affect the graphs.

Find the amplitudes and periods of trigonometric graphs, and write the function given a graph.

Example

Lesson 17-1:

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Example

Lesson 17-2:

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Resources:

SpringBoard-

PreCalculus

Online

Resources:

PreCalculus

Springboard Text

Unit 3

Vocabulary:




Unit Overview


Student Focus

Main Ideas for success in lesson 18-1:

Graph the reciprocal trigonometric functions, and determine the domain and range.

Find the period and locate asymptotes for the reciprocal trigonometric functions.

Example

Lesson 18-1:

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Resources:

SpringBoard-

PreCalculus

Online

Resources:

PreCalculus

Springboard Text

Unit 3

Vocabulary:




Unit Overview


Student Focus

Main Ideas for success in lesson 19-1, 19-2, and 19-3:

Define and apply inverse trigonometric functions to real-world situations.

Find values of inverse trigonometric functions.

Example

Lesson 19-1:

To find the values for , interchange the

values of so that the x-values become the y

-values, and the y-values become the x-values.

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Example

Lesson 19-2:

To find the values for , interchange the values of so that the

x-values become the y -values, and the y-values become the x-values.

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Example

Lesson 19-3:

To find the values for , interchange the values of so that

the x-values become the y -values, and the y-values become the x-values.

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Resources:

SpringBoard-

PreCalculus

Online

Resources:

PreCalculus

Springboard Text

Unit 3

Vocabulary:




Unit Overview


Student Focus

Main Ideas for success in lessons 20-1 and 20-2:

Use inverse functions and reference angles to solve trigonometric equations.

Determine when solutions are limited to a given interval.

Example

Lesson 20-1:

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Example

Lesson 20-2:

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1© 2015 College Board. All rights reserved. SpringBoard Precalculus, Unit 3 Practice

LeSSon 14-1 1. An angle of 1998 is drawn on a coordinate plane

with its vertex at the origin and its initial side on the positive x-axis. In which quadrant does the terminal side lie?

A. Quadrant I

B. Quadrant II

C. Quadrant III

D. Quadrant IV

2. Find an angle between 08 and 3608 that is coterminal with the given angle.

a. 6728

b. 21948

3. Model with mathematics. The second hand of a clock is 6 inches long.

a. How many degrees does the second hand rotate in 23 minutes?

b. How far does the tip of the second hand travel in 23 minutes?

4. Make sense of problems. Find the perimeter of the given figure.

7 inches

368

5. A end-pivot irrigation pipe that is 113 feet long runs with an arc length of 65 feet. How many degrees does the pipe pivot?

LeSSon 14-2 6. An angle of 15

13p radians is drawn on a coordinate

plane with its vertex at the origin and its initial side on the positive x-axis. In which quadrant does the terminal side lie?

A. Quadrant I

B. Quadrant II

C. Quadrant III

D. Quadrant IV

7. Find an angle between 0 and 2p that is coterminal with the given angle.

a. 235p

b. 61

7p

8. Model with mathematics. The face on the clock of Big Ben has a diameter of 7 meters. What is the maximum linear speed of the tip of the second hand in meters per hour?

9. Make sense of problems. Explain why the answer in Item 8 is a maximum.

Precalculus Unit 3 Practice

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© 2015 College Board. All rights reserved. SpringBoard Precalculus, Unit 3 Practice

10. Find the coordinates of P to three decimal places.

588

P

11

LeSSon 15-1 Model with mathematics. Dexter’s new tricycle has a front wheel with an 11-inch diameter and two back wheels, each with a 5.5-inch diameter. The horizontal difference between the center of the front wheel and the center of a back wheel is 18 inches. Dexter is riding at a steady pace, and the 11-inch wheel rotates once every 1.5 seconds. As Dexter is riding down the street, his bike runs over a freshly painted parking stripe, and each wheel picks up a narrow strip of fresh paint that leaves marks on the pavement. 11. Let t 5 0 seconds represent the time when Dexter’s

front wheel starts to move. Sketch a graph of the height above the pavement of the paint spot on the front wheel as a function of the number of seconds for the first 8 seconds after t 5 0.

1 2 3 4 5 6 7 8

c

2

4

6

8

10

12

y in

ches

t seconds

12. What distance will the front wheel travel in 1.5 seconds?

A. 22p inches

B. 11p inches

C. 5.5p inches

D. 2.75p inches

13. Make use of structure. How long does it take for the back wheel to make one complete revolution?

14. Dexter draws a mark on the front tire with chalk. Sketch a figure that shows this mark at 1458 from the ground. How many seconds after the chalk mark touches the ground will the mark first be in this position?

15. What will be the approximate height of the chalk mark at the time stated in Item 14?

LeSSon 15-2 16. Given the graph of f(x) below, which of the

following is the period of the graph?

x

f(x)

25210

25

22.5

2.5

5

5 10

A. 1

B. 2.5

C. 5

D. 10

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17. Which of the following is the amplitude of the graph of f(x)?

A. 1

B. 2.5

C. 5

D. 10

18. Use appropriate tools strategically. Use a graphing calculator to find a sine or cosine function that best matches the periodic graph in Item 16.

19. Make use of structure. Let g(x) be a vertical shift down 0.5 of f (x) from Item 16. Graph g(x).

x

g(x)

22242628210

210

28

26

24

22

2

4

6

8

10

2 4 6 8 10

20. Do the period and amplitude change for g(x) with the shift? Explain.

LeSSon 15-3Model with mathematics. Suny’s unicycle has a front wheel with a 36-inch diameter. She is riding at a steady pace, and the wheel rotates once every 2 seconds. She also made a mark the front tire with chalk. Suppose that the height of the chalk mark is measured as a vertical distance above or below the center of the wheel. The mark starts at a point on the same horizontal line as the center of the wheel at t 5 0. The wheel turns in the direction shown by the arrow in the figure at the given rate.

Chalk markwhen t 5 0

21. Draw a graph of the height of the mark as a function of time for 0 # t # 8.

22. Make use of structure. Instead of defining the function as height versus time, consider defining it as the height of the chalk mark in feet versus the angle of rotation, measured in degrees, of spoke s. Through how many degrees will the spoke rotate in 3 seconds?

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23. Redraw the graph from Item 21. Label the axes so that the graph illustrates the height of the chalk mark, in feet, as a function of the angle of rotation of spoke s, measured in radians.

24. Use a graphing calculator to graph the function y 5 sin(x). What is the period and amplitude of the graph of y 5 sin(x)?

25. How does the graph of the function in Item 23, including period and amplitude, compare to the graph of y 5 sin(x)?

LeSSon 16-1Attend to precision. For Items 26–28, use the unit circle to give the exact value.

26. sin

34p

27. cos

12

p

28. tan 758

29. Given tan

7

p 5 0.482, find tan p

87

.

30. Make use of structure. Given sin 858 5 0.848, explain how you can use the Pythagorean Theorem to find the coordinates of the point representing 858 on the unit circle.

LeSSon 16-2Attend to precision. For Items 31–33, use the unit circle and the definitions of the reciprocal trigonometric functions to give the exact value.

31. cot

43p

32. csc 2708

33. sec p

Make use of structure. For Items 34–35, find the values of the six trigonometric functions of u.

34. Point P (21, 5) on the terminal side of u, an angle in standard position

35. csc u 5 273

LeSSon 17-1 36. Attend to precision. Complete the table using the

unit circle.

x g(x) 5 cos 3x h(x)5 3 cos x

0

4p

2p

34p

p

54p

32p

74p

2p

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37. Use the axes below to graph the functions in Item 36 from 0 # x # 4p.

x

y

38. What is the period of g(x)?

A. 2p

B. 32p

C. 23p

D. 3p

39. If f (x) is the parent function, what are the period and amplitude of f (x), g(x), and h(x)?

40. Based on your work in Items 36–39, what can you conclude about A and B in the equation y 5 A cos Bx?

LeSSon 17-2Make use of structure. For Items 41–43, state the period and amplitude of each function and describe any phase shifts. Sketch the graph of each function over one period. Carefully label the scale on each axis.

41. y 5 2.5 cos x

x

y

42. y 5 sin

x3

x

y

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43. y 5 12

sin −

x

4p

x

y

44. For the function y 5 22 cos(5x 2 2p) 1 34

, state

the period and amplitude. Describe any horizontal or vertical shifts relative to the parent graph.

45. Construct viable arguments. Explain why the function in Item 44 is not a shift of the parent function to the right by 2p.

LeSSon 18-1 46. Which of the following is the reciprocal function of

the cosine function?

A. cosecant

B. secant

C. tangent

D. cotangent

47. Attend to precision. Complete the table of values using the unit circle.

x f (x) 5 cos x g(x) 5 sec x

0

4p

2p

34p

p

54p

32p

74p

2p

48. Sketch the graph of f(x) and g(x) from the values found in Item 47.

x

y

49. Reason abstractly. Explain how to locate vertical asymptotes of the graph of g(x) in Item 47.

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50. Describe the period, vertical asymptotes, domain, range, and zeros for g(x) in Item 47.

LeSSon 18-2Make use of structure. For Items 51–53, state the period and amplitude of each function and describe any phase shifts. Sketch the graph of each function over one period. Carefully label the scale on each axis.

51. y 5 34

sec x

x

y

52. y 5 csc px

x

y

53. y 5 2 cot x

6p

1

x

y

54. Describe the transformation of y 5 sec x to y 5 sec(x 2 3p) in words.

55. Describe the transformation of y 5 tan x to y 5 2tan(x 1 2) in words.

LeSSon 19-1 56. Attend to precision. Find the exact value of each

expression using the unit circle.

a. cos21 ( 32

2 ) b. cos21 (1

2) 57. Use appropriate tools strategically. Find the

approximate value for each expression using a calculator in radian measure, correct to three decimal places.

a. cos21(20.48)

b. cos21(1.32)

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58. The adjacent side of angle θ in a right triangle has a length of 6. The hypotenuse has a length of 11. Which of the following is the correct equation for finding the measure of angle θ?

A. cos

611

5 θ

B. cos θ 5 6

11

C. cos21

611

5 θ

D. cos21 θ 5 6

11

59. Regina is looking at the top of the flagpole. The line-of-sight distance between Regina and the top of the flagpole is 42 feet. If the horizontal distance between Regina and the flagpole is 23 feet, calculate the approximate angle of elevation (in degrees).

dRegina

42 ft

�

60. Kalem says that the reciprocal and inverse function of y 5 cos x are the same. Is he correct? If not, explain why they are not the same.



a. sin21 ( 32 )

b. sin21 (2 12 )

62. Use appropriate tools strategically. Find the approximate value for each expression using a calculator in radian measure, correct to three decimal places.

a. sin21 (0.957)

b. sin21 (2.01)

63. The opposite side of angle θ in a right triangle has a length of 4. The hypotenuse has a length of 5. Which of the following is the correct equation for finding the measure of angle θ?

A. sin21 (45) 5 θ

B. sin21 θ 5 45

C. sin21 (54) 5 θ

D. sin θ 5 54

64. Reason quantitatively. Explain why sin21 4 displays an error when you use a calculator to find its value.

65. A pole of a volleyball net is tethered to the ground with a rope that is 12 feet long. If the length of the pole is 6 feet, what is the measure, in degrees, of the angle of elevation θ?

12 ft6 ft

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a. tan21 (2 3)

b. tan21 (21)

67. Use appropriate tools strategically. Find the approximate value for each expression using a calculator in radian measure, correct to three decimal places.

a. tan21(1.795)

b. tan21(20.74)

68. Without using a calculator, find the exact value of

cos cos

41 p2 .

A. 0.707

B. 0.667

C. 4p

D. 4p

69. Reason abstractly and quantitatively. Explain how you found your answer to Item 68.

70. Dion is at the top of a lighthouse looking at a boat that is 600 meters from the lighthouse. Calculate the approximate angle of elevation if the height of the lighthouse is 85 meters.

85 m

Dion

600 mboat�

LeSSon 20-1 71. Reason abstractly. Find the general solutions of

each expression.

a. 2 sin x 5 2

b. 4 cos2 x 2 3 5 0

72. express regularity in repeated reasoning. Explain why 2kp is part of the solution to the functions in Item 71.

73. Find the exact solutions of each equation over the interval [0, 2p).

a. csc2 x 2 1 5 3

b. 2 tan x 5 22

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74. Find the exact solutions of each equation over the interval [08, 3608).

a. 2 cos θ 5 3

b. tan3 θ 5 2 3 tan2 θ

75. Which are the exact solutions of 4 sec u 5 8 over the interval [08, 3608)?

A. 458, 2258

B. 458, 1358

C. 308, 1508

D. 308, 3308

LeSSon 20-2Use appropriate tools strategically. For Items 76–77, find the solutions of each equation over the interval [0, 2p). Give answers to the nearest tenth of a radian.

76. 4 csc x 1 11 5 0

77. sin x 1 4 5 4.6

Make use of structure. For Items 78–79, find the solutions of each equation over the interval [08, 3608). Give answers to the nearest tenth of a degree.

78. 8 sin θ 5 3

79. cos2 θ 5 9

16

80. Suppose θ lies in Quadrant II and its reference angle is a 5 41.88. What is the measure of θ?

A. 41.88

B. 138.28

C. 221.88

D. 318.28

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