Graphing the Sine Function › ... › TB › ch5 › s52 › 1330_Ch5_Section2.pdf · 2011-10-14 · Graphing the Sine Function: Graphing the Cosine Function: SECTION 5.2 Graphs

CHAPTER 5 Graphing Trigonometric Functions

University of Houston Department of Mathematics 446

Section 5.2: Graphs of the Sine and Cosine Functions

Basic Graphs

Transformations

Basic Graphs

Graphing the Sine Function:

Graphing the Cosine Function:

SECTION 5.2 Graphs of the Sine and Cosine Functions

MATH 1330 Precalculus 447

Definition of a Periodic Function and Its Period:

Sine Function:



Cosine Function:

Definition of Amplitude:

Sine Function:



Cosine Function:

Example:

Solution:



Additional Example 1:

Solution:


Solution:




Solution:


Solution:



Transformations

Vertical Scaling:

Graph of y = Asin(x):



Graph of y = Acos(x):



Vertical Translations:

Graph of y = sin(x) + D:

Graph of y = cos(x) + D:



Example:

Solution:



Horizontal Scaling:

Graph of y = sin(Bx):



Example:

Solution:

Graph of y = cos(Bx):



Example:

Solution:



Horizontal Translations:

Graph of y = sin(x - C):

Graph of y = cos(x - C):



Example:

Solution:

Graphs of y = Asin(Bx - C) and y = Acos(Bx - C):



Example:

Solution:



Graphs of y = Asin(Bx - C) + D and y = Acos(Bx - C) + D:



Example:

Solution:




Solution:




Solution:






Solution:

Part (a):



Part (b):




Solution:

Part (a):



Part (b):

Exercise Set 5.2: Graphs of the Sine and Cosine Functions


For each of the following functions,

(a) Find the period.

(b) Find the amplitude.

1.

2.

3.

4.

5.

6.

Answer the following.

7. (a) Use your calculator to complete the

following chart. Round to the nearest

hundredth.

x siny x x siny x

0

6

7

6

4

5

4

3

4

3

2

3

2

2

3

5

3

3

4

7

4

5

6

11

6

2

(b) Plot the points from part (a) to discover the

graph of sinf x x .

(c) Use the graph from part (b) to sketch an

extended graph of sinf x x , where

4 4x . Be sure to show the

intercepts as well as the maximum and

minimum values of the function.

x

y

x

y

x

y

x

y

x

y

x

y

x

y



(d) State the domain and range of

sinf x x . Do not base this answer

simply on the limited domain from part (c),

but the entire graph of sinf x x .

(e) State the amplitude and period of

sinf x x .

(f) Give an interval on which sinf x x is

increasing.

8. (a) Use your calculator to complete the

following chart. Round to the nearest

hundredth.

x cosy x x cosy x

0

6

7

6

4

5

4

3

4

3

2

3

2

2

3

5

3

3

4

7

4

5

6

11

6

2

(b) Plot the points from part (a) to discover the

graph of cosg x x .

(c) Use the graph from part (b) to sketch an

extended graph of cosg x x , where

4 4x . Be sure to show the

intercepts as well as the maximum and


(d) State the domain and range of

cosg x x . Do not base this answer

simply on the limited domain from part (c),

but the entire graph of cosg x x .

(e) State the amplitude and period of

cosg x x .

(f) Give an interval on which cosg x x is

decreasing.


9. Use the graph of cosf x x below to

determine the coordinates of points A, B, C, and

D.

10. Use the graph of sinf x x below to

determine the coordinates of points A, B, C, and

D.

x

y

x

y

A

B

C

D

f

x

y

A

C

f

B

D



A. Shrink horizontally by a

factor of 13 .

B. Shift left 3 units, then

reflect in the y-axis.

C. Reflect in the x-axis,

then shift downward 3

units.

D. Shift right 3 units.

E. Shift right 2 units, then

reflect in the x-axis, then

shift upward 3 units.

F. Shift upward 3 units.

G. Stretch horizontally by a

factor of 3.

H. Shift left 3 units, then

shift upward 2 units.

I. Shift left 3 units.

J. Shift downward 3 units.

K. Stretch vertically by a

factor of 3.

L. Shrink vertically by a

factor of 13 .

Use the graphs of sinf x x and cosg x x

to answer the following.

11. Use the appropriate graph to find the following

function values.

(a) cos 0

(b) 3

cos2

(c) 5

sin2

(d) sin 3

12. Use the appropriate graph to find the following

function values.

(a) sin 0

(b) sin2

(c) cos

(d) 7

cos2

13. For 0 2x , use the graph of sinf x x

to find the value(s) of x for which:

(a) sin 1x

(b) sin 0x

(c) sin 1x

14. For 0 2x , use the graph of cosg x x

to find the value(s) of x for which:

(a) cos 0x

(b) cos 1x

(c) cos 1x

15. For 2 0x , use the graph of

cosg x x to find the value(s) of x for

which:

(a) cos 1x

(b) cos 1x

(c) cos 0x

16. For 2 0x , use the graph of

sinf x x to find the value(s) of x for

which:

(a) sin 1x

(b) sin 1x

(c) sin 0x

Graph each of the following functions over the interval

2 2x .

17. cosf x x

18. sing x x

Matching. The left-hand column contains equations

that represent transformations of sinf x x .

Match the equations on the left with the description on

the right of how to obtain the graph of y g x from

the graph of f .

19. sin 3y x

20. sin 3y x

21. sin 3y x

22. 1

sin3

y x

23. 3siny x

24. sin 3y x

25. sin 3y x

26. 1

sin3

y x

27. sin 3 2y x

28. sin 2 3y x

29. sin 3y x

30. sin 3y x



A. Stretch horizontally by a

factor of 2, then shift

right 16 units.

B. Shrink horizontally by a

factor of 12 , then shift

right 4 units.

C. Shrink horizontally by a


downward 8 units.

D. Stretch horizontally by a


right 8 units.

E. Stretch horizontally by a


downward 8 units.

F. Shrink horizontally by a


right 8 units.

E. Stretch horizontally by a


down 8 units.

F. Shrink horizontally by a


right 8 units.

Matching. The left-hand column contains equations

that represent transformations of cosf x x .

Match the equations on the left with the description on

the right of how to obtain the graph of y g x from

the graph of f .

31. cos 2 8y x

32. cos 2 8y x

33. cos 2 8y x

34. 12

cos 8y x

35. 12

cos 8y x

36. 12

cos 8y x

For each of the following functions,

(a) State the period.

(b) State the amplitude.

(c) State the phase shift.

(d) State the vertical shift.

(e) Use transformations to sketch the graph of

the function over one period. For consistency

in solutions, perform the appropriate

transformations on the function

sing x x or cosh x x where

0 2x . (Note: The resulting graph may

not fall within this same interval.) The x-axis

should be labeled to show the transformations

of the intercepts of sing x x or

cosh x x , as well as any x-values where

a maximum or minimum occurs. The y-axis

should be labeled to reflect the maximum and


37. 4sinf x x

38. 3cosf x x

39. 5cosf x x

40. 2sinf x x

41. 6sin 2f x x

42. 4cos 3f x x

43. cos 2f x x

44. sin 3f x x

45. cos4

xf x

46. 2

cos5 3

xf x

47. 4

sin3

f x x

48. cos 43

xf x

49. sin 12

xf x

50. 4cos 23

xf x

51. cos2

f x x

52. sinf x x

53. 3cosf x x

54. 5sin 4f x x

55. sin 2 3f x x

56. cos 3f x x

57. 7cos 4f x x

58. 1 3

5sin2 4

f x x



x

y59.

23sin 2 5

3f x x

60. 5cos 23

f x x

61. 4sin 22 2

xf x

62. 3cos 2 42

f x x

63. 7sinf x x Omit part (c)

64. 10sinf x x Omit part (c)

65. 2sin 2f x x Omit part (c)

66. 6sin 3 4f x x Omit part (c)

67. 4cos 32

f x x

Omit part(c)

68. 5sinf x x Omit part(c)

For each of the following graphs,

(a) Give an equation of the form

sinf x A Bx C D which could be

used to represent the graph. (Note: C or D

may be zero. Answers vary.)

(b) Give an equation of the form

cosf x A Bx C D which could be

used to represent the graph. (Note: C or D

may be zero. Answers vary.)

69.

70.

71.

72.

73.

74.

x

y

x

y

x

y

x

y

x

y



75.

76.


77. The depth of the water at a certain point near the

shore varies with the ocean tide. Suppose that the

high tide occurs today at 2AM with a depth of 6

meters, and the low tide occurs at 8:30AM with a

depth of 2 meters.

(a) Write a trigonometric equation that models

the depth, D, of the water t hours after

midnight.

(b) Use a calculator to find the depth of the

water at 11:15AM. (Round to the nearest

hundredth.)

78. The depth of the water at a certain point near the

shore varies with the ocean tide. Suppose that the

high tide occurs today at 4AM with a depth of 9

meters, and the low tide occurs at 11:45AM with

a depth of 3 meters.

(a) Write a trigonometric equation that models

the depth, D, of the water t hours after

midnight.

(b) Use a calculator to find the depth of the

water at 2:30PM. (Round to the nearest

hundredth.)

x

y

x

y

Documents

Graphing the Sine Function › ... › TB › ch5 › s52 › 1330_Ch5_Section2.pdf · 2011-10-14 · Graphing the Sine Function: Graphing the Cosine Function: SECTION 5.2 Graphs