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2 Graphs of y = a sin bx and y = a cos bx

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10.2 Graphs of y = a sin bx and y = a cos bx

LEARNING TIP

E X A M P L E 1 Finding the period of a function

If the period of a function F 1x 2 is P,

then the period of F 1bx 2 is P>b. Since

each of the functions sin x and cos x

has a period of 2p, each of the functions sin bx and cos bx has a period of

2p>b.

Practice Exercise

p

(a) The period of cos 4x is 2p

4 = 2.

2p

(c) The period of sin 12x is 1 = 4p.

2

(b) The period of sin 3px is 2p

3p = 3 .

2p

(d) The period of cos p4 x is p = 8.

4

2

In (a), the period tells us that the curve of y = cos 4x will repeat every p>2 (approximately

1.57) units of x. In (b), we see that the curve of y = sin 3px will repeat every 2>3 of a unit.

In (c) and (d), the periods are longer than those of y = sin x and y = cos x.

Combining the value of the period with the value of the amplitude from Section

10.1, we conclude that the functions y = a sin bx and y = a cos bx have an amplitude

of 0 a 0 and a period of 2p>b. These properties are very useful in sketching these

functions.

Find the period of each function.

1. y = sin px

301

2. y = cos 13x

E X A M P L E 2 Sketching the graph of y = a sin bx

Sketch the graph of y = 3 sin 4x for 0 … x … p.

Since a = 3, the amplitude is 3. The 4x tells us that the period is 2p>4 = p>2.

This means that y = 0 for x = 0 and for y = p>2. Since this sine function is zero

halfway between x = 0 and x = p>2, we find that y = 0 for x = p>4. Also, the fact

that the graph of the sine function reaches its maximum and minimum values halfway

between zeros means that y = 3 for x = p>8, and y = -3 for x = 3p>8. Note that

the values of x in the following table are those for which 4x = 0, p>2, p, 3p>2, 2p,

and so on, which correspond to the key values listed in Tables 10.1 and 10.2.

y

3

Max.

0

−3

Max.

p

_

2

p

Min.

Min.

x

Period

Fig. 10.11

LEARNING TIP

By finding one-fourth of the period,

we can easily find the important values

for sketching the curve.

x

0

p

8

p

4

3p

8

p

2

5p

8

3p

4

7p

8

p

y

0

3

0

-3

0

3

0

-3

0

Using the values from the table and the fact that the curve is sinusoidal in form, we

sketch the graph of this function in Fig. 10.11. We see that the key values where the

function has zeros, maxima, and minima occur when x is a multiple of p>8, which is

exactly one-fourth of the period.

Note from Example 2 that an important distance in sketching a sine curve or a

cosine curve is one-fourth of the period. For y = a sin bx, it is one-fourth of the period

from the origin to the first value of x where y is at its maximum (or minimum) value.

Then we proceed another one-fourth period to a zero, another one-fourth period to the

next minimum (or maximum) value, another to the next zero (this is where the period is

completed), and so on.

Similarly, one-fourth of the period is useful in sketching the graph of y = cos bx.

For this function, the maximum (or minimum) value occurs for x = 0. At the following one-fourth-period values, there is a zero, a minimum (or maximum), a zero, and a

maximum (or minimum) at the start of the next period.

We now summarize the important values for sketching the graphs of y = a sin bx

and y = a cos bx.

Important Values for Sketching y = a sin bx and y = a cos bx

1. The amplitude: 0 a 0

2. The period: 2p>b

3. Values of the function for each one-fourth period

302

CHAPTER 10 Graphs of the Trigonometric Functions

E X A M P L E 3 Using important values to sketch a graph

y

2

0

p

3

2p

3

p

4p

3

2p

5p

3

x

Sketch the graph of y = -2 cos 3x for 0 … x … 2p.

Note that the amplitude is 2 and the period is 2p

3 . This means that one-fourth of the

p

=

.

Since

the

cosine

curve

is

at

a maximum or minimum for x = 0,

period is 14 * 2p

3

6

we find that y = -2 for x = 0 (the negative value is due to the minus sign before the

function), which means it is a minimum point. The curve then has a zero at x = p6 , a

maximum value of 2 at x = 2 1 p6 2 = p3 , a zero at x = 3 1 p6 2 = p2 , and its next value of

- 2 at x = 4 1 p6 2 = 2p

3 , and so on. Therefore, we have the following table:

x

0

p

6

p

3

p

2

2p

3

5p

6

p

7p

6

4p

3

3p

2

5p

3

11p

6

2p

y

-2

0

2

0

-2

0

2

0

-2

0

2

0

-2

−2

Using this table and the sinusoidal shape of the cosine curve, we sketch the function in

Fig. 10.12.

Fig. 10.12

For a periodic function, a cycle is any section of the graph that includes exactly one

period. Fig. 10.11 shows two cycles, whereas Fig. 10.12 shows three.

E X A M P L E 4 Graph of y = a cos bx—application

A generator produces a voltage V = 200 cos 50pt, where t is the time in seconds (50p

is angular velocity, so it has units of rad>s; thus, 50pt is an angle in radians). Graph V

as a function of t for 0 … t … 0.06 s.

The amplitude is 200 V and the period is 2p> 150p2 = 0.04 s. Since one-fourth of

the period is 0.01 s, the function has zeros, maxima, and minima when x is a multiple of

0.01. Thus, we have the following table of key values:

V

200

0.06

t

t1s 2

−200

V1V 2

Fig. 10.13

0

0.01

0.02

0.03

0.04

0.05

0.06

200

0

-200

0

200

0

-200

The graph is shown in Fig. 10.13. Note that between 0 and 0.06 s, the function completes 1.5 cycles. We do not consider negative values of t, for they have no real meaning in this problem.

E XE R C I SES 1 0 .2

In Exercises 1 and 2, graph the function if the given changes are made

in the indicated examples of this section.

1. In Example 2, if the coefficient of x is changed from 4 to 6, sketch

the graph of the resulting function.

2. In Example 3, if the coefficient of x is changed from 3 to 4, sketch

the graph of the resulting function.

In Exercises 3–22, find the period of each function.

3. y = 2 sin 6x

4. y = 4 sin 2x

5. y = 3 cos 8x

6. y = 28 cos 10x

7. y = -2 sin 12x

8. y =

- 15

sin 5x

9. y = -cos 16x

10. y = -4 cos 2x

11. y = 520 sin 2px

12. y = 2 sin 3px

13. y = 3 cos 4px

14. y = 4 cos 10px

15. y = 15 sin

1

3x

16. y = - 25 sin

2

5x

17. y = - 12 cos 23x

19. y = 0.4 sin

2px

3

21. y = 3.3 cos p2x

18. y = 13 cos 14x

20. y = 1.5 cos

px

10

22. y = - 12.5 sin 2x

p

In Exercises 23–42, sketch the graphs of the given functions. (These

are the same functions as in Exercises 3–22.)

23. y = 2 sin 6x

24. y = 4 sin 2x

25. y = 3 cos 8x

26. y = 28 cos 10x

27. y = -2 sin 12x

28. y = - 15 sin 5x

29. y = -cos 16x

30. y = - 4 cos 2x

31. y = 520 sin 2px

32. y = 2 sin 3px

33. y = 3 cos 4px

34. y = 4 cos 10px

35. y = 15 sin 13x

36. y = - 25 sin 25x

303

10.3 Graphs of y = a sin 1bx + c2 and y = a cos 1bx + c2

37. y = - 12 cos 23x

39. y = 0.4 sin

38. y =

2px

3

1

3

56. Find the function and graph it for a function of the form

y = -2 cos bx that passes through 1p>2, 22 and for which b has

the smallest possible positive value.

cos 14x

40. y = 1.5 cos

px

10

42. y = - 12.5 sin 2x

p

41. y = 3.3 cos p2x

In Exercises 43–46, the period is given for a function of the form

y = sin bx. Write the function corresponding to the given period.

p

2p

1

44.

45.

46. 6

3

5

3

In Exercises 47–50, graph the given functions. In Exercises 47 and 48,

first rewrite the function with a positive angle, and then graph the

resulting function. (Refer to Eq. 8.7 for trigonometric functions of

negative angles.)

43.

48. y = -5 cos 1 -4px2

47. y = 3 sin1 -2x2

49. y = 8 ͉ cos

1 p2 x 2 ͉

50. y = 0.4 ͉ sin 6x ͉

In Exercises 51–60, solve the given problems.

51. By noting the periods of sin 2x and sin 3x, find the minimum

period of the function y = sin 2x + sin 3x.

57. The standard electric voltage in a 60-Hz alternating-current circuit is given by V = 170 sin 120pt, where t is the time in seconds. Sketch the graph of V as a function of t for 0 … t … 0.05 s.

58. To tune the instruments of an orchestra before a concert, an A

note is struck on a piano. The piano wire vibrates with a displacement y (in mm) given by y = 3.2 cos 880pt, where t is in seconds. Sketch the graph of y vs. t for 0 … t … 0.01 s.

59. The velocity v (in cm>s) of a piston is v = 450 cos 3600t, where

t is in seconds. Sketch the graph of v vs. t for 0 … t … 0.006 s.

60. On a certain day in St. John, New Brunswick, the difference

between high tide and low tide was 6.4 m. The period was about

12.4 h. Find a cosine function that describes these tides if high

tide was at midnight.

In Exercises 61–64, the graph of a function of the form y = a sin bx

or y = a cos bx is shown. Determine the specific function of each.

61.

52. By noting the period of cos 12x and cos 13x, find the minimum

period of the function y = cos 12x + cos 13x.

53. Find the function and graph it for a function of the form

y = -2 sin bx that passes through 1p>4, -22 and for which b

has the smallest possible positive value.

0

54. Find the function and graph it for a function of the form

y = 2 sin bx that passes through 1p>6, 22 and for which b has

the smallest possible positive value.

62.

y

0.5

8

x

p

0

64.

y

y

4

55. Find the function and graph it for a function of the form

y = 2 cos bx that passes through 1p, 02 and for which b has the

smallest possible positive value.

x

p

4

−8

−0.5

63.

y

0.1

x

2

0

0

Ϫ4

1

2

Ϫ0.1

1. 2

2. 6p

10.3 Graphs of y = a sin (bx + c) and y = a cos (bx + c)

In the function y = a sin 1bx + c2 , c represents the phase angle. It is another very

important quantity in graphing the sine and cosine functions. Its meaning is illustrated

in the following example.

1IBTF"OHMF t %JTQMBDFNFOU t

Graphs of y = a sin (bx + c) and

y = a cos (bx + c)

E X A M P L E 1 Sketch of a function with phase angle

Sketch the graph of y = sin 1 2x + p4 2 .

Here, c = p>4. Therefore, in order to obtain values for the table, we

assume a value for x, multiply it by 2, add p>4 to this value, and then

find the sine of the result. The values shown are those for which

2x + p>4 = 0, p>4, p, 2, 3p>4, p, and so on, which are the important

values for y = sin 2x.

y

1

p

−8 0

p

4

p

2

3p

4

p

x

x

- p8

0

p

8

p

4

3p

8

p

2

5p

8

3p

4

7p

8

p

y

0

0.7

1

0.7

0

-0.7

-1

-0.7

0

0.7

−1

Fig. 10.14

Solving 2x + p>4 = 0, we get x = -p>8, which gives y = sin 0 = 0. The other

values for y are found in the same way. See Fig. 10.14.

x

304

CHAPTER 10 Graphs of the Trigonometric Functions

Note from Example 1 that the graph of y = sin 1 2x + p4 2 is precisely the same as

the graph of y = sin 2x, except that it is shifted p>8 units to the left. Fig. 10.15 shows

the graphs of y = sin 2x and y = sin 1 2x + p4 2 . We see that the shapes are the same

and that the graph of y = sin 1 2x + p4 2 is about 0.4 unit (p>8 ≈ 0.39) to the left of

the graph of y = sin 2x.

In general, the effect of c in the equation y = sin1bx + c2 can be understood

x

when we write bx + c = b1x + bc 2. This means that the function y = sin1 bx + c2

is obtained as a result of adding the constant c>b to x in the function y = sin bx. As we

discussed in Section 3.5, by adding a constant to x, the graph of y = a sin bx is shifted

to the left or to the right. In this case, the graph is shifted to the left if c 7 0 and to the

right if c 6 0. The direction and magnitude of the shift is called the displacement (or

phase shift), and it is given by –c>b. Note that the displacement can be obtained by

solving for x in the equation bx + c = 0. In Example 1, the displacement is -p>8.

We can verify that the displacement is -c>b by noting corresponding points on the graphs

of y = sin bx and y = sin1bx + c2. For y = sin bx, when x = 0, then y = 0. For

y = sin1bx + c2, when x = -c>b, then y = 0. The point 1 -c>b, 02 on the graph of

y = sin1bx + c2 is -c>b units to the left of the point 10, 02 on the graph of y = sin bx.

We can use the displacement combined with the amplitude and the period along

with the other information from the previous sections to sketch curves of the functions

y = a sin1bx + c2 and y = a cos 1bx + c2, where b 7 0.

y

Period = p

1

0

p

y = sin (2x + 4 )

p

4

3p

4

p

2

5p

4

p

y = sin 2x

−1

p

shift

8

Fig. 10.15

LEARNING TIP

Carefully note the difference

between y = sin1bx + c 2 and

y = sin bx + c. Writing sin1bx + c 2

means to find the sine of the quantity bx + c, whereas sin bx + c means

to find the sine of bx and then add

the value c.

Important Quantities for Sketching Graphs of y = a sin (bx + c) and

y = a cos (bx + c)

Amplitude = 0 a 0

2p

Period =

b

c

Displacement = b

LEARNING TIP

Note that the constant c and the displacement - c>b differ in sign:

t *Gc 7 0, the graph is shifted to the

left, and displacement is negative.

(10.1)

These quantities allow us to evaluate the function at key values each one-fourth

period. Table 10.3 summarizes these key values for the cycle that starts at x = -c>b

and is completed at x = -c>b + period. A general illustration of the graph of

y = a sin1bx + c2 is shown in Fig. 10.16.

t *Gc 6 0, the graph is shifted to the

right, and displacement is positive.

Table 10.3

Key values

(one cycle)

- bc

- bc +

- bc +

- bc +

y = a sin 1bx + c2

0

y = a cos 1 bx + c2

a

0

0

-a

-a

0

0

a

period

4

period

2

3 # period

4

- bc + period

y

a

c

−b

y

2p

b

Since c < 0,

−c/b is positive

c

0

−b +

−a

y = a sin (bx + c), c > 0

Fig. 10.16

a

(a)

2p

b

x

0

For each

a > 0, b > 0

2p

b

a

c

c

−b

−b +

−a

y = a sin (bx + c), c < 0

(b)

2p

b

x

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