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908 Chapter 14 Trigonometric Graphs, Identities, and Equations908 Chapter 14 Trigonometric Graphs, Identities, and Equations

Key Vocabularyamplitude

periodic function

cycle

period

frequency

14.1 Graph Sine, Cosine,and Tangent Functions

In this lesson, you will learn to graph functions of the formy5asin bxandy5acos bxwhere aand bare positive constants and xis in radian measure.The graphs of all sine and cosine functions are related to the graphs of theparent functionsy5sinxandy5cosx, which are shown below.

y

x

21

1

2 2

2

3

22

22

3

2

period:2

amplitude: 1

range:21 y 1

M51

m521

y5sin x

y

x

period:2

amplitude: 1range:

21 y 1

M51

m52121

2 222

2

3

22

22

3

2

y5cos x

Before You evaluated sine, cosine, and tangent functions.

Now You will graph sine, cosine, and tangent functions.

Why? So you can model oscillating motion, as in Ex. 31.

KEY CONCEPT For Your Notebook

Characteristics of y5sin xand y5cos x

The domain of each function is all real numbers.

The range of each function is 21 y1. Therefore, the minimumvalue of each function is m521 and the maximum value is M51.

The amplitudeof each functions graph is half the difference of the

maximumMand the minimum m, or 1}2

(M2m) 5 1}2

[12(21)] 51.

Each function is periodic, which means that its graph has a repeatingpattern. The shortest repeating portion of the graph is called a cycle.The horizontal length of each cycle is called the period. Each graphshown above has a period of 2.

Thex-intercepts fory5sinxoccur whenx50, 6, 62, 63, . . . .

Thex-intercepts fory5cosxoccur whenx56p}2

,63p}2

, 65p}2

, 67p}2

, . . . .

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14.1 Graph Sine, Cosine, and Tangent Functions 9014.1 Graph Sine, Cosine, and Tangent Functions 90

GRAPHING KEY POINTS Each graph below shows five keyx-values on the interval

0 x 2p}b that you can use to sketch t he graphs ofy5asin bxandy5acos bxfor

a> 0 and b> 0. These are the x-values where the maximumand minimumvaluesoccur and the x-intercepts.

y

x

s , 0d12 2bs , 2ad34 2b

s , ad14 2

b

s , 0d2b(0, 0)

y5asin bx

y

x

s , ad2b(0, a)

s , 2ad12 2

b

s , 0d34 2b

s , 0d14 2b

y5acos bx

E X A M P L E 1 Graph sine and cosine functions

Graph (a) y54 sin xand (b) y5cos 4x.

Solution

a. The amplitude is a54 and the period is 2p}b 5 2p}

1 52.

Intercepts: (0, 0); 11}2p2, 0 25(p, 0); (2p, 0)

Maximum: 11}4p2, 4 251p

}

2, 42

Minimum: 13}4p2, 242513p

}

2 ,242

b. The amplitude is a51 and the period is 2p}b 5 2p}

4 5p}

2.

Intercepts: 11}4pp}

2, 0251p}8, 02; 1

3}4

pp}2

, 02513p}8 , 02

Maximums: (0, 1); 1p}2, 12

Minimum: 11}2pp}

2,21 251

p

}

4,212

KEY CONCEPT For Your Notebook

Amplitude and Period

The amplitude and period of the graphs ofy5asin bxandy5acos bx,where aand bare nonzero real numbers, are as follows:

Amplitude 5a Period 5 2p}b

GUIDEDPRACTICE for Example 1

Graph the function.

1. y52 cosx 2. y55 sinx 3. f(x) 5sin x 4. g(x) 5cos 4x

y

2

3

2

4

y

8

2

3

8

VARY CONSTANTS

Notice how changesin aand baffect thegraphs of y5asin bxand y5acos bx. When

the value of aincreases,the amplitude increases.When the value of bincreases, the perioddecreases.

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910 Chapter 14 Trigonometric Graphs, Identities, and Equations910 Chapter 14 Trigonometric Graphs, Identities, and Equations

EX A M P L E 3 Model with a sine function

AUDIO TEST A sound consisting of a single f requency iscalled a pure tone. An audiometerproduces pure tones totest a persons auditory functions. Suppose an audiometerproduces a pure tone with a frequencyfof 2000 hertz (cyclesper second). The maximum pressure Pproduced from thepure tone is 2 millipascals. Write and graph a sine model that

gives the pressure Pas a function of the time t(in seconds).

Solution

STEP 1 Find the values of aand bin the model P5asin bt. The maximumpressure is 2, so a52. You can use the f requencyfto find b.

frequency 5 1}period

2000 5 b}2p

40005b

The pressure Pas a function of time tis given by P52 sin 4000t.

STEP 2 Graph the model. The amplitude is a52 and the period is 1}f5 1}

2000.

Intercepts: (0, 0);

11}2p1}

2000, 0 251 1}4000, 02; 1

1}

2000, 02

Maximum: 11}4p1}

2000, 2 251 1}8000, 22

Minimum: 13}4p1}

2000, 22 251 3}8000,222

MODELING WITH TRIGONOMETRIC FUNCTIONS The periodic nature oftrigonometric functions is useful for modeling oscillatingmotions or repeating

patterns that occur in real life. Some examples are sound waves, the motion ofa pendulum, and seasons of the year. In such applications, the reciprocal of theperiod is called the frequency, which gives the number of cycles per unit of time.

EX A M P L E 2 Graph a cosine function

Graph y5 1}2

cos 2px.

Solution

The amplitude is a5 1}2

and the period is 2p}b 5 2p}

2p51.

Intercepts:

1

1}4

p1, 0

25

1

1}4

, 0

2;

13}4p1, 0 2513}4

, 02

Maximums: 10, 1}22; 11,1}2

2

Minimum: 11}2p1, 21}2

2511}2,21}2

2

y

x

1

1 2

P

t

2

1

8000

SKETCH A GRAPH

After you have drawnone complete cycle ofthe graph in Example 2

on the interval 0

x

1,you can extend thegraph by copying thecycle as many times asdesired to the left and

right of 0 x1.

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14.1 Graph Sine, Cosine, and Tangent Functions 9114.1 Graph Sine, Cosine, and Tangent Functions 91

GRAPH OF Y5TANX The graphs of all tangent functions are related to the graphof the parent functiony5tanx, which is shown below.

y

x

period:

1

2 222

2

3

22

22

3

2

y5tan x

The functiony5tanxhas the following characteristics:

1. The domain is all real numbers except odd multiples of p}2

. At these

x-values, the graph has vertical asymptotes.

2. The range is all real numbers. So, the functiony5tanxdoes not havea maximum or minimum value, and therefore the graph of y5tanxdoes not have an amplitude.

3. The graph has a period of .

4. Thex-intercepts of the graph occur whenx50, 6, 62, 63, . . . .

GUIDEDPRACTICE for Examples 2 and 3

Graph the function.

5. y5 1}4

sin x 6. y5 1}3

cos x 7. f(x) 52 sin 3x 8. g(x) 53 cos 4x

9. WHAT IF? In Example 3, how would the function change if t he audiometerproduced a pure tone with a frequency of 1000 hertz ?

KEY CONCEPT For Your Notebook

Characteristics ofy5atan bx

The period and vertical asymptotes of the graph of y5atan bx, whereaand bare nonzero real numbers, are as follows:

The period is p}b

.

The vertical asymptotes are at odd multiples of p}2b

.

GRAPHING KEY POINTS The graph at the rightshows fivekeyx-values that can help you sketchthe graph ofy5atan bxfor a> 0 and b> 0.These are the x-intercept, thex-values wherethe asymptotesoccur, and thex-values halfwaybetweenthex-intercept and the asymptotes. Ateach halfway point, the functions value is eitheraor 2a.

y

x

a

2b

4b2

2b

FIND ODD

MULTIPLES

Odd multiples of p}2 are

values such as these:

61 pp}256p}

2

63 pp}

2563

p

}

2

65 pp}2565p}

2

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912 Chapter 14 Trigonometric Graphs, Identities, and Equations912 Chapter 14 Trigonometric Graphs, Identities, and Equations

EXAMPLE 1

on p. 909for Exs. 314

14.1 EXERCISES

GUIDEDPRACTICE for Example 4

Graph one period of the function.

10. y53 tanx 11. y5tan 2x 12. f(x) 52 tan 4x 13. g(x) 55 tan x

1. VOCABULARY Copy and complete: The graphs of the functionsy5sinxandy5cosxboth have a(n) ? of 2.

2. WRITING Comparethe domains and ranges of the functions y5asin bx,y5acos bx, andy5atan bxwhere aand bare positive constants.

ANALYZING FUNCTIONS Identify the amplitude and the period of the g raph of

the function. 3. y

x

1

8

5

8

3

8

4. y

x

2

5. y

x

1

1

2

5

2

3

2

E X A M P L E 4 Graph a tangent function

Graph one period of the function y52 tan 3x.

Solution

The period is p}b5p}

3.

Intercept: (0, 0)

Asy mptotes:x5 p}2b5 p}

2 p3, or x5p}

6;

x52p}2b

52 p}2 p3

, or x52p}6

Halfway points: 1p}4b, a251p}

4 p3, 2 251p}12, 22;

12p}4b, 2a2512p}

4 p3, 22 2512p}12,222

at classzone.com

y

x

2

6

122

6

HOMEWORKKEY

5WORKED-OUT SOLUTIONSon p. WS1 for Exs. 5, 17, and 31

5STANDARDIZED TEST PRACTICEExs. 2, 15, 24, 25, and 31

5MULTIPLE REPRESENTATIONS

Ex. 32

SKILLPRACTICE

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14.1 Graph Sine, Cosine, and Tangent Functions 9114.1 Graph Sine, Cosine, and Tangent Functions 91

EXAMPLE 3

on p. 910for Exs. 2930

GRAPHING Graph the function.

6. y5sin 1}5

x 7. y54 cosx 8. f(x) 5cos 2}5

x 9. y5sin x

10. f(x) 5 2}3

sin x 11. f(x) 5sin p}2

x 12. y5p}4

cos x 13. f(x) 5cos 24x

14. ERROR ANALYSIS Describeand correctthe error in finding the period of the

function y5

sin2

}

3x.

15. MULTIPLE CHOICE The graph of which function has an amplitude of 4 anda period of 2?

A y54 cos 2x B y52 sin 4x C y54 sin x D y52 cos 1}2

x

GRAPHING Graph the function.

16. y52 sin 8x 17. f(x) 54 tan x 18. y53 cos x 19. y55 sin 2x

20. f(x) 52 tan 4x 21. y52 cos 1}4

x 22. f(x) 54 tan x 23. y5cos 4x

24. MULTIPLE CHOICE Which of the following is an asymptote of the graph ofy52 tan 3x?

A x5p}6

B x52 C x5 1}6

D x52p}12

25. OPEN-ENDED MATH Describea real-life situation that can be modeled by aperiodic function.

CHALLENGE Sketch the graph of the function by plotting points. Then state thefunctions domain, range, and period.

26. y5csc x 27. y5secx 28. y5cot x

29. PENDULUMS The motion of a certain pendulum can be modeled by thefunction d54 cos twhere dis the pendulums horizontal displacement(in inches) relative to its position at rest and tis the time (in seconds).Graph the function. What is the greatest horizontal distance the pendulumwill t ravel f rom its position at rest?

30. TUNING FORKS A tuning fork produces asound pressure wave that ca n be modeled by

P50.001 sin 880t

where Pis the pressure (in pascals) and tisthe time (in seconds). Find the period andfrequency of this function. Then graph thefunction.

PROBLEMSOLVING

EXAMPLES

2, 3, and 4

on pp. 910912for Exs. 1624

Period 5b

}

2p 5

2}3}

2p 5

1}

3p

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914 Chapter 14 Trigonometric Graphs, Identities, and Equations914 Chapter 14 Trigonometric Graphs, Identities, and Equations

31. SHORT RESPONSE A buoy oscil lates up and down as waves go past.The buoy moves a total of 3.5 feet from its low point to its high point, andthen returns to its high point every 6 seconds.

a. Write a n equation that gives the buoys vertica l posit ionyat time tif thebuoy is at its highest point when t50.

b. Explainwhy you chose y5asin btor y5acos btfor part (a).

32. MULTIPLE REPRESENTATIONSYou are standing on a br idge, 140 feet

above the ground. You look down at a car traveling away from the underpass.

a. Writing an EquationWrite an equation that gives the cars distance dfrom the base of the bridge as a function of the angle u.

b. Drawing a GraphGraph the function found in part (a). Explainhow thegraph relates to the given situation.

c. Making a TableMake a table of va lues for the function. Use the table tofind the cars distance from the bridge when u5208, 408, and 608.

33. CHALLENGE The motion of a spring can be modeled by y5Acos ktwhereyis the springs vertical displacement (in feet) relative to its position atrest, Ais the initial displacement (in feet), kis a constant that measuresthe elasticity of the spring, and tis the time (in seconds).

a. Suppose you have a spring whose motion can be modeled by thefunction y50.2 cos 6t. Find the initial displacement and the period ofthe spring. Then graph the given function.

b. Graphing CalculatorIf a damping force is applied to the spring, themotion of the spring can be modeled by the function y50.2e24.5tcos 4t.Graph this function. What effect does damping have on the motion?

EXTRA PRACTICE for Lesson 14.1, p. 1023 ONLINE QUIZ at classzone.com

140 ft

d

u

Graph the function. Label the vertex and the axis of symmetry. (p. 245)

34. y52(x22)211 35. y55(x21)217 36. f(x) 52(x16)223

37. y523(x13)212 38. y50.5(x12)215 39. g(x) 522(x14)223

Let f(x)5x225, g(x)527x, and h(x)5x1/3. Find the indicated value. (p. 428)

40. g(f(9)) 41. h(g(3)) 42. f(g(27)) 43. f(h(8))

Sketch the angle. Then find its reference angle. (p. 866)

44. 2508 45. 2908 46. 21458 47. 24308

48. 13p}

3 49. 21p

}

4 50. 213p}

6 51. 216p}

3

MIXEDREVIEW

PREVIEW

Prepare forLesson 14.2in Exs. 3439.