Homework, Page 392

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1

Homework, Page 392Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x.

1. y = 2 sin x

The graph of y = 2 sin x may be obtained from the graph of y = sin x by applying a vertical stretch of 2.

x

y


Homework, Page 392Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sin x.

5. y = 0.73 sin x

The graph of y = 2 sin x may be obtained from the graph of y = sin x by applying a vertical shrink of 0.73.

x

y


Homework, 92Find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cos x.

9.

The graph of y = cos (–7 x) may be obtained from the graph of y = cos x by applying a horizontal shrink of 1/7.

cos 7y x

2cos 7 cos 7

7y x x p

x

y


Homework, Page 392

Find the amplitude, period, and frequency of the function and use this information to sketch a graph of the function in the window [–3π, 3π] by [–4,4].

13. 3sin2

xy

3

24

121

4

a

p

f

x

y


Homework, Page 392Graph one period of the function. Show the scale on both axes

17. 2siny x

2sin

2

22

1

y x

a

p

x

y


Homework, Page 392Graph one period of the function. Show the scale on both axes

21. 0.5siny x

0.5sin

0.5

22

1

y x

a

p

x

y


Homework, Page 392Graph three period of the function. Show the scale on both axes.

25. 0.5cos3y x

0.5cos3

0.5

2

3

y x

a

p

x

y


Homework, Page 392Specify the period and amplitude of each function. Give the viewing window in which the graph is shown.

29. 1.5sin 2y x

1.5cos 2

1.5

2

2The viewing window is 2 ,2 by 2,2 .

y x

a

p


Homework, Page 392Specify the period and amplitude of each function. Give the viewing window in which the graph is shown.

33. 4sin3

y x

4sin3

4

26

3The viewing window is 3,3 by 5,5 .

y x

a

p


Homework, Page 392Identify the maximum and minimum values and the zeroes of the function in the interval [–2π, 2π].

37. cos 2y x

cos 2

1

2

2The function has a maximum y-value of 1 and a minimum y-value of 1.

7 5 3 3 5 7The zeroes of the function are at , , , , , , ,

4 4 4 4 4 4 4 4

y x

a

p

x

x

y


Homework, Page 39241. Write the functon sin as a phase shift of sin .y x y x

sin sin .y x x

x

y


Homework, Page 392Describe the transformations required to obtain the graph of the given function from a basic trigonometric graph.

45.2

cos3 3

xy

To obtain the graph of from the graph of cos ,

2apply a vertical shrink of , a horizontal stretch of 3, and reflect

3about

2cos

the

3

-

3

axis.

y

x

y xx

x

y


Homework, Page 392Describe the transformations required to obtain the graph of y2 from the graph of y1.

49. 21 cos 2 and5

cos 23

y x y x

2 1To obtain the graph of from the graph of ,

5apply a vertical stretch of .

3

y y

x

y


Homework, Page 392Select the pair of functions that have identical graphs..

53.

cos

sin2

cos2

a y x

b y x

c y x

and a b

x

y


Homework, Page 392Construct a sinusoid with the given amplitude that goes through the given point.

57. Amplitude 3, period , point 0, 0

23; 2

3sin 2 0 3sin 2 0 3sin 2

a p bb

y x c c y x


Homework, Page 392State the amplitude and period of the sinusoid and (relative to the basic function) the phase shift and vertical translation.

61.

The function has an amplitude of 2, a period of 2 π, a phase shift of 3π/4, and a vertical translation of +1.

2sin 14

y x


Homework, Page 392State the amplitude and period of the sinusoid and (relative to the basic function) the phase shift and vertical translation.

65.

The function has an amplitude of 2, a period of 1, no phase shift, and a vertical translation of +1.

2cos 2 1y x


Homework, Page 392Find values of a, b, h, and k so that the graph of the function y = a sin (b(x – h)) + k.

69.

2sin 2 2, 2, 0, 0y x a b h k


Homework, Page 39273. A Ferris wheel 50 ft in diameter makes one revolution every 40 sec. If the center of the wheel is 30 ft above the ground, how long after reaching the low point is a rider 50 ft above the ground?

2 5040 25, 30

20 2

25cos 3020

The rider will be 50-ft above the

ground 15.903 sec after passing

the low point.

p b a kb

xy


Homework, Page 39277. A block mounted on a spring is set into motion directly above a motion detector, which registers the distance to the block in 0.1 sec intervals. When the block is released, it is 7.2 cm above the detector. The table shows the data collected by the motion detector during the first two sec, with distance d measured in cm.

a. Make a scatterplot of d as a function of t and estimate the maximum value of d visually. Use this number and the stated minimum of 7.2 to compute the amplitude.

t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

d 9.2 13.9 18.8 21.4 20.0 15.6 10.5 7.4 8.1 12.1

t 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

d 17.3 20.8 20.8 17.2 12.0 8.1 7.5 10.5 15.6 19.9


Homework, Page 39277. a. Make a scatterplot of d as a function of t and estimate the maximum value of d visually. Use this number and the stated minimum of 7.2 to compute the amplitude.

b. Estimate the period of the motion from the scatter plot.

21.4 7.2 14.27.1

2 2 2

M ma

1.25 0.4 0.85secp


Homework, Page 39277. c. Model the motion of the block as a sinusoidal function d (t).

d. Graph the function with the scatterplot to support the model graphically.

2 27.1; 0.85 2.353

0.8521.4 7.2

14.3 7.1cos 2.353 14.32 2

a p bb

M mk d t t


Homework, Page 39281. The graph of y = sin 2x has half the period of the graph of y = sin 4x. Justify your answer.

False, the graph of y = sin 2x has twice the period of the graph of y = sin 4x because 2 2

22 4 2 2

p p


Homework, Page 39285. The period of the function f (x) = 210 sin (420x +840) isa.

b.

c.

d.

e.

840

420

210

210

420

2 2

420 210p

b


Homework, Page 39289. A piano tuner strikes a tuning fork for the note middle C and creates a sound wave modeled by y = 1.5 sin 524 πt, where t is the time in seconds.

(a) What is the period of the function?

(b) What is the frequency f = 1/p of this note?

(c) Graph the function.

2 2 1

524 262p

b

1 1262

1

262

fp

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

4.5

Graphs of Tangent, Cotangent, Secant, and Cosecant


What you’ll learn about

The Tangent Function The Cotangent Function The Secant Function The Cosecant Function

… and whyThis will give us functions for the remaining trigonometric ratios.


Asymptotes of the Tangent Function


Zeros of the Tangent Function


Asymptotes of the Cotangent Function


Zeros of the Cotangent Function


The Secant Function


The Cosecant Function


Basic Trigonometry Functions


Example Analyzing Trigonometric Functions

Analyze the function for domain, range, continuity, increasing or decreasing, symmetry, boundedness, extrema, asymptotes, and end behavior

secf x x


Example Transformations of Trigonometric Functions

Describe the transformations required to obtain the graph of the given function from a basic trigonometric function.

12sec

2f x x


Example Solving Trigonometric Equations

Solve the equation for x in the given interval.3sec 2, 2x x


Example Solving Trigonometric Equations With a Calculator

Solve the equation for x in the given interval.3csc 1.5, 2x x


Example Solving Trigonometric Word Problems

A hot air balloon is being blow due east from point P and traveling at a constant height of 800 ft. The angle y is formed by the ground and the line of vision from point P to the balloon. The angle changes as the balloon travels.

a. Express the horizontal distance x as a function of the angle y.

b. When the angle is , what is the horizontal distance from P?

c. An angle of is equivalent to how many degrees?

20

20


Homework Homework Assignment #30 Read Section 4.6 1, Exercises: 1 – 65 (EOO)


Quick Review

-3

2

State the domain and range of the function.

1. ( ) -3sin 2

2. ( ) | | 2

3. ( ) 2cos3

4. Describe the behavior of as .

5. Find and , given ( ) 3 and ( )

x

f x x

f x x

f x x

y e x

f g g f f x x g x x


Quick Review Solutions

State the domain and range of the function.

1. ( ) -3sin 2

2. ( ) | | 2

Domain: , Range: 3,3

Domain: , Rang

3. ( ) 2cos3

e: 2,

Domain:

4. Describe t

, Range:

he behavior o

2,2

f

f x x

f x x

f x x

2

3

2

3- as .

5. Find and , given ( ) 3 and (

lim 0

3; 3

)

x

x

xy e x

f g g f f x x g x x

e

f g x g f x


What you’ll learn about

Combining Trigonometric and Algebraic Functions Sums and Differences of Sinusoids Damped Oscillation

… and why

Function composition extends our ability to model

periodic phenomena like heartbeats and sound waves.


Example Combining the Cosine Function with x2

2Graph cos and state whether the function

appears to be periodic.

y x


Sums That Are Sinusoidal Functions

1 1 1 2 2 2

1 2 1 1 2 2

If sin( ( )) and cos( ( )), then

y sin( ( )) cos( ( )) is a

sinusoid with period 2 / | |.

y a b x h y a b x h

y a b x h a b x h

b


Sums That Are Not Sinusoidal Functions

1 1 1 2

2 2 2 2

1 2

2

If sin( ( )) and ( ) where ( ) is not

sin( ( )) or cos( ( )), but another

trigonometric function, then y is a periodic

function, but not a sinusoid.

If ( ) is not a trigon

y a b x h y f x f x

a b x h a b x h

y

y f x

1 2

ometric function, then y

is neither periodic nor sinusoidal.

y


Example Identifying a Sinusoid

Determine whether the following function is or is not

a sinusoid: ( ) 3cos 5sinf x x x


Example Identifying a Sinusoid


a sinusoid: ( ) cos3 sin5f x x x


Example Identifying a Non-Sinusoid


a sinusoid: ( ) 3 sin5f x x x


Damped Oscillation

The graph of ( )cos (or ( )sin ) oscillates

between the graphs of ( ) and - ( ). When this

reduces the amplitude of the wave, it is called

. The factor ( ) is calle

y f x bx y f x bx

y f x y f x

f x

damped

oscillation d the .damping factor


Example Working with Damped Oscillation

0.55

The oscillations of a spring subject to friction

are modeled by the equation 0.43 cos1.8 .

a Graph y and its two damping curves in the same

viewing window for 0 12.

b Approximately how long d

ty e t

t

oes it take for the spring

to be damped so that 0.2 0.2?y

Homework, Page 392

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