Download Slide 4- 1 Homework, Page 392

Homework, Page 392
Find 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.

y



x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley








Slide 4- 1
Homework, Page 392
Find 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.

y



x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley








Slide 4- 2
Homework, Page 392
Find 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.
y  cos  7 x 
2
y  cos  7 x   cos  7 x   p 
7
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.

y



x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley








Slide 4- 3
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.
y
x
y  3sin
2





a3
2
p
 4
1
2
1
f 
4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


x






Slide 4- 4
Homework, Page 392
Graph one period of the function. Show the scale on both axes
17. y  2sin x

y  2sin x
a2
2
p
 2
1
y

x




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 5
Homework, Page 392
Graph one period of the function. Show the scale on both axes
21. y  0.5sin x

y  0.5sin x
a  0.5
2
p
 2
1
y

x




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 6
Homework, Page 392
Graph three period of the function. Show the scale on both axes.
25. y  0.5cos 3x

y  0.5cos 3x
a  0.5
2
p
3
y

x




Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 7
Homework, Page 392
Specify the period and amplitude of each function. Give the
viewing window in which the graph is shown.
29. y  1.5sin 2 x
y  1.5cos 2 x
a  1.5
2
p

2
The viewing window is  2 , 2  by  2, 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 8
Homework, Page 392
Specify the period and amplitude of each function. Give the
viewing window in which the graph is shown.
33. y  4sin
y  4sin

3

3
x
x
a4
2
p
6

3
The viewing window is  3,3 by  5,5.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 9
Homework, Page 392
Identify the maximum and minimum values and the zeroes of the
function in the interval [–2π, 2π].
37. y  cos 2 x
y

y  cos 2 x
a 1








2
p


2
The function has a maximum y-value of 1 and a minimum y-value of  1.
x

 7 5 3   3 5 7 
The zeroes of the function are at x  
, , , , , , , 
4
4
4 4 4 4 4 
 4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 10
Homework, Page 392
41. Write the functon y   sin x as a phase shift of y  sin x.
y   sin x  sin  x    .

y



x








Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 4- 11
Homework, Page 392
Describe the transformations required to obtain the graph of the
given function from a basic trigonometric graph.
45.
2
x
y   cos
3
3
2
x
To obtain the graph of y   cos from the graph of y  cos x,
3
3
2
apply a vertical shrink of , a horizontal stretch of 3, and reflect
3
about the x-axis.
 y






x
         
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 12
Homework, Page 392
Describe the transformations required to obtain the graph of y2
from the graph of y1.
5
49. y1  cos 2 x and y2  cos 2 x
3
To obtain the graph of y2 from the graph of y1 ,
5
apply a vertical stretch of .
3
y

x











Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 13
Homework, Page 392
Select the pair of functions that have identical graphs..
53.  a  y  cos x


 b  y  sin  x  
2



 c  y  cos  x  
2

 a  and b

y
x











Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 14
Homework, Page 392
Construct a sinusoid with the given amplitude that goes through
the given point.
57. Amplitude 3, period  , point  0, 0
2
a  3; p   
b  2
b
y  3sin  2 x  c   0  3sin  2  0   c   y  3sin  2 x 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 15
Homework, Page 392
State the amplitude and period of the sinusoid and (relative to the
basic function) the phase shift and vertical translation.
61.y  2sin  x     1



4
The function has an amplitude of 2, a period of 2 π, a
phase shift of 3π/4, and a vertical translation of +1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 16
Homework, Page 392
State the amplitude and period of the sinusoid and (relative to the
basic function) the phase shift and vertical translation.
65. y  2 cos 2 x  1
The function has an amplitude of 2, a period of 1, no
phase shift, and a vertical translation of +1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 17
Homework, Page 392
Find values of a, b, h, and k so that the graph of the function
y = a sin (b(x – h)) + k.
69.
y  2sin 2 x  a  2, b  2, h  0, k  0
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 18
Homework, Page 392
73. 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

50
p  40 
b
b
20
a
2
 25, k  30
x 
y  25cos 
  30
 20 
The rider will be 50-ft above the
ground 15.903 sec after passing
the low point.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 19
Homework, Page 392
77. 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.
t
d
t
0.1
9.2
1.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
13.9 18.8 21.4 20.0 15.6 10.5 7.4
1.2 1.3 1.4 1.5 1.6 1.7 1.8
d
17.3 20.8 20.8 17.2 12.0 8.1
7.5
0.9
8.1
1.9
1.0
12.1
2.0
10.5 15.6 19.9
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 20
Homework, Page 392
77. 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.
M  m 21.4  7.2 14.2
a


 7.1
2
2
2
b. Estimate the period of the motion from the scatter plot.
p  1.25  0.4  0.85sec
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 21
Homework, Page 392
77. c. Model the motion of the block as a sinusoidal function d (t).
2
2
a  7.1; p  0.85 
b
 2.353
b
0.85
M  m 21.4  7.2
k

 14.3  d  t   7.1cos  2.353 t   14.3
2
2
d. Graph the function with the scatterplot to support the model
graphically.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 22
Homework, Page 392
81. 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
2
2 
y = sin 4x because
 
p
2
  p 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
4

 2   
2
2
Slide 4- 23
Homework, Page 392
85. The period of the function f (x) = 210 sin (420x +840)
is

a.
840
b.
c.
d.
e.

420

210
2 2

p


b
420 210
210

420

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 24
Homework, Page 392
89. 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.
2
2
1
(a) What is the period of the function? p 


b
524 262
(b) What is the frequency f = 1/p of this note? 1
1
f  
 262
1
p
(c) Graph the function.
262
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 25
4.5
Graphs of Tangent, Cotangent,
Secant, and Cosecant
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about




The Tangent Function
The Cotangent Function
The Secant Function
The Cosecant Function
… and why
This will give us functions for the remaining
trigonometric ratios.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 27
Asymptotes of the Tangent Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 28
Zeros of the Tangent Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 29
Asymptotes of the Cotangent Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 30
Zeros of the Cotangent Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 31
The Secant Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 32
The Cosecant Function
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 33
Basic Trigonometry Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 34
Example Analyzing Trigonometric
Functions
Analyze the function for domain, range, continuity, increasing or
decreasing, symmetry, boundedness, extrema, asymptotes, and
end behavior
f  x   sec x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 35
Example Transformations of
Trigonometric Functions
Describe the transformations required to obtain the graph of the
given function from a basic trigonometric function.
1
f  x   2sec x
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 36
Example Solving Trigonometric
Equations
Solve the equation for x in the given interval.
sec x   2,
  x  3 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 37
Example Solving Trigonometric
Equations With a Calculator
Solve the equation for x in the given interval.
csc x  1.5,
  x  3 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 38
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
20
P?
c. An angle of  20 is equivalent to how many degrees?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 39
Homework
Homework Assignment #30
 Read Section 4.6
 Page 401, Exercises: 1 – 65 (EOO)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 40
4.6
Graphs of Composite Trigonometric
Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
State the domain and range of the function.
1. f ( x)  -3sin 2 x
2. f ( x) | x | 2
3. f ( x)  2 cos 3 x
4. Describe the behavior of y  e as x  .
-3 x
5. Find f g and g f , given f ( x)  x  3 and g ( x)  x
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 42
Quick Review Solutions
State the domain and range of the function.
1. f ( x)  -3sin 2 x Domain:  ,   Range:  3,3
2. f ( x) | x | 2
Domain:  ,   Range:  2,  
3. f ( x)  2 cos 3 x Domain:  ,   Range:  2, 2
4. Describe the behavior of y  e as x  .
-3 x
lim e
x 
3 x
0
5. Find f g and g f , given f ( x)  x  3 and g ( x )  x
2
f g  x  3; g f  x  3
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 43
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 44
Example Combining the Cosine Function
with x2
Graph y   cos x  and state whether the function
2
appears to be periodic.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 45
Example Combining the Cosine Function
with x2
 
Graph y  cos x 2 and state whether the function
appears to be periodic.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 46
Sums That Are Sinusoidal Functions
If y  a sin(b( x  h )) and y  a cos(b( x  h )), then
1
1
1
2
2
2
y  y  a sin(b( x  h ))  a cos(b( x  h )) is a
1
2
1
1
2
2
sinusoid with period 2 / | b|.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 47
Sums That Are Not Sinusoidal Functions
If y  a sin(b( x  h )) and y  f ( x) where f ( x) is not
1
1
1
2
a sin(b( x  h )) or a cos(b( x  h )), but another
2
2
2
2
trigonometric function, then y  y is a periodic
1
2
function, but not a sinusoid.
If y  f ( x) is not a trigonometric function, then y  y
2
1
2
is neither periodic nor sinusoidal.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 48
Example Identifying a Sinusoid
Determine whether the following function is or is not
a sinusoid: f ( x)  3cos x  5sin x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 49
Example Identifying a Sinusoid
Determine whether the following function is or is not
a sinusoid: f ( x)  cos3x  sin 5x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 50
Example Identifying a Non-Sinusoid
Determine whether the following function is or is not
a sinusoid: f ( x)  3x  sin 5x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 51
Damped Oscillation
The graph of y  f ( x)cos bx (or y  f ( x)sin bx) oscillates
between the graphs of y  f ( x ) and y  - f ( x ). When this
reduces the amplitude of the wave, it is called damped
oscillation. The factor f ( x) is called the damping factor.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 52
Example Working with Damped
Oscillation
The oscillations of a spring subject to friction
are modeled by the equation y  0.43e cos1.8t.
0.55 t
 a  Graph y and its two damping curves in the same
viewing window for 0  t  12.
b
Approximately how long does it take for the spring
to be damped so that  0.2  y  0.2?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4- 53