mcgrawhill functions 11 unit 5
TRANSCRIPT
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describe key properties of periodic functions arising
from real-world applications, given a numerical or
graphical representation
predict, by extrapolating, the future behaviour of a
relationship modelled using a numeric or graphical
representation of a periodic function
make connections between the sine ratio and the
sine function and between the cosine ratio and the
cosine function by graphing the relationship between
angles from 0 to 360 and the corresponding sine
ratios or cosine ratios, with or without technology,
defining this relationship as the functionf(x) = sin x
orf(x) = cos x, and explaining why the relationship is
a function
sketch the graphs off(x) = sin xandf(x) = cos x for
angle measures expressed in degrees, and determine
and describe key properties
determine, through investigation using technology,and describe the roles of the parameters a, k, d, and c
in functions of the form y= af[k(x d)] + cin terms
of transformations of the graphs off(x) = sin x
andf(x) = cos xwith angles expressed in degrees
determine the amplitude, period, phase shift, domain,
and range of sinusoidal functions whose equations
are given in the formf(x) = asin [k(x d)] + c or
f(x) = acos [k(x d)] + c
sketch graphs of y= af[k(x d)] + c by applying on
or more transformations to the graphs off(x) = sin
andf(x) = cos x, and state the domain and range o
the transformed functions
represent a sinusoidal function with an equation,
given its graph or its properties
collect data that can be modelled as a sinusoidal
function from primary sources or from secondary
sources, and graph the data
identify sinusoidal functions, including those that
arise from real-world applications involving periodic
phenomena, given various representations, and
explain any restrictions that the context places on
the domain and range
determine, through investigation, how sinusoidal
functions can be used to model periodic phenomen
that do not involve angles
predict the effects on a mathematical model of an
application involving sinusoidal functions when the
conditions in the application are varied
pose and solve problems based on applications
involving a sinusoidal function by using a given
graph or a graph generated with technology from it
equation
Trigonometric
Functions
CHAP
5
By the end of this chapter, you will
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Refer to the Prerequisite Skills Appendix on
pages 496 to 516 for examples of the topics and
further practice.
Use the Cosine Law 1.
02
y
x
4
4
2
4 2
30
42
2
a)
b)
c)
Find Trigonometric Ratios of Special Angles
2.
a)
b)
c)
d)
Determine the Domain and Range of a Function
3. f x x
4.
x x
y y
Shift Functions
5. a)
i)yx
ii)yx
iii)yx
b)
6. a)
i)yx
ii)yx
iii)yx
b)
7. a) yx
b) yx
8. yx
Stretch Functions
9. a)
i)yx
ii)y
x
iii)y
x
b)
282 MHR Functions 11 Chapter 5
Prerequisite Skills
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10. a)
i)yx
ii)yx
iii)y
x
b)
11. yx
12. yx
Reflect Functions
13. a)
i)yx
ii)yx
b)
14. a)
b)
Combine Transformations
15. a) yx
b)
16.
yx
yx
Solve Equations Involving Rational Expressio
17. k
18.
k
Chapter Problem
Prerequisite Skills MHR 2
0
4
246 42
6y
2
iii
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5.1
284 MHR Functions 11 Chapter 5
Modelling Periodic Behaviour
Investigate
How can you model periodic behaviour mathematically?
1.
2.
Tools
grid paper
protractor
ruler
compasses
or
graphing calculator
JohnSuzanne
30
30
30
5 m
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3. Reflect
4.
5. Reflect
6. a)
b)
7. a)
b)
8. cycles
9. period
10. Reflect periodic function
11.
12. amplitude
13. Reflect
cycle
one complete repetit
of a pattern
period
the horizontal length
one cycle on a graph
periodic function
a function that has a
pattern of y-valuesthat repeats at regula
intervals
amplitude
half the distance
between the maximu
and minimum values
a periodic function
5.1 Modelling Periodic Behaviour MHR 2
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Example 1
Classify Functions
a)
i) ii)
b)
Solution
a) i)
y
y
x
x
x
ii) y
x0
4
4
2
6 4 62
2
y
4 2 x0
4
4
2
246 4 62
2
y
x0
4
4
2
246 4 62
2
y
x0
4
4
2
6 42
2
y
One PeriodOne Period
24 6
x0
4
4
2
246 4 62
2
y
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b)
y
Example 2
Predicting With Periodic Functions
a)
b) f f
c) ff f
d)
e) x
fx
Solution
a)
x
b) f f
c) f f f f f f
f f f
d)
e) f x
x
x x x x x
x
From part a), the period
of the function is 6. Th
value of the function a
is the same as the valu
at xplus or minus any
multiple of 6.
x
4
4
2
246 4 6Minimum
Amplitude
Maximum
2
2
y
0
x0
4
4
2
46 4 62
2
y
2
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Example 3
Natural Gas Consumption in Ontario
a)
b)
c)
d)
e)
f)
Solution
a)
b)
c)
d)
0
400 000
200 000
600 000
800 000
1 000 0001 200 000
1 400 000
1 600 000
Residential Natural Gas Consumption
Month Starting January 2001
ThousandsofCubicMetres
12 24 36 48 60
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Key Concepts
p fxp fx x
fx p
fxnp fx p n
x0
4
2
6
2
y
6 2
PeriodAmplitude
4
24 4
Communicate Your Understanding
C1
C2
a) fxq fx
b)
C3
5.1 Modelling Periodic Behaviour MHR 2
e) t g
t t
g g
f)
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A Practise
For help with questions 1 to 5, refer to
Example 1.
1.
a)
x0
4
4
2
24 6
2
y
6 42
b)
x0
4
6
4
2
2 4 62
2
6y
6 4
c)
x0
4
2
2 6
2
y
6 44 2
d)
x0
4
2
2 6
2
y
6 44 2
2.
3.
4.
5.
For help with questions 6 to 8, refer to
Example 2.
6.
fx
ff f
a)f b) f
c) f d) f
7. a)
fx
b) a x fa
c) b c
fa fb fc
B Connect and Apply
8.
fp fq
p q
9.
a)
b)
c)
d)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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10.
a)
b)
c)
11.
a)
b)
c)
d)
12. Use Technology
13.
14.
a)
b)
c)
15.
d
t
6400 kmQuito atMidnight
Location
at Time t
Rotation of Earth
d
a) d t
b)
c)
d) d t
5.1 Modelling Periodic Behaviour MHR 2
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating Connecting
Problem Solving
Reasoning and Proving
Reflectin
Selecting Representing
Communicating
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16.
a)
b)
17.
18. Chapter Problem
t (s)0
2
0.002 0.004 0.006
2
a)
b)
c)
19. Use Technology
20.
DateDaylight
(Hours:Minutes)
Jan. 1 9:09
Feb. 1 10:00
Mar. 1 11:14
Apr. 1 12:43
May 1 14:04
Jun. 1 15:04
Jul. 1 15:14
Aug. 1 14:28
Sep. 1 13:10
Oct. 1 11:45
Nov. 1 10:21
Dec. 1 9:19
a)
b)
c)
Achievement Check
21.
Connections
One of the oldest purely electronicinstruments is the theremin,
invented in 1919 by Leon Theremin,
a Russian engineer. Players control
the instruments sound by moving
their hands toward or away from the
instruments two antennas. One antenna controls the
pitch of the sound; the other controls the volume. You
have probably heard the eerie, gliding, warbling sounds
of a theremin in science fiction or horror films.
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C Extend
22.
100 m
Lighthouse
North
Sweep
a)
b)
c)
d)
e)
f)
g)
23.
y
x0
2
2
300200100
a) y
y
b) y x
c)
d)
e)
24. Math Contest
A B C D
25. Math Contest
A B C D
26. Math Contest
A B C D
27. Math Contest
A B C D
5.1 Modelling Periodic Behaviour MHR 2
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The Sine Function and
the Cosine Function
Investigate A
How can you use a table and grid paper with the sine ratio to construct
a function?
x
xfx x gx x
1. x
x
x
sin x
Exact ValueRounded to
One Decimal Place
0 0 0.0
301
2 0.5
360
Tools
calculator
grid paper
5.2
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2. a) x x x
x x
b)
3. Reflect
4.
5. a)
b)
6.
y x
Property y= sin x
maximumminimum
amplitude
period
domain
range
y-intercept
x-intercepts
intervals of
increase
intervals of
decrease
7. Reflect y x
8. y x
sinusoidal
sinusoidal
having the curved fo
of a sine wave
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Investigate B
How can you use technology with the sine ratio to construct a
function?
Method 1: Use a Graphing Calculator
1. 2nd TABLE
SETUP
TblStart Tbl
Indpnt Depend
AUTO
2. MODE DEGREE
Y=
Y1
3.
GRAPH
4. Reflect
2nd 1:
value y x
5. 2nd
6.
7.
8. Reflect fx x
x x
9.
fx x x x
Tools
TI-83 Plus or TI-84 Plus
graphing calculator
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Method 2: Use a TI-Nspire CAS Graphing Calculator
1. a) c 8:System Info 2:System Settings...
e Angle
Degree Auto or Approx
Auto OK x
b) c 6:New Document
2:Add Graphs & Geometryc) f1
d) b 4:Window
1:Window Settings XMin
XMax Ymin
YMax OK
2. Reflect
b) b 6:Points & Lines 2:Point On
c) /x
3. a) c Lists & Spreadsheet
b) b 5:Function Table
1:Switch to Function Table
c) b 5:Function Table 3:Edit Function Table Settings
Table Start Table Step
OK
4.
5.
6.
Tools
TI-NspireTMCAS
graphing calculator
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7. a)/
b) fx x
x x
8. Reflect
fx x x x
Investigate C
How can you use the cosine ratio to construct a function?
Key Concepts
Properties y= sin x y= cos x
sketch of graph
maximum value 1 1
minimum value 1 1
amplitude 1 1
domain {x} {x}
range {y, 1 y1} {y, 1 y1}
x-intercepts 0, 180, and 360 over one cycle 90 and 270 over one cycle
y-intercept 0 1intervals of
increase (over
one cycle)
{x,0x90, 270 x360} {x,180 x360}
intervals of
decrease (over
one cycle)
{x,90 x270} {x,0x180}
y
x
1
1
270 36090 1800
y
x
1
1
270 36090 1800
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Communicate Your Understanding
C1 x y xy x
C2 x n x n
C3
a)
b)
c)
B Connect and Apply1.
x
a)
b)
2. Chapter Problem
y x
y x x
a)
y x
y x x
b)
c) x
Connections
In musical terms, you have added the second harmon
sin 2x, to the fundamental, sin x. An electronics
engineer can mimic the sounds of conventional
instruments electronically by adding harmonics, or
overtones. This process is known as music synthesis
and is the basic principle behind the operation of
synthesizers. To learn more about how the addition
harmonics changes a sound, go to the Functions 11
page of the McGraw-Hill Ryerson Web site and follow
the links to Chapter 5.
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3.
a)
b)
c)
d)
C Extend
4. y x
x
a) x
x
b)
x x
c)
d) y x
y
x
e)
x x
f)
x x
x x
g)
x
x
x
h) y
x
5. a)
y x
b)
c) x
d) Use Technology
x x
e)
x
2nd 4:Vertical
x
300 MHR Functions 11 Chapter 5
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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f)
6. a) fx x
x x
b)
y x
c)
d) y x
e)
7.
y x
8.
y x
9. y x x
a) y
b) x
c)
10. Math Contest
11. Math Contest
12. Math Contest
A B C D
Career Connection
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Dynamically Unwrap the Unit Circle
y
1. MODE
PAR
SIMUL
2. xy
SIMUL
Y=
X1T Y1T
3. Reflect
4.
X2T Y2T
Tools
graphing calculator
Technology TipWhen you are in
parametric mode,
pressing X, T, , n will
return a T.
Use Technology
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5. WINDOW
6. GRAPH
Note:
x
x
7. Reflect
8.
Extend9.
a)
b)
Technology Tip
If you want to watch
the graphs be drawn
again, you cannot just
press QUIT and the
press GRAPH . Thegraphing calculator
remembers the last
graph that you asked
for, and will just
display it, provided
that you have not
made any changes
that affect the graph.
There are several
ways to get around
this feature. One isto select PlotsOn
from the STATPLOT
menu and then select
PlotsOff. When you
press GRAPH , the unicircle and sine functio
will be drawn again.
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Investigate
Transformations of Sineand Cosine Functions
Investigate
How can you investigate transformations of sine and cosine functions
using technology?
A: Graph y= asin x
1. a) y x
y x x x
b)
2. a) y x
b)
3. a) y xy x
y x
b) y
x y x
4. a) y
x
y x
b) y x
y x
c) y x
304 MHR Functions 11 Chapter 5
Tools
graphing calculator
Optional
graphing software
Technology Tip
Another way to
compare two graphs
is to toggle them on
and off. In the Y=
editor, move the cursorover the equal sign
and press ENTER .
When the equal sign is
highlighted, the graph
is displayed. When
the equal sign is not
highlighted, the graph
is not displayed.
5.3
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5. Reflect
aya x
a) a b) a
c) a d) a
B: Graph y= sin kx
1. a) y xy x
x x
b)
c)
2. a) y x
y x
b) y x
3. a) y x
y
x
b) y x
y
x.
4. Reflect
x ky kx
a) y kx y x
b) k
k
c) k
k
C: Graph y= sin (x d)
1 a) y xy x
x x
b)
2. a) y x
b) y x
3. a) y x
b) y x
4. Reflect
d xy xd
a) d b) d
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D: Graph y= sin x+ c
1. a) y xy x
x x
b)
2. a) y x
b) y x
3. a) y x
b) y x
4. Reflect
c y xc
a) c b) c
5. Reflect
Factor Value Effect
a
a> 1 amplitude is greater than 1
0 < a< 1
1 < a< 0
a< 1
k
k> 10 < k< 1
1 < k< 0
k< 1
dd> 0
d< 0
cc> 0
c< 0
Example 1Functions of the Form y= asin kx
y x
a)
b)
c) x x
d) x x
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Solution
y xya kx
a) a y
b) k k
c) k
d)y
x0
3
3
90 180 270 360
Example 2
Functions of the Form y= asin (x d) + c
y x
a)
b)
c)
d)
e) x x
Solution
y x ya xd c
a) a
b) k
c) d
d) c
e)y
x
2
0 360 540 720180
2
4
Connections
In a sinusoidal functio
a horizontal translatio
is also known as a
phase shift. A vertica
translation is also
known as a vertical
shift.
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Example 3
Functions of the Form y= asin [k(x d)] + c
y x
a)
b)
c)
d)
e) x x
Solution
y x ya kxd c
a) a
b) k
c) d
d) c
e)y
x
2
2
0 180 270 36090
The value of dis negative.
This indicates a horizontal
translation to the left.
Key Concepts
a k d cya kxd c
a a
k p p k
d d d
c c
c
ya kxd c.
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Communicate Your Understanding
C1 y xc c y
C2
a) b)
C3 y xy x
x180180
y
2
2
0 x180 360180360
y
2
0
2
A Practise
For help with questions 1 to 3, refer to
Example 1.
1.
a)y x b) y x
c) y x d) y
x
2.
a)y x b) y x
c) y x d) y
x
3.
a)y x b) y
x
c)
x d) y
x
e)y x f) y x
g) y
x h) y
x
For help with Questions 4 to 8, refer to
Examples 2 and 3.
4.
ya kx
ya kx d
a)
x90 270 360
4
2
6
0
6
4
2
y
180
b)
x90 270 360
3
0
3
y
180
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5.
ya kx
ya kx d
a)
180
2
0
2
4
4
y
270 360 x90
b)
x180 360
2
0
2
y
27090
6.
y x
a)y x
b)y x
c) y x
d)y x
7.
y x
a)y x
b)y x
c) y x
d)y x
8. a)
i)y x
ii)y x
iii)y x
iv)y x
b)
9. a)
i)y x
ii)y x
iii)y x
iv)y x
b)
B Connect and Apply
10.
y
t
y t
a)
b)
c)
d)
11. a)
i)y x
ii)y x
iii)y x
iv)y x
b) Use Technology
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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12.
a)
b)
c)
d)
13. Chapter Problem
y kt
y kt
y kt
a
y kta)
b)
c)
y kt
14.
y
x
x
a)
b)
y
c) y
15. a)
b)
c)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting TRepresenting
Communicating
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16. a)
b)
c)
Achievement Check
17. a)
y x
i)
ii)
iii)
iv)
b)
C Extend
18.
y x
a)
x x
b)
x x
c)
d)
x
e)
f)
19. y x
ya kxd c
a k d c
y x
20. Math Contest
A B
C D
21. Math Contest
A
D
C
B
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Graphing and Modelling Withy= asin [k(x d)] + candy= acos [k(x d)] + c
Example 1
Determine the Characteristics of a Sinusoidal Function From an
Equation
y x
y x
a)
b)
c) Use Technology
d)
Solution
a) y x ya kxd c a k d
c
a
5.4
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k
d
c
b)
c) Method 1: Use a Graphing Calculator
2nd 4:maximum
3:minimum
Method 2: Use a TI-NspireTMCAS Graphing Calculator
d) y x x
y y
y x x
y y
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Example 2
Sketch a Graph
a)
fx x gx x
gx
b) fx gx c) gx
hx
gx hx
Solution
a)
fx x
y
b) fx x x
y y gx x
x y y
c)
hx x
gx hx
1.
2.
3.
4.
5.
x90
4
2
0
2
4
y
180 270 360
ii
iiiiv
i
x
4
2
0
2
yg(x)
h(x)
90 270 360180
5.4 Graphing and Modelling With y= asin [k(x d)] + cand y= acos [k(x d)] + c MHR 3
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Example 3
Represent a Sinusoidal Function Given Its Properties
a)
b)
Solution
a) Method 1: Use a Cosine Function
a
k
k
x x
c
fx x
Method 2: Use a Sine Function
a k c
x
y
gx x
b) Y1 Y2
Y2 GRAPH
ENTER
ENTER
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Example 4
Determine a Sinusoidal Function Given a Graph
Solution
y
a
y c
y
x
d
x
k
k
a k d c
ya kxd c
y x
x
2
0
4
2
y
60 120 240 300 360180
5.4 Graphing and Modelling With y= asin [k(x d)] + cand y= acos [k(x d)] + c MHR 3
x
2
0
4
2
y
60 120 240 300 360180
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Key Concepts
fx a kxd cfx a kxd c
x
Communicate Your Understanding
C1
y x
C2 y x
a)
b)
c) y
d)
C3
A Practise
For help with questions 1 and 2, refer to
Example 1.
1.
y x
a)y x
b)y x
c) y x
d)y
x
2.
y x
a)y x b)y x
c) y x
d)y x
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For help with questions 3 and 4, refer to
Example 2.
3. a)
fx x
gx x
gx
b) fx
gx
c) gx
hx
gx
hx
4. a)
fx x gx x
b) fx
gx
c) gx
hx
gx
hx
For help with questions 5 and 6, refer to
Example 3.
5.
a)
b)
6.
a)
b)
For help with question 7, refer to Example 4.
7. a)
b)
B Connect and Apply
8.
fx x
a)
y x
b)
c) x
d) y
9.
gx x
a)
y x
b)
c) x
d) y
10. Use Technology
11. a) fx x
gx x
b) fx
gx
c) Use Technology
5.4 Graphing and Modelling With y= asin [k(x d)] + cand y= acos [k(x d)] + c MHR 3
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12. a) fx x
gx x
b) fx
gx
c) Use Technology
13. a)
fx x
b) Use Technology
14. a)
x
4
2
0
2
y
60 120 240 300180
b) Use Technology
15. Chapter Problem
y x
a)
y x x
x
y
x
b)
y x x
y x
c)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
Connections
Robert Moog invented the electronic synthesizer in
1964. Although other electronic instruments existed
before this time, Moog was the first to control the
electronic sounds using a piano-style keyboard. This
allowed musicians to make use of the new technology
without first having to learn new musical skills. Visit
the Functions 11page on the McGraw-Hill Ryerson
Web site and follow the links to Chapter 5 to find out
more about the Moog synthesizer.
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16.
a)
b)
c)
d)
e)
Achievement Check
17. a)
fx x
gx x
b) gx
c) gx
C Extend
18.
p q
a
ya x
19. y
x
a)
y x
b)
y
x
c)
d)
y
x
y
x
e) y
x
20. a)
y x
x y x
b)
y x
x y x
21. Math Contest
y x
x
y.
22. Math Contest
y x
A B C D
23. Math Contest
A B C D
5.4 Graphing and Modelling With y= asin [k(x d)] + cand y= acos [k(x d)] + c MHR 3
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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Data Collecting and
Modelling
Investigate
How can you collect data on the motion of a pendulum and use the
data to construct a sinusoidal model?
1.
2. a)
b) APPS 2:CBL/CBR
c) CBL/CBR ENTER 3:Ranger ENTER
d) 1:SETUP/SAMPLE
START NOW ENTER
3.
ENTER
4. Reflect
ENTER
REPEAT SAMPLE
Tools
graphing calculator
motion sensor
pendulum
Technology Tip
These instructions
assume the use of a
CBRTMmotion sensor
with a TI-83 Plus or
TI-84 Plus graphing
calculator. If youare using different
technology, refer to
the manual.
Technology Tip
The motion sensor
cannot measure
distances less than
about 0.5 m. Ensure
that your pendulum isnever closer than this.
The maximum distance
that it can measure is
about 4 m, but your
pendulum may be too
small a target to return
a usable signal from
this distance.
5.5
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5. a) ENTER SHOW PLOT
b) ENTER QUIT L1 L2
6. a) GRAPH TRACE
a
b)
c Y= DRAW
3:Horizontal
c)
d
d)
k
7. a) a c d k
b) Y=
8. Reflect
9. a)
i)
ii)
b)
Connections
For help in determinin
an equation given a
graph, refer to
Example 4 in Section 5
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Example 1
Retrieve Data from Statistics Canada
Solution
Sinusoidal
table 051-0001
Geography Canada
Sex Both sexes
Age group 20 to 24 years
from 1976 to 2005
Retrieve as individual Time Series
CSV (comma-separated values) Time as rows
Time
as rows Retrieve Now
Population of Canada, Aged 20 to 24 Years, Both Sexes, By Year
Year Population Year Population Year Population
1976 2 253 367 1986 2 446 250 1996 2 002 036
1977 2 300 910 1987 2 363 227 1997 2 008 307
1978 2 339 362 1988 2 257 415 1998 2 014 301
1979 2 375 197 1989 2 185 706 1999 2 039 468
1980 2 424 484 1990 2 124 363 2000 2 069 868
1981 2 477 137 1991 2 088 165 2001 2 110 324
1982 2 494 358 1992 2 070 089 2002 2 150 370
1983 2 507 401 1993 2 047 334 2003 2 190 876
1984 2 514 313 1994 2 025 846 2004 2 224 652
1985 2 498 510 1995 2 009 474 2005 2 243 341
0
1 000 000
500 000
1 500 000
2 000 000
2 500 000
3 000 000
Population Aged 20 to 24 in Canada,1976 to 2005
Year Relative to 1976
Population
10 20 30
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Example 2
Make Predictions
a)
b)
Solution
a)
b)
Example 3
Use a Sinusoidal Model to Determine Values
h
t
ht t
a)
b)
i)
ii)
iii)
Platform
PhaseShift
Platform
PhaseShift
Directionof
Rotation
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Solution
a)
b)
Method 1: Use the Equation
i)
ii) t
ht t
iii) k
k
k
Method 2: Use the Graph
i) 2nd maximum CALCULATE
Technology Tip
If you are using a
TI-NspireTMCAS
graphing calculator,
refer to the
instructions onpage 33 to determine
the maximum and
minimum values and
the period.
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minimum CALCULATE
ii) value CALCULATE
iii)
Key Concepts
Communicate Your Understanding
C1
C2
C3
Connections
A maximum and an
adjacent minimum are
half a cycle apart. To
obtain the value of
the full period, it is
necessary to multiply
by 2.
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A Practise
For help with questions 1 and 2, refer to the
Investigate.
1.
a)
a
b)
c
c)
d
d)
k
e)
f) Use Technology
2.
t
3
2
0
1
d
2 4Time (seconds)
8 106
Distan
ce(metres)
a)
a
b)
c
c)
d
d)
k
e)
f) Use Technology
For help with questions 3 and 4, refer to
Example 3.
3. h
t
ht t
a) h
b)
c)
d)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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4. P
Pt t t
a)
b)
c)
d)
For help with question 5, refer to Example 1.
5. a)
b)
MonthDaily Sales
($) MonthDaily Sales
($)
1 45 7 355
2 115 8 285
3 195 9 205
4 290 10 105
5 360 11 42
6 380 12 18
For help with Questions 6 and 7, refer to
Example 2.
6.
a)
b)
c)
7.
8.
y t
y
t
a)
b)
c)
d)
t
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B Connect and Apply
9. T
T
g
g
a)
b)
c)
d)
e)
10.
a)
b)
c)
11.
a)
b)
c)
d)
e)
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
Connections
You cannot actually expel all of the air from your lungs.
Depending on the size of your lungs, 1 L to 2 L of air
remains even when you think your lungs are empty.
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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12.
I t
I t
a)
b)
c)
d)
13. a)
b)
c)
14. Chapter Problem
a)
y x
y x x x
x
b)
15.
Sinusoidal
International travellers intoCanada table 387-0004
table 075-0013
Geography Canada
Travel category Inboundinternational travel
Sex Both sexes
International Travellers Total travel
Seasonal adjustment Unadjusted
from Mar 198to Mar 2006
Retrieve as a Tab Retrieve Now
Retrieve as individual Time Series
CSV (comma-separated values)
Time as rows
a)
b)
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c)
d)
16.
Sinusoidal
a)
b)
c)
d)
e)
Achievement Check
17.
Time, t(s)
Height, y(m)
Time, t(s)
Height, y(m)
0 110 8 35
1 103 9 60
2 85 10 85
3 60 11 103
4 35 12 110
5 17 13 103
6 10 14 85
7 17 15 60
C Extend
18.
s
s t t
a)
b)
19. Math Contest
A B C D
20. Math Contest
A B C D
21. Math Contest
xyz
xyz
A B C D
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Use Sinusoidal
Functions to ModelPeriodic PhenomenaNot Involving Angles
Investigate
How can you use sinusoidal functions to model the tides?
1. ht t
2.
3.
4.
5.
6. Reflect
Tools
graphing calculator
Optional
graphing software or
other graphing tools
Connections
Tides are caused
principally by the
gravitational pull of t
moon on Earths ocea
The physics of rotatin
systems predicts that
one high tide occurs
when the water is
facing the moon and
another occurs when
the water is on the
opposite side of Earth
away from the moon.
This results in two tid
cycles each day. To le
more about tides, visi
the Functions 11pag
on the McGraw-Hill
Ryerson Web site and
follow the links to
Chapter 5.
5.6
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Example 1
Model Alternating Electric Current (AC)
Va ktd c
a)
b) k
c)
d)
e) Use Technology
Solution
a)
b)
k
k
k
c)
d) V t
e) x
x
x
y
y y
Technology Tip
You can enter rational
expressions for the
window variables. To
set the maximum
value for xto1
30,
you can just type
1 30. When
you press ENTER , thecalculator will calculate
the desired value,
0.03333.
Connections
The adoption of AC
power transmission
in North America was
spearheaded by the
inventor Nikola Tesla.
He demonstrated that
AC power was easier
to transmit over longdistances than the
direct current favoured
by rival inventor Thomas
Edison. There is a statue
honouring Tesla on Goat
Island in Niagara Falls,
New York.
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Example 2
Model the Angle of the Sun Above the Horizon on the Summer Solstice
in Inuvik
Hour Past Midnight 0 1 2 3 4 5 6 7 8 9 10 11
Angle Above theHorizon ()
2.4 1.8 2.4 4.2 7.2 11 16 22 27 33 38 42
Hour Past Midnight 12 13 14 15 16 17 18 19 20 21 22 23
Angle Above theHorizon ()
45 46 45 42 38 33 27 22 17 12 7.5 4.3
a)
b)
c)
Solution
a)
a
c
d
k
k
ht t
b)
c)
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Example 3
Predator-Prey Populations
N
Nt t t
a)
b)
c)
d)
e)
f)
Solution a) t
b) Use Technology
maximum
CALCULATE
c)
d) minimum
CALCULATE
e)
f)
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Key Concepts
Communicate Your Understanding
C1 x a c
C2 ya kxd c d
C3 k
A Practise
For help with questions 1 and 2, refer to the
Investigate.
1. ht t
a)
b)
2.
a)
b)
For help with question 3, refer to Example 1.
3.
5.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles MHR 3
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For help with question 4, refer to Example 3.
4.
0
800
8 124 16 20
600
1000
200
400Population
Time (years)t
N
a)
b) c
c) d
d)
k
e)
f)
B Connect and Apply
5. d
d t t
a)
b)
c)
d)
e)
6.
a)
b)
c)
7.
a)
b) k
c)
d)
e) Use Technology
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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8.
a)
b) k
c)
d)
e) Use Technology
9.
Year
(since1970)
Sunspots
(AnnualAverage)
Year
(since1970)
Sunspots
(AnnualAverage)
0 107.4 19 162.2
1 66.5 20 145.1
2 67.3 21 144.3
3 36.7 22 93.5
4 32.3 23 54.5
5 14.4 24 31.0
6 11.6 25 18.2
7 26.0 26 8.4
8 86.9 27 20.39 145.8 28 61.6
10 149.1 29 96.1
11 146.5 30 123.3
12 114.8 31 123.3
13 64.7 32 109.4
14 43.5 33 65.9
15 16.2 34 43.3
16 11.0 35 30.2
17 29.0 36 15.4
18 100.9
a)
b)
c)
10.
a)
b)
c)
11.
Nt t
a)
b)
c)
12. Use Technology
a)
b)
c)
d)
5.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles MHR 3
Connections
Sunspots were observed as early as 165 B.C.E. Like
many phenomena in the sky, they were thought to
have a mystical significance to humans.
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13.
a)
b)
c)
14.
Month Hours of Daylight
1 8:30
2 10:07
3 11:48
4 13:44
5 15:04
6 16:21
7 15:38
8 14:33
9 12:42
10 10:47
11 9:06
12 8:05
a)
b)
c)
d)
15. Use Technology
Functions 11
a)
b)
Duration of Daylight Table for One Year
c)
d)
e)
f)
16.
a)
b)
c)
0
400 000
200 000
600 000
800 000
1 000 0001 200 000
1 400 000
1 600 000
Residential Natural Gas Consumption
Month Starting January 2001
ThousandsofCubicMetres
12 24 36 48 60
Connecting
Problem Solving
Reasoning and Proving
Reflecting
Selecting ToolsRepresenting
Communicating
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17. Chapter Problem
x
x
y x x
a) Use Technology
x
y
b)
c)
18.
a)
b)
c)
d)
e)
f) Use Technology
19.
a)
b)
c)
5.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles MHR 3
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C Extend
20.
Px x x x
Px
a)
b)
c)
21.
a)
b)
c) Use Technology
22. Math Contest n
n
A B C D
23. Math Contest y
x
x y
Connections
Reflection from a film that is
much thinner than the
wavelength of the light
being used forms a
perfect window. No
light is reflected; it
is all transmitted.
This is why images
viewed through non-
reflective eyeglasses
appear brighter than
when viewed through
lenses without the coating.
You can use a soap bubble kit
to see another example of this perfect window. Dip
the bubble blower into the solution, and hold it so
that the soap film is vertical. Orient the film to reflectlight. You will see a series of coloured bands as the
soap drains to the bottom. Then, you will see what
looks like a break in the film forming at the top. This
is the perfect window. You can test to see if there is
still soap there by piercing the window with a pin or a
sharp pencil.
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63/68Use Technology: Create a Scatter Plot and a Function Using a TI-Nspire CAS Graphing Calculator MHR 3
Use Technolog
Create a Scatter Plot and a Function Using a TI-NspireTMCASGraphing Calculator
Hour Past Midnight 0 1 2 3 4 5 6 7 8 9 10 11
Angle Above the Horizon () 2.4 1.8 2.4 4.2 7.2 11 16 22 27 33 38 42
Hour Past Midnight 12 13 14 15 16 17 18 19 20 21 22 23
Angle Above the Horizon () 45 46 45 42 38 33 27 22 17 12 7.5 4.3
1.
Lists & Spreadsheet
2.
3.
4. Graphs & Geometry
b Scatter Plot
5. hour x ang
y
6. b Window
x
y
7. Function
8.
ht t
f1
9.
Tools
TI-Nspire CASgraphing calculator
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Chapter 5 Review
344 MHR Functions 11 Chapter 5
5.1 Modelling Periodic Behaviour,
pages 284 to 293
1.
x0
2
180
4
y
360360 180
2
a)
b)
c)
d)
e)
2.
a)
b)
c)
5.2 The Sine Function and the Cosine
Function, pages 294 to 301
3. x
4.
x
5.3 Investigate Transformations of Sine and
Cosine Functions, pages 304 to 312
5. y x
a)
b)
c)
d)
e) x
f)
5.4 Graphing and Modelling With
y= asin [k(x d)] + cand
y= acos [k(x d)] + c, pages 313 to 321
6.
y
t
y t
a)
b)
c)
d)
5.5 Data Collecting and Modelling, pages
322 to 332
7.
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Month (21st day) Time (EST)
1 7:40
2 7:04
3 6:17
4 5:25
5 4:47
6 4:37
7 4:55
8 5:28
9 6:01
10 6:36
11 7:15
12 7:43
a)
b)
c)
8.
5.6 Use Sinusoidal Functions to Model
Periodic Phenomena Not Involving Angles,
pages 333 to 341
9.
a)
b)
c)
d)
e)
f)
Chapter Problem WRAP-UP
a)
b)
c)
Chapter 5 Review MHR 3
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For questions 1 to 8, select the best answer.
For questions 1 to 3, refer to the graph of the
periodic function shown.
x0
4
2
2 4 62
y
46
2
1.
A B C D
2.
A B C D
3. f
A B C D
For questions 4 to 7, consider the function
y = 38cos [5(x 30)] 3
4.
4.
A B C D
5.
A
B C
D
6. y x
A B
C D
7. y x
A
B
C
D
8.
y x
A B
C D
9.
y x
a)
b)
c)
d)
e)
f)
10.
a)
b)
11.
a)
b)
12.
fx x
a)
y x
b)
c) x
d) y
Chapter 5Practice Test
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13. a)
x
3
2
0
1
1
y
1206030 90
b)
14.
Month Employment (%)
1 62
2 67
3 75
4 80
5 87
6 92
7 96
8 93
9 89
10 79
11 72
12 65
a)
b)
c)
d)
15.
a)
b)
c)
Chapter 5 Practice Test MHR 3
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Modelling a Rotating Object
a)
b)
c)
d)
e)
f)
Task
Tools
string
large paper clip
tape measure
grid paper