McGrawHill Functions 11 Unit 5

7/23/2019 McGrawHill Functions 11 Unit 5

1/682

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


2/68

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


3/68

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


4/68

5.1


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


5/68

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


6/68286 MHR Functions 11 Chapter 5

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


7/685.1 Modelling Periodic Behaviour MHR 2

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



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


9/68

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


e) t g

t t

g g

f)



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.


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


11/68

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


Connecting

Problem Solving


Reflecting


Communicating Connecting

Problem Solving


Reflectin

Selecting Representing

Communicating



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.


13/68

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




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


15/685.2 The Sine Function and the Cosine Function MHR 2

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



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



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



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




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.


20/68

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


Connecting

Problem Solving


Reflecting


Communicating

Connecting

Problem Solving


Reflecting


Communicating



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



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


23/68Use Technology: Dynamically Unwrap the Unit Circle MHR 3

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.


24/68

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


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


25/685.3 Investigate Transformations of Sine and Cosine Functions MHR 3

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



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



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.



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.




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


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



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


Reflecting


Communicating



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


Reflecting

Selecting TRepresenting

Communicating



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


33/685.4 Graphing and Modelling With y= asin [k(x d)] + cand y= acos [k(x d)] + c MHR 3

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



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


35/68

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



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


37/68

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


x

2

0

4

2

y

60 120 240 300 360180



Key Concepts

fx a kxd cfx a kxd c

x


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


39/68


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


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




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


Reflecting


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.


41/68

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


Connecting

Problem Solving


Reflecting


Communicating



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


43/685.5 Data Collecting and Modelling MHR 3

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



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



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



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.



minimum CALCULATE

ii) value CALCULATE

iii)

Key Concepts


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.



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


Example 3.

3. h

t

ht t

a) h

b)

c)

d)

Connecting

Problem Solving


Reflecting


Communicating



4. P

Pt t t

a)

b)

c)

d)


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



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


Reflecting


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


Reflecting


Communicating



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)



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


53/685.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles MHR 3

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



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.


55/685.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles MHR 3

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


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)



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)


57/68

Key Concepts


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)


3.

5.6 Use Sinusoidal Functions to Model Periodic Phenomena Not Involving Angles MHR 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


Reflecting


Communicating


59/68

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)


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.



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


Reflecting


Communicating


61/68

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)




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.


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


Angle Above the Horizon () 2.4 1.8 2.4 4.2 7.2 11 16 22 27 33 38 42


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


64/68

Chapter 5 Review


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.


65/68

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



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


67/68

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


68/68

Modelling a Rotating Object

a)

b)

c)

d)

e)

f)

Task

Tools

string

large paper clip

tape measure

grid paper

McGrawHill Functions 11 Unit 5

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Transcript of McGrawHill Functions 11 Unit 5