k., i ) . .- - - 0€¦ · homework assignment part a construct accurate sketches of each of the...

5.1 Graphing Sine and Cosine Functions T4 (p.222-232)

Periodic Functions = A function that repeats itself over regular intervals (cycles) of its domain.

Period = The length of the interval over which a function travels one full rotation or cycle. (The distance from one point to that exact same point again.)

Amplitude = the vertical distance from the horizontal central axis the maximum or the minimum. The central axis is not always the x-axis.

Sinusoidal Functions = Oscillating periodic functions, like y = sin x or y = cos x . cod)

o) o)

y = sin x 0,11

x f ii- a -3Tr- lir

Lt I T5:, k., i ) _._.- - - 0

i: ( 7Til,D. IT-Yz -3- 4 I i w

—if ai: a

:7„ T

,.,,., I a , -"ri'' -"rii- 0 41. •

Note: 7r

• The two graphs are identical except for a translation of —2

• The two graphs are periodic and continuous. They are considered sinusoidal graphs.

• The period of each of these two graph is 27r.

Properties of y = sin x

Domain: (-00,00)

Range: [-1, 1]

Zeros: x = kg, k E I

y-intercept y = 0

Maximum value: y = 1

Minimum value: y = —1

Amplitude: 1

Period: 27-c

Properties of y = cos x

Domain: (-00,00)

Range: [-1, 1]

Zeros: x = TC —2

+ Kg, ke I

y-intercept: y = 1

Maximum value: y = 1

Minimum value: y = —1

Amplitude: 1

Period: 22-c

Quick Sketches

y = sin x y = cos x

2

!=ir:

by y = sin 4x Toctor of kti

bLf /, pen.dd 21r: on Tol cyc it in

y = cos — 2 I)

Two transformations uzci 1R---i-eftro'oe)

/ -thc p

y = a sin (bx) cri6 Cii0420h+61 ( &fit-0)

ompliftt ver-titot )

= amplitude (vertical stretch)

2g- 360° or = period

Note: The b value is not the period. The b value is used to find the period, but it is not the period.

1 yew -•)-re+ct byji

Refer to Examples 2, 3 & 4 on pages 226-231 of the textbook for examples. Homework: Page 233 #1-11, 14, 15, C5

4. Determine the values of a, b, c, and d.

a

b

5.2 Transformations of Sinusoidal Functions — Part 1 T4 (p.238-249)

Before graphing sinusoidal functions, they must be in the form

ua.A. horizoi-del period r--ht ot v point)

y = a sin (b (x — c)) ± d pho iff

rfsp h 114 de, Vtr-koJ f+- (hi cd i ga )

The procedure for transforming sinusoidal functions in the form y = a sin(b(x c))+ d is as follows:

22-c Example: Sketch the graph of y = 3 sin 2x — +2.

3 (

1. First, re-write the equation in factored form: y = 3 sin 2 X - - 4- 2 3

2. Then recall that graph of y = sin x looks like:

3. Transformations

D — vertical displacement/shift d 7:4;11

C — phase shift

y

',eh=

•

73'

VW' T.

A — amplitude GI 3

imasnersoaa

.1011211•1101.. •• 0,1161061... anowammu,

wiactratm 4.1.4. • ' ,i714MILV,

B — used to find the period 6= a. y yvic d1 mow/ kylccifinft\,iy\cix 71-- +-Tr

3

Qrr= Tr

Try these: ir —

+. .t.„, (x —2

41;01.0

`et

Igertg15,

.10615., ■,010,311,1111., .4(0.0043,

• .,1,411,AfrOV,

(( y = 4 cos 2x + +3

4)) C

•

'

• ,qufp.gr,...,

•

Summary of sketching a sinusoidal function given an equation:

Properties of the graph Calculations

Central axis CI

Amplitude I I

Perio d (.9 7

I b I

Phase shift C

Maximum I CI ± I

Minimum d - 1 0 1

Range E d- 0 di-icil ) C'' tni'n , Moue 1

Homework: Page 250 #1, 2, 3

- s e

ri XX.

tna)( = Cc

tyv'r) .3, Li

tt: let it)

d = ryvA)('-t Pni)

a)

CcP ine C 0

trY) CA.,k4 Ptak or tvkini. irnrrs

PaY-4, trir) ,„ per) od

bE=1rll

5.2 Transformations of Sinusoidal Functions — Part 2 T4 (p.238-249)

A sinusoidal function is expressed in the form:

y = a sin(b(x — c)) d or y = a cos(b(x—c)) d

Properties of the graph Calculations .

a rn ct k -1 m 1,4 ilf) —rnIrti m 1.4 ry \

ad

d mak i m (-4 rn 1- liTk 1vn10

trt u 4,Y\

,

Period all- b

b w

o

period

c = sine funcion , s-i-cl r 1 f) i ..tTWd $C

C = cosine function 7 t ji c , kr f : r ' 1, fnitv,,

Write the equation of the following graphs in the form y = a sin(b(x c)) d or y = a cos(b(x —c)) d

A

4

2

"

4

2

-4 V

A

y - ))

A

4

-2

5.2 Writing Equations: Transformations of Sinusoidal Functions

T4

Write an equation for each of the following in terms of y = asin(b(x - c)) + d

•

Q('

e4,

p 6,Tvc

A 4-

I I I

-4 \,/

, M1,1( 1. I r

( ( 1

2

t..4

-2 7C

4,4

A

4 :

I I I I P.\ I

4

V col •

A

Write an equation for each of the following in terms of y = acos(b(x — c)) + d

www.4....

0 mow.

w

... .... ...

....p. 4

. 411 . . . WO . a, .

N. OP . 06 Mb 611 .

. Oft. X

. 0,410

4

Homework Assignment

Part A Construct accurate sketches of each of the following:

1) y=-2sin(x-21.4)+1

2)

Y = --03sin(0.5(x— it))+.

4) -3-r=

5) y = sin(70 0.5)) 4- 3

Each graph MUST be neat and labeled appropriately with the .scales clearly marked on the x and y axes

Part

Identify the equations for each of the following graphs In terms of sine and cosine.

dirwarompromman10.0.....msimr

40S Graphing Assignment

,41,11p, • .aLitaaa. •

." •

ZNI+Mar.,

,

•

5.3 The Tangent Function

0(6) .,(cosai sin 9 )

Q(1

T4 (p.256-262)

sin Recall that tan 0 =

Note that tan 19 is the slope of the line through the point P(0).

cos e •

rise sin 0 tan 0 Slope=

run cos 0 1

sin (9 Since tan 0 = cos ' there is a non-permissible value when cos x = 0.

This is represented by an asymptote. (o t9

(I)

x 16\ ----r if ( ff- \`‘

.9-, 3r

1 —4 _Sr if

air e ''‘ iv ti (r\

c) 1\\I„Artip} — I I LIND •— I

Propertied of y = tanx

71" Domain: xERIx#—+rur,neI

2

Range: (-00,00)

Zeros: x=nic,n EI

y-intercept: y = 0

Period: 7C

Equation of asymptotes: X = + /yr, n E

Homework: Page 262 #2, 3, 8, Cl

Page 12

7r y = 2 cos — (x+ 1)-1

3

2

1

-2

-3

-4

44007,, .

,

. • : ••,: , ..

.: .. • ,

, • -5

iitS141, ItEa, •

N.,0„,„,,, •

Page 13 WE,

b) Solve the following equations algebraically: (

i) 1 = 2 cos — + 1) –1 ■ 3

(x)

CA' ,4

i% 0= (6)( °'"‘L‘

IT TrOcti

-TT

0 =Ai-

(71- – 2 = 2 cos — ± –1

3 I icoz(

c) Solve the same equations graphically.

3

5.4 Equations and Graphs of Trigonometric Functions T4 (p.266-274)

Exl: a) Sketch the following graph over the interval 0 x < :

-0 tas,...11ggEttft

1 ...4, ,....,,, OW,

d) Use the graph to explain why the following equation has no solution:

2 cos —(x+ 1) —1 3

moo( ---- H+'%-) -

A WO 1 htva ca)

Whet+ iS" Ve4 y

r11r)t: i3

-

Ex2: The graph of y = 3 sin 0 4 2 is sketched below.

, - 2 Graphically solve the equation sin u = - over the interval 0 ._ x < 22-t- . 4 3

al.-1, int"

,

,,, I ' 4,--/

0) ,If r Cttf g 3 - 0 .‘, 1'

„„, a _ ( -1-41,k- are e vet I ,/, tv'e Call kit '1414/ P WO ,.,

Note: _,,, 0 re/P 46 )

This equation simplifies to sin u = - 3

. Thus, the zeros are the solutions.

To find the zeros of y

fi

= 3 sin 0 1-2 , we produce the equation 0 = 3 sin 0 - 2 . , - 2

Page 14

)9'

a) Write a sinusoidal equation to represent this scenario using the cosine function.

4,1,„pec)

dor d

4 4 S.APRWRIVIMPWRINN,110,14,....12,

rib

t:\ Lf 55"

Homework: Page 275 #5, 9, 15, 18, 19, 20, 21

'

.11,..AKIMIF

1r

314-2:M 311 4- 3 +301-15- •--- (a

Ex3: The average daily maximum temperature in Winnipeg follows a sinusoidal pattern. The lowest value of-14°C is found on January 15 and the highest temperature is 26°C on July 15.

Key to this question: January 15 —4 15th day of the year July lOnd day I 044` ciCiy There are 365 days in one year

co5i n e

v,"

,

b) Find the average temperature on October 27. (300th day) 300

Page 15

Applications of Periodic Functions T4

Problem 1 At a seaport the depth of the water, h meters, at time, t hours, during a certain day is given by this formula.

h = 1.8 sin (

27z- (t — 4.00)

+3.1 12.4 j

a) State the:

i) period

ii) amplitude ii) phase shift

b) What is the maximum depth of the water? When does it occur?

c) What is the depth of the water at 5:00 am?

Hi) phase shift

y g— Real Life I CZ LP,

At a seaport, the depth of the water, h (in meters), at time, t (in hours), during a certain day is given by the formula:

h= 1.8 sin2p,(--- 4.01 t 3.1

b) What is the maximum depth of the water? When does it ocv teit

C) What is the depth of the water at 5:00 am?

12.4.= 3.1 oparf Lt

7.1 ht3 (c)r

an= 3.1+L8

h 1.8 gin 2.

sir) 27703 f3.1 h 11.4

2. The period of a tidal wave is 16 Mnutes with amplitude of 8 meters, The normal depth of water at Crescent Beach is 6 meters.

a) What is the maximum and minimum height of water caused by the tidal wave at Crescent Beach?

14 Yv) /4- tin —a., min no a4c.r •

b) Write a periodic model of the tided wave when it first reachea Crescent Beach.

h o eyn a I

7-Th

124

a) State the: 0 Period b = ampDtude

I 2,, 1.1—r (123 \

2n- /

11- 30-• . •

\

71'

in4 nOtryn al —firs+ wal do 14 bacL

-1--b in I -fhe.6 1)1'3

tik

#1

Solve the following equation graphcially.

a) sin (

#2

Solve the following equation algebraically. Calcualtor not permitted.

a in <

#3

Solve the following equation graphcially.

b} 4 cos(x — 47°) 7 10 0' 61r

#4

Solve the following equation algebraically. Calculator permitted.

#5

A err.s xvlie 1 with a radius of 10 in rotates once every 60 s. Passengers get on board at a point in above the ground a the bottom of the Ferris wheel A sketch for the first 150 chon.

a) Write an equatio n to he pat f o çassi ger on the Ferri.s heel here

.the hoitht is a :ftint t.ion. of time.

b) if En-nix is at the bottom of tht Ferris wheel when it beFins to move determine her hei.ht above t o the nearest tenth f a Ii: et wheel has been i bon for

Deter:rime Ole an ou time ...hat passes 'before a. r7d3.7.- a 1..ei.ght oflo ni for th e i rt. tinit. Detern in

me :other timethi i.der vi 1 he tt

Ile wi..th1.13, tilt .1:r:A th.at

round, en the

Calculator permitted. You may want to re-draw the diagram.

is + 1000. 00 sin nionths.

our

OW Many .niont...hs Duid it hike for .fox population to drop to 650? Round.

inswerto the nearest -:month.

#6 The, Arctic fox is coin hroughout the Arctic. tundra. Suppc. se the population, of fOxes in a :re - ion of :northern 1 initolia

nodelled bvI function

#7 electric. heater turns cii and off on a

c.iir basis as it h.e the xvater in a o tub. T :e V a[ei tenperatu.re, .e Celsius, varies sin..uso daily With time n minutes. The, heater turns on viien the

temperature of the water reaches and turn off when the water temperature

43 T.:. Suppose the water t& inperaturt drops to 34 QC and the h:aJLCr ta.irns on.

er another 30 min the heater turns off. arid then iftfr another 30 min the heater

air.

a Write the equation that : expresses mp( rature as function of

tt) Determ ne the teniperature .10 min a -ter •the heater first turns on.

#8

717he Can.adian National llistori.c: iVindpover Ce.ntre, :Et..zi..korp, Alberta,

.h i ff10 P t\ies of wi..n.dmi..11.s on (1...isplay. 'The tip of t..h.:6 blade ci ine reac..h.es itt niininuni height of 8 in above 1.e ground at a time of Its in.aximu.m. eight is 11 above the xniil The tip

of the blace rot.a.t(s I 2 times per mi.nute.

a) \\>rite t sine -or :a. cosine .u..n..cti.on to model tile rotation. ot t.,716 lip a: the bl..(id.e,„

b) 'What is the height of the tp of the blade after 4.

C) For how :1..onois the tip of the blade above a hc..ii.ht of :ra in the first 10 sl

#9

Solve the following equation over the interval [0,24

6 t os2 0 cos 9

Calculator permitted.

#10

Solve the following equation over the interval [-1800, 1801. sec' 9 =

Calculator not permitted.

Solve the following equation where the domain is the reals. .... in x 412 x

Calculator not permitted.

#11

Sketch the graphs of:

a) y = sinx

b) y = cosx

c) y = tanx

/2,

)

••••• , 57r Okc—

ir IT X—(.42 )(—(a

)(

Ltcoz =i0 Co kr 5' T-33

Cos()(- )

.6Yr xc 90*

SO1 f'2- 30'

h(-6:) io ozp: 'ff?'-t: -361)) i- iai 3 inih /3k -13S Fcc, 9i d

1) (13&j - 10 zlh( tTi(/3t?---/s))4.-- -= I.s'in ( I 2, ) f

is, Ociol(9,„ riv

CrA ICA tC4 ih 40 12

io d r b . 21-1,_= _Tr .7-7

Knool 00 3D

C?sl C 7-30 co 1) (ti) izz los)h

— 'Pp —12d TE)T- ; a"-7 r,1-77) al

" " •

5.,LL 11.1)_p_b

147 Lfr = 1 0 Lf cos' Oci-LI ) -3

pfrioci= 3E70"

0-) fr)b e)) =

iI (y; (x---G)) =- I

,s')6 (316,----. 0(-),1 -1 =0

OP= b=76:

?L=r=

124-61=-Ie

Cos

COS-1

az' LH , Oci6,„

x-LfS Ltoci‘n Z(0,1-10%,.:3

((t-rits-))

kt /0 = 4 5

„FL i? CP f

6:*J14)-1 f/1 ",to =:ô, L35

30 ± 91. iliqR„,,x,c3Act

6) y co,r 5;1_ r t s wu d t a, a 7. r y o.tut en ?A o a • War w 7,0

41-zi) 1-14S

(p CoZ2-0'. -(r*CoSt, rz I Eol3o'.]

("0,3r2. +Cod 6- -0

CoS =-31— cc)

61,7 Cocf' i) L0,

_7-11)-15

70!52-81989,_,147

Sit\ )(-:7-31h2 )( .NieR

sit /-1 2)(--S5Nx sih (3)h = °

4„ 21.6.1) L

0-='s 120

tz -45- cos ( irz 4-3R.5 **a"

I ) +3g,

BO

CCO4 0 (

36,4, aa

,; LL d s

32/5 b = 2.1r 12:

46 30

SI

2/mi6 12 per scc.

-- ,$)cco A ck perib d. 12,

o11T1r

I IC

Se-0 —Lf -0 ec

Sec; Cosa ------

or

10 y .= cos

c) X

is#e

k., i ) . .- - - 0€¦ · homework assignment part a construct accurate sketches of each of the...

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