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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
Page 2
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
Page 3
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
Page 4
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
Page 5
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,...,
•
Page 6
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
Page 7
- 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
Page 8
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.,
,
•
.n..
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