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**Unformatted text preview: **\ 4.4 Graphs of the Sine and Cosine Functions A. Basic Graphs of Sine and Cosine Def. A function f (x) is said to be periodic if there is a positive number p, such that f(x+p) = f(x) In other words, the function will repeat itself over an interval of length p. i
\ Sine is Periodic: sin(x + 21:) = sinx mnx=0 Soc %=O,Tr,?:u,.... 1
:7 '+ = rm J:or omj ‘mecger m ‘ ﬁned 2 2'“ _
Odd gov-Chm '. and (-Jc) = 3W? i
_ _ ”W =
ﬁa‘ Ex SW1 (-2 —- \ 3‘0 i —
— (3m ’“’/23 = y 3=+ermanx \ntm um‘r
F“ ’93 owe as \ooedmo/ i. POIQB ‘ ,
P Cx, x33 =i<cos<zﬂ sxnebx
CM $733=Q<©30ﬁpﬁ1> Q(‘><,77 =(Coe’if5‘m 33
amcc (£33: >0
‘m CH Cosine is Periodic: cos(x + 21:) = cos x. one ”Pedoda \ C05a1=0 %\’+=’E\3g)5-33"’ 2, Z. 2.
:7 3:: CZn+\)'%
‘ Zh'“ + (E
2. 2.
" “TY 1- :1
2.
S‘or at»? h
’Domin ~. 000,003
Ra '- [—\,\']
’Pexi 1 2m
Even Rimhcn 2 mac-+3 =
CO3?
For Home: cosC—m = —\ = coshr) ll. Amplitude and Period: y = a sin bx and y = a co la] is called the amplitude. The amplitude changes the range of the graph to
[-a, a]. If a > 0 and a whole number, we stretch the graph vertically.
If 0 < a < 1, we shrink the graph vertically. If a < 0, reﬂect the graph over the X—axis. b affects the period of the graph by compressing it or stretching it
horizontally so that they complete one-cycle over any interval of length 27”. If b > 0, then y = sin bx and y = cos bx have a period of 31,2 If b < 0, then use the properties of even/odd. CHANGING THE AMPLITUDE AND PERlOD
OF THE SINE AND COSINE FUNCTIONS The functions y *- 4: sin In and y - (1 ans bit (I) '2' 0} have amplitude in? and
27.?
period T If a I?» 0. the graphs ofy * a sun In and; ,-, (1 cos bx are snmlar to the
J
graphs of y = sin .t and r: em .\‘. respectively. with three changes. I. The range is {th. til. 21: 7.
2. One cycle is completed over the interval 0. I i.
V , J
: 27 77 3:.- 2n"
' 3. The .t- ‘ s u‘ ﬁnale». 1‘ W the kc -‘ tints are (l. —. -. —. and —. The \'-value< in
m ‘ ‘ 5 p‘ 2b h 21, h ‘ each of these numbers is one of: —a. (i. or a. M: )' 11':- a sin bx (a 23* 0) “t “i ncus in (u 23> 0)
if a ‘71 O. the graphs are the reﬂections of the graph of y —., in lsin in and
, v ‘= la {cos bx. respectively. about the taxis.
If h ~< {3. ﬁrst use even~odd properties then graph the function. See page 3 l9. \ Ex 4- Graph y - 3 cos 1x over a one period interval
. . — — ' . ‘ p _ Zr“, —
\a\=3 Pm git-3,51 b=‘/2. Penod. 2‘1 ’ ‘T‘ 'LW 5W*Ch\n% \Ie com! 3 own: ‘3 'Z
I566 RDWV‘TJ: O) "E. ) r‘l ,QlT 3 21 are-‘Ch‘ng homEOPﬁOQLk
2);) b 2b b O _\
= o :L 1 .33. 21: 005T "_
3 2a; ‘/7_ ’3/2) V; C0523 ‘0
— (:03 3: = ‘\
‘0) «)23’W3‘23W)U‘ﬁ‘ b
‘ -- \‘3 C039(
LX
— \‘b 30-05 z. C. Phase Shift EX. 5: Graph 32 = sin (x — g) over a one-period interval.
\( = 3m (x An) \hd\QQT€5 a mr\'ton’c0\ sh'G‘V +0 we ricsh‘c bY ’h ‘ un’ﬁ-s. \' '= fun CX-H'ﬂ \nChCOJCS q mmwnxok 80634 *0 me \ewr h urnb.
ThKS ‘13 b \s Smidouf Qa- cosCx-hﬁ :6 cosCx+h3
\' Note mm: onto Shi‘H's ﬁne Pcncd 09 «he lane‘hcn . ‘ ‘(T/ mﬁ’s
=s\n( -'rr SWWﬂ V‘ N ‘33 Z
T x A) For aﬁxﬁxmod Eoll'ﬂ mm, arm-1g) rm <1 P6003 o? ”1751:1413
z 2 7- EX. 6: Find the amplitude, period, and phase shift of the graph y= 3 sin [2 ( -— 5)].
\(1—3 3W3 [1 CX" 1;)1 \’ = a :1anbe «3] a” New Range. [-3.31
AmPYrtude = \od = \3\= 5
Period= 2% = 2.3 -.- '11 com 5531 honzm’rQU )
? z ( £5;ch emd m mm, Pmee Shs¥+ 31609: +0 +he. mgm “0% 33:3: mos;
“ID 2121’ 2:1 — o'ﬁ _ __
\hetéigrﬁ: o 20 , w )
wﬁh m : ,n ”W q. Tl
0+1; 1*: :"z‘trﬂszw Z : "TV/:- 3 ETT/H 3 "ﬁ st/H 3% .. ‘5 a 3\\"\EZ CX (Tr/731 EX. 8: Find the period and the phase shift of each function. 11'
a) y— 2cos("3\ —;) ”Romero
\) = Z cos 32>
=2. cos 3;— X'W/Q
QMod .21 = 2:5. mm 311% 2 CE om: +o*he 61%th 3
b 5 so two: C. b) y = 3 sin(1rx + 2) Y = e sme (x Ari/Try] ”Per\cd=2'\_T—. 211:2)
b 11' Phase thQ+ _: 77,“, and?) *0 WE \egi") so 41/“ = C EX. 9: Write an equation for each sinusoidal graph. 8%. 32; BE m ﬂu EWﬁ bx (a. 4:633 bx we emriwwca “:3; Na? - Wyimuwmwamwwm ‘31 xaeQiecﬁzeGi ac: (“03%; Xmmxﬁfg ...

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