00020002010012Trig_Shifting2.ACT0001020012Trig_Shifting2.EAC0100000018c4 y=3sin(x+0.35)-2.1 2Qvc6 `!g`` YU) dx29  <vKr  A 'L'pC uxy qq  3sin(x+0.35)-2.1/  y=0.8403868cos(x)+0.5       0.8403868sin(x+0.54)+0.53  6a`bP ! bI >vKr  A 'LrC uxy qq @! 0.8403868cos(x)+0.5؈: Trig_Shifting2.EACACT,/9<@S `% 'I[4LExpansion, Compres  and  techniques for Sine!Cos FunctBsT\\ AuthorswlProfuX Wei-Chi Yang e-mail: wyang@radd.edu @!URL: http://www.#/5S Objeves:[V5(1) When a circle is drawn (where center and radius aknown), we would like to Jthe respective sineHco curves thatTassociated with ?.[RD(2 e will explofgraphs of y=r*i(x+a)+bsc)+dLL(3Lsee ntabotwog3b+dentical[ ExampN1.  Aaby following steps 1C4,nswz)}b)slow Step1. Draw a circle[S2. Pickpoint P on the+ and form= vector from"enter to0?.[,[ 3. Animate! P alongc4544. Colljangles, (x,y) coordinUof P.[ a) What isgraph# when we dravtable ich consistsLcolumn1=t'$2=x valux ,i?SKb܆(ۦS\ESuppose this is5e circle: Center=[-1.35,-0.225] and radius=1.179248ΈMX^cghG `@Gs$ . DvKr  A 'LrC uxy qq@ (.H E@>628RQ&BPY%Yq UBʗ+ N NB#` C@65 `% Y7@! M+)a+"*H*w kJ``1-[,"* eR2UW. Y8RQ&A3X  PS+e Y"Th7SJ@ YɈ!Ɗ*"  @6 & %@  #RH Fpu#XI0.Y:Y:a$H BXnĘn &8@ '֔` ywR 8RQ&AF(z!Ę\.'W2gf)*E"EQ %+>k,   @65`% YGd= -F `9BGP; ]d.C%H!Y,/@o#0Ř1 #rdii Y5&h ' sG@3K2a0"q;5ĘÈ,g7`xi _HVAVA$xh"q1#"'#(3 [('Answer to a): y=1.179248*cos(x)+(-1.35)/(/b/sin/0.220[pYGeneralization: If the cent!and radius for a circle are (a,b)" respectively,An .M point [x(t),y] onfNs|sfyinga) .=rt)+a b) Ct)+b:\B Try your ownhere->Έ*4.`@Gs$ E  1v$>r  A 'L' C uxy qqv [[ Example 2.!G (1) If f(x)=r1*sin(x-a)+b and g2*cos(x+c)+d with a c>0. Then i explain wwillXF.[FS(2r+r-rkr [ Exploration 1. [ (IWe use the following geometry link to eD'graph of y=r*sin(x+a)+bQ]}^Note. O(a) Openostrip and drag[p left or right( observe howequchanges.pN(bpupgdownf_;f\K!Ήs2Q*1. We will explore that y=r*cos(x+a)+b andsinby ug0e followGeometry links: [ *2. Noticge-/2hbs.[K V3. Start fixdrag to guess wˈa b should be .in orderrese tw?raphsI+idenal.] \3D one curvematch%othi equation.ΈC CH)Question: Do you know when these two equa% s represent same graph?5 HNFinaForm$NGraph2D, 3@ LISTSYSL4< Modify P<STATCALC d< \x S:equence,xSheetO4| olveEq`wr0(Up<tupFLG1H(<Lis{pDPicViewWind_osvevxy^0iy2` P %  fWȒԒdx(4@FLXdp ,|!@"T#h$|%&č'І  (?ܒ)*+PY,d -x.$0% %1<2H3T4`5lExF nCIJKLM ̒N؆Ά0OPQR<SPT ] , ^0_4`8ab D͑\Α,h Б@t בTrkّ| ڑ Xۑ " FinancialFormat  ! system]"^_` a "bxR @R @x0"a+c=3*/2. Then Y%Q? implies that r1=r2[b=dP(2+-$,Algebraically:R- Clear_a_z Rdone<>-Checking for (1): We assum',c>0, a+c=3*/2, r1=r2 and b=d.Ra:=0.21R21100, c:=(R)-a233?2\-Qsin(x-a)-cos(x+c)0[GD2چ Clear_a_z odone#  -8sin(x+a)-cos(x-c)R0R eActa c<<  e5y010008main.ACT0001020012eActivity Save.EAC010000001e8d y=3sin(x+0.35)-2.1 2Qvc6 `!g`` YU) dx29  <vKr  A 'L'pC uxy qq  3sin(x+0.35)-2.1/  y=0.8403868cos(x)+0.5       0.8403868sin(x+0.54)+0.53  6a`bP ! bI >vKr  A 'LrC uxy qq @! 0.8403868cos(x)+0.5؈: Trig_Shifting2.EACACT,/9<@S `% 'I[4LExpansion, Compres  and  techniques for Sine!Cos FunctBsT\\ AuthorswlProfuX Wei-Chi Yang e-mail: wyang@radd.edu @!URL: http://www.#/5S Objeves:[V5(1) When a circle is drawn (where center and radius aknown), we would like to Jthe respective sineHco curves thatTassociated with ?.[RD(2 e will explofgraphs of y=r*i(x+a)+bsc)+dLL(3Lsee ntabotwog3b+dentical[ ExampN1.  Aaby following steps 1C4,nswz)}b)slow Step1. Draw a circle[S2. Pickpoint P on the+ and form= vector from"enter to0?.[,[ 3. Animate! P alongc4544. Colljangles, (x,y) coordinUof P.[ a) What isgraph# when we dravtable ich consistsLcolumn1=t'$2=x valux ,i?SKb܆(ۦS\ESuppose this is5e circle: Center=[-1.35,-0.225] and radius=1.179248ΈMX^ `h# `@Gs$ .  Dv r  A kL'C uxy qqH ( sin(x) ."..cos.>-[(as` "!F1  2FpDBb0 Sew  fd8@wQ (XRP De;as`$w!G02xp< C H%UATAf slWwls$6l1hl8l"!F1l82SblTC`pTh%leuWlv 4RQg!2FC&xlT4 1eAAGvHrDcVDyc6D pgD w,(10D( (% YX57SYHB  #7` B80%Gb$EP0@9 < hH EdPT S)i 0AX`waV %TEtxVX0&udx7"WxrC&0`@!$2p<YUTYi`Q<`$bXA`2vi`8,`@2'^`9VIY\`5vq )yTUx\`Tv(` 3tYsYBYF$` %"H! (as> "!F1 2FpDBb0$Sew0 fd8@<wQ (XRP Delw!Gl2xl C%UAAf slWwls$6l1hl8l82SblTC`pTh%leuWlv 4RQg!DC&xlT4 1eAA@ GvHrcVP$yc6/0 pg`< w H(10pT(^(pu#XDY3 Y1D8p3Y%$3d9Y hYDFS<q` xW@`9tQ`XXy`vf`#` X q$#&ySH4fAHCHX`IITRT#pl $0=5%e YGdp]1% `B&P; 2%H! 03*@ 1'4HŘ#5^ #Nr@n6udiiP Y5&h  ' sG@nYK6a4"q7*5ĘÈ0g7`x  VAVA$x" &+"v(I['/Answer to a): y=1.179248*cos(x)+(-1.35)[(/b/sin/0.220[9YGeneralization: If the cent!and radius for a circle are (a,b)" respectively,An .M point [x(t),y] onfNs|sfyinga) .=rt)+ab) Ct)+b\ Try your ownhere->Έ*8.7<H `@Gs$ .E DvKrA L  C uxy Vc [`[k Example 2.!G (1) If f(x)=r1*sin(x-a)+b and g2*cos(x+c)+d with a c>0. Then i explain wwillXF.[S(2r+r-rkrE oration 1 &I,We use the following geometry link to explor' graph of y=r*sin(x+a)+b[] Note. [ O(a) Openostrip and drag[p left or right( observe howrequation changes.pN(bpupgdownf_f\!Ή"+EU2Q*1. We will explore that y=r*cos(x+a)+b andsinby ug0e followGeometry links: [ *2. Noticge-/2hbs.[K V3. Start fixdrag to guess wˈa b should be .in orderrese tw?raphsI+idenal.] \3D one curvematch%othi equation.ΈC CH)Question: Do you know when these two equa% s represent same graph?9 HNFinaForm$NGraph2D, 3@ LISTSYSL4< Modify P<STATCALC d< \x S:equence,xSheetO4| olveEq`wr0(Up<tupFLG1H(<Lis{pDPicViewWind_osvevxy^0iy2` P %  fWȒԒdx(4@FLXdp ,|!@"T#h$|%&č'І  (?ܒ)*+PY,d -x.$0% %1<2H3T4`5lExF nCIJKLM ̒N؆Ά0OPQR<SPT ] , ^0_4`8ab D͑\Α,h Б@t בTrkّ| ڑ Xۑ " FinancialFormat  ! system]"^_` a "bxR @R @x0"a+c=3*/2. 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