{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 19 256 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 259 12 "Solution s to" }}{PARA 257 "" 0 "" {TEXT 257 0 "" }{TEXT 260 18 "Graphing Exerc ises" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" }{TEXT -1 17 " 13 September 2001" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 262 37 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 263 2 "1." }{TEXT -1 96 " Find a funct ion whose graph is a parabola passing through the points (3,0), (6,0), and (0,12)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Solution: The quadratic function " }{XPPEDIT 18 0 "f(x) = (x-3 )(x-6)" "6#/-%\"fG6#%\"xG-,&F'\"\"\"\"\"$!\"\"6#,&F'F*\"\"'F," }{TEXT -1 5 " has " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 76 "-intercepts at the points (3,0) and (6,0). We define f to be this function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> (x-3)*(x-6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&,&9$\"\"\"!\"$F/F/,&F.F/!\"'F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "We find that f(0) = 18, whereas w e want our function to pass through the point (0.12). To get the func tion we want, we have to multiply f by 12/18 = 2/3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g := x -> (2/3)*f(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%\"fG6#9$ #\"\"#\"\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(g (x), x=-1..7);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6%-%'CURVESG6$7S7$$!\"\"\"\"!$\"1mmmmmmm=!#97$$!1nmmmwAc#)!#;$\"1@#*=! ><3u\"F-7$$!1LLL$o!)*QnF1$\"1E-5[ZhM;F-7$$!1nmmmxnK]F1$\"1,aL'*e%)=:F- 7$$!1nmmmV1:LF1$\"1HM]'HIiS\"F-7$$!1MLL$[9cg\"F1$\"1ggCMb0)H\"F-7$$!1B nmm;ct?!#=$\"1mQkBWC,7F-7$$\"1******\\YJ?;F1$\"1.+r+9`/6F-7$$\"1KLL$= \"\\<>Jm'Fgn7$$\"1++++z,u6Fgn$\"1SpM'GrZ(eFgn7$$\"1+++SP)4M\"Fg n$\"1eiH([AH:&Fgn7$$\"1LLL=Zg#\\\"Fgn$\"1`Rz'e<'HXFgn7$$\"1nmmEn*Gn\"F gn$\"1*p%RX>MGQFgn7$$\"1mmm1xiD=Fgn$\"1H$**)3\"y\"oKFgn7$$\"1+++X,H.?F gn$\"19%pLt1dl#Fgn7$$\"1mmm1:bg@Fgn$\"1qI(>]z'[@Fgn7$$\"1+++X@4LBFgn$ \"1MrqMkKI;Fgn7$$\"1+++N;R(\\#Fgn$\"1\\'yB%oit6Fgn7$$\"1nmm;4#)oEFgn$ \"1yLe,*yZN(F17$$\"1nmm6lCEGFgn$\"18+sk#Qjn$F17$$\"1LLL$G^g*HFgn$\"1A# 4ACGy!zFK7$$\"1LLL=2VsJFgn$!1E.]#o)R]KF17$$\"1+++N&pfK$Fgn$!1&)eM[k,6e F17$$\"1LLLjcz\"\\$Fgn$!10N%oV$\\B#)F17$$\"1+++!G5Jm$Fgn$!19APkq2L5Fgn 7$$\"1******4#32$QFgn$!1j!flml8?\"Fgn7$$\"1*****\\#y'G*RFgn$!1zJ0QYaG8 Fgn7$$\"1******H%=H<%Fgn$!1BU/J%y'G9Fgn7$$\"1mmm1>qMVFgn$!14pAgVy\"[\" Fgn7$$\"1++++.W2XFgn$!1SRi%4j**\\\"Fgn7$$\"1LLLep'Rm%Fgn$!15mrdl2#[\"F gn7$$\"1+++S>4N[Fgn$!1\"pk6EU^U\"Fgn7$$\"1mmm6s5'*\\Fgn$!1^:-B%=fL\"Fg n7$$\"1+++lXTk^Fgn$!13Ra!>-d?\"Fgn7$$\"1mmmmd'*G`Fgn$!1(\\'zrr(=/\"Fgn 7$$\"1+++DcB,bFgn$!1+C([:[oJ)F17$$\"1MLLt>:ncFgn$!1.2'49v$=fF17$$\"1LL L.a#o$eFgn$!1vWo1c)f3$F17$$\"1nmm^Q40gFgn$\"1un)\\:+0-\"!#<7$$\"1+++!3 :(fhFgn$\"1r$H&y4OkLF17$$\"1nmmc%GpL'Fgn$\"1W&ygOu`\\(F17$$\"1LLL8-V& \\'Fgn$\"1uc&3\\%\\a6Fgn7$$\"1+++XhUkmFgn$\"1uj5e.;B;Fgn7$$\"1+++:o " 0 "" {MPLTEXT 1 0 28 "f := x -> (x-2)*(x-4)*(x-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*(,&9$\"\"\"! \"#F/F/,&F.F/!\"%F/F/,&F.F/!\"&F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#S" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(0) = -40 " "6#/-%\"fG6#\"\"!,$\"#S!\"\"" }{TEXT -1 30 " and we want a function \+ whose " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 59 "-intercept is 10, w e will have to multiply our function by " }{XPPEDIT 18 0 "-(1/4)" "6#, $*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g := x -> -(1/4)*f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%\"fG6#9$#!\"\"\"\"% F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(g(x), x=-1.. 6, -3..12);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%- %'CURVESG6$7S7$$!\"\"\"\"!$\"1++++++]A!#97$$!1LLL3#*>u%)!#;$\"16fI4eu< ?F-7$$!1nm\"z43m9(F1$\"1%\\vN[$FS7$$\"1++]s2O[5FS$\"1Hj9$eN\\x #FS7$$\"1n;aG\"H5=\"FS$\"1#*G6z:FS$\"1..;a\"p=M\"F17$$\"1,]i! o<-1#FS$!1En_[*R[e)FP7$$\"1ML3-$=-@#FS$!1kH6GE5CEF17$$\"1M$3xplzM#FS$! 1'))Ry?88\"QF17$$\"1nm\"H([a'\\#FS$!1<.Cf#zAn%F17$$\"1n;ayo(3l#FS$!19J S\\\"*)p:&F17$$\"1+]7VLA&y#FS$!1C!eG*z_\"G&F17$$\"1nmT07KIHFS$!1I:gkb2 \\^F17$$\"1+++&\\@-3$FS$!17IU#[f&oZF17$$\"1++v$opoA$FS$!1-C,(*)oY?%F17 $$\"1,](oMf(oLFS$!1:*R#zsaBNF17$$\"1++DEOIENFS$!1xK$)[FsjEF17$$\"1LLLo T'ym$FS$!1SgvHk'[%=F17$$\"1,+]i-,>QFS$!1M/J\"p+-s*FP7$$\"1m;a)3rf&RFS$ !1%4>$)RvxC#FP7$$\"1,+]Zaq0TFS$\"1h**\\kMSw\\FP7$$\"1L$3-\"QfYUFS$\"1- )H#z-YV5F17$$\"1+]PWF'QR%FS$\"12>IbjuG9F17$$\"1LL$e/Xy`%FS$\"1*QF\")4m qd\"F17$$\"1+](=<\"e)o%FS$\"1Ept`\"H8W\"F17$$\"1ommwzvL[FS$\"1=P/P!p$> )*FP7$$\"1nm\"zAAA)\\FS$\"1>E[a%o=I\"FP7$$\"1M$3-7d%H^FS$!1!yD#QQ%R9\" F17$$\"1,++&p]ZE&FS$!1rHp(R_Ht#F17$$\"1MLe*R7)>aFS$!1`s!3U*)f4&F17$$\" 1nmmO9]ebFS$!1-27DT_VxF17$$\"1+](o(GP1dFS$!1/w\"zrbo6\"FS7$$\"1,]78Z!z %eFS$!1A!y-gps]\"FS7$$\"\"'F*$!\"#F*-%'COLOURG6&%$RGBG$\"#5F)F*F*-%+AX ESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fgz;$!\"$F*$\"#7F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 2 "3." }{TEXT -1 21 " Plot the graphs of " } {XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y=x^4" "6#/%\"yG*$%\"xG\"\"%" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y=x^6" "6#/%\"yG*$%\"xG\"\"'" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "y=x^8" "6#/%\"yG*$%\"xG\"\")" }{TEXT -1 11 ". Comment. 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For " }{XPPEDIT 18 0 "abs(x);" "6#-%$absG6#%\" xG" }{TEXT -1 97 ">1, the highers powers are larger. As the exponent \+ increases, the curve gets closer to zero for " }{XPPEDIT 18 0 "-1" "6# ,$\"\"\"!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 " x " "6#%\"xG" } {TEXT -1 46 " < 1, and increases ever more steeply through " } {XPPEDIT 18 0 "x = 1 " "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 94 ". For very high p owers of x, we'd expect a curve that almost has corners at (1,0) and ( -1,0)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 3 "4. " }{TEXT -1 6 " Plot " }{XPPEDIT 18 0 "y = x^3;" "6#/%\"yG*$%\"xG\"\"$" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = x^5" "6#/%\"yG*$%\"xG\"\"&" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y = x^7" "6#/%\"yG*$%\"xG\"\"(" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "y = x^9" "6#/%\"yG*$%\"xG\"\"*" }{TEXT -1 11 ". Commen t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "plot([x^3, x^5, x^7, x^9], x=-1.5..1.5, -2..2, color=[red, orange, green, blue], scaling=C ONSTRAINED);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6) -%'CURVESG6$7S7$$!1+++++++:!#:$!1++++++vLF*7$$!1++]P&3YV\"F*$!1g4b1-d_ HF*7$$!1+]ivF*7$$!1+]7V0@&=\"F*$!1JfNKP*[m\"F*7$$!1+]i&exd7\" F*$!1q![:c#yE9F*7$$!1+]i+#QU1\"F*$!1+/0B$f`?\"F*7$$!1+]i!3%f+5F*$!1P+) 3I$y,5F*7$$!1++D\"oS:P*!#;$!1CCT3#G1B)FX7$$!1+++v@)*=()FX$!18UYPpAGmFX 7$$!1++](G3U9)FX$!1zI%)[4!>S&FX7$$!1*****\\-\\r\\(FX$!1A*o#33%R@%FX7$$ !1+++vGVZoFX$!1?$=k#*y0@$FX7$$!1+++v4J@iFX$!1Q$zkQSzS#FX7$$!1++D1Bt_cF X$!1mP*pmPPFX$!1v*\\eWd:A&F]q7$$!1,++]=$z9$FX$!1$)))fv^V>JF]q 7$$!1***\\iX/4]#FX$!1oN2pk>k:F]q7$$!1***\\(o8y%)=FX$!1plAd))\\&p'!#=7$ $!1****\\i:#>C\"FX$!1gALE:]:>Fbr7$$!1!***\\7ev:lF]q$!1AUNR*oiw#!#>7$$! 1G++](o2[\"Fbr$!1L/RMi%oC$!#C7$$\"1)***\\P>:mkF]q$\"1asz-1d.FF]s7$$\"1 ++DcdQA7FX$\"1\"=&\\C<_E=Fbr7$$\"1,+]PPBW=FX$\"1W6NWUgsiFbr7$$\"1***** *\\Nm'[#FX$\"1_1s9tiP:F]q7$$\"1****\\(yb^6$FX$\"1f)HD53I-$F]q7$$\"1++v VVDBPFX$\"18A*e429;&F]q7$$\"1++]7TW)R%FX$\"1b2Y+mO4&)F]q7$$\"1)*****\\ @80]FX$\"1(H'pkI&QD\"FX7$$\"1,++D6!Hl&FX$\"1^(H<,,k!=FX7$$\"1***\\P4w) RiFX$\"1;S^/:cHCFX7$$\"1-++vZf\")oFX$\"1!*>^w=()eKFX7$$\"1)**\\P/-a[(F X$\"1JLHlQ;%>%FX7$$\"1++v=Yb;\")FX$\"1Gr\"p4hqM&FX7$$\"1*****\\i@Ot)FX $\"1'>=>/q;m'FX7$$\"1***\\PfL'z$*FX$\"1vlv;'p>D)FX7$$\"1+++!*>=+5F*$\" 13rOpga+5F*7$$\"1++DE&4Q1\"F*$\"11#)3cL!R?\"F*7$$\"1+]P%>5p7\"F*$\"1%= \"3;K4J9F*7$$\"1+++bJ*[=\"F*$\"1nF*7$$\"1+++Ijy58F*$\"1ME?$pT@D#F*7$$\"1+]P/)fTP\"F*$\"1u@!=!o%[f#F* 7$$\"1+]i0j\"[V\"F*$\"1B?6;K&Q&HF*7$$\"1+++++++:F*$\"1++++++vLF*-%'COL OURG6&%$RGBG$\"*++++\"!\")\"\"!Fc[l-F$6$7W7$F($!1+++++v$f(F*7$$!1++voU In9F*$!1&H;,`S9!oF*7$F.$!1D(o6j*owgF*7$$!1+Dcc,;19F*$!1$3$Gf,h(\\&F*7$ F3$!1Sr(osVN'\\F*7$F8$!1Zlh*f=J\"RF*7$F=$!1(Rt]5/M/$F*7$FB$!11=zNirQBF *7$FG$!1@/x%zo#3=F*7$FL$!1TyGAN>l8F*7$FQ$!17-aMR(H+\"F*7$FV$!1PQ.!*HhG sFX7$Ffn$!1s'oC@@)Q]FX7$F[o$!1]P#)4)zHe$FX7$F`o$!1i%y6(*R&oBFX7$Feo$!1 eD;g\\N0:FX7$Fjo$!1A5B&\\j)>$*F]q7$F_p$!1P]hN^ardF]q7$Fdp$!1xD-v0n_IF] q7$Fip$!1%*RXgmZc;F]q7$F_q$!1!\\S9zwXH(Fbr7$Fdq$!1je`nk>\"4$Fbr7$Fiq$! 1=-lZHI$y*F]s7$F^r$!1^)G]b4&yBF]s7$Fdr$!1*=FV\\5W&H!#?7$Fir$!1QT$*>:Uu 6!#@7$F_s$!1aaA4@G>r!#I7$Fes$\"12-ysJRI6Fj`l7$Fjs$\"1\"y2>'zBHFFf`l7$F _t$\"1W'G.&pVL@F]s7$Fdt$\"1XHx^<\"z]*F]s7$Fit$\"1un!o='eLHFbr7$F^u$\"1 COjCS1brFbr7$Fcu$\"16c`$\\[ik\"F]q7$Fhu$\"1YdEE42TJF]q7$F]v$\"1!=fD-2C x&F]q7$Fbv$\"1)G.Br`(f%*F]q7$Fgv$\"1FEjiEGV:FX7$F\\w$\"1jF\"*y@/]BFX7$ Faw$\"1$ovfWhD_$FX7$Ffw$\"1\\&)38_E\"3&FX7$F[x$\"1fQAW(y)fsFX7$F`x$\"1 $)QE\"G54+\"F*7$Fex$\"1%)QeTiWi8F*7$Fjx$\"1`BTALQ<=F*7$F_y$\"1mg^#['eN BF*7$Fdy$\"1^vr=0DoIF*7$Fiy$\"1\"4r)z:apQF*7$F^z$\"1#QvVd)))**[F*7$$\" 1+++b!)[/9F*$\"1NZ7\\5+laF*7$Fcz$\"18&p(374\"3'F*7$$\"1+D\"G:3uY\"F*$ \"1m#))fb[Q!oF*7$Fhz$\"1+++++v$f(F*-F][l6&F_[l$\")+++!)Fb[l$\")Vyg>Fb[ lFefl-F$6$7W7$F($!1+++]Pf3n?E@$Fbr7$F_q$!1Xg1041>5 Fbr7$Fdq$!1;*o0R8K1$F]s7$Fiq$!10TMF%*)*=hFf`l7$F^r$!1z&*3l#>%\\%)Fj`l7 $Fdr$!1b/Ia;zcX!#A7$Fir$!1-d1J_,')\\Fcs7$F_s$!1V;emz-h:!#N7$Fes$\"1.Bx e0IEZFcs7$Fjs$\"1%z\"R!)35ySFf[m7$F_t$\"1f&HhlTiD(Fj`l7$Fdt$\"1>**eeI@ zeFf`l7$Fit$\"1N<(3T4o%GF]s7$F^u$\"1q/zXaz=**F]s7$Fcu$\"1Jy%)pM)[=$Fbr 7$Fhu$\"1\")Q2Og!)oyFbr7$F]v$\"1V6@p%*eW=F]q7$Fbv$\"11E.#yaKo$F]q7$Fgv $\"175kuGU3tF]q7$F\\w$\"1v*[;'yv;8FX7$Faw$\"1^*zO>41K#FX7$Ffw$\"1icd2L zvQFX7$F[x$\"1XcRN<1(Q'FX7$F`x$\"1n&Qdiu7+\"F*7$Fex$\"1)3T>vn=a\"F*7$F jx$\"1OIU)RVzI#F*7$F_y$\"1JanJv4zKF*7$Fdy$\"1nf3I$)[/[F*7$Fiy$\"1$y$yn Y\\[mF*7$F^z$\"1KmI;R`_#*F*7$Fbel$\"1V9mO)=!y5F]gl7$Fcz$\"12A4GH\">D\" F]gl7$Fjel$\"1^pb^O1l9F]gl7$Fhz$\"1+++]Pf3G[(F]q7$Feo$!1Ul[ h(>Fdfm7$Fjs$\"1TGi(=3O4'Fcs 7$F_t$\"1e+#y^\"*zY#Ff[m7$Fdt$\"1[GL9)3aj$Fj`l7$Fit$\"1k2KI&*fiFFf`l7$ F^u$\"1H4p4_+v8F]s7$Fcu$\"1uLo'[u:;'F]s7$Fhu$\"1J-(3(>Cr>Fbr7$F]v$\"17 y@]$RW*eFbr7$Fbv$\"1()QceS6M9F]q7$Fgv$\"1*4m42-5Y$F]q7$F\\w$\"1QUWr!ez P(F]q7$Faw$\"1nUSt:yG:FX7$Ffw$\"1ufj)y0j&HFX7$F[x$\"13.W@*y\">cFX7$F`x $\"1?%RG5R;+\"F*7$Fex$\"1CG03u\"\\u\"F*7$Fjx$\"1v02\"o=4$HF*7$F_y$\"1` S3F/w.YF*7$Fdy$\"1ol#3$\\@BvF*7$Fiy$\"1hul$H=B9\"F]gl7$$\"1+v=nIZU8F*$ \"1KD\"oi$G;9F]gl7$F^z$\"1eDi8+Bfh&[E@F]gl7$Fcz$\"1 )GB&H/JxDF]gl7$Fjel$\"1b\"=k0-Z:$F]gl7$Fhz$\"1++]PfLWQF]gl-F][l6&F_[lF c[lFc[lF`[l-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q\"x6\"%!G-%%VIE WG6$;$F*!\"\"$\"#:F]]n;$!\"#Fc[l$\"\"#Fc[l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The higher powers are close r to zero in " }{XPPEDIT 18 0 "-1 < x" "6#2,$\"\"\"!\"\"%\"xG" }{TEXT -1 31 " < 1 and further from zero for " }{XPPEDIT 18 0 "abs(x) > 1" "6 #2\"\"\"-%$absG6#%\"xG" }{TEXT -1 36 ". The graph of a huge odd power of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 148 " would look (up clos e) like a vertical line rising to (-1,0), a horizontal segment from (- 1,0) to (1,0), and then a vertical line rising from (1,0)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 3 "5. " }{TEXT -1 17 " Plot the curves " } {XPPEDIT 18 0 "y = x^(1/2)" "6#/%\"yG)%\"xG*&\"\"\"F(\"\"#!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "y = x^(1/4)" "6#/%\"yG)%\"xG*&\"\"\"F( \"\"%!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = x^(1/6)" "6#/%\"yG)% \"xG*&\"\"\"F(\"\"'!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "y = x^( 1/8)" "6#/%\"yG)%\"xG*&\"\"\"F(\"\")!\"\"" }{TEXT -1 10 ". Comment" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([x^(1/2), x^(1/4), x^ (1/6), x^(1/8)], x=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7W7$\"\"!F(7$$\"1LLL3x&)*3\"!#<$\"1)o3=F/7$$ \"1LLLL3VfVF,$\"1w`j1]#z3#F/7$$\"1++]i&*)fD'F,$\"1m&Qci(>,DF/7$$\"1nmm \"H[D:)F,$\"194mjnEbGF/7$$\"1LLLe0$=C\"F/$\"1PE$>jhR_$F/7$$\"1LLL3RBr; F/$\"15r^$ys!)3%F/7$$\"1mm;zjf)4#F/$\"1\\%zI'R/\"e%F/7$$\"1LL$e4;[\\#F /$\"1]tv0M\"[*\\F/7$$\"1++]i'y]!HF/$\"1Cu?k\"y)*Q&F/7$$\"1LL$ezs$HLF/$ \"1&ewjur+x&F/7$$\"1++]7iI_PF/$\"1'Gf@32c7'F/7$$\"1nmm;_M(=%F/$\"1V+jb *p4Z'F/7$$\"1LLL3y_qXF/$\"12#><)ycgnF/7$$\"1+++]1!>+&F/$\"1!Htmk6C2(F/ 7$$\"1+++]Z/NaF/$\"1AQ[)fvAP(F/7$$\"1+++]$fC&eF/$\"1!pT)Go8]wF/7$$\"1L L$ez6:B'F/$\"19kKBD*R*yF/7$$\"1mmm;=C#o'F/$\"1bjfP!)\\u\")F/7$$\"1mmmm #pS1(F/$\"1WZ[iCXJ kK@5Fdt7$$\"1++v$Q#\\\"3\"Fdt$\"1V78g#[*R5Fdt7$$\"1LL$e\"*[H7\"Fdt$\"1 6PKSGpf5Fdt7$$\"1+++qvxl6Fdt$\"1![4&*z6(z5Fdt7$$\"1++]_qn27Fdt$\"1te^2 O%*)4\"Fdt7$$\"1++Dcp@[7Fdt$\"18-y-jB<6Fdt7$$\"1++]2'HKH\"Fdt$\"1*zIe` -s8\"Fdt7$$\"1nmmwanL8Fdt$\"1TT7Fdt7$$\"1nmmT9C#e\"Fdt$\"1o:j!zryD\"Fdt7$$\"1++D1*3`i\"Fdt$\"1G.k Ng([F\"Fdt7$$\"1LLL$*zym;Fdt$\"1=m\\P9/\"H\"Fdt7$$\"1LL$3N1#4#)yL\"Fdt7$$\"1nm;9@BM=Fdt$\"1IN#pDQVN\"Fdt7$$\"1LLL`v&Q(=Fdt$ \"1-(ew;*))o8Fdt7$$\"1++DOl5;>Fdt$\"1T-P+]B%Q\"Fdt7$$\"1++v.Uac>Fdt$\" 1&HWDCl()R\"Fdt7$$\"\"#F($\"1&4tBc8UT\"Fdt-%'COLOURG6&%$RGBG$\"#5!\"\" F(F(-F$6$7gnF'7$$\"1nmTN@Ki8!#=$\"13MD*Q)=@>F/7$$\"1LL$3FWYs#Fg\\l$\"1 p2w,4p%G#F/7$$\"1++D1k'p3%Fg\\l$\"1oC38hUGDF/7$$\"1mmmT&)G\\aFg\\l$\"1 !*e&fnqpr#F/7$$\"1++]7G$R<)Fg\\l$\"1XjVBB#o+$F/7$F*$\"1(\\+\"f3/JKF/7$ $\"1++]ilyM;F,$\"19&z0]Mdd$F/7$F1$\"1;o]ynPUQF/7$F6$\"1g)\\%4*)G_UF/7$ F;$\"1R:_.=QpXF/7$F@$\"1&>UA[(>,]F/7$FE$\"1j)4`ipMM&F/7$FJ$\"1I&[a^'HO fF/7$FO$\"10'QF'Q!QR'F/7$FT$\"1;Jo!)4MonF/7$FY$\"1ShK%Q*RnqF/7$Fhn$\"1 ^05h*y:M(F/7$F]o$\"1z])=$))4'f(F/7$Fbo$\"1SQ?idiEyF/7$Fgo$\"1#H@A$RBW! 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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 3 "6. " }{TEXT -1 6 " Plot " } {XPPEDIT 18 0 "y = 2^x" "6#/%\"yG)\"\"#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = (1/2)^x" "6#/%\"yG)*&\"\"\"F'\"\"#!\"\"%\"xG" } {TEXT -1 48 ". Compare the graphs, and explain the symmetry." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([2^x, (1/2)^x], x=-2..2 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"#\"\"!$\"1+++ ++++D!#;7$$!1LLL$Q6G\">!#:$\"1z?I`^ubEF-7$$!1nm;M!\\p$=F1$\"1i**H/68*z #F-7$$!1LLL))Qj^'***F-$\"10iw.wJ,]F-7$$!1++++0\"*H\"*F-$\"1R**)*y)G3J&F-7$$ !1++++83&H)F-$\"1go2G1@FcF-7$$!1LLL3k(p`(F-$\"1ppbV`\"3$fF-7$$!1nmmmj^ NmF-$\"1N([+*=A8jF-7$$!1ommm9'=(eF-$\"1[/Jb\"*RcmF-7$$!1,++v#\\N)\\F-$ \"1_G?*QN\"zqF-7$$!1pmmmCC(>%F-$\"1#)Ha*)\\nvuF-7$$!1*****\\FRXL$F-$\" 1$G:s$=MOzF-7$$!1+++D=/8DF-$\"16'RV'fO,%)F-7$$!1mmm;a*el\"F-$\"1e'H(p6 k:*)F-7$$!1pmm;Wn(o)!#<$\"14RcE!*e:%*F-7$$!1qLLL$eV(>!#=$\"1Bb&3:Cj)** F-7$$\"1Mmm;f`@')Fjr$\"1S=Ol;eh5F17$$\"1)****\\nZ)H;F-$\"1\\dVn5g>6F17 $$\"1lmm;$y*eCF-$\"1eml=0$e=\"F17$$\"1*******R^bJ$F-$\"1c!=74p$e7F17$$ \"1'*****\\5a`TF-$\"1&yf7(fiL8F17$$\"1(****\\7RV'\\F-$\"1cJ'G>A2T\"F17 $$\"1'*****\\@fkeF-$\"1p$>C(pb,:F17$$\"1JLLL&4Nn'F-$\"1?)HH:a\")e\"F17 $$\"1*******\\,s`(F-$\"1SA%p6Nho\"F17$$\"1lmm\"zM)>$)F-$\"16.X\"3J,y\" F17$$\"1*******pfa<*F-$\"1J/gL&**)))=F17$$\"1HLLeg`!)**F-$\"1'H'4WNI(* >F17$$\"1++]#G2A3\"F1$\"1$H?>(HF<@F17$$\"1LLL$)G[k6F1$\"1U@7j&G:C#F17$ $\"1++]7yh]7F1$\"1%\\>$oHVzBF17$$\"1nmm')fdL8F1$\"1*H]*efE?DF17$$\"1nm m,FT=9F1$\"1WN$)oE\"Hn#F17$$\"1LL$e#pa-:F1$\"1#y\"[C[ULGF17$$\"1+++Sv& )z:F1$\"1;G$=H.%*)HF17$$\"1LLLGUYo;F1$\"1phu#Hg(yJF17$$\"1nmm1^rZF^zG3c$F17$$\"1++]2%)38>F1$\"1NF@Q Y9mPF17$$\"\"#F*$\"\"%F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-F$6$7S7$F(Fiz 7$F/$\"1b%>\"Q:UlPF17$F5$\"1(***y+s`sNF17$F:$\"1Sx2krRnLF17$F?$\"12.3, EzsJF17$FD$\"1et'))H!G!*HF17$FI$\"1Ad*4[g/$GF17$FN$\"1&>a6$=(Rn#F17$FS $\"1NOh\"*fA@DF17$FX$\"1`sD^RlxBF17$Fgn$\"1&)zO()y\\QAF17$F\\o$\"1@XR& 4\"pA@F17$Fao$\"1.LM(4t%**>F17$Ffo$\"1\"Q!)yCXH)=F17$F[p$\"1pvg%Qzqx\" F17$F`p$\"1k%)y4)3ho\"F17$Fep$\"1Y-2us(Re\"F17$Fjp$\"1W(GNv8B]\"F17$F_ q$\"17Ar&)=g79F17$Fdq$\"1)\\,4&=nP8F17$Fiq$\"1(fv/PE+E\"F17$F^r$\"1>6! Hj#G!>\"F17$Fcr$\"1Ts*RKC;7\"F17$Fhr$\"1'R.aWo?1\"F17$F^s$\"1FUwd%p8+ \"F17$Fds$\"1xr!>Z1*>%*F-7$Fis$\"1!4Jx#GvJ*)F-7$F^t$\"1)Qe/G3HV)F-7$Fc t$\"1:(*fWSzYzF-7$Fht$\"1V`'GFa$)\\(F-7$F]u$\"1z\\.jyc)3(F-7$Fbu$\"1*) \\]kRvfmF-7$Fgu$\"1eyb$)zh'H'F-7$F\\v$\"16or5GsIfF-7$Fav$\"15meVQco-UF-7$F_x$\"1n(Qe?Ny'RF-7$Fdx$\"1;!*R%3P7u$ F-7$Fix$\"15L<+yHHNF-7$F^y$\"1OP\"**=\\^M$F-7$Fcy$\"1sO)R\\!)e9$F-7$Fh y$\"1?Do'3Ix(HF-7$F]z$\"1=iEi\\L3GF-7$Fbz$\"1e]+]_BbEF-7$FgzF+-F\\[l6& F^[lF*F_[lF*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "x" "6#% \"xG" }{TEXT -1 6 " > 0, " }{XPPEDIT 18 0 "2^x" "6#)\"\"#%\"xG" } {TEXT -1 16 " is larger than " }{XPPEDIT 18 0 "(1/2)^x" "6#)*&\"\"\"F% \"\"#!\"\"%\"xG" }{TEXT -1 6 "; for " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 52 " < 0, the reverse is true. The symmetry across the " } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 36 "-axis is explained by the fa ct that " }{XPPEDIT 18 0 "2^x = (1/2)^(-x)" "6#/)\"\"#%\"xG)*&\"\"\"F) F%!\"\",$F&F*" }{TEXT -1 76 ". That is, one of these functions is a h orizontal reflection of the other. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 4 "7-8." }{TEXT -1 122 " We're given a function whose graph is a \+ semicircle, and asked to translate it and scale it into two different \+ positions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := x -> sqr t(4*x - x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(-%%sqrtG6#,&9$\"\"%*$)F0\"\"#\"\"\"!\"\"F(F(F(" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "For the first given graph, we ne ed to compress the function horizontally by a factor of two, expand it vertically by a factor of two, translate it to the left one unit, and translate it up one unit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g1 := x -> 1 + 2*f(2*(x+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #g1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*&\"\"#F--%\"fG6#,&9$ F/F/F-F-F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "For the second given graph, we ne ed to expand vertically and horizontally by a factor of 3/2, reflect a cross the " }{XPPEDIT 256 1 "x" "6#%\"xG" }{TEXT -1 43 "-axis, and tra nslate to the right 2 units. 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It is shifted up 3/2 units and back (to the left) 2 units." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "g1 := x -> (3/2) + (3/2)* f((2*Pi/8)*(x+2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1Gf*6#%\"xG 6\"6$%)operatorG%&arrowGF(,&#\"\"$\"\"#\"\"\"*&F-F0-%\"fG6#,$*&%#PiGF0 ,&9$F0F/F0F0#F0\"\"%F0F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The second given curve has amplitude 2 and period 4, and is not shift ed vertically or horizontally." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g2 := x -> 2 * f((2*Pi/4)*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%\"fG6#,$*&%#PiG\"\"\"9 $F3#F3\"\"#F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot ([g1(x), g2(x)], x=-4..10, color=[green, blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7[q7$$!\"%\"\"!$F*F*7 $$!3vmmTg*4P#R!#<$\"39`d%fnC=p#!#?7$$!3]LL$3#*>u%QF/$\"3SJ,qZPwv5!#>7$ $!3E++D\"))H6x$F/$\"3S[FzR*[oT#F87$$!3cmmmT)R[p$F/$\"3Q#**)ebZi(G%F87$ $!3'***\\iI23iNF/$\"33n20?#Q]y)F87$$!3PLLe>;KHMF/$\"3Yr.&=nM;[\"!#=7$$ !3ymm;4'=28$F/$\"3GP;VE2AiLFM7$$!3)pmmmVF/)HF/$\"3S4jaGCpdXFM7$$!3vmm; ki8IGF/$\"3s(yDNk)[)*eFM7$$!3')**\\P*Rf0o#F/$\"3iK9qzpqetFM7$$!3+LLeMD )4`#F/$\"3xVjV,6EC*)FM7$$!3#****\\Pj0BR#F/$\"3-EZ#H(R5X5F/7$$!3)om;HtG OD#F/$\"3B()ez`G<.7F/7$$!38LLeMo.5@F/$\"3-1@[=s_q8F/7$$!3%)***\\i$\\Wm >F/$\"3;\\XM_l_R:F/7$$!34LLe*)>%z\"=F/$\"3i)Go\">>v87@7F/$\"3A>#zE4E9O#F/7$$!3%HLL$[$e)o5F/$\" 3.\\v08cx,DF/7$$!3o'***\\7WWZ$*FM$\"3nWCFM$\"3W@=RF#eB)HF/7$$!3G)******>6UA \"FM$\"3qAxm?=2$*HF/7$$!3hy*****\\ct$\\F8$\"3Gl\"[nMs))*HF/7$$\"3cD+++ !*RnBF8$\"3/gxb=2u**HF/7$$\"3uH+++X:s'*F8$\"33\\DlySn&*HF/7$$\"3+pmT5! 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