0002000201000feActProject.ACT0002020013ExponentialFunc.EAC010000003be4<$[Exponential Functions[\AuthorBarry Kissane School of Education Murdoch University Murdoch, WA, Australia kissane@murdoch.edu.au http//wwwstaff.murdoch. edu.au/~kissane [[ 1. Objective:[To explore exponential[functions and their[graphs.[[2. Exploration 1[An exponential function[has a variable exponent.[Here is an example:[y=1.2x[[Tap the geometry window[below. Highlight the[function above and then[remove the stylus. [Drag the function to the[geometry window to[ graph it.[\Exploring graphs -%dQa&`a)"WP`BXE gt#vrALCuxy[[You can compare the[graphs of many functions[by dragging them to the[ same screen.[[Change the base of the[exponential function[y=5x[and drag each new[function to the graph[window. For now, choose[ bases > 1.[\base >1DGraph2Dh Graph3D| LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind y10Hy2HHy3`Hx               ,  8 !D "P #\ $h %t & ' ( ) * + , - . 0 1 2 3 4 5( E4 F@ HL IX Jd Kp L| M N O P Q R S T ] ^ _ ` a b    $ 0 system]listsystem^]=system_^system`_systema`systemb~.......................................................................................................................................................................................... aseq_histbNewFolde system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $  3x2^x 3x3^x 3x7^x`P`P`p`p `(10qy`#YuY ` R[Q1. Through which point[do all the graphs pass?[[Q2. What happens to [the graph as the base[ increases?[[Close the geometry[window from the [menu when finished.[[3. Exploration 2[Another way way to see[graphs of exponential[functions is to use a [geometry link. Highlight[and then drag this link[to the geometry window[below.[]=2^[[Edit the function in the[link and press EXE.[Notice the graph[changes accordingly.[\Geometry window33333)`SE0`fffffe t'g0vrALCuxy[ [Q3. Which function [contains the point (3,8)?[[Close the geometry[window from the [menu when finished.[[4. Exploration 3[The previous graphs have[increased rapidly, since[the bases have been[greater than 1. What[happens with positive[bases less than one?[[Edit the expression below[(change the base or the[exponent and press EXE)[to see some examples:[R5^-2R0.04R[Keep the base positive,[but try both positive and[negative exponents.[[To explore graphs when[the base is between 0[and 1, drag the link[below to the geometry[window[ ] =0.4^[[Q4. What is the effect[on graphs of changes[to the base (0 1, graph curves downwards, like exponential graphs before with 0 < base < 1. For 0 < base < 1, however, graphs of negative exponentials look like graphs of exponentials with base > 1. [eAct02000bexp/log.EAC010000002c0c<$[Exponential and [Logarithmic Functions[RDefine f(x)=a*^(x)RdoneR\compare graphs Graph2DT Graph3Dh LISTSYSt4NModify $ STATCALC NSTATSYS \NSequence0, Sheet\| Sheet3D| SolveEqTSolveLwrX SolveUprd StupFLG1p(StupListD StupPict$ ViewWind y1Hy24HP \ h t               ! "( #4 $@ %L &X 'd (p )| * + , - . 0 1 2 3 4 5 E F H$ I0 J< KH LT M` Nl Ox P Q R S T ] ^ _ ` a b      system]listsystem^]=system_^system`_systema`systemb~.......................................................................................................................................................................................... aseq_histbNewFolde system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $  3x^x  3x2*^x`P`P`p`p `(10qy`#YuY ` Q[Question 1: Are these [graphs related by [ shiftings???[Question 2: What happens[if a gets larger or [smaller?[[Let's find the inverse[for y=a*^(x)Rsolve(y=a*^(x),x)Rx=lnya[ So, if we Rdefine g(x)=ln(x/a)Rdone[then f and g will be [inverse of each other.[ Let's checkRf(g(x))RxRg(f(x))R1xR\f and gHGraph2Dh Graph3D| LISTSYS4NModify $ STATCALC NSTATSYS \NSequenceD, Sheetp| Sheet3D| SolveEqhSolveLwrl SolveUprx StupFLG1(StupListD StupPict$ ViewWind y10Hy2LHy3hH|              $ 0  < !H "T #` $l %x & ' ( ) * + , - . 0 1 2 3 4 5, E8 FD HP I\ Jh Kt L M N O P Q R S T ] ^ _ ` a b    ( 4 system]listsystem^]=system_^system`_systema`systemb~.......................................................................................................................................................................................... a(x/a)b;1 system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.system]listsystem^]=system.  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $   3xa*^x  3x_(x/a) 3xx`P`P`p`p `(10qy`#YuY ` R[eActa b fH xa*^(x)gH x_(x/a)010008main.ACT0001020012eActivity Save.EAC010000001bf1 ExponentialFunc.EACeActProject.ACT+  Q <$[(^ _tions \% AuthorFapG^Barry Kissane School of Educa\ Murdoch University, WA, Australia kF@m2.edu.au http//wwwstaff. /~7 [1. Ob8:  To explorerf and theirP graphs. i,2. Exploration 1 An exponential func" has a variable,. @Here i$H ample: _y=1.2mxu=Tap the geometry window below. Highlt- above andKŽrem`stylus. Dragy to to graph i \Siing #s -Έ$(y %dQa& `a)"WP BXE gt# v !&r  A 'L'<C uxy qv [ You can compare the $graphs of many functions F by dragging9m toA same screen.q^Changp baseg; exponential{zy=5RxZand each newĎ window. For now, choose  bases > 1. \ >1ЈʊUDS Graph2Dh 3| LISTSYS4NModify; ($< STATCALC U(< \< Sequence},xSheetp|olveEqLwrl (Upx tupFLG1<(<Lis{dDPic ViewWind,yJ0EH23`(x   ̒ؒ ,8!D"P#,\$@h%Tt&h'|()*  +,Ȓ-Ԓ.012345(E2F@HLIXJ,dK@pFW|MhN|OPQ RĒSВTܒ]U ^_`ab̎ ͐ΐ$0! systemlist]=~.Ž.ˆ 8p0p.† @@x. @@x. @ a seq_histb NewFolde system]l0^]=  asKbase increaseWVClose~eometryz1window from the menu when finished. &'3. Exploration 2 !AnoPr wayy to seegraphs of exponential{funcOs is9use a ~=geometry link. Highltandn dragis*Gbelow.] =2^[EditB inRO press EXEOkNoticeEchanges accordingly\Geometry window  33333) `SE0 fffffeT  t'g0DWv uQri^A 'L'C uxy qv [ Q3. Which function %contains the point (3,8)?"FClose'gR k[ fromI|menu when finished.Z 4. Explora3-The previous graphs have increased idly, se tCb 9 been <greater than 1. Whathappens with positiyvRless=one?aEdit^e ex ssion below(change$ o8[nt and pEXE) to see somdamp:R!' 5^-2R70.04 [RKeep,bury boWnegative exponents.  Tolore graphs when (the base is between 0= and 1, drag )linkb below togeometrywindow ] =0.4^[Q4. Whateffecton of changes(0<<1)? \G Έ#; L!86Gy `'g)TP W5! SE DvQr ^AL  C uxy IV [`\ base<1Ј ʊ?HS Graph2Dh 3| LISTSYS4NModify; ($< STATCALC U(< \< Sequence},xSheetp|olveEqLwrl (Upx tupFLG1<(<Lis{dDPic ViewWind,yJ0EH23`(hhx   ĒВܒ n$0 <!H"T#,`nCl%Tx&h'|()*  +,̒-ؒ.01234 5E8FDHPI\J,hK@tLTMhN|OQ RȒSԒT]U ^_`ab͐ΐ(Б4]a! systemlist]=~.Ž.ˆ 8p0p.† @@x. @@x. @ a seq_histb NewFolde system]l0^]= Close the geometry  window from! 5menu when finished.6 5. Exploration 4=What happens with neg- ve exponents?X*Edit the expression below (change&base or2 9 onent and D EXE) ? to see someamples:=Rbh 4^-2Rx0.0625"[KeeppositivՊ nega).Drag link to$getry wind ?lorf## ials. Experim with differJ values forF }:] y=3^(-x)[\Geometry windowΈ )F:GyB`W#'@@ 5"%x` acP Dvv{Qr^A'L'C uxy  v [ Q5. Describe the graphs (of exponential functions@with negative,s.aJClosbt } frommenu when finished.   Return to ClassPad 300MC ithout savingD\L Answers eGv'Q1. (0,1), as raising a number to pow 'zero gives 1. Q2. Graphs become steep3Gthe base increases23. y=2^xi8 3 4. A smallm3 leads xa RghB5. 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