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.EAC01000000101d  exp/log.EACeActProject.ACT#  I <$[(Exponential and Logarithmic Functions 6RXiDefine f(x)=a*^ R d`0\8compare graphs Ј#S Graph2DT 3h LISTSYSt4NModify; ($< STATCALC ̆U(< \< Sequence0<,xShee]\|ؒolveEqLwrX (Upd tupFLG1p<(<LisdDPicܐViewWinL0,yJEH24PG||x   ȒԒ !n"(#4$,@%@L&TX'hdnp)|*+  ,-.0Ē1В2ܒ345E FH$IJ<K,FL@TMT`NhlO|xPQR ST]A ^Ē_Ȓ`̒aВbԒؒ͒Βn! systemlist]=~.Ž . @@x.. @@x.Ž @@x.Š @ an seq_histb NewFolde3 system]l0^]= Question 1: Are these graphs related by 9 shiftings???/M2: What happens oif a gets larger orm smaller?cl Let's find inversefD y=a*^(x)R#solve(y=a*^(x),x)Rx=lny$a[- So, if we R=C define g`=ln(x/aedone[othen f and g will be inverse of each o/r. 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