0002000101000feActProject.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)0301c7002000080000000009d4xKl\;yAAhu!28 *Q$.b%{XrhR$H] ,` $- $ǂ]E] x>/DV~ҙsd.,-fY:Y67{jx!Ճۋβn}Unv7뱣Z_[{JݥzO]KuT7Jui|5Kk,Y_4fi|5Jk(Q_4Fi|5Jkwo-KSKuT7Qj>ۚ?6ËsgҖWö쇝lÓEv9)_=}5[ÎbɩS)JUY5{l#}~=?βkK+Ꝿg}5=<Yg,E0x{|c޴=co#[^9whph୔6fcKsrtʮ;'IYj-Eb]ҙwgv_>л2?.._F^mű7DVd7FNd_쯑m˶m"bdDvKdȾȾپȾXd};F6٭ȞlGd?k=푽#{)ZdD3"FdoFvGdٻo: E5Fd#kF#Ȇ"}Hd#MEv(l.Ⱦٷ"{:#{6~xdl"_Gv4W#;=ȎGG6{ME~dӷlS߈D5"dd"kE=Ss X~Y=E7Gv07#;ۑ=DȖ#Sd"sdlC:6f"\J<+tϷOv[,S}{mgϞ={?}l->N[[˭sdks*o ݼMjWcj r-@j /c΍:uTݥzON{>kgnLo^_ݥzO'a^*JX׻,e#x\=SoY߿Wf~Uot|Υv)gS{1S{57S{'6cL-ԿWRJ_I+ُ}Om}نn:PY[|kY\}a㥾_m5~0t"|?Zy~OgOjOʕ^Jj-?v{~rso@vZsӫŋ}0˞KwU_,G'G&3o4+ӭb O++>{b.L'+懻Y1-S٦b.n֙3N^:s?>]щuZ=ޚY8;x5ҮR )nX?~v;o̓Zgig=b~G=z>\w=ƮӍZw=- 7|0Y=d$1uVΞm-^֖ kb{j~zf֛_x;m139mz1ͦZz^꫓ugGWGzYWj~CH=?^'T=?PwMLǼ>ؙ٥tvkO_=U?48>P#;cxkv~H}gɩwNj-}xljdZ텳ᑣugON<=zS t׍zRj>)~o5շQg W|=YX9^VղSv~qvZYOZ< GG/kV斫t=kˋ 3kʦJeÆJ'_RK04b9