0002000201000feActProject.ACT0002020016Polynomials-Yang07.EAC010000001997 y=2x^(3)+3x^(2)-2x-2  `E(0   @v PQr ^A 'L'pC uxy qq@! u 2x^3+3 2+(-2)+-29  y=x^(3)-2x+1      `F(x  @v$Qr ^A 'L'pC uxy qq @! ux^3+(-2) +1, Polynomials-Yang07.EACeActProject.ACT.2 U `E '[4n functions` Ob{ive!To explore cubic and > other higord Ypu.n1. AWer#\+ Er· ]SChXe coefficientto see a variety ofgraph shapes.)'Add further terms to[explore higorder polynomial2 han cubics.<D Close _Ugraph window from s menu when finishedU2. Co shapes Let's look mc|lyat s. Wellexaminwith n quadratic&. Tapbel to start. \ Έ +Leav+L andconRntfixed change only the linear [term to get differentshapes of graphs.6][ CKQ1. WhaH sortskGQarere?DNow constant , leavingo r two fixed:Q2is econ 9$?Closۍ1 windowlfrol menu when finishedd 3. Special types of polynoms\&9P that can be factored=>Cyou find a pattern?Јgkt Nd%FinaForm$NGraph2D 3, LISTSYS8@4< Modify lP<STATCALC d< \x S:equence,xSheetO | olveEq`wr(Up(tupFLG14(<Lis{\DPicViewWind_osvev xy^(<H2x(T,(hP p+  lMĒВܒ܊ь  $!,0 "@< #THFkT%|` &l !'x(  )~*+,-.̆؆1234 5E F҆H8ID JP Kh\ Lh Mt N.OƆ PQRS!Ȇ]q Ԇ} ^_܆=ab쒌͑ Α Б, ב@(ҋW4ّh@ ڑ|L 6ۑX FinancialFormat  ! system]"^_` a bxR @0` `à6 @x !% a. MatDatab:.EACAF    C<^    <0 Band&<0.hpRedo the problems ifKa)-b(223c:}4. Curitting Polynomial c s maygigood apxima2sCto& me data. Twopoints determine a unique~lAthree 0a4unique parabola. In the[same way, four points determine a Hcubic.>FThe List window showFan example, with q, (1,3), (2,8),(3,23) and (4,54). TapƐpto seeYdata\FrveԈ!%S Graph2D̆ 3 LISTSYS4NModify; ($< STATCALC U(< L\< Sequence<,xSheetԆ|PolveEqLwrІ (Upܒ tupFLG1<(<Lis{dDPicT ViewWindx,yJE(HFȐ||x   (4@F{LXdp |!"#$%&Ē'В"kܒ)*+,  -.$0 1<2H3T4`5lExFnIJKLM̒NؒOPQRS T ],-D ^p_`abdĒdeLXZȌ|F,͑@ΑTБ|! systemlistem^]=syst_`a b~ @@ю @@x @@x @ a seq_histb NewFolde   system]l0^]= 188<Bs[ exactly (as r2=1). Then tap graph iconne curveted to the data.[Q. Give a formul%6cubicr%thatAfitsWsYChang3y seeor\sAn%way=p polynomialYa[t ofpoinio usequence command witha lisK2fir fewx values. NotiChow[U<belmatche": (1,3), (2,8),(3,23) (4,54)R sequence({3,8,23,54},k)[Press EXE to fit the$C , using k as A( variable this time.` Check thaR formula~ works by tryZIous values in expionN y1 below, ngeach:y1(3 Nows se sorts of0proc*es yourself withive poin5instead=fourz7Q. What sort of curve[is needed for five points?"*eAct02000fPolynomials.EAC010000009daby=2x^(3)+3x^(2)-2x-2`E(0`E(0vrALCuxy@! 2x^3+3x^2-2x-2 y=x^(3)-2x+1    ``vrALCuxy @! x^3-2x+1  <$#[Polynomial functions[\AuthorBarry Kissane School of Education Murdoch University Murdoch, WA, Australia kissane@murdoch.edu.au http//wwwstaff.murdoch. edu.au/~kissane[[ Objective[To explore cubic and [other higher order [polynomial functions.[[1. A cubic explorer[Cubic functions have a [3rd power of the [variable. A good way to[explore them involves[studying their graphs.[Tap the Cubic Explorer[window to start.[\Polynomial Explorer[The graph shows a[cubic function. You can[edit the function and tap[EXE to see how this [affects the graph shape[ and position:[][[Change the coefficients[to see a variety of[graph shapes. [[Add further terms to[explore higher order[polynomials than cubics.[[ Close the[graph window from the [ menu when finished.[[2. Cubic shapes[Let's look more closely[at cubic graphs. We will[examine those with no[quadratic term. Tap the[window below to start. [ \ Cubic shapes [Leave the cubic and[constant terms fixed and[change only the linear [term to get different[shapes of graphs.[] [[Q1. What different sorts[of shapes are there?[[Now change only the[constant term, leaving[the other two fixed:[[Q2. What is the effect[on the graph of changes[to the constant term?[[Close the graph window[from the menu when[ finished.[[3. A cubic equation[There are several ways[of solving a cubic [equation. Consider for[example the equation[x3=5x+2[Open the numerical[solver window to get an[approximate numerical[result.\Numerical solverGraph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr< SolveUprH StupFLG1T(StupList|D StupPict$ ViewWind    $ 0 < H T ` l x        ! " # $ % & ' ( ), *8 +D ,P -\ .h 0t 1 2 3 4 5 E F H I J K L M N O( P4 Q@ RL SX Td ]p ^t _x `| a b      system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3Dx^(3)=5x+2c \i \<$ $ `P`P`p`p `(10qy`#YuY ` O[Highlight and then drag[the equation above into[the space under Equation[below. Press EXE.[Enter a starting value[for x and tap Solve in[ the toolbar.[[Try starting with positive[and with negative values[for x.[[Q3. How many different[solutions can you find?[[4.Graphical solution 1[It's a good idea to draw[a graph, so that you[can see how many[solutions there are to an[equation. There are[often several ways of[ doing this.[Consider again x3=5x+2.[[One way is to graph the[ function f(x)=x3-5x-2 [and find the roots. Drag[the right side of the[function to the Graph 1[window, select the box[and tap the graph icon. [\Graph 1 Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $ `P`P`p`p `(10qy`#YuY `` O[To find approximations[to the roots of the[function, you can use[the menu command:[Analysis>G-Solve>Root[and the cursor. [Use the[Resize comand at the [bottom of the screen to[show more screen.] [[Q4. Compare your[solutions with the earlier[numerical solutions. What[has the graph shown[that you did not know[ previously?[[4.Graphical solution 2[Another possibility is to[draw two functions and[find the intersections of[their graphs. To do this[drag each of the two[ sides of the equation[x3=5x+2 into the [function list in Graph 2,[select the two boxes[and tap the graph icon. [\Graph 2Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $ `P`P``aa `(10qy`#YuY ABV#s@qx`&1WG7 O[To find approximations[to the points of[intersection, you can use[the Analysis menu:[G-Solve>Intersect[and the cursor.[[Q5. What are the[solutions to the [equation? How are these[different from what the[G-Solve command gives?[How do the solutions[compare with earlier[ results? [[4.Graphical solution 3[Another way to think of[x3=5x+2 is to regard it[as x3-5x=2. So you can[graph the function[f(x)=x3-5x and see if it[has a value of 2 for any[ values of x.[[Highlight and then drag[the right side of the[function to the Graph 3[function list and draw [the graph. Use the [Analysis menu to trace [the graph to see if y=2[ anywhere. [\Graph 3Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $ `P`P`p`p `(10qy`#YuY `` O[To find approximations[to the required points, [you can use the Analysis[menu: G-Solve>x-Cal[and enter y=2. Use the [cursor to locate all the[values.[[Q6. What solutions can[be found in this way? [How do they compare[with earlier results?[[5. Exact solutions[So far, solutions have[been numerical[aproximations. Some [higher order polynomial [equations can be solved[exactly, but most [can not. Highlight and [then drag the [ equation x3 =5x+2 into[the parentheses of the[solve command in the[window below and press[EXE to solve. [\ Solve commandRsolve()Graph2D, Graph3D@ LISTSYSL4NModify $ STATCALC NSTATSYS \NSequence, Sheet4| Sheet3D| SolveEq,SolveLwr0 SolveUpr< StupFLG1H(StupListpD StupPict$ ViewWind     $ 0 < H T ` l x       ! " # $ % & ' ( ) *, +8 ,D -P .\ 0h 1t 2 3 4 5 E F H I J K L M N O P( Q4 R@ SL TX ]d ^h _l `p at bx |     system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $ `P`P`p`p `(10qy`#YuY ` O[[Q7. How do the solutions[compare with the earlier[numerical solutions?[[Q8. Change the constant[(i.e. 2) in the equation[above. How often are [equations solved only[approximately with this [ command? [[6. Curve fitting[Polynomial curves may [give good approximations[to some data. Two [points determine a unique[line and three points a[unique parabola. In the[same way, four points[determine a unique cubic.[[The List window shows[an example, with the [points, (1,3), (2,8),[(3,23) and (4,54). Tap[the window to see the [data.[\Four points curveGraph2D Graph3D LISTSYS4NModify $ STATCALC DNSTATSYS L\NSequence, Sheet| Sheet3DP| SolveEqSolveLwr SolveUpr StupFLG1(StupListD StupPictT$ ViewWindx y1(H         ( 4 @ L X d p  | ! " # $ % & ' ( ) * + , - .$ 00 1< 2H 3T 4` 5l Ex F H I J K L M N O P Q R S T ],D ^pD _ ` a b dD eD L X d p |      system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde  system]listsystem^]=system1system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i <$ $  3x1*x^3+-1*x^2+1*x+2`P`P` 03vb3vb  `B(10qy`#YuY @ $ $0@ $ $0@`` 9G6! W[ [Use the Calc menu to [select Cubic Reg to fit a[cubic model to the data.[You will see that it fits[ exactly (as r2=1). [[Then tap the graph icon[to see the curve fitted[ to the data.[[Q9. Give a formula for[the cubic curve that[fits these data.[[Change the data to see[other cubic curves.[[Another way to fit a [polynomial to a set of[points is to use the [sequence command with[a list of the first few [values. Notice how the[list below matches the[points: (1,3), (2,8),[(3,23) and (4,54).[Rsequence({3,8,23,54},k)[[Press EXE to fit the[sequence, using k as [the variable this time.[Check that the formula[works by trying various[values in the expression[for y1 below, pressing[EXE each time:[Ry1(3)[[Now try these sorts of[processes yourself with[five points instead of[four.[[Q10. What sort of curve[is needed for five points?[\ Solutions(Q1.There are 2 shapes; one has two 'loops', the other doesn't. There are two loops when the linear term is negative. Q2. The constant term affects the vertical position of the graph. Q3. You will find two solutions. If you start with positive x, you will get x=2.4142... . Starting with (most) negative values will give x=-2. Q4. The graph shows the function has three roots, and so the equation has three solutions. Two are the same as before, and there is now another at x=-0.4142... Q5. The solutions are x=2.4142... x=-0.4142... x=-2 They are the x-values of the points of intersection: G-Solve gives (x,y) values. These are the same as the other graphical solutions. Q6. The same three solutions are found. Only the x-values are used, as before. Q7. The three solutions given are all exact. The previous solutions were approximations. Q8. Most equations like this are not solved exactly by this command. Q9. The coefficients are 1, -1, 1 and 2, so a formula is y=x^3-x^2+x+2 Q10.Five points determinea unique quartic curve. [eAct010008main.ACT0001020012eActivity Save.EAC010000001997 y=2x^(3)+3x^(2)-2x-2  `E(0   @v PQr ^A 'L'pC uxy qq@! u 2x^3+3 2+(-2)+-29  y=x^(3)-2x+1      `F(x  @v$Qr ^A 'L'pC uxy qq @! ux^3+(-2) +1, Polynomials-Yang07.EACeActProject.ACT.2 U `E '[4n functions` Ob{ive!To explore cubic and > other higord Ypu.n1. AWer#\+ Er· ]SChXe coefficientto see a variety ofgraph shapes.)'Add further terms to[explore higorder polynomial2 han cubics.<D Close _Ugraph window from s menu when finishedU2. Co shapes Let's look mc|lyat s. Wellexaminwith n quadratic&. Tapbel to start. \ Έ +Leav+L andconRntfixed change only the linear [term to get differentshapes of graphs.6][ CKQ1. WhaH sortskGQarere?DNow constant , leavingo r two fixed:Q2is econ 9$?Closۍ1 windowlfrol menu when finishedd 3. Special types of polynoms\&9P that can be factored=>Cyou find a pattern?Јgkt Nd%FinaForm$NGraph2D 3, LISTSYS8@4< Modify lP<STATCALC d< \x S:equence,xSheetO | olveEq`wr(Up(tupFLG14(<Lis{\DPicViewWind_osvev xy^(<H2x(T,(hP p+  lMĒВܒ܊ь  $!,0 "@< #THFkT%|` &l !'x(  )~*+,-.̆؆1234 5E F҆H8ID JP Kh\ Lh Mt N.OƆ PQRS!Ȇ]q Ԇ} ^_܆=ab쒌͑ Α Б, ב@(ҋW4ّh@ ڑ|L 6ۑX FinancialFormat  ! system]"^_` a bxR @0` `à6 @x !% a. MatDatab:.EACAF    C<^    <0 Band&<0.hpRedo the problems ifKa)-b(223c:}4. Curitting Polynomial c s maygigood apxima2sCto& me data. Twopoints determine a unique~lAthree 0a4unique parabola. In the[same way, four points determine a Hcubic.>FThe List window showFan example, with q, (1,3), (2,8),(3,23) and (4,54). 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