(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 11688, 326]*) (*NotebookOutlinePosition[ 12743, 362]*) (* CellTagsIndexPosition[ 12699, 358]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData["Experimenting with Lines"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Initialization"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True, FontSize->10], Cell[TextData["Needs[\"Graphics`Colors`\"];"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Building a Line."], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The following will explain to students how to form a line by plotting \ points."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Let's make a table of eleven points by finding y when x goes from 0 to 10."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[f,x]\nf[x_]=2x;\ncoord1=Table[{x,f[x]},{x,0,10}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Now we will graph these coordinate pairs to see what they look like."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "grcoord1=ListPlot[coord1,\n\t\t\t\ PlotStyle->{PointSize[.02],RGBColor[0,1,0]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Now lets find more points by solving for y as x goes from 0 to 10, but this \ time by going up by 0.5. This will give us 20 points from 0 to 10."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["coord2=Table[{x,f[x]},{x,0,10,0.5}]"], "Input", AspectRatioFixed->True], Cell[TextData["Again, we will graph to see what is happening."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "grcoord2=ListPlot[coord2,\n\t\t\t\ PlotStyle->{PointSize[.008],RGBColor[1,0,0]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "We can now combine the two graphs and see if they are related."], "SmallText",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[grcoord1,grcoord2];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Now let's make even more points from 0 to 10!! First look at the table, \ then look at the graph."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["coord3=Table[{x,f[x]},{x,0,10,0.25}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "grcoord3=ListPlot[coord3,\n\t\t\t\ PlotStyle->{PointSize[.006],RGBColor[0,0,1]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Again, we want to look at all three graphs together."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[grcoord1,grcoord2,grcoord3];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Lastly, we want to look at more points. We will make a table of coordinates \ as x goes from -40 to 40. First look at the table then look at the graph."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["coord4=Table[{x,f[x]},{x,-40,40}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "grcoord4=ListPlot[coord4,\n\t\t\t\ PlotStyle->{PointSize[.006],RGBColor[0,0,1]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Finally, we want to combine all of the graphs together. Can we make any \ predictions based on the graphs we have seen?"], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[grcoord4,grcoord1,grcoord2,grcoord3];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Exploring Slope"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We want to start with a simple line equation y=x. Let's see what it looks \ like."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[f,x]\nf[x_]=x;\nline1=Show[Plot[f[x],{x,-10,10},\n\t \ PlotStyle->Green,\n\t PlotRange->{{-10,10},{-10,10}},\n\t \ AxesLabel->{\"x\",\"y\"},\n\t AspectRatio->Automatic,\n \t\ \tTicks->{{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,\n \t\t \t\t \ 0,1,2,3,4,5,6,7,8,9,10},\n \t \t\t{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,\n\ \t\t \t\t 0,1,2,3,4,5,6,7,8,9,10}},\n\t\t \ DisplayFunction->Identity],\n\t\t Graphics[{Green,Text[\"y = x\",{11,10.5}, \ {1,0}]}],\n\t DisplayFunction->$DisplayFunction]; \n "], "Input", AspectRatioFixed->True], Cell[TextData["Now see what happens when we double x. (y = 2x)"], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "f[x_]=2x;\nline2=Show[Plot[f[x],{x,-10,10},\n\t \ PlotStyle->VioletRed,\n\t PlotRange->{{-10,10},{-10,10}},\n\t \ AxesLabel->{\"x\",\"y\"},\n\t AspectRatio->Automatic,\n \ \t\tTicks->{{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,\n \t\t \t\t \ 0,1,2,3,4,5,6,7,8,9,10},\n \t \t\t{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,\n\ \t\t \t\t 0,1,2,3,4,5,6,7,8,9,10}},\n\t\t \ DisplayFunction->Identity],\n\t\t Graphics[{VioletRed,Text[\"y = \ 2x\",{7,10.5}, {1,0}]}],\n\t DisplayFunction->$DisplayFunction]; \n "], "Input", AspectRatioFixed->True], Cell[TextData["Let's look at both graphs together."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[line1,line2];\n"], "Input", AspectRatioFixed->True], Cell[TextData["Now, double the x value again. (y = 4x)"], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "f[x_]=4x;\nline3=Show[Plot[f[x],{x,-10,10},\n\t PlotStyle->Blue,\n\ \t PlotRange->{{-10,10},{-10,10}},\n\t \ AxesLabel->{\"x\",\"y\"},\n\t AspectRatio->Automatic,\n \t\t\ Ticks->{{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,\n \t\t \t\t \ 0,1,2,3,4,5,6,7,8,9,10},\n \t \t\t{-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,\n\ \t\t \t\t 0,1,2,3,4,5,6,7,8,9,10}},\n\t\t \ DisplayFunction->Identity],\n\t\t Graphics[{Blue,Text[\"y = \ 4x\",{3.5,10.5}, {1,0}]}],\n\t DisplayFunction->$DisplayFunction]; \n \n"], "Input", AspectRatioFixed->True], Cell[TextData[ "We will now look at all three graphs. Can you make any general comments \ about why the graphs are different by looking at the graphs and the \ equations?"], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[line1,line2,line3];\n\n"], "Input", AspectRatioFixed->True], Cell[TextData[ "Now we can look at many lines going through the origin. We will start with \ y = 0.5 x, and increase the slope in increments of 0.5 to see what happens."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[f,x,c]\nf[x_,c_]:= c x\nDo[Show[Plot[f[x,c],{x,-5,5}, \n \ PlotRange->{{-6,6},{-6,6}},\n DisplayFunction->Identity],\n \ Graphics[{Text[\"Slope = \",{-1,5}, {1,0}],\n \ Text[c,{-1,5},{-1,0}]}],\n DisplayFunction->$DisplayFunction], \n \ {c,0.5,4,0.5}];\n "], "Input", AspectRatioFixed->True], Cell[TextData["Select the cell and annimate these graphs."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Lastly, we will use the same line but this time give it a negative slope. \ We will decrease the slope in increments of -0.5."], "SmallText", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[f,x,c]\nf[x_,c_]:= c x\nDo[Show[Plot[f[x,c],{x,-5,5},\n\t \ PlotStyle->Magenta, \n PlotRange->{{-6,6},{-6,6}},\n \ DisplayFunction->Identity],\n Graphics[{Text[\"Slope = \",{2,5}, \ {1,0}],\n Text[c,{2,5},{-1,0}]}],\n \ DisplayFunction->$DisplayFunction], \n {c,-0.5,-4,-0.5}];\n "], "Input",\ AspectRatioFixed->True], Cell[TextData["Again, select the cells and annimate these graphs."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Experimenting with y-intercepts."], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "We will be looking at how changing the constant term (y-intercept) in an \ equation will change the graph. 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