(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; currentKernel; ] :[font = title; inactive; preserveAspect; fontName = "CalcMath"; startGroup] Parabolas :[font = subsubtitle; inactive; preserveAspect; fontName = "CalcMath"] By: Debra Woods 12/1993 :[font = smalltext; inactive; Cclosed; preserveAspect; startGroup] initialization :[font = input; initialization; preserveAspect; endGroup] *) Needs["Graphics`Colors`"]; (* :[font = section; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] The Basic Graph of a Parabola :[font = text; inactive; preserveAspect; fontName = "CalcMath"] The purpose of this lesson is to demonstrate the properties of the graphs of quadratic functions of one independent variable. We will begin with the basic graph of y = xÛ. :[font = input; preserveAspect] Clear[x]; :[font = input; preserveAspect; startGroup] Plot[x^2,{x,-4,4}, PlotStyle->{Red}, AspectRatio->1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.0238095 0.0595238 [ [(-4)] .02381 .02381 0 2 Msboxa [(-2)] .2619 .02381 0 2 Msboxa [(2)] .7381 .02381 0 2 Msboxa [(4)] .97619 .02381 0 2 Msboxa [(2.5)] .4875 .17262 1 0 Msboxa [(5)] .4875 .32143 1 0 Msboxa [(7.5)] .4875 .47024 1 0 Msboxa [(10)] .4875 .61905 1 0 Msboxa [(12.5)] .4875 .76786 1 0 Msboxa [(15)] .4875 .91667 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .02381 m .02381 .03006 L s P [(-4)] .02381 .02381 0 2 Mshowa p .002 w .2619 .02381 m .2619 .03006 L s P [(-2)] .2619 .02381 0 2 Mshowa p .002 w .7381 .02381 m .7381 .03006 L s P [(2)] .7381 .02381 0 2 Mshowa p .002 w .97619 .02381 m .97619 .03006 L s P [(4)] .97619 .02381 0 2 Mshowa p .001 w .07143 .02381 m .07143 .02756 L s P p .001 w .11905 .02381 m .11905 .02756 L s P p .001 w .16667 .02381 m .16667 .02756 L s P p .001 w .21429 .02381 m .21429 .02756 L s P p .001 w .30952 .02381 m .30952 .02756 L s P p .001 w .35714 .02381 m .35714 .02756 L s P p .001 w .40476 .02381 m .40476 .02756 L s P p .001 w .45238 .02381 m .45238 .02756 L s P p .001 w .54762 .02381 m .54762 .02756 L s P p .001 w .59524 .02381 m .59524 .02756 L s P p .001 w .64286 .02381 m .64286 .02756 L s P p .001 w .69048 .02381 m .69048 .02756 L s P p .001 w .78571 .02381 m .78571 .02756 L s P p .001 w .83333 .02381 m .83333 .02756 L s P p .001 w .88095 .02381 m .88095 .02756 L s P p .001 w .92857 .02381 m .92857 .02756 L s P p .002 w 0 .02381 m 1 .02381 L s P p .002 w .5 .17262 m .50625 .17262 L s P [(2.5)] .4875 .17262 1 0 Mshowa p .002 w .5 .32143 m .50625 .32143 L s P [(5)] .4875 .32143 1 0 Mshowa p .002 w .5 .47024 m .50625 .47024 L s P [(7.5)] .4875 .47024 1 0 Mshowa p .002 w .5 .61905 m .50625 .61905 L s P [(10)] .4875 .61905 1 0 Mshowa p .002 w .5 .76786 m .50625 .76786 L s P [(12.5)] .4875 .76786 1 0 Mshowa p .002 w .5 .91667 m .50625 .91667 L s P [(15)] .4875 .91667 1 0 Mshowa p .001 w .5 .05357 m .50375 .05357 L s P p .001 w .5 .08333 m .50375 .08333 L s P p .001 w .5 .1131 m .50375 .1131 L s P p .001 w .5 .14286 m .50375 .14286 L s P p .001 w .5 .20238 m .50375 .20238 L s P p .001 w .5 .23214 m .50375 .23214 L s P p .001 w .5 .2619 m .50375 .2619 L s P p .001 w .5 .29167 m .50375 .29167 L s P p .001 w .5 .35119 m .50375 .35119 L s P p .001 w .5 .38095 m .50375 .38095 L s P p .001 w .5 .41071 m .50375 .41071 L s P p .001 w .5 .44048 m .50375 .44048 L s P p .001 w .5 .5 m .50375 .5 L s P p .001 w .5 .52976 m .50375 .52976 L s P p .001 w .5 .55952 m .50375 .55952 L s P p .001 w .5 .58929 m .50375 .58929 L s P p .001 w .5 .64881 m .50375 .64881 L s P p .001 w .5 .67857 m .50375 .67857 L s P p .001 w .5 .70833 m .50375 .70833 L s P p .001 w .5 .7381 m .50375 .7381 L s P p .001 w .5 .79762 m .50375 .79762 L s P p .001 w .5 .82738 m .50375 .82738 L s P p .001 w .5 .85714 m .50375 .85714 L s P p .001 w .5 .8869 m .50375 .8869 L s P p .001 w .5 .94643 m .50375 .94643 L s P p .001 w .5 .97619 m .50375 .97619 L s P p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p 1 0 0 r p .004 w .02381 .97619 m .06349 .82407 L .10317 .68519 L .14286 .55952 L .18254 .44709 L .22222 .34788 L .2619 .2619 L .30159 .18915 L .34127 .12963 L .38095 .08333 L .40079 .06515 L .42063 .05026 L .44048 .03869 L .4504 .03414 L .46032 .03042 L .47024 .02753 L .4752 .02639 L .48016 .02546 L .48512 .02474 L .4876 .02446 L .49008 .02422 L .49256 .02404 L .4938 .02397 L .49504 .02391 L .49628 .02387 L .49752 .02384 L .49876 .02382 L .5 .02381 L .50124 .02382 L .50248 .02384 L .50372 .02387 L .50496 .02391 L .50744 .02404 L .50992 .02422 L .5124 .02446 L .51488 .02474 L .51984 .02546 L .5248 .02639 L .52976 .02753 L .53968 .03042 L .5496 .03414 L .55952 .03869 L .57937 .05026 L .59921 .06515 L .61905 .08333 L .65873 .12963 L .69841 .18915 L .7381 .2619 L .77778 .34788 L .81746 .44709 L Mistroke .85714 .55952 L .89683 .68519 L .93651 .82407 L .97619 .97619 L Mfstroke P P % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] The graph y = c xÛ :[font = text; inactive; preserveAspect; fontName = "CalcMath"] Now, while keeping our base graph of y = xÛ in red, let's investigate what happens if we multiply the xÛ by a constant. The new plot will be in blue, and will in this case be y = 2xÛ. :[font = input; preserveAspect; startGroup] Plot[{2x^2,x^2},{x,-4,4}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.0238095 0.0423378 [ [(-4)] .02381 .02381 0 2 Msboxa [(-2)] .2619 .02381 0 2 Msboxa [(2)] .7381 .02381 0 2 Msboxa [(4)] .97619 .02381 0 2 Msboxa [(5)] .4875 .2355 1 0 Msboxa [(10)] .4875 .44719 1 0 Msboxa [(15)] .4875 .65888 1 0 Msboxa [(20)] .4875 .87057 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .02381 m .02381 .03006 L s P [(-4)] .02381 .02381 0 2 Mshowa p .002 w .2619 .02381 m .2619 .03006 L s P [(-2)] .2619 .02381 0 2 Mshowa p .002 w .7381 .02381 m .7381 .03006 L s P [(2)] .7381 .02381 0 2 Mshowa p .002 w .97619 .02381 m .97619 .03006 L s P [(4)] .97619 .02381 0 2 Mshowa p .001 w .07143 .02381 m .07143 .02756 L s P p .001 w .11905 .02381 m .11905 .02756 L s P p .001 w .16667 .02381 m .16667 .02756 L s P p .001 w .21429 .02381 m .21429 .02756 L s P p .001 w .30952 .02381 m .30952 .02756 L s P p .001 w .35714 .02381 m .35714 .02756 L s P p .001 w .40476 .02381 m .40476 .02756 L s P p .001 w .45238 .02381 m .45238 .02756 L s P p .001 w .54762 .02381 m .54762 .02756 L s P p .001 w .59524 .02381 m .59524 .02756 L s P p .001 w .64286 .02381 m .64286 .02756 L s P p .001 w .69048 .02381 m .69048 .02756 L s P p .001 w .78571 .02381 m .78571 .02756 L s P p .001 w .83333 .02381 m .83333 .02756 L s P p .001 w .88095 .02381 m .88095 .02756 L s P p .001 w .92857 .02381 m .92857 .02756 L s P p .002 w 0 .02381 m 1 .02381 L s P p .002 w .5 .2355 m .50625 .2355 L s P [(5)] .4875 .2355 1 0 Mshowa p .002 w .5 .44719 m .50625 .44719 L s P [(10)] .4875 .44719 1 0 Mshowa p .002 w .5 .65888 m .50625 .65888 L s P [(15)] .4875 .65888 1 0 Mshowa p .002 w .5 .87057 m .50625 .87057 L s P [(20)] .4875 .87057 1 0 Mshowa p .001 w .5 .06615 m .50375 .06615 L s P p .001 w .5 .10849 m .50375 .10849 L s P p .001 w .5 .15082 m .50375 .15082 L s P p .001 w .5 .19316 m .50375 .19316 L s P p .001 w .5 .27784 m .50375 .27784 L s P p .001 w .5 .32017 m .50375 .32017 L s P p .001 w .5 .36251 m .50375 .36251 L s P p .001 w .5 .40485 m .50375 .40485 L s P p .001 w .5 .48953 m .50375 .48953 L s P p .001 w .5 .53186 m .50375 .53186 L s P p .001 w .5 .5742 m .50375 .5742 L s P p .001 w .5 .61654 m .50375 .61654 L s P p .001 w .5 .70122 m .50375 .70122 L s P p .001 w .5 .74355 m .50375 .74355 L s P p .001 w .5 .78589 m .50375 .78589 L s P p .001 w .5 .82823 m .50375 .82823 L s P p .001 w .5 .9129 m .50375 .9129 L s P p .001 w .5 .95524 m .50375 .95524 L s P p .001 w .5 .99758 m .50375 .99758 L s P p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w s s .09607 1 m .10317 .96465 L s .10317 .96465 m .14286 .78589 L .18254 .62595 L .22222 .48482 L .2619 .36251 L .30159 .25902 L .34127 .17434 L .38095 .10849 L .40079 .08261 L .42063 .06144 L .44048 .04498 L .4504 .03851 L .46032 .03322 L .47024 .0291 L .4752 .02748 L .48016 .02616 L .48512 .02513 L .4876 .02473 L .49008 .0244 L .49256 .02414 L .4938 .02404 L .49504 .02396 L .49628 .02389 L .49752 .02385 L .49876 .02382 L .5 .02381 L .50124 .02382 L .50248 .02385 L .50372 .02389 L .50496 .02396 L .50744 .02414 L .50992 .0244 L .5124 .02473 L .51488 .02513 L .51984 .02616 L .5248 .02748 L .52976 .0291 L .53968 .03322 L .5496 .03851 L .55952 .04498 L .57937 .06144 L .59921 .08261 L .61905 .10849 L .65873 .17434 L .69841 .25902 L .7381 .36251 L .77778 .48482 L .81746 .62595 L .85714 .78589 L .89683 .96465 L Mistroke Mfstroke .90393 1 m .89683 .96465 L s s s P P p 0 0 1 r p .02 w .02381 .70122 m .06349 .59302 L .10317 .49423 L .14286 .40485 L .18254 .32488 L .22222 .25432 L .2619 .19316 L .30159 .14141 L .34127 .09908 L .38095 .06615 L .40079 .05321 L .42063 .04263 L .44048 .03439 L .4504 .03116 L .46032 .02851 L .47024 .02646 L .4752 .02565 L .48016 .02499 L .48512 .02447 L .4876 .02427 L .49008 .0241 L .49256 .02397 L .4938 .02392 L .49504 .02388 L .49628 .02385 L .49752 .02383 L .49876 .02381 L .5 .02381 L .50124 .02381 L .50248 .02383 L .50372 .02385 L .50496 .02388 L .50744 .02397 L .50992 .0241 L .5124 .02427 L .51488 .02447 L .51984 .02499 L .5248 .02565 L .52976 .02646 L .53968 .02851 L .5496 .03116 L .55952 .03439 L .57937 .04263 L .59921 .05321 L .61905 .06615 L .65873 .09908 L .69841 .14141 L .7381 .19316 L .77778 .25432 L .81746 .32488 L Mistroke .85714 .40485 L .89683 .49423 L .93651 .59302 L .97619 .70122 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect] The plot of y = xÛ is the thicker of the two plots. :[font = text; inactive; preserveAspect; fontName = "CalcMath"] It appears that multiplying by 2 "squished" the graph. Change the constant above and see if this rule is general. Answer the following questions: 1) What does multiplying by a constant greater than one do? 2) What does multiplying by a constant less than one do? 3) Why? (Hint: try plotting points) 4) What does multiplying by a negative constant do? :[font = text; inactive; preserveAspect; fontName = "CalcMath"] Now let's demonstrate the function y = cxÛ with several constants. :[font = input; preserveAspect; startGroup; animationSpeed = 30; infiniteLoop; loopDistance = 1] Table[Show[GraphicsArray[ Plot[{c x^2, x^2},{x,-4,4}, PlotRange->{{-4.0,4.0},{-4.0,4.0}}, AxesLabel->{"x","y"}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1, DisplayFunction->$DisplayFunction, Epilog->Text[c,{2.5,1}]]]],{c,-3,-1}]; Table[Show[GraphicsArray[ Plot[{c x^2, x^2},{x,-4,4}, PlotRange->{{-4.0,4.0},{-4.0,4.0}}, AxesLabel->{"x","y"}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1, DisplayFunction->$DisplayFunction, Epilog->Text[c,{2.5,1}]]]],{c,-1,0,.2}]; Table[Show[GraphicsArray[ Plot[{c x^2, x^2},{x,-4,4}, PlotRange->{{-4.0,4.0},{-4.0,4.0}}, AxesLabel->{"x","y"}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1, DisplayFunction->$DisplayFunction, Epilog->Text[c,{2.5,1}]]]], {c,.1,.9,.2}]; Table[Show[GraphicsArray[ Plot[{c x^2, x^2},{x,-4,4}, PlotRange->{{-4.0,4.0},{-4.0,4.0}}, AxesLabel->{"x","y"}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1, DisplayFunction->$DisplayFunction, Epilog->Text[c,{2.5,1}]]]],{c,1,3}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; startGroup; animationSpeed = 30; infiniteLoop; loopDistance = 1] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 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.84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(2)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup; endGroup; animationSpeed = 30; infiniteLoop; loopDistance = -1] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w s s s s s s s s s .35714 1 m .375 .875 L s .375 .875 m .39583 .76042 L .41667 .66667 L .4375 .59375 L .44792 .5651 L .45833 .54167 L .46875 .52344 L .47396 .51628 L .47917 .51042 L .48437 .50586 L .48698 .50407 L .48958 .5026 L .49219 .50146 L .49349 .50102 L .49479 .50065 L .49609 .50037 L .4974 .50016 L .4987 .50004 L .5 .5 L .5013 .50004 L .5026 .50016 L .50391 .50037 L .50521 .50065 L .50781 .50146 L .51042 .5026 L .51302 .50407 L .51562 .50586 L .52083 .51042 L .52604 .51628 L .53125 .52344 L .54167 .54167 L .55208 .5651 L .5625 .59375 L .58333 .66667 L .60417 .76042 L .625 .875 L s .64286 1 m .625 .875 L s s s s s s s s s s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(3)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"] Select the above cells and press apple-y simultaneously to see the animation. The graph of y = xÛ is the thicker of the two. ;[s] 3:0,0;79,1;125,0;126,-1; 2:2,24,16,CalcMath,0,12,0,0,65535;1,24,16,CalcMath,0,12,65535,0,65535; :[font = text; inactive; preserveAspect; fontName = "CalcMath"; endGroup] The above animation shows the effect of the c constant in the graph of the function y = cxÛ. :[font = section; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup] The function y = xÛ + c :[font = text; inactive; preserveAspect; fontName = "CalcMath"] In this section, we study the effect of adding a constant to the y value of the function. Let's begin with y = xÛ in red and y = xÛ + 2 in blue. :[font = input; preserveAspect; startGroup] Plot[{x^2 + 2,x^2},{x,-4,4}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.119048 0.0238095 0.0529101 [ [(-4)] .02381 .02381 0 2 Msboxa [(-2)] .2619 .02381 0 2 Msboxa [(2)] .7381 .02381 0 2 Msboxa [(4)] .97619 .02381 0 2 Msboxa [(2.5)] .4875 .15608 1 0 Msboxa [(5)] .4875 .28836 1 0 Msboxa [(7.5)] .4875 .42063 1 0 Msboxa [(10)] .4875 .55291 1 0 Msboxa [(12.5)] .4875 .68519 1 0 Msboxa [(15)] .4875 .81746 1 0 Msboxa [(17.5)] .4875 .94974 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .02381 m .02381 .03006 L s P [(-4)] .02381 .02381 0 2 Mshowa p .002 w .2619 .02381 m .2619 .03006 L s P [(-2)] .2619 .02381 0 2 Mshowa p .002 w .7381 .02381 m .7381 .03006 L s P [(2)] .7381 .02381 0 2 Mshowa p .002 w .97619 .02381 m .97619 .03006 L s P [(4)] .97619 .02381 0 2 Mshowa p .001 w .07143 .02381 m .07143 .02756 L s P p .001 w .11905 .02381 m .11905 .02756 L s P p .001 w .16667 .02381 m .16667 .02756 L s P p .001 w .21429 .02381 m .21429 .02756 L s P p .001 w .30952 .02381 m .30952 .02756 L s P p .001 w .35714 .02381 m .35714 .02756 L s P p .001 w .40476 .02381 m .40476 .02756 L s P p .001 w .45238 .02381 m .45238 .02756 L s P p .001 w .54762 .02381 m .54762 .02756 L s P p .001 w .59524 .02381 m .59524 .02756 L s P p .001 w .64286 .02381 m .64286 .02756 L s P p .001 w .69048 .02381 m .69048 .02756 L s P p .001 w .78571 .02381 m .78571 .02756 L s P p .001 w .83333 .02381 m .83333 .02756 L s P p .001 w .88095 .02381 m .88095 .02756 L s P p .001 w .92857 .02381 m .92857 .02756 L s P p .002 w 0 .02381 m 1 .02381 L s P p .002 w .5 .15608 m .50625 .15608 L s P [(2.5)] .4875 .15608 1 0 Mshowa p .002 w .5 .28836 m .50625 .28836 L s P [(5)] .4875 .28836 1 0 Mshowa p .002 w .5 .42063 m .50625 .42063 L s P [(7.5)] .4875 .42063 1 0 Mshowa p .002 w .5 .55291 m .50625 .55291 L s P [(10)] .4875 .55291 1 0 Mshowa p .002 w .5 .68519 m .50625 .68519 L s P [(12.5)] .4875 .68519 1 0 Mshowa p .002 w .5 .81746 m .50625 .81746 L s P [(15)] .4875 .81746 1 0 Mshowa p .002 w .5 .94974 m .50625 .94974 L s P [(17.5)] .4875 .94974 1 0 Mshowa p .001 w .5 .05026 m .50375 .05026 L s P p .001 w .5 .07672 m .50375 .07672 L s P p .001 w .5 .10317 m .50375 .10317 L s P p .001 w .5 .12963 m .50375 .12963 L s P p .001 w .5 .18254 m .50375 .18254 L s P p .001 w .5 .20899 m .50375 .20899 L s P p .001 w .5 .23545 m .50375 .23545 L s P p .001 w .5 .2619 m .50375 .2619 L s P p .001 w .5 .31481 m .50375 .31481 L s P p .001 w .5 .34127 m .50375 .34127 L s P p .001 w .5 .36772 m .50375 .36772 L s P p .001 w .5 .39418 m .50375 .39418 L s P p .001 w .5 .44709 m .50375 .44709 L s P p .001 w .5 .47354 m .50375 .47354 L s P p .001 w .5 .5 m .50375 .5 L s P p .001 w .5 .52646 m .50375 .52646 L s P p .001 w .5 .57937 m .50375 .57937 L s P p .001 w .5 .60582 m .50375 .60582 L s P p .001 w .5 .63228 m .50375 .63228 L s P p .001 w .5 .65873 m .50375 .65873 L s P p .001 w .5 .71164 m .50375 .71164 L s P p .001 w .5 .7381 m .50375 .7381 L s P p .001 w .5 .76455 m .50375 .76455 L s P p .001 w .5 .79101 m .50375 .79101 L s P p .001 w .5 .84392 m .50375 .84392 L s P p .001 w .5 .87037 m .50375 .87037 L s P p .001 w .5 .89683 m .50375 .89683 L s P p .001 w .5 .92328 m .50375 .92328 L s P p .001 w .5 .97619 m .50375 .97619 L s P p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w .02381 .97619 m .06349 .84098 L .10317 .71752 L .14286 .60582 L .18254 .50588 L .22222 .4177 L .2619 .34127 L .30159 .2766 L .34127 .22369 L .38095 .18254 L .40079 .16637 L .42063 .15315 L .44048 .14286 L .4504 .13882 L .46032 .13551 L .47024 .13294 L .4752 .13193 L .48016 .1311 L .48512 .13046 L .4876 .1302 L .49008 .13 L .49256 .12984 L .4938 .12977 L .49504 .12972 L .49628 .12968 L .49752 .12965 L .49876 .12964 L .5 .12963 L .50124 .12964 L .50248 .12965 L .50372 .12968 L .50496 .12972 L .50744 .12984 L .50992 .13 L .5124 .1302 L .51488 .13046 L .51984 .1311 L .5248 .13193 L .52976 .13294 L .53968 .13551 L .5496 .13882 L .55952 .14286 L .57937 .15315 L .59921 .16637 L .61905 .18254 L .65873 .22369 L .69841 .2766 L .7381 .34127 L .77778 .4177 L .81746 .50588 L Mistroke .85714 .60582 L .89683 .71752 L .93651 .84098 L .97619 .97619 L Mfstroke P P p 0 0 1 r p .02 w .02381 .87037 m .06349 .73516 L .10317 .6117 L .14286 .5 L .18254 .40006 L .22222 .31188 L .2619 .23545 L .30159 .17078 L .34127 .11787 L .38095 .07672 L .40079 .06055 L .42063 .04733 L .44048 .03704 L .4504 .033 L .46032 .02969 L .47024 .02712 L .4752 .02611 L .48016 .02528 L .48512 .02464 L .4876 .02438 L .49008 .02418 L .49256 .02402 L .4938 .02395 L .49504 .0239 L .49628 .02386 L .49752 .02383 L .49876 .02382 L .5 .02381 L .50124 .02382 L .50248 .02383 L .50372 .02386 L .50496 .0239 L .50744 .02402 L .50992 .02418 L .5124 .02438 L .51488 .02464 L .51984 .02528 L .5248 .02611 L .52976 .02712 L .53968 .02969 L .5496 .033 L .55952 .03704 L .57937 .04733 L .59921 .06055 L .61905 .07672 L .65873 .11787 L .69841 .17078 L .7381 .23545 L .77778 .31188 L .81746 .40006 L Mistroke .85714 .5 L .89683 .6117 L .93651 .73516 L .97619 .87037 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; fontName = "CalcMath"] What do you suppose adding or subtracting a constant from the y value of the function will do in general? Why does this happen? Try changing the constant above to different values to test your hypothesis. :[font = text; inactive; preserveAspect] Run the following and press apple-y to animate the effect of the c in the function y = xÛ + c. ;[s] 1:0,0;95,-1; 1:1,24,16,CalcMath,0,12,0,0,0; :[font = input; preserveAspect; startGroup] Table[Show[GraphicsArray[ Plot[{x^2 + c,x^2 },{x,-4,4}, PlotRange->{{-4.0,4.0},{-4.0,4.0}}, AxesLabel->{"x","y"}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1, DisplayFunction->$DisplayFunction, Epilog->Text[c,{2.5,1}]]]],{c,-3,3}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; startGroup; animationSpeed = 30; infiniteLoop; loopDistance = 1] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 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L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w s s s s s s s s .32407 1 m .33333 .97222 L s .33333 .97222 m .375 .875 L .39583 .83681 L .41667 .80556 L .4375 .78125 L .44792 .7717 L .45833 .76389 L .46875 .75781 L .47396 .75543 L .47917 .75347 L .48437 .75195 L .48698 .75136 L .48958 .75087 L .49219 .75049 L .49349 .75034 L .49479 .75022 L .49609 .75012 L .4974 .75005 L .4987 .75001 L .5 .75 L .5013 .75001 L .5026 .75005 L .50391 .75012 L .50521 .75022 L .50781 .75049 L .51042 .75087 L .51302 .75136 L .51562 .75195 L .52083 .75347 L .52604 .75543 L .53125 .75781 L .54167 .76389 L .55208 .7717 L .5625 .78125 L .58333 .80556 L .60417 .83681 L .625 .875 L .66667 .97222 L s .67593 1 m .66667 .97222 L s s s s s s s s s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(2)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup; endGroup; endGroup; animationSpeed = 30; infiniteLoop; loopDistance = -1] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w s s s s s s s s s s .375 1 m .39583 .96181 L s .39583 .96181 m .41667 .93056 L .4375 .90625 L .44792 .8967 L .45833 .88889 L .46875 .88281 L .47396 .88043 L .47917 .87847 L .48437 .87695 L .48698 .87636 L .48958 .87587 L .49219 .87549 L .49349 .87534 L .49479 .87522 L .49609 .87512 L .4974 .87505 L .4987 .87501 L .5 .875 L .5013 .87501 L .5026 .87505 L .50391 .87512 L .50521 .87522 L .50781 .87549 L .51042 .87587 L .51302 .87636 L .51562 .87695 L .52083 .87847 L .52604 .88043 L .53125 .88281 L .54167 .88889 L .55208 .8967 L .5625 .90625 L .58333 .93056 L .60417 .96181 L .625 1 L s .625 1 m .625 1 L s s s s s s s s s s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(3)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; startGroup] The function y = (x + c)Û ;[s] 1:0,0;28,-1; 1:1,32,22,CalcMath,1,18,0,0,0; :[font = text; inactive; preserveAspect] This section studies the effects of adding a constant to the x value of the function y = xÛ and then squaring it, that is, y = (x + c)Û. Let's begin with adding 3, thus the equation y = (x + 3)Û. ;[s] 1:0,0;199,-1; 1:1,24,16,CalcMath,0,12,0,0,0; :[font = input; preserveAspect; startGroup] Plot[{(x + 3)^2,x^2},{x,-6,4}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.595238 0.0952381 0.0238095 0.0379986 [ [(-6)] .02381 .02381 0 2 Msboxa [(-4)] .21429 .02381 0 2 Msboxa [(-2)] .40476 .02381 0 2 Msboxa [(2)] .78571 .02381 0 2 Msboxa [(4)] .97619 .02381 0 2 Msboxa [(5)] .58274 .2138 1 0 Msboxa [(10)] .58274 .4038 1 0 Msboxa [(15)] .58274 .59379 1 0 Msboxa [(20)] .58274 .78378 1 0 Msboxa [(25)] .58274 .97377 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .02381 m .02381 .03006 L s P [(-6)] .02381 .02381 0 2 Mshowa p .002 w .21429 .02381 m .21429 .03006 L s P [(-4)] .21429 .02381 0 2 Mshowa p .002 w .40476 .02381 m .40476 .03006 L s P [(-2)] .40476 .02381 0 2 Mshowa p .002 w .78571 .02381 m .78571 .03006 L s P [(2)] .78571 .02381 0 2 Mshowa p .002 w .97619 .02381 m .97619 .03006 L s P [(4)] .97619 .02381 0 2 Mshowa p .001 w .0619 .02381 m .0619 .02756 L s P p .001 w .1 .02381 m .1 .02756 L s P p .001 w .1381 .02381 m .1381 .02756 L s P p .001 w .17619 .02381 m .17619 .02756 L s P p .001 w .25238 .02381 m .25238 .02756 L s P p .001 w .29048 .02381 m .29048 .02756 L s P p .001 w .32857 .02381 m .32857 .02756 L s P p .001 w .36667 .02381 m .36667 .02756 L s P p .001 w .44286 .02381 m .44286 .02756 L s P p .001 w .48095 .02381 m .48095 .02756 L s P p .001 w .51905 .02381 m .51905 .02756 L s P p .001 w .55714 .02381 m .55714 .02756 L s P p .001 w .63333 .02381 m .63333 .02756 L s P p .001 w .67143 .02381 m .67143 .02756 L s P p .001 w .70952 .02381 m .70952 .02756 L s P p .001 w .74762 .02381 m .74762 .02756 L s P p .001 w .82381 .02381 m .82381 .02756 L s P p .001 w .8619 .02381 m .8619 .02756 L s P p .001 w .9 .02381 m .9 .02756 L s P p .001 w .9381 .02381 m .9381 .02756 L s P p .002 w 0 .02381 m 1 .02381 L s P p .002 w .59524 .2138 m .60149 .2138 L s P [(5)] .58274 .2138 1 0 Mshowa p .002 w .59524 .4038 m .60149 .4038 L s P [(10)] .58274 .4038 1 0 Mshowa p .002 w .59524 .59379 m .60149 .59379 L s P [(15)] .58274 .59379 1 0 Mshowa p .002 w .59524 .78378 m .60149 .78378 L s P [(20)] .58274 .78378 1 0 Mshowa p .002 w .59524 .97377 m .60149 .97377 L s P [(25)] .58274 .97377 1 0 Mshowa p .001 w .59524 .06181 m .59899 .06181 L s P p .001 w .59524 .09981 m .59899 .09981 L s P p .001 w .59524 .13781 m .59899 .13781 L s P p .001 w .59524 .1758 m .59899 .1758 L s P p .001 w .59524 .2518 m .59899 .2518 L s P p .001 w .59524 .2898 m .59899 .2898 L s P p .001 w .59524 .3278 m .59899 .3278 L s P p .001 w .59524 .3658 m .59899 .3658 L s P p .001 w .59524 .44179 m .59899 .44179 L s P p .001 w .59524 .47979 m .59899 .47979 L s P p .001 w .59524 .51779 m .59899 .51779 L s P p .001 w .59524 .55579 m .59899 .55579 L s P p .001 w .59524 .63179 m .59899 .63179 L s P p .001 w .59524 .66979 m .59899 .66979 L s P p .001 w .59524 .70778 m .59899 .70778 L s P p .001 w .59524 .74578 m .59899 .74578 L s P p .001 w .59524 .82178 m .59899 .82178 L s P p .001 w .59524 .85978 m .59899 .85978 L s P p .001 w .59524 .89778 m .59899 .89778 L s P p .001 w .59524 .93578 m .59899 .93578 L s P p .002 w .59524 0 m .59524 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w .02381 .3658 m .06349 .2774 L .10317 .20219 L .14286 .14018 L .18254 .09136 L .20238 .0719 L .22222 .05574 L .24206 .04287 L .25198 .03768 L .2619 .03331 L .27183 .02976 L .27679 .0283 L .28175 .02704 L .28671 .02599 L .29167 .02515 L .29415 .0248 L .29663 .02451 L .29911 .02426 L .30159 .02407 L .30283 .024 L .30407 .02393 L .30531 .02388 L .30655 .02385 L .30779 .02382 L .30903 .02381 L .31027 .02381 L .31151 .02383 L .31275 .02385 L .31399 .02389 L .31523 .02395 L .31647 .02401 L .32143 .0244 L .32391 .02468 L .32639 .025 L .33135 .02581 L .34127 .02803 L .35119 .03108 L .36111 .03496 L .38095 .04518 L .40079 .05871 L .42063 .07553 L .46032 .11907 L .5 .1758 L .53968 .24573 L .57937 .32885 L .61905 .42517 L .65873 .53468 L .69841 .65738 L .7381 .79328 L .77778 .94237 L Mistroke Mfstroke .79187 1 m .77778 .94237 L s s s s s s P P p 0 0 1 r p .02 w s s s .11282 1 m .14286 .88115 L s .14286 .88115 m .18254 .73734 L .22222 .60672 L .2619 .48929 L .30159 .38506 L .34127 .29402 L .38095 .21618 L .42063 .15153 L .46032 .10007 L .48016 .07929 L .5 .06181 L .51984 .04762 L .52976 .04177 L .53968 .03674 L .5496 .03253 L .55952 .02915 L .56448 .02777 L .56944 .0266 L .5744 .02563 L .57937 .02487 L .58433 .02431 L .58681 .02411 L .58805 .02403 L .58929 .02396 L .59053 .0239 L .59177 .02386 L .59301 .02383 L .59425 .02381 L .59549 .02381 L .59673 .02382 L .59797 .02384 L .59921 .02388 L .60045 .02392 L .60169 .02398 L .60417 .02414 L .60665 .02435 L .60913 .02462 L .61409 .0253 L .61905 .02618 L .62897 .02858 L .63889 .03179 L .65873 .0407 L .67857 .0529 L .69841 .06841 L .7381 .10931 L .77778 .1634 L .81746 .23069 L .85714 .31117 L .89683 .40485 L .93651 .51172 L Mistroke .97619 .63179 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] What happened? It appears that the graph was displaced three units to the left. Try this with other values. Be sure to subtract some values. What can you say in general about the effect of adding a constant in this way? Below is another animation of various constants effect on the parabola. ;[s] 1:0,0;297,-1; 1:1,24,16,CalcMath,0,12,0,0,0; :[font = input; preserveAspect; startGroup] Table[Show[GraphicsArray[ Plot[{(x + c)^2,x^2},{x,-4,4}, PlotRange->{{-4.0,4.0},{-4.0,4.0}}, AxesLabel->{"x","y"}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1, DisplayFunction->$DisplayFunction, Epilog->Text[c,{2.5,1}]]]],{c,-3,3}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; startGroup; animationSpeed = 30; infiniteLoop; loopDistance = 1] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w s s s s s s s s s s s s s s s s .625 1 m .66667 .84722 L s .66667 .84722 m .70833 .72222 L .75 .625 L .77083 .58681 L .79167 .55556 L .8125 .53125 L .82292 .5217 L .83333 .51389 L .84375 .50781 L .84896 .50543 L .85417 .50347 L .85938 .50195 L .86198 .50136 L .86458 .50087 L .86719 .50049 L .86849 .50034 L .86979 .50022 L .87109 .50012 L .8724 .50005 L .8737 .50001 L .875 .5 L .8763 .50001 L .8776 .50005 L .87891 .50012 L .88021 .50022 L .88281 .50049 L .88542 .50087 L .88802 .50136 L .89063 .50195 L .89583 .50347 L .90104 .50543 L .90625 .50781 L .91667 .51389 L .92708 .5217 L .9375 .53125 L .95833 .55556 L .97917 .58681 L 1 .625 L s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(-3)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w s s s s s s s s s s s s s .5 1 m .54167 .84722 L s .54167 .84722 m .58333 .72222 L .625 .625 L .64583 .58681 L .66667 .55556 L .6875 .53125 L .69792 .5217 L .70833 .51389 L .71875 .50781 L .72396 .50543 L .72917 .50347 L .73438 .50195 L .73698 .50136 L .73958 .50087 L .74219 .50049 L .74349 .50034 L .74479 .50022 L .74609 .50012 L .7474 .50005 L .7487 .50001 L .75 .5 L .7513 .50001 L .7526 .50005 L .75391 .50012 L .75521 .50022 L .75651 .50034 L .75781 .50049 L .76042 .50087 L .76302 .50136 L .76563 .50195 L .77083 .50347 L .77604 .50543 L .78125 .50781 L .79167 .51389 L .80208 .5217 L .8125 .53125 L .83333 .55556 L .85417 .58681 L .875 .625 L .91667 .72222 L .95833 .84722 L 1 1 L s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(-2)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s 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s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w 0 1 m .04167 .84722 L .08333 .72222 L .125 .625 L .14583 .58681 L .16667 .55556 L .1875 .53125 L .19792 .5217 L .20833 .51389 L .21875 .50781 L .22396 .50543 L .22917 .50347 L .23437 .50195 L .23698 .50136 L .23958 .50087 L .24219 .50049 L .24349 .50034 L .24479 .50022 L .24609 .50012 L .2474 .50005 L .2487 .50001 L .25 .5 L .2513 .50001 L .2526 .50005 L .25391 .50012 L .25521 .50022 L .25781 .50049 L .26042 .50087 L .26302 .50136 L .26562 .50195 L .27083 .50347 L .27604 .50543 L .28125 .50781 L .29167 .51389 L .30208 .5217 L .3125 .53125 L .33333 .55556 L .35417 .58681 L .375 .625 L .41667 .72222 L .45833 .84722 L .5 1 L s .5 1 m .5 1 L s s s s s s s s s s s s s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(2)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup; endGroup; endGroup; animationSpeed = 30; infiniteLoop; loopDistance = -1] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.125 0.5 0.125 [ [(-4)] 0 .5 0 2 Msboxa [(-3)] .125 .5 0 2 Msboxa [(-2)] .25 .5 0 2 Msboxa [(-1)] .375 .5 0 2 Msboxa [(1)] .625 .5 0 2 Msboxa [(2)] .75 .5 0 2 Msboxa [(3)] .875 .5 0 2 Msboxa [(4)] 1 .5 0 2 Msboxa [(x)] 1.025 .5 -1 0 Msboxa [(-4)] .4875 0 1 0 Msboxa [(-3)] .4875 .125 1 0 Msboxa [(-2)] .4875 .25 1 0 Msboxa [(-1)] .4875 .375 1 0 Msboxa [(1)] .4875 .625 1 0 Msboxa [(2)] .4875 .75 1 0 Msboxa [(3)] .4875 .875 1 0 Msboxa [(4)] .4875 1 1 0 Msboxa [(y)] .5 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .5 m 0 .50625 L s P [(-4)] 0 .5 0 2 Mshowa p .002 w .125 .5 m .125 .50625 L s P [(-3)] .125 .5 0 2 Mshowa p .002 w .25 .5 m .25 .50625 L s P [(-2)] .25 .5 0 2 Mshowa p .002 w .375 .5 m .375 .50625 L s P [(-1)] .375 .5 0 2 Mshowa p .002 w .625 .5 m .625 .50625 L s P [(1)] .625 .5 0 2 Mshowa p .002 w .75 .5 m .75 .50625 L s P [(2)] .75 .5 0 2 Mshowa p .002 w .875 .5 m .875 .50625 L s P [(3)] .875 .5 0 2 Mshowa p .002 w 1 .5 m 1 .50625 L s P [(4)] 1 .5 0 2 Mshowa p .001 w .025 .5 m .025 .50375 L s P p .001 w .05 .5 m .05 .50375 L s P p .001 w .075 .5 m .075 .50375 L s P p .001 w .1 .5 m .1 .50375 L s P p .001 w .15 .5 m .15 .50375 L s P p .001 w .175 .5 m .175 .50375 L s P p .001 w .2 .5 m .2 .50375 L s P p .001 w .225 .5 m .225 .50375 L s P p .001 w .275 .5 m .275 .50375 L s P p .001 w .3 .5 m .3 .50375 L s P p .001 w .325 .5 m .325 .50375 L s P p .001 w .35 .5 m .35 .50375 L s P p .001 w .4 .5 m .4 .50375 L s P p .001 w .425 .5 m .425 .50375 L s P p .001 w .45 .5 m .45 .50375 L s P p .001 w .475 .5 m .475 .50375 L s P p .001 w .525 .5 m .525 .50375 L s P p .001 w .55 .5 m .55 .50375 L s P p .001 w .575 .5 m .575 .50375 L s P p .001 w .6 .5 m .6 .50375 L s P p .001 w .65 .5 m .65 .50375 L s P p .001 w .675 .5 m .675 .50375 L s P p .001 w .7 .5 m .7 .50375 L s P p .001 w .725 .5 m .725 .50375 L s P p .001 w .775 .5 m .775 .50375 L s P p .001 w .8 .5 m .8 .50375 L s P p .001 w .825 .5 m .825 .50375 L s P p .001 w .85 .5 m .85 .50375 L s P p .001 w .9 .5 m .9 .50375 L s P p .001 w .925 .5 m .925 .50375 L s P p .001 w .95 .5 m .95 .50375 L s P p .001 w .975 .5 m .975 .50375 L s P [(x)] 1.025 .5 -1 0 Mshowa p .002 w 0 .5 m 1 .5 L s P p .002 w .5 0 m .50625 0 L s P [(-4)] .4875 0 1 0 Mshowa p .002 w .5 .125 m .50625 .125 L s P [(-3)] .4875 .125 1 0 Mshowa p .002 w .5 .25 m .50625 .25 L s P [(-2)] .4875 .25 1 0 Mshowa p .002 w .5 .375 m .50625 .375 L s P [(-1)] .4875 .375 1 0 Mshowa p .002 w .5 .625 m .50625 .625 L s P [(1)] .4875 .625 1 0 Mshowa p .002 w .5 .75 m .50625 .75 L s P [(2)] .4875 .75 1 0 Mshowa p .002 w .5 .875 m .50625 .875 L s P [(3)] .4875 .875 1 0 Mshowa p .002 w .5 1 m .50625 1 L s P [(4)] .4875 1 1 0 Mshowa p .001 w .5 .025 m .50375 .025 L s P p .001 w .5 .05 m .50375 .05 L s P p .001 w .5 .075 m .50375 .075 L s P p .001 w .5 .1 m .50375 .1 L s P p .001 w .5 .15 m .50375 .15 L s P p .001 w .5 .175 m .50375 .175 L s P p .001 w .5 .2 m .50375 .2 L s P p .001 w .5 .225 m .50375 .225 L s P p .001 w .5 .275 m .50375 .275 L s P p .001 w .5 .3 m .50375 .3 L s P p .001 w .5 .325 m .50375 .325 L s P p .001 w .5 .35 m .50375 .35 L s P p .001 w .5 .4 m .50375 .4 L s P p .001 w .5 .425 m .50375 .425 L s P p .001 w .5 .45 m .50375 .45 L s P p .001 w .5 .475 m .50375 .475 L s P p .001 w .5 .525 m .50375 .525 L s P p .001 w .5 .55 m .50375 .55 L s P p .001 w .5 .575 m .50375 .575 L s P p .001 w .5 .6 m .50375 .6 L s P p .001 w .5 .65 m .50375 .65 L s P p .001 w .5 .675 m .50375 .675 L s P p .001 w .5 .7 m .50375 .7 L s P p .001 w .5 .725 m .50375 .725 L s P p .001 w .5 .775 m .50375 .775 L s P p .001 w .5 .8 m .50375 .8 L s P p .001 w .5 .825 m .50375 .825 L s P p .001 w .5 .85 m .50375 .85 L s P p .001 w .5 .9 m .50375 .9 L s P p .001 w .5 .925 m .50375 .925 L s P p .001 w .5 .95 m .50375 .95 L s P p .001 w .5 .975 m .50375 .975 L s P [(y)] .5 1 0 -4 Mshowa p .002 w .5 0 m .5 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p 1 0 0 r p .004 w 0 .625 m .02083 .58681 L .04167 .55556 L .0625 .53125 L .07292 .5217 L .08333 .51389 L .09375 .50781 L .09896 .50543 L .10417 .50347 L .10937 .50195 L .11198 .50136 L .11458 .50087 L .11719 .50049 L .11849 .50034 L .11979 .50022 L .12109 .50012 L .1224 .50005 L .1237 .50001 L .125 .5 L .1263 .50001 L .1276 .50005 L .12891 .50012 L .13021 .50022 L .13281 .50049 L .13542 .50087 L .13802 .50136 L .14062 .50195 L .14583 .50347 L .15104 .50543 L .15625 .50781 L .16667 .51389 L .17708 .5217 L .1875 .53125 L .20833 .55556 L .22917 .58681 L .25 .625 L .29167 .72222 L .33333 .84722 L .375 1 L s .375 1 m .375 1 L s s s s s s s s s s s s s s s s P P p 0 0 1 r p .02 w s s s s s s s .25 1 m .29167 .84722 L s .29167 .84722 m .33333 .72222 L .375 .625 L .39583 .58681 L .41667 .55556 L .4375 .53125 L .44792 .5217 L .45833 .51389 L .46875 .50781 L .47396 .50543 L .47917 .50347 L .48437 .50195 L .48698 .50136 L .48958 .50087 L .49219 .50049 L .49349 .50034 L .49479 .50022 L .49609 .50012 L .4974 .50005 L .4987 .50001 L .5 .5 L .5013 .50001 L .5026 .50005 L .50391 .50012 L .50521 .50022 L .50781 .50049 L .51042 .50087 L .51302 .50136 L .51562 .50195 L .52083 .50347 L .52604 .50543 L .53125 .50781 L .54167 .51389 L .55208 .5217 L .5625 .53125 L .58333 .55556 L .60417 .58681 L .625 .625 L .66667 .72222 L .70833 .84722 L .75 1 L s .75 1 m .75 1 L s s s s s s s P P P [(3)] .8125 .625 0 0 Mshowa % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; startGroup] Putting it all together ;[s] 1:0,0;24,-1; 1:1,32,22,CalcMath,1,18,0,0,0; :[font = text; inactive; preserveAspect] Let's take this opportunity to put two and two together and determine for ourselves what the graph of y = a(x + c)Û + d would look like, given that a, c and d are constants. Try graphing the given parabolas on your own graph paper and then check your work using mathematica. ;[s] 1:0,0;276,-1; 1:1,24,16,CalcMath,0,12,0,0,0; :[font = text; inactive; preserveAspect] 1) y = 0.5 xÛ + 4 2) y = (x - 6)Û - 3 3) y = 2 (x + 5)Û + 6 Make up your own problems. The answers to these can be obtained by running the code below. ;[s] 1:0,0;155,-1; 1:1,24,16,CalcMath,0,12,0,0,0; :[font = input; preserveAspect; startGroup] Plot[{0.5 x^2 + 4,x^2},{x,-6,6}, PlotStyle->{{Red},{Thickness[0.02],Blue}}, AspectRatio->1]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0793651 0.0238095 0.026455 [ [(-6)] .02381 .02381 0 2 Msboxa [(-4)] .18254 .02381 0 2 Msboxa [(-2)] .34127 .02381 0 2 Msboxa [(2)] .65873 .02381 0 2 Msboxa [(4)] .81746 .02381 0 2 Msboxa [(6)] .97619 .02381 0 2 Msboxa [(5)] .4875 .15608 1 0 Msboxa [(10)] .4875 .28836 1 0 Msboxa [(15)] .4875 .42063 1 0 Msboxa [(20)] .4875 .55291 1 0 Msboxa [(25)] .4875 .68519 1 0 Msboxa [(30)] .4875 .81746 1 0 Msboxa [(35)] .4875 .94974 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .02381 m .02381 .03006 L s P [(-6)] .02381 .02381 0 2 Mshowa p .002 w .18254 .02381 m .18254 .03006 L s P [(-4)] .18254 .02381 0 2 Mshowa p .002 w .34127 .02381 m .34127 .03006 L s P [(-2)] .34127 .02381 0 2 Mshowa p .002 w .65873 .02381 m .65873 .03006 L s P [(2)] .65873 .02381 0 2 Mshowa p .002 w .81746 .02381 m .81746 .03006 L s P [(4)] .81746 .02381 0 2 Mshowa p .002 w .97619 .02381 m .97619 .03006 L s P [(6)] .97619 .02381 0 2 Mshowa p .001 w .05556 .02381 m .05556 .02756 L s P p .001 w .0873 .02381 m .0873 .02756 L s P p .001 w .11905 .02381 m .11905 .02756 L s P p .001 w .15079 .02381 m .15079 .02756 L s P p .001 w .21429 .02381 m .21429 .02756 L s P p .001 w .24603 .02381 m .24603 .02756 L s P p .001 w .27778 .02381 m .27778 .02756 L s P p .001 w .30952 .02381 m .30952 .02756 L s P p .001 w .37302 .02381 m .37302 .02756 L s P p .001 w .40476 .02381 m 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.68519 L s P [(25)] .4875 .68519 1 0 Mshowa p .002 w .5 .81746 m .50625 .81746 L s P [(30)] .4875 .81746 1 0 Mshowa p .002 w .5 .94974 m .50625 .94974 L s P [(35)] .4875 .94974 1 0 Mshowa p .001 w .5 .05026 m .50375 .05026 L s P p .001 w .5 .07672 m .50375 .07672 L s P p .001 w .5 .10317 m .50375 .10317 L s P p .001 w .5 .12963 m .50375 .12963 L s P p .001 w .5 .18254 m .50375 .18254 L s P p .001 w .5 .20899 m .50375 .20899 L s P p .001 w .5 .23545 m .50375 .23545 L s P p .001 w .5 .2619 m .50375 .2619 L s P p .001 w .5 .31481 m .50375 .31481 L s P p .001 w .5 .34127 m .50375 .34127 L s P p .001 w .5 .36772 m .50375 .36772 L s P p .001 w .5 .39418 m .50375 .39418 L s P p .001 w .5 .44709 m .50375 .44709 L s P p .001 w .5 .47354 m .50375 .47354 L s P p .001 w .5 .5 m .50375 .5 L s P p .001 w .5 .52646 m .50375 .52646 L s P p .001 w .5 .57937 m .50375 .57937 L s P p .001 w .5 .60582 m .50375 .60582 L s P p .001 w .5 .63228 m .50375 .63228 L s P p .001 w .5 .65873 m .50375 .65873 L s P p 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Conduct your own mathematica investigation of the translation properties using different functions. For example y = xÜ, y = ln x, y = sin x, or any old function will do. Hopefully, you will find that the properties will be the same, and if you understand these properties, sketching graphs of functions will become easy once you know the graph of the base function. ^*)