(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 74287, 1740] NotebookOptionsPosition[ 73785, 1709] NotebookOutlinePosition[ 74194, 1725] CellTagsIndexPosition[ 74151, 1722] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Function Transformations", "Title", CellChangeTimes->{{3.544218964673705*^9, 3.5442189676518755`*^9}, { 3.5970608088894444`*^9, 3.5970608108944473`*^9}, {3.6489890833051977`*^9, 3.6489890845492687`*^9}}, Background->RGBColor[0.88, 1, 0.88]], Cell[CellGroupData[{ Cell[TextData[{ "We have seen how the parameters ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["h", FontSlant->"Italic"], ", and ", StyleBox["k", FontSlant->"Italic"], " affected the graph of the basic ", StyleBox["quadratic", FontVariations->{"Underline"->True}], " graph ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], StyleBox["a", FontSize->24, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["\[CenterDot]", FontSize->24], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", " ", "-", " ", StyleBox["h", FontColor->RGBColor[1, 0, 0]]}], ")"}], "2"], TraditionalForm]], FontSize->24, FontWeight->"Bold"], StyleBox[" ", FontSize->24], StyleBox["+ ", FontSize->24, FontSlant->"Italic"], StyleBox["k", FontSize->24, FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[". ", FontSize->24], "\n\nThe parameter ", StyleBox["h", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], " represents a HORIZONTAL SHIFT: it shifts the graph of", StyleBox[" y", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], StyleBox[" ", FontSize->18], "to the ", StyleBox["right", FontVariations->{"Underline"->True}], StyleBox[" ", FontSize->18], StyleBox["h", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], " units (or ", StyleBox["left", FontVariations->{"Underline"->True}], ", if ", StyleBox["h", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], " < 0).\nThe parameter ", StyleBox["k", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], " represents a VERTICAL SHIFT: it shifts the graph of", StyleBox[" y", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], StyleBox[" ", FontSize->18], StyleBox["up", FontVariations->{"Underline"->True}], StyleBox[" ", FontSize->18], StyleBox["k", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], " units (or ", StyleBox["down", FontVariations->{"Underline"->True}], ", if ", StyleBox["k", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], " < 0).", StyleBox["\n", FontSlant->"Italic"], "The parameter ", StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " represents a STRETCH/COMPRESSION: stretches (if ", StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " > 1) or compresses (if 0 < ", StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " < 1) the graph of", StyleBox[" y", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], ".\n\n", StyleBox["Example 1:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Graph the function ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], StyleBox["2", FontSize->24, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", " ", "+", "3"}], ")"}], "2"], "-", "1"}], TraditionalForm]], FontSize->24, FontWeight->"Bold"], " by hand. Use the interactive frame below to verify your graph." }], "Section", CellChangeTimes->{{3.5582689511406293`*^9, 3.558268952466632*^9}, { 3.558268982842684*^9, 3.5582691676094007`*^9}, {3.5582692031582623`*^9, 3.55826930374883*^9}, {3.558269337451288*^9, 3.5582693791057596`*^9}, { 3.597060921654602*^9, 3.597061025034747*^9}, {3.5970610647398024`*^9, 3.5970610681348076`*^9}, {3.5970615941055436`*^9, 3.5970616526256256`*^9}, { 3.59706405536302*^9, 3.5970640576030235`*^9}, {3.6489890870514116`*^9, 3.6489890955198965`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 2, $CellContext`b$$ = -1, $CellContext`c$$ = -3, $CellContext`fcn$$ = \ #^2& , Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fcn$$], #^2& , "original function"}, { Identity -> "linear", Abs -> "absolute value", (#^2& ) -> "quadratic", (#^3& ) -> "cubic", (#^(1/2)& ) -> "square root", (Sign[#] Abs[#]^(1/3)& ) -> "cube root", Log -> "logarithmic", Exp -> "exponential"}}, {{ Hold[$CellContext`a$$], 1, "stretch or compression, a"}, -4, 4, 0.01}, {{ Hold[$CellContext`b$$], 0, "vertical shift, k"}, -5, 5, 0.01}, {{ Hold[$CellContext`c$$], 0, "horizontal shift, h"}, -5, 5, 0.01}}, Typeset`size$$ = {400., {197., 203.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fcn$7701$$ = False, $CellContext`a$7702$$ = 0, $CellContext`b$7703$$ = 0, $CellContext`c$7704$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 0, $CellContext`c$$ = 0, $CellContext`fcn$$ = #^2& }, "ControllerVariables" :> { Hold[$CellContext`fcn$$, $CellContext`fcn$7701$$, False], Hold[$CellContext`a$$, $CellContext`a$7702$$, 0], Hold[$CellContext`b$$, $CellContext`b$7703$$, 0], Hold[$CellContext`c$$, $CellContext`c$7704$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> (If[$CellContext`a$$ == 0, $CellContext`a$$ = 0.001]; Plot[$CellContext`b$$ + $CellContext`a$$ \ $CellContext`fcn$$[$CellContext`x - $CellContext`c$$], {$CellContext`x, -6, 6}, PlotRange -> {-6, 6}, BaseStyle -> {Medium, "Label"}, AxesLabel -> {$CellContext`x, $CellContext`y}, PlotStyle -> { Thickness[0.005], Hue[0.6]}, ImageSize -> {400, 400}, AspectRatio -> Automatic, Ticks -> { Range[-6, 6], Range[-6, 6]}, GridLines -> { Table[{$CellContext`n, RGBColor[0, 0.6, 1.]}, {$CellContext`n, -6, 6, 1}], Table[{$CellContext`n, RGBColor[0, 0.6, 1.]}, {$CellContext`n, -6, 6, 1}]}]), "Specifications" :> {{{$CellContext`fcn$$, #^2& , "original function"}, { Identity -> "linear", Abs -> "absolute value", (#^2& ) -> "quadratic", (#^3& ) -> "cubic", (#^(1/2)& ) -> "square root", (Sign[#] Abs[#]^(1/3)& ) -> "cube root", Log -> "logarithmic", Exp -> "exponential"}, ControlType -> SetterBar}, {{$CellContext`a$$, 1, "stretch or compression, a"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Left, ImageSize -> Tiny}, {{$CellContext`b$$, 0, "vertical shift, k"}, -5, 5, 0.01, Appearance -> "Labeled", ControlPlacement -> Left, ImageSize -> Tiny}, {{$CellContext`c$$, 0, "horizontal shift, h"}, -5, 5, 0.01, Appearance -> "Labeled", ControlPlacement -> Left, ImageSize -> Tiny}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{759., {244., 251.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]], "a"}]], "Input", CellChangeTimes->{{3.597061761745778*^9, 3.597061763405781*^9}}] }, Open ]], Cell[TextData[{ "\[MathematicaIcon]Consider the general equation ", StyleBox["y = ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["\[CenterDot]", FontColor->GrayLevel[0]], StyleBox["f(", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["b", FontSlant->"Italic", FontColor->RGBColor[0, 0.67, 0]], StyleBox["(x - ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["c", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox[")) + ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["d", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[". 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2)^2 + 4, $CellContext`x >= 1}}]}; $CellContext`buttonText = { "linear", "quadratic", "cubic", "absolute value", "square root", "piecewise"}; $CellContext`functionButtons = Map[Part[#, 1] -> Part[#, 2]& , Transpose[{ Range[ Length[$CellContext`buttonText]], $CellContext`buttonText}]]; Show[{ Graphics[{ If[ MemberQ[$CellContext`options$$, $CellContext`sp], Text[ ToString[ Style[{ Round[$CellContext`xval$$, 0.01], Part[ $CellContext`function[ Round[$CellContext`xval$$, 0.01]], $CellContext`fff$$]}, $CellContext`color1], TraditionalForm], {4.75, -4.25}, {1, 0}], $CellContext`color1], If[ MemberQ[$CellContext`options$$, $CellContext`sp], Text[ ToString[ Style[ If[$CellContext`b$$ =!= 0., {(1/$CellContext`b$$) Round[$CellContext`xval$$, 0.01] + $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[ Round[$CellContext`xval$$, 0.01]], $CellContext`fff$$] + $CellContext`d$$}, ""], $CellContext`color2], TraditionalForm], { 4.75, -4.75}, {1, 0}], $CellContext`color1], AbsolutePointSize[9], $CellContext`color2, Dashing[{0.01}], If[$CellContext`b$$ =!= 0., Point[{(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}], $CellContext`color2], If[$CellContext`b$$ =!= 0., Line[{{0, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}, {( 1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}}], $CellContext`color2], If[$CellContext`b$$ =!= 0., Line[{{(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, 0}, {(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}}], $CellContext`color2], \ $CellContext`color1, Dashing[{0.02}], Point[{$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}], Line[{{0, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}, {$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}}], Line[{{$CellContext`xval$$, 0}, {$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}}]}], Plot[$CellContext`a$$ Part[ $CellContext`function[$CellContext`b$$ ($CellContext`x - \ $CellContext`c$$)], $CellContext`fff$$] + $CellContext`d$$, {$CellContext`x, Which[$CellContext`b$$ < 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`b$$ == 0., -5, $CellContext`b$$ > 0., -5], Which[$CellContext`b$$ < 0., 5, $CellContext`b$$ == 0., 5, $CellContext`b$$ > 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$]}, PlotStyle -> {$CellContext`color2, AbsoluteThickness[3]}, PlotRange -> All], Plot[$CellContext`a$$ Part[ $CellContext`function[$CellContext`b$$ ($CellContext`x - \ $CellContext`c$$)], $CellContext`fff$$] + $CellContext`d$$, {$CellContext`x, Which[$CellContext`b$$ < 0., -5, $CellContext`b$$ == 0., -5, $CellContext`b$$ > 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$], Which[$CellContext`b$$ < 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`b$$ == 0., 5, $CellContext`b$$ > 0., 5]}, PlotStyle -> $CellContext`color2, PlotRange -> All], Plot[ Part[ $CellContext`function[$CellContext`x], $CellContext`fff$$], \ {$CellContext`x, -5, $CellContext`xval$$}, PlotStyle -> {$CellContext`color1, AbsoluteThickness[3]}, PlotRange -> All], Plot[ Part[ $CellContext`function[$CellContext`x], $CellContext`fff$$], \ {$CellContext`x, $CellContext`xval$$, 5}, PlotStyle -> $CellContext`color1, PlotRange -> All]}, PlotRange -> {{-5, 5}, {-5, 5}}, Axes -> True, AxesStyle -> AbsoluteThickness[ If[ MemberQ[$CellContext`options$$, $CellContext`grid], 1.25, 1]], PlotLabel -> If[ MemberQ[$CellContext`options$$, $CellContext`pl], Grid[{{ ToString[ Style[ HoldForm[$CellContext`f][ HoldForm[$CellContext`x]], $CellContext`color1], TraditionalForm]}, { ToString[ Style[If[Round[$CellContext`a$$, 0.1] == 1., 1, If[ Round[$CellContext`a$$, 0.1] == 0., 0, $CellContext`a$$]] HoldForm[$CellContext`f][ If[Round[$CellContext`b$$, 0.1] == 1., 1, If[ Round[$CellContext`b$$, 0.1] == 0., 0, $CellContext`b$$]] (HoldForm[$CellContext`x] - If[ Round[$CellContext`c$$, 0.1] == 0., 0, $CellContext`c$$])] + If[Round[$CellContext`d$$, 0.1] == 0., 0, $CellContext`d$$], $CellContext`color2], TraditionalForm]}}]], ImageSize -> If[ MemberQ[$CellContext`options$$, $CellContext`format], 500, 380], GridLines -> If[ MemberQ[$CellContext`options$$, $CellContext`grid], { Range[-5, 5], Range[-5, 5]}, None]]], "Specifications" :> {{{$CellContext`a$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`b$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`c$$, 0}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`d$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, Delimiter, {{$CellContext`xval$$, 0, "x-value"}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, Delimiter, {{$CellContext`reset$$, False, "reset"}, {False, True}}, Delimiter, {{$CellContext`fff$$, 1, " "}, { 1 -> "linear", 2 -> "quadratic", 3 -> "cubic", 4 -> "absolute value", 5 -> "square root", 6 -> "piecewise"}, ControlType -> RadioButtonBar, Appearance -> "Vertical"}, Delimiter, {{$CellContext`options$$, {$CellContext`sp, \ $CellContext`pl}, ""}, {$CellContext`pl -> " plot label", $CellContext`sp -> " point location", $CellContext`grid -> " grid lines", $CellContext`format -> " large format"}, ControlType -> CheckboxBar, Appearance -> "Vertical"}}, "Options" :> { ControlPlacement -> Left, AutorunSequencing -> {{1, 3}, {3, 3}, {5, 3}, {7, 6}}}, "DefaultOptions" :> {}], ImageSizeCache->{595., {255., 262.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.648988964322392*^9, {3.6489901367224493`*^9, 3.648990145197934*^9}, 3.6489903208039784`*^9}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 9:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Consider f(x) = ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"-", RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}]}]], TraditionalForm]]], "+ 3. Identify the parent function and the transformations applied. Sketch \ the graph.\nRemember that ", StyleBox["y = ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["\[CenterDot]", FontColor->GrayLevel[0]], StyleBox["f(", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["b", FontSlant->"Italic", FontColor->RGBColor[0, 0.67, 0]], StyleBox["(x - ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["c", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox[")) + ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["d", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], StyleBox[". ", FontColor->GrayLevel[0]], "\n\nIn the interactive frame, enter the following parameters: \n", StyleBox["a = 1\t(vertical stretch)", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontSize->18], StyleBox["b = -1 (reflection about y-axis)", FontSize->18, FontColor->RGBColor[0, 0.67, 0]], StyleBox["\n", FontSize->18], StyleBox["c = -2 (horizontal shift)", FontSize->18, FontColor->RGBColor[1, 0, 0]], StyleBox["\n", FontSize->18], StyleBox["d = 3 (vertical shift)", FontSize->18, FontColor->RGBColor[0.5, 0, 0.5]], "\n\nNotice that the function may be rewritten as f(x) = ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ RowBox[{"-", "x"}], "-", "2"}]], TraditionalForm]]], "+ 3. \n\nUsing the Order of Operations, here\[CloseCurlyQuote]s how the \ point (0, 0) is affected:\n refl\t horiz\t vert\n \ y-axis\t shift\t shift\n(0, 0) ---> (0, 0) -----> (-2, 0) ---> \ (-2, 3) \n\nUsing the Order of Operations, here\[CloseCurlyQuote]s how the \ point (1, 1) is affected:\n refl\t horiz\t vert\n \ y-axis\t shift\t shift\n(1, 1) ---> (-1, 1) -----> (-3, 1) ---> \ (-3, 4) \n" }], "Subsubsection", CellChangeTimes->{{3.5878173714504366`*^9, 3.587817550060687*^9}, { 3.5878175828007326`*^9, 3.5878176297507987`*^9}, {3.5878177207529273`*^9, 3.5878177380129514`*^9}, {3.587817925033213*^9, 3.587818195103591*^9}, { 3.597063936237853*^9, 3.5970639399078584`*^9}, {3.5970684532513013`*^9, 3.5970685032563715`*^9}, {3.5970685496514363`*^9, 3.5970685738314705`*^9}, { 3.5970686575215874`*^9, 3.597068660426592*^9}, {3.5970688611168723`*^9, 3.5970689363469777`*^9}, {3.5970689820170417`*^9, 3.59706900212707*^9}, { 3.5970690948921995`*^9, 3.5970691261272435`*^9}, {3.5970691590172896`*^9, 3.5970693161425095`*^9}, {3.5970693628225746`*^9, 3.597069429527668*^9}, { 3.5998606266093216`*^9, 3.599860638154338*^9}}, FontSize->24], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, $CellContext`b$$ = 1, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`fff$$ = 5, $CellContext`options$$ = {$CellContext`sp, $CellContext`pl}, \ $CellContext`reset$$ = False, $CellContext`xval$$ = 0, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`a$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`b$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`c$$], 0}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`d$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`xval$$], 0, "x-value"}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`reset$$], False, "reset"}, {False, True}}, {{ Hold[$CellContext`fff$$], 1, " "}, { 1 -> "linear", 2 -> "quadratic", 3 -> "cubic", 4 -> "absolute value", 5 -> "square root", 6 -> "piecewise"}}, {{ Hold[$CellContext`options$$], {$CellContext`sp, $CellContext`pl}, ""}, {$CellContext`pl -> " plot label", $CellContext`sp -> " point location", $CellContext`grid -> " grid lines", $CellContext`format -> " large format"}}}, Typeset`size$$ = {380., {203., 208.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$21484$$ = 0, $CellContext`b$21485$$ = 0, $CellContext`c$21486$$ = 0, $CellContext`d$21487$$ = 0, $CellContext`xval$21488$$ = 0, $CellContext`reset$21489$$ = False, $CellContext`fff$21490$$ = False, $CellContext`options$21491$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 1, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`fff$$ = 1, $CellContext`options$$ = {$CellContext`sp, $CellContext`pl}, \ $CellContext`reset$$ = False, $CellContext`xval$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$21484$$, 0], Hold[$CellContext`b$$, $CellContext`b$21485$$, 0], Hold[$CellContext`c$$, $CellContext`c$21486$$, 0], Hold[$CellContext`d$$, $CellContext`d$21487$$, 0], Hold[$CellContext`xval$$, $CellContext`xval$21488$$, 0], Hold[$CellContext`reset$$, $CellContext`reset$21489$$, False], Hold[$CellContext`fff$$, $CellContext`fff$21490$$, False], Hold[$CellContext`options$$, $CellContext`options$21491$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> DynamicModule[{$CellContext`function, $CellContext`buttonText, \ $CellContext`color1 = RGBColor[1, 0, 0], $CellContext`color2 = RGBColor[0, 0, 1]}, If[$CellContext`reset$$ == True, {$CellContext`a$$, $CellContext`b$$, $CellContext`c$$, \ $CellContext`d$$, $CellContext`xval$$} = {1, 1, 0, 0, 0}; $CellContext`reset$$ = False]; $CellContext`function[ Pattern[$CellContext`x, Blank[]]] := {$CellContext`x, $CellContext`x^2, $CellContext`x^3, Abs[$CellContext`x], Sqrt[$CellContext`x], Piecewise[{{3, $CellContext`x < -3}, { Abs[$CellContext`x], Inequality[-3, LessEqual, $CellContext`x, Less, 1]}, {-($CellContext`x - 2)^2 + 4, $CellContext`x >= 1}}]}; $CellContext`buttonText = { "linear", "quadratic", "cubic", "absolute value", "square root", "piecewise"}; $CellContext`functionButtons = Map[Part[#, 1] -> Part[#, 2]& , Transpose[{ Range[ Length[$CellContext`buttonText]], $CellContext`buttonText}]]; Show[{ Graphics[{ If[ MemberQ[$CellContext`options$$, $CellContext`sp], Text[ ToString[ Style[{ Round[$CellContext`xval$$, 0.01], Part[ $CellContext`function[ Round[$CellContext`xval$$, 0.01]], $CellContext`fff$$]}, $CellContext`color1], TraditionalForm], {4.75, -4.25}, {1, 0}], $CellContext`color1], If[ MemberQ[$CellContext`options$$, $CellContext`sp], Text[ ToString[ Style[ If[$CellContext`b$$ =!= 0., {(1/$CellContext`b$$) Round[$CellContext`xval$$, 0.01] + $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[ Round[$CellContext`xval$$, 0.01]], $CellContext`fff$$] + $CellContext`d$$}, ""], $CellContext`color2], TraditionalForm], { 4.75, -4.75}, {1, 0}], $CellContext`color1], AbsolutePointSize[9], $CellContext`color2, Dashing[{0.01}], If[$CellContext`b$$ =!= 0., Point[{(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}], $CellContext`color2], If[$CellContext`b$$ =!= 0., Line[{{0, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}, {( 1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}}], $CellContext`color2], If[$CellContext`b$$ =!= 0., Line[{{(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, 0}, {(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}}], $CellContext`color2], \ $CellContext`color1, Dashing[{0.02}], Point[{$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}], Line[{{0, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}, {$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}}], Line[{{$CellContext`xval$$, 0}, {$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}}]}], Plot[$CellContext`a$$ Part[ $CellContext`function[$CellContext`b$$ ($CellContext`x - \ $CellContext`c$$)], $CellContext`fff$$] + $CellContext`d$$, {$CellContext`x, Which[$CellContext`b$$ < 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`b$$ == 0., -5, $CellContext`b$$ > 0., -5], Which[$CellContext`b$$ < 0., 5, $CellContext`b$$ == 0., 5, $CellContext`b$$ > 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$]}, PlotStyle -> {$CellContext`color2, AbsoluteThickness[3]}, PlotRange -> All], Plot[$CellContext`a$$ Part[ $CellContext`function[$CellContext`b$$ ($CellContext`x - \ $CellContext`c$$)], $CellContext`fff$$] + $CellContext`d$$, {$CellContext`x, Which[$CellContext`b$$ < 0., -5, $CellContext`b$$ == 0., -5, $CellContext`b$$ > 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$], Which[$CellContext`b$$ < 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`b$$ == 0., 5, $CellContext`b$$ > 0., 5]}, PlotStyle -> $CellContext`color2, PlotRange -> All], Plot[ Part[ $CellContext`function[$CellContext`x], $CellContext`fff$$], \ {$CellContext`x, -5, $CellContext`xval$$}, PlotStyle -> {$CellContext`color1, AbsoluteThickness[3]}, PlotRange -> All], Plot[ Part[ $CellContext`function[$CellContext`x], $CellContext`fff$$], \ {$CellContext`x, $CellContext`xval$$, 5}, PlotStyle -> $CellContext`color1, PlotRange -> All]}, PlotRange -> {{-5, 5}, {-5, 5}}, Axes -> True, AxesStyle -> AbsoluteThickness[ If[ MemberQ[$CellContext`options$$, $CellContext`grid], 1.25, 1]], PlotLabel -> If[ MemberQ[$CellContext`options$$, $CellContext`pl], Grid[{{ ToString[ Style[ HoldForm[$CellContext`f][ HoldForm[$CellContext`x]], $CellContext`color1], TraditionalForm]}, { ToString[ Style[If[Round[$CellContext`a$$, 0.1] == 1., 1, If[ Round[$CellContext`a$$, 0.1] == 0., 0, $CellContext`a$$]] HoldForm[$CellContext`f][ If[Round[$CellContext`b$$, 0.1] == 1., 1, If[ Round[$CellContext`b$$, 0.1] == 0., 0, $CellContext`b$$]] (HoldForm[$CellContext`x] - If[ Round[$CellContext`c$$, 0.1] == 0., 0, $CellContext`c$$])] + If[Round[$CellContext`d$$, 0.1] == 0., 0, $CellContext`d$$], $CellContext`color2], TraditionalForm]}}]], ImageSize -> If[ MemberQ[$CellContext`options$$, $CellContext`format], 500, 380], GridLines -> If[ MemberQ[$CellContext`options$$, $CellContext`grid], { Range[-5, 5], Range[-5, 5]}, None]]], "Specifications" :> {{{$CellContext`a$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`b$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`c$$, 0}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`d$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, Delimiter, {{$CellContext`xval$$, 0, "x-value"}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, Delimiter, {{$CellContext`reset$$, False, "reset"}, {False, True}}, Delimiter, {{$CellContext`fff$$, 1, " "}, { 1 -> "linear", 2 -> "quadratic", 3 -> "cubic", 4 -> "absolute value", 5 -> "square root", 6 -> "piecewise"}, ControlType -> RadioButtonBar, Appearance -> "Vertical"}, Delimiter, {{$CellContext`options$$, {$CellContext`sp, \ $CellContext`pl}, ""}, {$CellContext`pl -> " plot label", $CellContext`sp -> " point location", $CellContext`grid -> " grid lines", $CellContext`format -> " large format"}, ControlType -> CheckboxBar, Appearance -> "Vertical"}}, "Options" :> { ControlPlacement -> Left, AutorunSequencing -> {{1, 3}, {3, 3}, {5, 3}, {7, 6}}}, "DefaultOptions" :> {}], ImageSizeCache->{595., {255., 262.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.648988964322392*^9, {3.6489901367224493`*^9, 3.648990145197934*^9}, 3.6489903208039784`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 10", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[":", FontWeight->"Bold"], " Now let\[CloseCurlyQuote]s use the graph of the parent function y = \ \[LeftBracketingBar]x\[RightBracketingBar] to graph the function \n\n\t\t\ty \ = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], RowBox[{"\[LeftBracketingBar]", RowBox[{"x", "+", "2"}], "\[RightBracketingBar]"}]}], "+", "4"}], TraditionalForm]]], " .\nBegin with the point (0, 0) on y = \[LeftBracketingBar]x\ \[RightBracketingBar], then apply the stretches, shifts, etc. using the Order \ of Operation. Repeat for (1, 1).\n\nRemember that for ", StyleBox["y = ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["\[CenterDot]", FontColor->GrayLevel[0]], StyleBox["f(", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["b", FontSlant->"Italic", FontColor->RGBColor[0, 0.67, 0]], StyleBox["(x - ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["c", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox[")) + ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["d ", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], " we use:\n", StyleBox["\n", FontSize->18], StyleBox["a = -0.5", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontSize->18], StyleBox["b = 1", FontSize->18, FontColor->RGBColor[0, 0.67, 0]], StyleBox["\n", FontSize->18], StyleBox["c = -2", FontSize->18, FontColor->RGBColor[1, 0, 0]], StyleBox["\n", FontSize->18], StyleBox["d = 4", FontSize->18, FontColor->RGBColor[0.5, 0, 0.5]] }], "Subsection", CellChangeTimes->{{3.558273676029143*^9, 3.5582736849211583`*^9}, { 3.55827373177444*^9, 3.558273886692696*^9}, {3.558273917076349*^9, 3.5582741159982905`*^9}, {3.5582742836825767`*^9, 3.558274306801814*^9}, { 3.558274341153474*^9, 3.558274341325074*^9}, {3.5582743948047667`*^9, 3.558274395506768*^9}, {3.5582744550456705`*^9, 3.558274458546476*^9}, { 3.587816131710621*^9, 3.5878162136807356`*^9}, {3.597069556677846*^9, 3.597069576552874*^9}, {3.5998607832895412`*^9, 3.599860788549548*^9}}, FontSize->24, Background->GrayLevel[1]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, $CellContext`b$$ = 1, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`fff$$ = 4, $CellContext`options$$ = {$CellContext`sp, $CellContext`pl}, \ $CellContext`reset$$ = False, $CellContext`xval$$ = 0, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`a$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`b$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`c$$], 0}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`d$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`xval$$], 0, "x-value"}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`reset$$], False, "reset"}, {False, True}}, {{ Hold[$CellContext`fff$$], 1, " "}, { 1 -> "linear", 2 -> "quadratic", 3 -> "cubic", 4 -> "absolute value", 5 -> "square root", 6 -> "piecewise"}}, {{ Hold[$CellContext`options$$], {$CellContext`sp, $CellContext`pl}, ""}, {$CellContext`pl -> " plot label", $CellContext`sp -> " point location", $CellContext`grid -> " grid lines", $CellContext`format -> " large format"}}}, Typeset`size$$ = {380., {203., 208.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$21484$$ = 0, $CellContext`b$21485$$ = 0, $CellContext`c$21486$$ = 0, $CellContext`d$21487$$ = 0, $CellContext`xval$21488$$ = 0, $CellContext`reset$21489$$ = False, $CellContext`fff$21490$$ = False, $CellContext`options$21491$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 1, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`fff$$ = 1, $CellContext`options$$ = {$CellContext`sp, $CellContext`pl}, \ $CellContext`reset$$ = False, $CellContext`xval$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$21484$$, 0], Hold[$CellContext`b$$, $CellContext`b$21485$$, 0], Hold[$CellContext`c$$, $CellContext`c$21486$$, 0], Hold[$CellContext`d$$, $CellContext`d$21487$$, 0], Hold[$CellContext`xval$$, $CellContext`xval$21488$$, 0], Hold[$CellContext`reset$$, $CellContext`reset$21489$$, False], Hold[$CellContext`fff$$, $CellContext`fff$21490$$, False], Hold[$CellContext`options$$, $CellContext`options$21491$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> DynamicModule[{$CellContext`function, $CellContext`buttonText, \ $CellContext`color1 = RGBColor[1, 0, 0], $CellContext`color2 = RGBColor[0, 0, 1]}, If[$CellContext`reset$$ == True, {$CellContext`a$$, $CellContext`b$$, $CellContext`c$$, \ $CellContext`d$$, $CellContext`xval$$} = {1, 1, 0, 0, 0}; $CellContext`reset$$ = False]; $CellContext`function[ Pattern[$CellContext`x, Blank[]]] := {$CellContext`x, $CellContext`x^2, $CellContext`x^3, Abs[$CellContext`x], Sqrt[$CellContext`x], Piecewise[{{3, $CellContext`x < -3}, { Abs[$CellContext`x], Inequality[-3, LessEqual, $CellContext`x, Less, 1]}, {-($CellContext`x - 2)^2 + 4, $CellContext`x >= 1}}]}; $CellContext`buttonText = { "linear", "quadratic", "cubic", "absolute value", "square root", "piecewise"}; $CellContext`functionButtons = Map[Part[#, 1] -> Part[#, 2]& , Transpose[{ Range[ Length[$CellContext`buttonText]], $CellContext`buttonText}]]; Show[{ Graphics[{ If[ MemberQ[$CellContext`options$$, $CellContext`sp], Text[ ToString[ Style[{ Round[$CellContext`xval$$, 0.01], Part[ $CellContext`function[ Round[$CellContext`xval$$, 0.01]], $CellContext`fff$$]}, $CellContext`color1], TraditionalForm], {4.75, -4.25}, {1, 0}], $CellContext`color1], If[ MemberQ[$CellContext`options$$, $CellContext`sp], Text[ ToString[ Style[ If[$CellContext`b$$ =!= 0., {(1/$CellContext`b$$) Round[$CellContext`xval$$, 0.01] + $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[ Round[$CellContext`xval$$, 0.01]], $CellContext`fff$$] + $CellContext`d$$}, ""], $CellContext`color2], TraditionalForm], { 4.75, -4.75}, {1, 0}], $CellContext`color1], AbsolutePointSize[9], $CellContext`color2, Dashing[{0.01}], If[$CellContext`b$$ =!= 0., Point[{(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}], $CellContext`color2], If[$CellContext`b$$ =!= 0., Line[{{0, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}, {( 1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}}], $CellContext`color2], If[$CellContext`b$$ =!= 0., Line[{{(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, 0}, {(1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`a$$ Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$] + $CellContext`d$$}}], $CellContext`color2], \ $CellContext`color1, Dashing[{0.02}], Point[{$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}], Line[{{0, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}, {$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}}], Line[{{$CellContext`xval$$, 0}, {$CellContext`xval$$, Part[ $CellContext`function[$CellContext`xval$$], \ $CellContext`fff$$]}}]}], Plot[$CellContext`a$$ Part[ $CellContext`function[$CellContext`b$$ ($CellContext`x - \ $CellContext`c$$)], $CellContext`fff$$] + $CellContext`d$$, {$CellContext`x, Which[$CellContext`b$$ < 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`b$$ == 0., -5, $CellContext`b$$ > 0., -5], Which[$CellContext`b$$ < 0., 5, $CellContext`b$$ == 0., 5, $CellContext`b$$ > 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$]}, PlotStyle -> {$CellContext`color2, AbsoluteThickness[3]}, PlotRange -> All], Plot[$CellContext`a$$ Part[ $CellContext`function[$CellContext`b$$ ($CellContext`x - \ $CellContext`c$$)], $CellContext`fff$$] + $CellContext`d$$, {$CellContext`x, Which[$CellContext`b$$ < 0., -5, $CellContext`b$$ == 0., -5, $CellContext`b$$ > 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$], Which[$CellContext`b$$ < 0., (1/$CellContext`b$$) $CellContext`xval$$ + \ $CellContext`c$$, $CellContext`b$$ == 0., 5, $CellContext`b$$ > 0., 5]}, PlotStyle -> $CellContext`color2, PlotRange -> All], Plot[ Part[ $CellContext`function[$CellContext`x], $CellContext`fff$$], \ {$CellContext`x, -5, $CellContext`xval$$}, PlotStyle -> {$CellContext`color1, AbsoluteThickness[3]}, PlotRange -> All], Plot[ Part[ $CellContext`function[$CellContext`x], $CellContext`fff$$], \ {$CellContext`x, $CellContext`xval$$, 5}, PlotStyle -> $CellContext`color1, PlotRange -> All]}, PlotRange -> {{-5, 5}, {-5, 5}}, Axes -> True, AxesStyle -> AbsoluteThickness[ If[ MemberQ[$CellContext`options$$, $CellContext`grid], 1.25, 1]], PlotLabel -> If[ MemberQ[$CellContext`options$$, $CellContext`pl], Grid[{{ ToString[ Style[ HoldForm[$CellContext`f][ HoldForm[$CellContext`x]], $CellContext`color1], TraditionalForm]}, { ToString[ Style[If[Round[$CellContext`a$$, 0.1] == 1., 1, If[ Round[$CellContext`a$$, 0.1] == 0., 0, $CellContext`a$$]] HoldForm[$CellContext`f][ If[Round[$CellContext`b$$, 0.1] == 1., 1, If[ Round[$CellContext`b$$, 0.1] == 0., 0, $CellContext`b$$]] (HoldForm[$CellContext`x] - If[ Round[$CellContext`c$$, 0.1] == 0., 0, $CellContext`c$$])] + If[Round[$CellContext`d$$, 0.1] == 0., 0, $CellContext`d$$], $CellContext`color2], TraditionalForm]}}]], ImageSize -> If[ MemberQ[$CellContext`options$$, $CellContext`format], 500, 380], GridLines -> If[ MemberQ[$CellContext`options$$, $CellContext`grid], { Range[-5, 5], Range[-5, 5]}, None]]], "Specifications" :> {{{$CellContext`a$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`b$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`c$$, 0}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, {{$CellContext`d$$, 1}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, Delimiter, {{$CellContext`xval$$, 0, "x-value"}, -4.9, 4.9, 0.1, ImageSize -> Dynamic[ If[ MemberQ[$CellContext`options$$, $CellContext`format], Medium, Tiny]]}, Delimiter, {{$CellContext`reset$$, False, "reset"}, {False, True}}, Delimiter, {{$CellContext`fff$$, 1, " "}, { 1 -> "linear", 2 -> "quadratic", 3 -> "cubic", 4 -> "absolute value", 5 -> "square root", 6 -> "piecewise"}, ControlType -> RadioButtonBar, Appearance -> "Vertical"}, Delimiter, {{$CellContext`options$$, {$CellContext`sp, \ $CellContext`pl}, ""}, {$CellContext`pl -> " plot label", $CellContext`sp -> " point location", $CellContext`grid -> " grid lines", $CellContext`format -> " large format"}, ControlType -> CheckboxBar, Appearance -> "Vertical"}}, "Options" :> { ControlPlacement -> Left, AutorunSequencing -> {{1, 3}, {3, 3}, {5, 3}, {7, 6}}}, "DefaultOptions" :> {}], ImageSizeCache->{595., {255., 262.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.648988964322392*^9, {3.6489901367224493`*^9, 3.648990145197934*^9}, 3.6489903208039784`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 11:", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" ", FontWeight->"Bold"], " \tSketch the graph of ", StyleBox["f(x) = -", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], "3"], "-", " ", "1"}], TraditionalForm]], FontSlant->"Italic"], ".\n\t \tIdentify the transformations applied to the point (0, 0) and (1, \ 1) and what the resulting points will be. \n\n(Remember that ", StyleBox["y = ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["a", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["\[CenterDot]", FontColor->GrayLevel[0]], StyleBox["f(", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["b", FontSlant->"Italic", FontColor->RGBColor[0, 0.67, 0]], StyleBox["(x - ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["c", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox[")) + ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["d", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], ")", StyleBox["\n\n", FontSize->16], StyleBox["a = -1", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontSize->18], StyleBox["b = 1", FontSize->18, FontColor->RGBColor[0, 0.67, 0]], StyleBox["\n", FontSize->18], StyleBox["c = -1", FontSize->18, FontColor->RGBColor[1, 0, 0]], StyleBox["\n", FontSize->18], StyleBox["d = -1", FontSize->18, FontColor->RGBColor[0.5, 0, 0.5]] }], "Subsection", CellChangeTimes->{{3.5582747179681187`*^9, 3.5582747348941483`*^9}, { 3.5582747700462065`*^9, 3.558274840765929*^9}, {3.5582748789735928`*^9, 3.558274906898839*^9}, {3.55827496580254*^9, 3.5582750622158985`*^9}, { 3.587816255570794*^9, 3.5878163160008793`*^9}, {3.587817080030029*^9, 3.5878171247400913`*^9}, {3.59706972382308*^9, 3.5970697410231047`*^9}}, FontSize->24], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, $CellContext`b$$ = 1, $CellContext`c$$ = 0, $CellContext`d$$ = 0, $CellContext`fff$$ = 3, $CellContext`options$$ = {$CellContext`sp, $CellContext`pl}, \ $CellContext`reset$$ = False, $CellContext`xval$$ = 0, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`a$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`b$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`c$$], 0}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`d$$], 1}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`xval$$], 0, "x-value"}, -4.9, 4.9, 0.1}, {{ Hold[$CellContext`reset$$], False, "reset"}, {False, True}}, {{ Hold[$CellContext`fff$$], 1, " "}, { 1 -> "linear", 2 -> "quadratic", 3 -> "cubic", 4 -> "absolute value", 5 -> "square root", 6 -> "piecewise"}}, {{ Hold[$CellContext`options$$], {$CellContext`sp, $CellContext`pl}, ""}, {$CellContext`pl -> " plot label", $CellContext`sp -> " point location", $CellContext`grid -> " grid lines", $CellContext`format -> " large format"}}}, Typeset`size$$ = {380., {203., 208.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$21484$$ = 0, $CellContext`b$21485$$ = 0, $CellContext`c$21486$$ = 0, $CellContext`d$21487$$ = 0, $CellContext`xval$21488$$ = 0, $CellContext`reset$21489$$ = False, $CellContext`fff$21490$$ = False, $CellContext`options$21491$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 1, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`fff$$ = 1, $CellContext`options$$ = {$CellContext`sp, $CellContext`pl}, \ $CellContext`reset$$ = False, $CellContext`xval$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$21484$$, 0], Hold[$CellContext`b$$, $CellContext`b$21485$$, 0], Hold[$CellContext`c$$, $CellContext`c$21486$$, 0], Hold[$CellContext`d$$, $CellContext`d$21487$$, 0], Hold[$CellContext`xval$$, $CellContext`xval$21488$$, 0], Hold[$CellContext`reset$$, $CellContext`reset$21489$$, False], Hold[$CellContext`fff$$, $CellContext`fff$21490$$, False], Hold[$CellContext`options$$, $CellContext`options$21491$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> DynamicModule[{$CellContext`function, $CellContext`buttonText, \ $CellContext`color1 = RGBColor[1, 0, 0], $CellContext`color2 = RGBColor[0, 0, 1]}, If[$CellContext`reset$$ == True, {$CellContext`a$$, $CellContext`b$$, $CellContext`c$$, \ $CellContext`d$$, $CellContext`xval$$} = {1, 1, 0, 0, 0}; $CellContext`reset$$ = False]; $CellContext`function[ Pattern[$CellContext`x, Blank[]]] := {$CellContext`x, $CellContext`x^2, $CellContext`x^3, Abs[$CellContext`x], Sqrt[$CellContext`x], Piecewise[{{3, $CellContext`x < -3}, { Abs[$CellContext`x], Inequality[-3, LessEqual, $CellContext`x, Less, 1]}, {-($CellContext`x - 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