(* 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[ 66853, 1571] NotebookOptionsPosition[ 65745, 1525] NotebookOutlinePosition[ 66156, 1541] CellTagsIndexPosition[ 66113, 1538] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Quadratic Functions and Transformations", "Title", CellChangeTimes->{{3.544218964673705*^9, 3.5442189676518755`*^9}, { 3.5963100352584996`*^9, 3.596310044968055*^9}, {3.6489915595948334`*^9, 3.648991560537887*^9}}, Background->RGBColor[0.88, 1, 0.88]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Background:", FontVariations->{"Underline"->True}], " We know that for linear functions, the form y = ", StyleBox["m", FontColor->RGBColor[0.5, 0, 0.5]], "x + ", StyleBox["b", FontColor->RGBColor[1, 0.5, 0]], " gives us the slope (", StyleBox["m", FontColor->RGBColor[0.5, 0, 0.5]], ") and y-intercept (", StyleBox["b", FontColor->RGBColor[1, 0.5, 0]], ") of the line. " }], "Section", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellFrameColor->RGBColor[0, 0, 1], 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}, { 3.5963101839730053`*^9, 3.5963103224189243`*^9}, {3.5964839489279113`*^9, 3.5964839568003616`*^9}, {3.5964840233321667`*^9, 3.596484143452038*^9}, { 3.5964842866862297`*^9, 3.59648442148494*^9}, {3.5964844611032057`*^9, 3.5964845142162437`*^9}, {3.5965586743577423`*^9, 3.5965586788599997`*^9}, 3.596559339579791*^9}, FontSize->24, FontWeight->"Bold"], Cell[TextData[{ StyleBox["Example 1:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " In the interactive frame below, use the ", StyleBox["m", FontColor->RGBColor[0.5, 0, 0.5]], " and ", StyleBox["b ", FontColor->RGBColor[1, 0.5, 0]], "sliders to graph the line y = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox["5", "2"], "x"}], "+", "3"}], TraditionalForm]]], ".\n\nNotice that:\n\[Bullet]\tThe parameter ", StyleBox["m", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], " affects how steep the line is and whether it is an \"uphill\" or \ \"downhill\" line (from left to right). \n\[Bullet]\tThe parameter ", StyleBox["b", FontSlant->"Italic", FontColor->RGBColor[1, 0.5, 0]], " gives the y-intercept of the line, but it is also worthy to note that the \ b-value represents a ", StyleBox["vertical shift", FontVariations->{"Underline"->True}], " of the entire line. \nTherefore, the parent graph of y = x is ", StyleBox["transformed", FontWeight->"Bold", FontSlant->"Italic"], " by the parameters ", StyleBox["m", FontColor->RGBColor[0.5, 0, 0.5]], " and ", StyleBox["b", FontColor->RGBColor[1, 0.5, 0]], "." }], "Section", CellGroupingRules->{GroupTogetherGrouping, 10000.}, 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}, { 3.5963101839730053`*^9, 3.5963103224189243`*^9}, {3.5964839489279113`*^9, 3.5964839568003616`*^9}, {3.5964840233321667`*^9, 3.596484143452038*^9}, { 3.5964842866862297`*^9, 3.59648442148494*^9}, {3.5964844611032057`*^9, 3.596484543377912*^9}, {3.596484575287737*^9, 3.596484616349086*^9}, { 3.596484658674506*^9, 3.5964847558740664`*^9}, {3.5964847969454155`*^9, 3.5964848972551527`*^9}, {3.5964849319271355`*^9, 3.596484946079945*^9}, { 3.596556794832239*^9, 3.5965568246979475`*^9}, 3.596559260382261*^9, 3.596559339579791*^9}, FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, $CellContext`b$$ = 0, $CellContext`c$$ = 2, $CellContext`DomainF$$ = False, $CellContext`FunctionF$$ = True, $CellContext`GraphF$$ = True, $CellContext`h$$ = 0, $CellContext`k$$ = 0, $CellContext`m$$ = 1, $CellContext`n$$ = 2, $CellContext`p$$ = 2, $CellContext`parent$$ = 3, $CellContext`po$$ = 3, $CellContext`RangeF$$ = False, $CellContext`resetabsolute$$ = False, $CellContext`resetcubic$$ = False, $CellContext`resetlinear$$ = False, $CellContext`resetquadratic$$ = False, $CellContext`resetroot$$ = False, $CellContext`w$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`parent$$], 5, "parent function"}, { 1 -> "constant", 2 -> "identity", 3 -> "linear", 4 -> "quadratic (or even power)", 5 -> "cubic (or odd power)", 6 -> "absolute value", 7 -> "square root (or \!\(\*SuperscriptBox[\(n\), \(th\)]\) root)"}}, {{ Hold[$CellContext`c$$], 2}, -5, 5, 0.1}, { Hold[ OpenerView[{"constant", Manipulate`Place[1]}, Dynamic[ If[$CellContext`parent$$ == 1, True, False]], Enabled -> False]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`m$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`b$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`resetlinear$$], False, "reset"}, {False, True}}, { Hold[ OpenerView[{"linear", Column[{ Manipulate`Place[2], Manipulate`Place[3], Manipulate`Place[4]}]}, Dynamic[ If[$CellContext`parent$$ == 3, True, False]], Enabled -> False]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`h$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`k$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`w$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`p$$], 2}, 2, 10, 2}, {{ Hold[$CellContext`resetquadratic$$], False, "reset"}, {False, True}}, { Hold[ OpenerView[{"quadratic (or even power)", Column[{ Item[ Style[ "f(x)=a[w(x-h)\!\(\*SuperscriptBox[\(]\), \(p\)]\)+k", Italic, 11], Alignment -> Center], Manipulate`Place[5], Manipulate`Place[6], Manipulate`Place[7], Manipulate`Place[8], Manipulate`Place[9], Manipulate`Place[10]}]}, Dynamic[ If[$CellContext`parent$$ == 4, True, False]], Enabled -> False]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`h$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`k$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`w$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`po$$], 3, $CellContext`p$$}, 1, 11, 2}, {{ Hold[$CellContext`resetcubic$$], False, "reset"}, {False, True}}, { Hold[ OpenerView[{"cubic (or odd power)", Column[{ Item[ Style[ "f(x)=a[w(x-h)\!\(\*SuperscriptBox[\(]\), \(p\)]\)+k", Italic, 11], Alignment -> Center], Manipulate`Place[11], Manipulate`Place[12], Manipulate`Place[13], Manipulate`Place[14], Manipulate`Place[15], Manipulate`Place[16]}]}, Dynamic[ If[$CellContext`parent$$ == 5, True, False]], Enabled -> False]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`h$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`k$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`w$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`resetabsolute$$], False, "reset"}, {False, True}}, { Hold[ OpenerView[{"absolute value", Column[{ Item[ Style["f(x)=a|w(x-h)|+k", Italic, 11], Alignment -> Center], Manipulate`Place[17], Manipulate`Place[18], Manipulate`Place[19], Manipulate`Place[20], Manipulate`Place[21]}]}, Dynamic[ If[$CellContext`parent$$ == 6, True, False]], Enabled -> False]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`h$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`k$$], 0}, -5, 5, 0.1}, {{ Hold[$CellContext`w$$], 1}, -3, 3, 0.1}, {{ Hold[$CellContext`n$$], 2}, 2, 8, 2}, {{ Hold[$CellContext`resetroot$$], False, "reset"}, {False, True}}, { Hold[ OpenerView[{ "square root (or \!\(\*SuperscriptBox[\(n\), \(th\)]\) root)", Column[{ Item[ Style[ "f(x)=a\!\(\*RadicalBox[\(w \((x - h)\)\), \(n\)]\)+k", Italic, 11], Alignment -> Center], Manipulate`Place[22], Manipulate`Place[23], Manipulate`Place[24], Manipulate`Place[25], Manipulate`Place[26], Manipulate`Place[27]}]}, Dynamic[ If[$CellContext`parent$$ == 7, True, False]], Enabled -> False]], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`FunctionF$$], True, "function"}, {False, True}}, {{ Hold[$CellContext`GraphF$$], True, "graph"}, {False, True}}, {{ Hold[$CellContext`DomainF$$], False, "domain"}, {False, True}}, {{ Hold[$CellContext`RangeF$$], False, "range"}, {False, True}}}, Typeset`size$$ = {400., {197., 203.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`parent$220760$$ = False, $CellContext`c$220761$$ = 0, $CellContext`m$220762$$ = 0, $CellContext`b$220763$$ = 0, $CellContext`resetlinear$220764$$ = False, $CellContext`a$220765$$ = 0, $CellContext`h$220766$$ = 0, $CellContext`k$220767$$ = 0, $CellContext`w$220768$$ = 0, $CellContext`p$220769$$ = 0, $CellContext`resetquadratic$220770$$ = False, $CellContext`po$220771$$ = 0, $CellContext`resetcubic$220772$$ = False, $CellContext`n$220773$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 2, StandardForm, "Variables" :> {$CellContext`a$$ = 1, $CellContext`b$$ = 0, $CellContext`c$$ = 2, $CellContext`DomainF$$ = False, $CellContext`FunctionF$$ = True, $CellContext`GraphF$$ = True, $CellContext`h$$ = 0, $CellContext`k$$ = 0, $CellContext`m$$ = 1, $CellContext`n$$ = 2, $CellContext`p$$ = 2, $CellContext`parent$$ = 5, $CellContext`po$$ = 3, $CellContext`RangeF$$ = False, $CellContext`resetabsolute$$ = False, $CellContext`resetcubic$$ = False, $CellContext`resetlinear$$ = False, $CellContext`resetquadratic$$ = False, $CellContext`resetroot$$ = False, $CellContext`w$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`parent$$, $CellContext`parent$220760$$, False], Hold[$CellContext`c$$, $CellContext`c$220761$$, 0], Hold[$CellContext`m$$, $CellContext`m$220762$$, 0], Hold[$CellContext`b$$, $CellContext`b$220763$$, 0], Hold[$CellContext`resetlinear$$, $CellContext`resetlinear$220764$$, False], Hold[$CellContext`a$$, $CellContext`a$220765$$, 0], Hold[$CellContext`h$$, $CellContext`h$220766$$, 0], Hold[$CellContext`k$$, $CellContext`k$220767$$, 0], Hold[$CellContext`w$$, $CellContext`w$220768$$, 0], Hold[$CellContext`p$$, $CellContext`p$220769$$, 0], Hold[$CellContext`resetquadratic$$, \ $CellContext`resetquadratic$220770$$, False], Hold[$CellContext`po$$, $CellContext`po$220771$$, 0], Hold[$CellContext`resetcubic$$, $CellContext`resetcubic$220772$$, False], Hold[$CellContext`n$$, $CellContext`n$220773$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Module[{$CellContext`tickset$}, $CellContext`tickset$ = Range[-5, 5]; $CellContext`fx = Row[{ Style["f", Italic], "(", Style["x", Italic], ") = "}]; $CellContext`do = If[$CellContext`DomainF$$, Row[{" ", Style["D", Italic], " = \[DoubleStruckCapitalR] "}], ""]; If[$CellContext`resetlinear$$, $CellContext`m$$ = 1; $CellContext`b$$ = 0; $CellContext`resetlinear$$ = False]; If[$CellContext`resetquadratic$$, $CellContext`a$$ = 1; $CellContext`h$$ = 0; $CellContext`k$$ = 0; $CellContext`w$$ = 1; $CellContext`p$$ = 2; $CellContext`resetquadratic$$ = False]; If[$CellContext`resetcubic$$, $CellContext`a$$ = 1; $CellContext`h$$ = 0; $CellContext`k$$ = 0; $CellContext`w$$ = 1; $CellContext`po$$ = 3; $CellContext`resetcubic$$ = False]; If[$CellContext`resetabsolute$$, $CellContext`a$$ = 1; $CellContext`h$$ = 0; $CellContext`k$$ = 0; $CellContext`w$$ = 1; $CellContext`resetabsolute$$ = False]; If[$CellContext`resetroot$$, $CellContext`a$$ = 1; $CellContext`h$$ = 0; $CellContext`k$$ = 0; $CellContext`w$$ = 1; $CellContext`n$$ = 2; $CellContext`resetroot$$ = False]; Which[$CellContext`parent$$ == 1, Plot[ If[$CellContext`GraphF$$, $CellContext`c$$, 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Text[ Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, $CellContext`c$$, "?"], "\n", $CellContext`do, If[$CellContext`RangeF$$, Row[{" ", Style["R", Italic], " = {", $CellContext`c$$, "} "}], ""]}], 25]]], $CellContext`parent$$ == 2, Plot[ If[$CellContext`GraphF$$, $CellContext`x, 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, $CellContext`x, "?"], "\n", If[$CellContext`DomainF$$, Row[{" ", Style["D", Italic], " = \[DoubleStruckCapitalR] "}], ""], If[$CellContext`RangeF$$, Row[{" ", Style["R", Italic], " = \[DoubleStruckCapitalR] "}], ""]}], 25]], $CellContext`parent$$ == 3, Plot[ If[$CellContext`GraphF$$, $CellContext`m$$ $CellContext`x + \ $CellContext`b$$, 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, Row[{ If[$CellContext`m$$ == 0, "", Row[{ If[$CellContext`m$$ == 1, "", Rationalize[$CellContext`m$$]], $CellContext`x}]], Which[$CellContext`b$$ > 0, Row[{" + ", $CellContext`b$$}], $CellContext`b$$ == 0, " ", $CellContext`b$$ < 0, $CellContext`b$$]}], "?"], "\n", $CellContext`do, If[$CellContext`RangeF$$, Row[{" ", Style["R", Italic], " = ", If[$CellContext`m$$ == 0, Row[{"{", $CellContext`b$$, "}"}], "\[DoubleStruckCapitalR]"], " "}], ""]}], 25]], $CellContext`parent$$ == 4, Plot[ If[$CellContext`GraphF$$, $CellContext`a$$ ($CellContext`w$$ \ ($CellContext`x - $CellContext`h$$))^$CellContext`p$$ + $CellContext`k$$, 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, Row[{ If[ Or[$CellContext`a$$ == 0, $CellContext`w$$ == 0], "0", Row[{ If[$CellContext`a$$ == 1, "", $CellContext`a$$], If[$CellContext`w$$ == 1, "", Row[{"[", $CellContext`w$$}]], If[$CellContext`h$$ === 0, "", "("], $CellContext`x - $CellContext`h$$, If[$CellContext`h$$ === 0, "", ")"], If[$CellContext`w$$ == 1, "", "]"], Superscript["", $CellContext`p$$]}]], Which[$CellContext`k$$ > 0, Row[{" + ", $CellContext`k$$}], $CellContext`k$$ == 0, " ", $CellContext`k$$ < 0, $CellContext`k$$]}], "?"], "\n", $CellContext`do, If[$CellContext`RangeF$$, Row[{" ", Style["R", Italic], " = ", Which[$CellContext`a$$ > 0, Row[{"[", $CellContext`k$$, ", " Infinity, ")"}], $CellContext`a$$ == 0, Row[{"{", $CellContext`k$$, "}"}], $CellContext`a$$ < 0, Row[{"(", (-Infinity) ", ", $CellContext`k$$, "]"}]], " "}], ""]}], 25]], $CellContext`parent$$ == 5, Plot[ If[$CellContext`GraphF$$, $CellContext`a$$ ($CellContext`w$$ \ ($CellContext`x - $CellContext`h$$))^$CellContext`po$$ + $CellContext`k$$, 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, Row[{ If[ Or[$CellContext`a$$ == 0, $CellContext`w$$ == 0], "0", Row[{ If[$CellContext`a$$ == 1, "", $CellContext`a$$], If[$CellContext`w$$ == 1, "", Row[{"[", $CellContext`w$$}]], If[$CellContext`h$$ === 0, "", "("], $CellContext`x - $CellContext`h$$, If[$CellContext`h$$ === 0, "", ")"], If[$CellContext`w$$ == 1, "", "]"], Superscript["", $CellContext`po$$]}]], Which[$CellContext`k$$ > 0, Row[{"+", $CellContext`k$$}], $CellContext`k$$ == 0, " ", $CellContext`k$$ < 0, $CellContext`k$$]}], "?"], "\n", $CellContext`do, If[$CellContext`RangeF$$, Row[{" ", Style["R", Italic], " = ", If[$CellContext`a$$ == 0, Row[{"{", $CellContext`k$$, "}"}], "\[DoubleStruckCapitalR]"], " "}], ""]}], 25]], $CellContext`parent$$ == 6, Plot[ If[$CellContext`GraphF$$, $CellContext`a$$ Abs[$CellContext`w$$ ($CellContext`x - $CellContext`h$$)] + \ $CellContext`k$$, 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, Row[{ If[ Or[$CellContext`a$$ == 0, $CellContext`w$$ == 0], "0", Row[{ If[$CellContext`a$$ == 1, "", $CellContext`a$$], "|", If[$CellContext`w$$ == 1, "", Row[{$CellContext`w$$, "("}]], $CellContext`x - $CellContext`h$$, If[$CellContext`w$$ == 1, "", ")"], "|"}]], Which[$CellContext`k$$ > 0, Row[{" + ", $CellContext`k$$}], $CellContext`k$$ == 0, " ", $CellContext`k$$ < 0, $CellContext`k$$]}], "?"], "\n", $CellContext`do, If[$CellContext`RangeF$$, Row[{" ", Style["R", Italic], " = ", Which[$CellContext`a$$ > 0, Row[{"[", $CellContext`k$$, ", ", Infinity, ")"}], Or[$CellContext`a$$ == 0, $CellContext`w$$ == 0], Row[{"{", $CellContext`k$$, "}"}], $CellContext`a$$ < 0, Row[{"(", -Infinity, ", ", $CellContext`k$$, "]"}]], " "}], ""]}], 25]], $CellContext`parent$$ == 7, Plot[ If[$CellContext`GraphF$$, If[ EvenQ[$CellContext`n$$], $CellContext`a$$ ($CellContext`w$$ \ ($CellContext`x - $CellContext`h$$))^(1/$CellContext`n$$) + $CellContext`k$$, Piecewise[{{( Sign[$CellContext`w$$] $CellContext`a$$) ( Abs[$CellContext`w$$] ($CellContext`x - \ $CellContext`h$$))^( 1/$CellContext`n$$) + $CellContext`k$$, $CellContext`x - \ $CellContext`h$$ >= 0}, {((-Sign[$CellContext`w$$]) $CellContext`a$$) ( Abs[$CellContext`w$$] (-$CellContext`x + \ $CellContext`h$$))^( 1/$CellContext`n$$) + $CellContext`k$$, $CellContext`x - \ $CellContext`h$$ < 0}}]], 10], {$CellContext`x, -6, 6}, PlotStyle -> {Red, Thick}, AxesStyle -> Thick, BaseStyle -> {14}, PlotRange -> {{-6, 6}, {-6, 6}}, Ticks -> {$CellContext`tickset$, $CellContext`tickset$}, GridLines -> {$CellContext`tickset$, $CellContext`tickset$}, AspectRatio -> Automatic, ImageSize -> {400, 400}, PlotLabel -> Style[ Row[{$CellContext`fx, If[$CellContext`FunctionF$$, Row[{ If[ Or[$CellContext`a$$ == 0, $CellContext`w$$ == 0], "0", Row[{ If[$CellContext`a$$ == 1, "", $CellContext`a$$], Row[{ If[$CellContext`w$$ == 1, "", Row[{$CellContext`w$$, "("}]], $CellContext`x - $CellContext`h$$, If[$CellContext`w$$ == 1, "", ")"]}]^( 1/$CellContext`n$$)}]], Which[$CellContext`k$$ > 0, Row[{" + ", $CellContext`k$$}], $CellContext`k$$ == 0, " ", $CellContext`k$$ < 0, $CellContext`k$$]}], "?"], "\n", If[$CellContext`DomainF$$, If[ EvenQ[$CellContext`n$$], Row[{" ", Style["D", Italic], " = ", Which[$CellContext`w$$ > 0, Row[{"[", $CellContext`h$$, ", ", Infinity, ")"}], $CellContext`w$$ == 0, "\[DoubleStruckCapitalR]", $CellContext`w$$ < 0, Row[{"(" - Infinity, ", ", $CellContext`h$$, "]"}]]}], " ", Row[{ Style["D", Italic], " = \[DoubleStruckCapitalR] "}]], ""], If[$CellContext`RangeF$$, If[ EvenQ[$CellContext`n$$], Row[{" ", Style["R", Italic], " = ", Which[$CellContext`a$$ > 0, Row[{"[", $CellContext`k$$, ", ", Infinity, ")"}], Or[$CellContext`a$$ == 0, $CellContext`w$$ == 0], Row[{"{", $CellContext`k$$, "}"}], $CellContext`a$$ < 0, Row[{"(", -Infinity, ", ", $CellContext`k$$, "]"}]], " "}], " ", Row[{ Style["R", Italic], " = \[DoubleStruckCapitalR] "}]], ""]}], 25]]]], "Specifications" :> {{{$CellContext`parent$$, 5, "parent function"}, { 1 -> "constant", 2 -> "identity", 3 -> "linear", 4 -> "quadratic (or even power)", 5 -> "cubic (or odd power)", 6 -> "absolute value", 7 -> "square root (or \!\(\*SuperscriptBox[\(n\), \(th\)]\) root)"}, ControlPlacement -> Top}, {{$CellContext`c$$, 2}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 1}, OpenerView[{"constant", Manipulate`Place[1]}, Dynamic[ If[$CellContext`parent$$ == 1, True, False]], Enabled -> False], {{$CellContext`m$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 2}, {{$CellContext`b$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 3}, {{$CellContext`resetlinear$$, False, "reset"}, {False, True}, ControlPlacement -> 4}, OpenerView[{"linear", Column[{ Manipulate`Place[2], Manipulate`Place[3], 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Manipulate`InterpretManipulate[1]]], "Output"], "\n" }], "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}, { 3.5963101839730053`*^9, 3.5963103224189243`*^9}, {3.5964839489279113`*^9, 3.5964839568003616`*^9}, {3.5964840233321667`*^9, 3.596484143452038*^9}, { 3.5964842866862297`*^9, 3.59648442148494*^9}, {3.5964844611032057`*^9, 3.596484549518263*^9}, {3.5964849063856745`*^9, 3.596484909959879*^9}}], Cell["\<\ \"A Library of Functions with Transformations\" written for Wolfram \ Demonstrations Project http://demonstrations.wolfram.com/ALibraryOfFunctionsWithTransformations/ is contributed by Ed Zaborowski and is licensed under CC-BY-NC-SA.\ \>", "Text", CellChangeTimes->{{3.6489914771821194`*^9, 3.6489915256348906`*^9}}, TextAlignment->Right] }, Open ]], Cell[CellGroupData[{ Cell["Transformations on Quadratic Functions", "Subtitle", CellFrame->1, CellFrameColor->RGBColor[0, 0, 1], CellChangeTimes->{{3.5965586383116803`*^9, 3.5965586498723416`*^9}}, FontSize->36, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[TextData[{ "\[MathematicaIcon]We can apply similar transformations to the basic \ quadratic graph ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontSize->24, FontWeight->"Plain"], " as well. Let\[CloseCurlyQuote]s begin by adding a constant ", StyleBox["k", FontWeight->"Bold"], " to the ", StyleBox["y-value", FontVariations->{"Underline"->True}], ", so we have ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontSize->24, FontWeight->"Plain"], " + ", StyleBox["k", FontSlant->"Italic", FontColor->RGBColor[0.6, 0.4, 0.2]], ". 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The graph of y = f(x) + k is graphed in ", StyleBox["blue", FontColor->RGBColor[0, 0, 1]], ". \n\na.\tUse the k slider to graph the functions ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontSize->24, FontWeight->"Plain"], " + ", StyleBox["3", FontSlant->"Italic", FontColor->RGBColor[0.6, 0.4, 0.2]], StyleBox[",", FontColor->RGBColor[0.6, 0.4, 0.2]], StyleBox[" ", FontSlant->"Italic", FontColor->RGBColor[0.6, 0.4, 0.2]], "Use the x-value slider to trace the coordinates on both graphs. Sketch \ both graphs and answer the following questions: \n\n\tHow are the \ coordinates of the ", StyleBox["blue", FontColor->RGBColor[0, 0, 1]], " graph different than those on the ", StyleBox["red", FontColor->RGBColor[1, 0, 0]], " graph?\n\tFor the function ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontSize->24, FontWeight->"Plain"], " + ", StyleBox["3,", FontSlant->"Italic", FontColor->RGBColor[0.6, 0.4, 0.2]], " how does the parameter ", StyleBox["3", FontSlant->"Italic", FontColor->RGBColor[0.6, 0.4, 0.2]], StyleBox[" ", FontColor->RGBColor[0.6, 0.4, 0.2]], "shift the graph of", StyleBox[" y", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " ?\n\nb.\tRepeat the process in (a) for the function ", StyleBox["y = ", FontSize->24, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontSize->24, FontWeight->"Plain"], " ", StyleBox["- 1.5", FontSlant->"Italic", FontColor->RGBColor[0.6, 0.4, 0.2]], ". 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