(* 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[ 55539, 1203] NotebookOptionsPosition[ 55000, 1170] NotebookOutlinePosition[ 55414, 1186] CellTagsIndexPosition[ 55371, 1183] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Polynomial Functions of Higher Degree", "Title", CellChangeTimes->{{3.5457550953305087`*^9, 3.5457551112654204`*^9}, { 3.558272773044202*^9, 3.5582727910878305`*^9}}, Background->RGBColor[1, 0.85, 0.85]], Cell[CellGroupData[{ Cell[TextData[{ "How are even-powered functions related? odd-powered functions? \n\n", StyleBox["Example 1:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " In the interactive frame below, choose the \ \[OpenCurlyDoubleQuote]quadratic (or even power)\[CloseCurlyDoubleQuote] \ parent function from the drop-down menu. Use the \[OpenCurlyDoubleQuote]p\ \[CloseCurlyDoubleQuote] slider to change the degree. \n\na.\tHow are the \ graphs of even-powered functions similar? different?\nb.\tIdentify the \ following for even-powered functions:\n\ti.\tShape of f:\n\tii. \tf is \ decreasing on the interval_______.\n\tiii.\tf is increasing on the \ interval_______.\n\tiv.\tf is constant at x = _____.\n\tv.\tDomain of f:\n\t\ vi.\tRange of f:\n\tvii.\tSymmetry of f:\n\tviii.\tf passes through the \ following coordinate points:\nc.\tRepeat questions (a) and (b) for the \ \[OpenCurlyDoubleQuote]cubic (or odd power)\[CloseCurlyDoubleQuote] parent \ function." }], "Section", CellChangeTimes->{{3.5582728232806845`*^9, 3.5582729325056663`*^9}, { 3.5582729855145593`*^9, 3.5582730148710055`*^9}, {3.558273117895185*^9, 3.55827312709*^9}, {3.5971882655162697`*^9, 3.597188294826311*^9}, { 3.5971883336913652`*^9, 3.5971884471965246`*^9}, {3.5971885700766964`*^9, 3.5971886106917534`*^9}, {3.597188654881815*^9, 3.5971887770969863`*^9}, { 3.597188809977032*^9, 3.5971888528720922`*^9}, {3.597189175517544*^9, 3.5971891780825477`*^9}, {3.5972388603196316`*^9, 3.597238885019666*^9}, { 3.6490854260935163`*^9, 3.649085434633005*^9}, {3.649085517567748*^9, 3.6490855258642225`*^9}}, FontSize->22, Background->RGBColor[0.88, 1, 0.88]], 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$$ = 2, $CellContext`parent$$ = 4, $CellContext`po$$ = 5, $CellContext`RangeF$$ = 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$$, False, "reset"}, {False, True}, ControlPlacement -> 4}, OpenerView[{"linear", Column[{ Manipulate`Place[2], Manipulate`Place[3], Manipulate`Place[4]}]}, Dynamic[ If[$CellContext`parent$$ == 3, True, False]], Enabled -> False], {{$CellContext`a$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 5}, {{$CellContext`h$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 6}, {{$CellContext`k$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 7}, {{$CellContext`w$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 8}, {{$CellContext`p$$, 2}, 2, 10, 2, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 9}, {{$CellContext`resetquadratic$$, False, "reset"}, {False, True}, ControlPlacement -> 10}, 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$$ == 4, True, False]], Enabled -> False], {{$CellContext`a$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 11}, {{$CellContext`h$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 12}, {{$CellContext`k$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 13}, {{$CellContext`w$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 14}, {{$CellContext`po$$, 3, $CellContext`p$$}, 1, 11, 2, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 15}, {{$CellContext`resetcubic$$, False, "reset"}, {False, True}, ControlPlacement -> 16}, 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$$ == 5, True, False]], Enabled -> False], {{$CellContext`a$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 17}, {{$CellContext`h$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 18}, {{$CellContext`k$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 19}, {{$CellContext`w$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 20}, {{$CellContext`resetabsolute$$, False, "reset"}, {False, True}, ControlPlacement -> 21}, 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$$ == 6, True, False]], Enabled -> False], {{$CellContext`a$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 22}, {{$CellContext`h$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 23}, {{$CellContext`k$$, 0}, -5, 5, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 24}, {{$CellContext`w$$, 1}, -3, 3, 0.1, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 25}, {{$CellContext`n$$, 2}, 2, 8, 2, Appearance -> "Labeled", ImageSize -> Tiny, ControlPlacement -> 26}, {{$CellContext`resetroot$$, False, "reset"}, {False, True}, ControlPlacement -> 27}, 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$$ == 7, True, False]], Enabled -> False], {{$CellContext`FunctionF$$, True, "function"}, { False, True}}, {{$CellContext`GraphF$$, True, "graph"}, { False, True}}, {{$CellContext`DomainF$$, False, "domain"}, { False, True}}, {{$CellContext`RangeF$$, False, "range"}, { False, True}}}, "Options" :> { ControlPlacement -> Left, AutorunSequencing -> {2}, TrackedSymbols :> {$CellContext`parent$$, $CellContext`c$$, \ $CellContext`m$$, $CellContext`b$$, $CellContext`a$$, $CellContext`h$$, \ $CellContext`k$$, $CellContext`w$$, $CellContext`p$$, $CellContext`po$$, \ $CellContext`n$$, $CellContext`FunctionF$$, $CellContext`GraphF$$, \ $CellContext`DomainF$$, $CellContext`RangeF$$, $CellContext`resetlinear$$, \ $CellContext`resetquadratic$$, $CellContext`resetcubic$$, \ $CellContext`resetabsolute$$, $CellContext`resetroot$$}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{615., {245., 252.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->502838831] }, Open ]] }, Open ]], Cell["\<\ \[OpenCurlyDoubleQuote]Catalog of Some Elementary Functions\ \[CloseCurlyDoubleQuote] from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/CatalogOfSomeElementaryFunctions/ contributed by German Vargas is licensed under CC-BY-NC-SA.\ \>", "Text", CellGroupingRules->{"GroupTogetherGrouping", 0}, CellChangeTimes->{ 3.649086280038359*^9, {3.6490863227988048`*^9, 3.649086336364581*^9}}, TextAlignment->Right], Cell["End Behavior of Polynomial Functions", "Section", CellGroupingRules->{"SectionGrouping", Inherited}, CellChangeTimes->{{3.5582732119849467`*^9, 3.558273222974965*^9}}, FontSize->24, Background->RGBColor[1, 0.9, 0.8]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 8:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Consider the function f(x) in the graph below. What happens to the \ y-values of f when x increases/decreases without bound? Use the slider to \ zoom out. Fill in the blanks:\n\nAs ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "\[LongRightArrow]\[Infinity], f(x) \[LongRightArrow]______.\nAs ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "\[LongRightArrow] -\[Infinity], f(x) \[LongRightArrow]______. \n\n\ Therefore, as |x| \[LongRightArrow]\[Infinity], f(x) \[LongRightArrow]______.\ \n\nDo the y-values of f follow any pattern when x increases/decreases \ without bound? How does f compare to the general monomial function of the \ same degree? Fill in the blank:\n\nAs |x| \[RightArrow] \[Infinity], f(x) \ behaves like the function_____________." }], "Section", CellChangeTimes->{{3.545755125850254*^9, 3.5457552220297556`*^9}, { 3.597250942952325*^9, 3.597250944832327*^9}, {3.597250986202385*^9, 3.5972510057374125`*^9}, {3.597251298522846*^9, 3.5972513205028768`*^9}, { 3.597251359117931*^9, 3.597251399137987*^9}, {3.597255544820793*^9, 3.597255568170826*^9}, {3.5972557361310616`*^9, 3.5972558392212057`*^9}, { 3.597255878631261*^9, 3.5972558968012857`*^9}, {3.597255965381382*^9, 3.5972559902664165`*^9}, {3.5972560509265018`*^9, 3.597256062231518*^9}, { 3.661000125423519*^9, 3.661000126230054*^9}}, FontSize->22, Background->RGBColor[0.88, 1, 0.88]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 0, $CellContext`b$$ = -1, $CellContext`c$$ = 1, $CellContext`d$$ = 1, $CellContext`e$$ = 1, $CellContext`f$$ = -1, $CellContext`n$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 5, "range"}, 1, 120, 1}, { Hold["coefficients of:"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 0, "\!\(\*SuperscriptBox[\(x\), \(5\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`b$$], 0, "\!\(\*SuperscriptBox[\(x\), \(4\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`c$$], 0, "\!\(\*SuperscriptBox[\(x\), \(3\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`d$$], 1, "\!\(\*SuperscriptBox[\(x\), \(2\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`e$$], 1, "\!\(\*SuperscriptBox[\(x\), \(1\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`f$$], 41, "\!\(\*SuperscriptBox[\(x\), \(0\)]\)"}, -100, 100, 1}}, Typeset`size$$ = {350., {131., 136.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`n$165624$$ = 0, $CellContext`a$165625$$ = 0, $CellContext`b$165626$$ = 0, $CellContext`c$165627$$ = 0, $CellContext`d$165628$$ = 0, $CellContext`e$165629$$ = 0, $CellContext`f$165630$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 0, $CellContext`b$$ = 0, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`e$$ = 1, $CellContext`f$$ = 41, $CellContext`n$$ = 5}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$165624$$, 0], Hold[$CellContext`a$$, $CellContext`a$165625$$, 0], Hold[$CellContext`b$$, $CellContext`b$165626$$, 0], Hold[$CellContext`c$$, $CellContext`c$165627$$, 0], Hold[$CellContext`d$$, $CellContext`d$165628$$, 0], Hold[$CellContext`e$$, $CellContext`e$165629$$, 0], Hold[$CellContext`f$$, $CellContext`f$165630$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> With[{$CellContext`func$ = $CellContext`a$$ $CellContext`x^5 + \ $CellContext`b$$ $CellContext`x^4 + $CellContext`c$$ $CellContext`x^3 + \ $CellContext`d$$ $CellContext`x^2 + $CellContext`e$$ $CellContext`x + \ $CellContext`f$$}, Plot[$CellContext`func$, {$CellContext`x, -$CellContext`n$$, \ $CellContext`n$$}, PlotRange -> All, PlotLabel -> $CellContext`func$, ImageSize -> 350, ImagePadding -> {{10, 10}, {10, 35}}]], "Specifications" :> {{{$CellContext`n$$, 5, "range"}, 1, 120, 1, Appearance -> "Labeled", ImageSize -> Tiny}, Delimiter, "coefficients of:", {{$CellContext`a$$, 0, "\!\(\*SuperscriptBox[\(x\), \(5\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`b$$, 0, "\!\(\*SuperscriptBox[\(x\), \(4\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`c$$, 0, "\!\(\*SuperscriptBox[\(x\), \(3\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`d$$, 1, "\!\(\*SuperscriptBox[\(x\), \(2\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`e$$, 1, "\!\(\*SuperscriptBox[\(x\), \(1\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`f$$, 41, "\!\(\*SuperscriptBox[\(x\), \(0\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny, Exclusions -> 0}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{569., {161., 168.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->73546283] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 9:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Consider the degree-5 function ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " below (in blue). The general function ", StyleBox["y = 20", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "5"], TraditionalForm]], FontColor->RGBColor[1, 0, 0]], " is graphed in red. What do you notice as you zoom out? Use the range \ slider. Fill in the blanks:\n\nAs ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "\[LongRightArrow]\[Infinity], ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " \[LongRightArrow]______.\nAs ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "\[LongRightArrow] -\[Infinity], ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " \[LongRightArrow]______. \n\nDo the y-values of ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " follow any pattern when x increases/decreases without bound? How does ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " compare to the graph of ", StyleBox["y = 20", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "5"], TraditionalForm]], FontColor->RGBColor[1, 0, 0]], "? Fill in the blank:\n\nAs |x| \[RightArrow] \[Infinity], ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " behaves like the function_____________." }], "Section", CellChangeTimes->{{3.545755125850254*^9, 3.5457552220297556`*^9}, { 3.56917116387004*^9, 3.569171181925065*^9}, {3.569171243520151*^9, 3.5691713416902885`*^9}, {3.5798060320096836`*^9, 3.5798061015616617`*^9}, { 3.597255355285528*^9, 3.5972553808255634`*^9}, {3.597255453055665*^9, 3.5972555314957743`*^9}, {3.59725560695088*^9, 3.597255607445881*^9}, { 3.5972560098314447`*^9, 3.5972560270514684`*^9}, {3.5972560766365376`*^9, 3.5972561064215794`*^9}, {3.6490031461192865`*^9, 3.649003153306697*^9}, { 3.64908645541539*^9, 3.6490864676750913`*^9}, {3.6490866018657665`*^9, 3.649086604034891*^9}, {3.661000139382786*^9, 3.661000140278381*^9}}, FontSize->22, Background->RGBColor[0.88, 1, 0.88]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 20, $CellContext`b$$ = -7, $CellContext`c$$ = 26, $CellContext`d$$ = -62, $CellContext`e$$ = 1, $CellContext`f$$ = 41, $CellContext`n$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 5, "range"}, 1, 120, 1}, { Hold["coefficients of:"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 0, "\!\(\*SuperscriptBox[\(x\), \(5\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`b$$], 0, "\!\(\*SuperscriptBox[\(x\), \(4\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`c$$], 0, "\!\(\*SuperscriptBox[\(x\), \(3\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`d$$], 1, "\!\(\*SuperscriptBox[\(x\), \(2\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`e$$], 1, "\!\(\*SuperscriptBox[\(x\), \(1\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`f$$], 41, "\!\(\*SuperscriptBox[\(x\), \(0\)]\)"}, -100, 100, 1}}, Typeset`size$$ = {350., {131., 136.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`n$3733$$ = 0, $CellContext`a$3734$$ = 0, $CellContext`b$3735$$ = 0, $CellContext`c$3736$$ = 0, $CellContext`d$3737$$ = 0, $CellContext`e$3738$$ = 0, $CellContext`f$3739$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 0, $CellContext`b$$ = 0, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`e$$ = 1, $CellContext`f$$ = 41, $CellContext`n$$ = 5}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$3733$$, 0], Hold[$CellContext`a$$, $CellContext`a$3734$$, 0], Hold[$CellContext`b$$, $CellContext`b$3735$$, 0], Hold[$CellContext`c$$, $CellContext`c$3736$$, 0], Hold[$CellContext`d$$, $CellContext`d$3737$$, 0], Hold[$CellContext`e$$, $CellContext`e$3738$$, 0], Hold[$CellContext`f$$, $CellContext`f$3739$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> With[{$CellContext`func$ = $CellContext`a$$ $CellContext`x^5 + \ $CellContext`b$$ $CellContext`x^4 + $CellContext`c$$ $CellContext`x^3 + \ $CellContext`d$$ $CellContext`x^2 + $CellContext`e$$ $CellContext`x + \ $CellContext`f$$, $CellContext`func1$ = $CellContext`a$$ $CellContext`x^5}, Plot[{$CellContext`func$, $CellContext`func1$}, {$CellContext`x, \ -$CellContext`n$$, $CellContext`n$$}, PlotRange -> All, PlotLabel -> $CellContext`func$, ImageSize -> 350, ImagePadding -> {{10, 10}, {10, 35}}]], "Specifications" :> {{{$CellContext`n$$, 5, "range"}, 1, 120, 1, Appearance -> "Labeled", ImageSize -> Tiny}, Delimiter, "coefficients of:", {{$CellContext`a$$, 0, "\!\(\*SuperscriptBox[\(x\), \(5\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`b$$, 0, "\!\(\*SuperscriptBox[\(x\), \(4\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`c$$, 0, "\!\(\*SuperscriptBox[\(x\), \(3\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`d$$, 1, "\!\(\*SuperscriptBox[\(x\), \(2\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`e$$, 1, "\!\(\*SuperscriptBox[\(x\), \(1\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny}, {{$CellContext`f$$, 41, "\!\(\*SuperscriptBox[\(x\), \(0\)]\)"}, -100, 100, 1, Appearance -> "Labeled", ImageSize -> Tiny, Exclusions -> 0}}, "Options" :> {ControlPlacement -> Left}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{569., {161., 168.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.35696210375764*^9, 3.390138721382368*^9, 3.569170946724736*^9, 3.5691710051448174`*^9}, CellID->538064199] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Example 10:", FontWeight->"Bold", FontVariations->{"Underline"->True}], " Consider the degree-4 function ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " below (in blue). The general function ", StyleBox["y = 2", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]], FontColor->RGBColor[1, 0, 0]], " is graphed in red. What do you notice as you zoom out? Use the range \ slider. Fill in the blanks:\n\nAs ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "\[LongRightArrow]\[Infinity], ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " \[LongRightArrow]______.\nAs ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "\[LongRightArrow] -\[Infinity], ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " \[LongRightArrow]______. \n\nDo the y-values of ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " follow any pattern when x increases/decreases without bound? How does ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " compare to the graph of ", StyleBox["y = 2", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "4"], TraditionalForm]], FontColor->RGBColor[1, 0, 0]], "? Fill in the blank:\n\nAs |x| \[RightArrow] \[Infinity], ", StyleBox["f(x)", FontColor->RGBColor[0, 0, 1]], " behaves like the function_____________." }], "Section", CellChangeTimes->{{3.545755125850254*^9, 3.5457552220297556`*^9}, { 3.56917116387004*^9, 3.569171181925065*^9}, {3.569171243520151*^9, 3.5691713416902885`*^9}, {3.5798060320096836`*^9, 3.5798061015616617`*^9}, { 3.597255355285528*^9, 3.5972553808255634`*^9}, {3.597255453055665*^9, 3.5972555314957743`*^9}, {3.59725560695088*^9, 3.597255607445881*^9}, { 3.5972560098314447`*^9, 3.5972560270514684`*^9}, {3.5972560766365376`*^9, 3.5972561064215794`*^9}, {3.5972561990617094`*^9, 3.5972562070817204`*^9}, { 3.6490864742174654`*^9, 3.6490864893373303`*^9}, {3.6490865718630505`*^9, 3.649086581793618*^9}, {3.661000144683305*^9, 3.661000144920462*^9}}, FontSize->22, Background->RGBColor[0.88, 1, 0.88]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 0, $CellContext`b$$ = 2, $CellContext`c$$ = 100, $CellContext`d$$ = -26, $CellContext`e$$ = 1, $CellContext`f$$ = 41, $CellContext`n$$ = 38, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 5, "range"}, 1, 120, 1}, { Hold["coefficients of:"], Manipulate`Dump`ThisIsNotAControl}, {{ Hold[$CellContext`a$$], 0, "\!\(\*SuperscriptBox[\(x\), \(5\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`b$$], 0, "\!\(\*SuperscriptBox[\(x\), \(4\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`c$$], 0, "\!\(\*SuperscriptBox[\(x\), \(3\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`d$$], 1, "\!\(\*SuperscriptBox[\(x\), \(2\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`e$$], 1, "\!\(\*SuperscriptBox[\(x\), \(1\)]\)"}, -100, 100, 1}, {{ Hold[$CellContext`f$$], 41, "\!\(\*SuperscriptBox[\(x\), \(0\)]\)"}, -100, 100, 1}}, Typeset`size$$ = {350., {131., 136.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`n$14569$$ = 0, $CellContext`a$14570$$ = 0, $CellContext`b$14571$$ = 0, $CellContext`c$14572$$ = 0, $CellContext`d$14573$$ = 0, $CellContext`e$14574$$ = 0, $CellContext`f$14575$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 0, $CellContext`b$$ = 0, $CellContext`c$$ = 0, $CellContext`d$$ = 1, $CellContext`e$$ = 1, $CellContext`f$$ = 41, $CellContext`n$$ = 5}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$14569$$, 0], Hold[$CellContext`a$$, $CellContext`a$14570$$, 0], Hold[$CellContext`b$$, $CellContext`b$14571$$, 0], Hold[$CellContext`c$$, $CellContext`c$14572$$, 0], Hold[$CellContext`d$$, $CellContext`d$14573$$, 0], Hold[$CellContext`e$$, $CellContext`e$14574$$, 0], Hold[$CellContext`f$$, $CellContext`f$14575$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, 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