(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 9.0' *) (*************************************************************************) (* *) (* 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. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1063, 20] NotebookDataLength[ 104741, 2314] NotebookOptionsPosition[ 104010, 2273] NotebookOutlinePosition[ 104605, 2295] CellTagsIndexPosition[ 104562, 2292] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["End Behavior of Rational Functions", "Title", CellChangeTimes->{{3.5450622213804007`*^9, 3.5450622330400677`*^9}, { 3.545755784099904*^9, 3.545755786889064*^9}, {3.590329468786028*^9, 3.590329469659629*^9}, {3.5903396103261094`*^9, 3.5903396181445565`*^9}, { 3.6483928415504723`*^9, 3.6483928433815775`*^9}}, Background->RGBColor[0.94, 0.88, 0.94]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["\[MathematicaIcon]How do the numerator and denominator of a \ rational function affect the overall graph? ", FontWeight->"Bold"], "\n\nSuppose that a rational function f(x) = ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"N", "(", "x", ")"}], RowBox[{"D", "(", "x", ")"}]], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"N", "(", "x", ")"}], RowBox[{"D", "(", "x", ")"}]], TraditionalForm]]], " is in simplest form. The solutions of N(x) = 0 are the ZEROS of f(x).\n\n\ Now, consider the degrees of N(x) and D(x). The following describes the ", StyleBox["End Behavior", FontWeight->"Bold", FontSlant->"Italic"], " of f(x), or how the y-values respond to extremely large values of |x|. In \ other words, what happens to the y-\tvalues of the function as |x| \ \[RightArrow]\[Infinity]? \n" }], "Section", CellChangeTimes->{{3.5903295552569795`*^9, 3.5903295803886237`*^9}, { 3.5903297216936717`*^9, 3.590329839707879*^9}, {3.5903299128876076`*^9, 3.5903301201339717`*^9}, {3.5903314608939266`*^9, 3.5903314728435473`*^9}, {3.592320870162917*^9, 3.5923208900529447`*^9}, { 3.5923209463180237`*^9, 3.592321006233107*^9}, {3.5925596476556044`*^9, 3.592559659535621*^9}, {3.592559930696001*^9, 3.5925599498560276`*^9}, { 3.592559985916078*^9, 3.59255998729108*^9}, {3.5927408204002123`*^9, 3.5927408315688515`*^9}, {3.592907792686829*^9, 3.592907804277492*^9}, 3.5938783988924737`*^9}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["There are THREE TYPES of ", FontWeight->"Bold"], StyleBox["End Behavior", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" in a rational function.\n", FontWeight->"Bold"], "First, find", StyleBox[" ", FontWeight->"Bold"], "the ", StyleBox["degree", FontSlant->"Italic"], " of each polynomial function, N(x) and D(x)." }], "Section", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.5903301713956614`*^9, 3.5903303074591007`*^9}, { 3.5903303476291714`*^9, 3.5903309586198444`*^9}, {3.5903311809982347`*^9, 3.5903313537685385`*^9}, {3.5903314308014736`*^9, 3.5903314363394833`*^9}, {3.592321071893199*^9, 3.592321107128248*^9}, { 3.5923211818483534`*^9, 3.5923211842233562`*^9}, {3.5925599798410697`*^9, 3.5925601276162767`*^9}, {3.5925602999405336`*^9, 3.5925603037355385`*^9}, {3.5925604175956984`*^9, 3.5925604775157824`*^9}, {3.5925605496648912`*^9, 3.5925606648800526`*^9}, {3.5925607086151133`*^9, 3.592560743615163*^9}, { 3.59256078469022*^9, 3.5925610405455785`*^9}, {3.5925611423657207`*^9, 3.5925612176858263`*^9}, {3.5925612507958727`*^9, 3.5925612696858993`*^9}, {3.592561347796008*^9, 3.5925614422511406`*^9}, { 3.592561489486207*^9, 3.59256162022639*^9}, {3.5925616893754897`*^9, 3.5925617448555675`*^9}, {3.5925618085656567`*^9, 3.5925619056957927`*^9}, {3.5925619891859093`*^9, 3.5925620301059666`*^9}, {3.5925622794633512`*^9, 3.592562289723366*^9}, { 3.5925624658836117`*^9, 3.592562505753668*^9}, {3.5925628033340845`*^9, 3.5925628440591416`*^9}, 3.5925628969142156`*^9, {3.592588780142481*^9, 3.592588781057482*^9}, {3.592588873242611*^9, 3.5925888890926332`*^9}, { 3.5925889386127024`*^9, 3.5925889521577215`*^9}, {3.5925889908477755`*^9, 3.592589097272925*^9}, {3.5925891357179785`*^9, 3.592589196923064*^9}, { 3.592740837705202*^9, 3.592741061262989*^9}, {3.592907957595261*^9, 3.592908042524119*^9}, {3.5929134517188373`*^9, 3.592913461491396*^9}}, TextAlignment->Left, FontColor->RGBColor[0, 0.67, 0]], Cell[TextData[{ "\n\n1.\tIf ", StyleBox["deg(N(x)) < deg(D(x))", FontColor->RGBColor[1, 0, 0]], ", then the ", StyleBox["denominator", FontSlant->"Italic"], " of the function will increase ", StyleBox["faster", FontSlant->"Italic"], " than the ", StyleBox["numerator", FontSlant->"Italic"], " for large values of |x|, which causes y \[RightArrow] 0. Consequently, \ the graph gets closer and closer to y = 0 for |x| \[RightArrow] \[Infinity], \ \tapproaching a horizontal asymptote at y = 0. \n\n\t", StyleBox["Example 1:", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" Consider the function f(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"3", "x"}], RowBox[{ SuperscriptBox["x", "3"], "-", "8"}]], TraditionalForm]], FontColor->GrayLevel[0]], ". ", StyleBox["Choose N(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{"3", "x"}], TraditionalForm]], FontWeight->"Plain", FontColor->GrayLevel[0]], " ", StyleBox["and D(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "-", "8"}], TraditionalForm]], FontColor->GrayLevel[0]], " ", StyleBox["from the drop-down menu in the interactive frame below. Check the \ \"Show Asymptotes\" box. \n\t\n\t", FontColor->GrayLevel[0]], StyleBox["Question 1:", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\t(a) \tProvide a sketch the graph of f. \n\t\t\t\t\t(b) Find \ the equations of any vertical and horizontal asymptotes. (Use the domain, \ range and zoom sliders, if necessary.)\n\t\t\t\t\t(c)\tFind f(10) and f(100), \ writing their answers in fractional form. Fill in the blank:\n\t\t\t\t\t\t\t\ As x \[RightArrow] \[Infinity], y \[RightArrow] ____.\n\t\t\t\t\t(d)\tFind \ f(-10) and f(-100), writing their answers in fractional form. Fill in the \ blank:\n\t\t\t\t\t\t\tAs x \[RightArrow] - \[Infinity], y \[RightArrow] ____.\ \n\t\t\t\t\t(e)\tFill in the blanks: \n\t\t\t\t\t\t\tThe degree of N(x) \ ______ (", FontColor->GrayLevel[0]], StyleBox["<, >, or =", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") the degree of D(x).\n\t\t\t\t\t\t\tThe D(x) increases ________(", FontColor->GrayLevel[0]], StyleBox["faster, slower, at the same rate", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") as N(x). \n\t\t\t\t\t\t", FontColor->GrayLevel[0]] }], "Section", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.5903301713956614`*^9, 3.5903303074591007`*^9}, { 3.5903303476291714`*^9, 3.5903309586198444`*^9}, {3.5903311809982347`*^9, 3.5903313537685385`*^9}, {3.5903314308014736`*^9, 3.5903314363394833`*^9}, {3.592321071893199*^9, 3.592321107128248*^9}, { 3.5923211818483534`*^9, 3.5923211842233562`*^9}, {3.5925599798410697`*^9, 3.5925601276162767`*^9}, {3.5925602999405336`*^9, 3.5925603037355385`*^9}, {3.5925604175956984`*^9, 3.5925604775157824`*^9}, {3.5925605496648912`*^9, 3.5925606648800526`*^9}, {3.5925607086151133`*^9, 3.592560743615163*^9}, { 3.59256078469022*^9, 3.5925610405455785`*^9}, {3.5925611423657207`*^9, 3.5925612176858263`*^9}, {3.5925612507958727`*^9, 3.5925612696858993`*^9}, {3.592561347796008*^9, 3.5925614422511406`*^9}, { 3.592561489486207*^9, 3.59256162022639*^9}, {3.5925616893754897`*^9, 3.5925617448555675`*^9}, {3.5925618085656567`*^9, 3.5925619056957927`*^9}, {3.5925619891859093`*^9, 3.5925620301059666`*^9}, {3.5925622794633512`*^9, 3.592562289723366*^9}, { 3.5925624658836117`*^9, 3.592562505753668*^9}, {3.5925628033340845`*^9, 3.5925628440591416`*^9}, 3.5925628969142156`*^9, {3.592588780142481*^9, 3.592588781057482*^9}, {3.592588873242611*^9, 3.5925888890926332`*^9}, { 3.5925889386127024`*^9, 3.5925889521577215`*^9}, {3.5925889908477755`*^9, 3.592589097272925*^9}, {3.5925891357179785`*^9, 3.592589196923064*^9}, { 3.592740837705202*^9, 3.592741061262989*^9}, {3.592907957595261*^9, 3.592908042524119*^9}, {3.5929134517188373`*^9, 3.592913461491396*^9}, { 3.6483929456114244`*^9, 3.6483929808274384`*^9}, 3.648394264108838*^9, 3.6483968432493563`*^9}, TextAlignment->Left, FontColor->RGBColor[0, 0.67, 0]] }, Open ]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`dom$$ = {-5, 5}, $CellContext`optns$$ = {$CellContext`asymflg}, $CellContext`p$$ = -6 + \ $CellContext`x + 2 $CellContext`x^2, $CellContext`q$$ = -4 + $CellContext`x^2, \ $CellContext`ran$$ = {-10, 10}, $CellContext`zoom$$ = 2, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`p$$], -4 - 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The leading coefficients of N(x) and D(x) affect \ the outcome in the long run as the other terms are eventually insignificant \ in comparison. The y-values of the graph get closer and closer to the ratio \ given by:\n\n\t\t\t\t\t\ty = ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"leading", " ", "coefficient", " ", "of", " ", "N", RowBox[{"(", "x", ")"}]}], RowBox[{"leading", " ", "coefficient", " ", "of", " ", RowBox[{"D", "(", "x", ")"}]}]], TraditionalForm]]], "\n\t\t\t\t\t\n\t", StyleBox["Example 2:", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" Consider the function g(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "+", "x", "-", "6"}], SuperscriptBox["x", "2"]], TraditionalForm]], FontColor->GrayLevel[0]], ". ", StyleBox["Choose N(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "+", "x", "-", "6"}], TraditionalForm]], FontWeight->"Plain", FontColor->GrayLevel[0]], " ", StyleBox["and D(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontColor->GrayLevel[0]], " ", StyleBox["from the drop-down menu in the interactive frame below. Check the \ \"Show Asymptotes\" box. \n\t\n\t", FontColor->GrayLevel[0]], StyleBox["Question 2:", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\t(a) \tProvide a sketch the graph of g. \n\t\t\t\t\t(b) Find \ the equations of any vertical and horizontal asymptotes. (Use the domain, \ range and zoom sliders, if necessary.)\n\t\t\t\t\t(c)\tFind f(10) and f(100), \ writing their answers in fractional form. Fill in the blank:\n\t\t\t\t\t\t\t\ As x \[RightArrow] \[Infinity], y \[RightArrow] ____.\n\t\t\t\t\t(d)\tFind \ f(-10) and f(-100), writing their answers in fractional form. Fill in the \ blank:\n\t\t\t\t\t\t\tAs x \[RightArrow] - \[Infinity], y \[RightArrow] ____.\ \n\t\t\t\t\t(e)\tFill in the blanks: \n\t\t\t\t\t\t\tThe degree of N(x) \ ______ (", FontColor->GrayLevel[0]], StyleBox["<, >, or =", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") the degree of D(x).\n\t\t\t\t\t\t\tThe D(x) increases ________(", FontColor->GrayLevel[0]], StyleBox["faster, slower, at the same rate", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") as N(x). \n\t\t\t\t\t(f)\tNotice that g(x) may be rewritten in \ mixed-polynomial form using division (review long division of polynomials, if \ necessary).\n\t\t\t\n\t\t\t\t\t\t\tf(x) = 2 + ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"x", " ", "-", " ", "6"}], SuperscriptBox["x", "2"]], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[" \n\t\t\t\t", FontColor->GrayLevel[0]], " \t \t\t\t \[UpArrow] \[UpArrow]\n\t\t\t\t \t\t\t", StyleBox["quotient remainder\n\t\t\t\t \t\t(polynomial) (rational \ function)\n\t\t\t\t\t\t\t ", FontSize->16], "\n\t\t\t\t\t", StyleBox["(g)", FontColor->GrayLevel[0]], "\t", StyleBox["For the ", FontColor->GrayLevel[0]], StyleBox["remainder function", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[", fill in the blank.", FontColor->GrayLevel[0]], "\n\t\t\t\n\t\t\t\t\t\t\t", StyleBox["As |x| \[RightArrow] \[Infinity], ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"x", " ", "-", " ", "6"}], SuperscriptBox["x", "2"]], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[" \[RightArrow] ______. \n\t\t\t\t\n\t\t\t\t\t(h)\tFill in the \ blanks for the overall function g(x).\n\t\t\n\t\t\t\t\t\t\tAs |x| \ \[RightArrow] \[Infinity], g(x) \[RightArrow] ______.\n\t\t\t\t\t\t\tThis \ means that g(x) approaches a _________asymptote, which has equation y = \ ______.\n\t\t\t\t", FontColor->GrayLevel[0]] }], "Section", CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.5903301713956614`*^9, 3.5903303074591007`*^9}, { 3.5903303476291714`*^9, 3.5903309586198444`*^9}, {3.5903311809982347`*^9, 3.5903313537685385`*^9}, {3.5903314308014736`*^9, 3.5903314363394833`*^9}, {3.592321071893199*^9, 3.592321107128248*^9}, { 3.5923211818483534`*^9, 3.5923211842233562`*^9}, {3.5925599798410697`*^9, 3.5925601276162767`*^9}, {3.5925602999405336`*^9, 3.5925603037355385`*^9}, {3.5925604175956984`*^9, 3.5925604775157824`*^9}, {3.5925605496648912`*^9, 3.5925606648800526`*^9}, {3.5925607086151133`*^9, 3.592560743615163*^9}, { 3.59256078469022*^9, 3.5925610405455785`*^9}, {3.5925611423657207`*^9, 3.5925612176858263`*^9}, {3.5925612507958727`*^9, 3.5925612696858993`*^9}, {3.592561347796008*^9, 3.5925614422511406`*^9}, { 3.592561489486207*^9, 3.59256162022639*^9}, {3.5925616893754897`*^9, 3.5925617448555675`*^9}, {3.5925618085656567`*^9, 3.5925619056957927`*^9}, {3.5925619891859093`*^9, 3.5925620301059666`*^9}, {3.5925622794633512`*^9, 3.592562289723366*^9}, { 3.5925624658836117`*^9, 3.592562505753668*^9}, {3.5925628033340845`*^9, 3.5925628440591416`*^9}, 3.5925628969142156`*^9, {3.592588780142481*^9, 3.592588781057482*^9}, {3.592588873242611*^9, 3.5925888890926332`*^9}, { 3.5925889386127024`*^9, 3.5925889521577215`*^9}, {3.5925889908477755`*^9, 3.592589097272925*^9}, {3.5925891357179785`*^9, 3.592589196923064*^9}, { 3.592740837705202*^9, 3.592741061262989*^9}, {3.592907957595261*^9, 3.592908042524119*^9}, {3.5929134517188373`*^9, 3.592913461491396*^9}, { 3.6483929456114244`*^9, 3.6483929808274384`*^9}, 3.648394264108838*^9, { 3.6483968432493563`*^9, 3.6483968555510597`*^9}}, TextAlignment->Left, FontColor->RGBColor[0, 0.67, 0]], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`dom$$ = {-5, 5}, $CellContext`optns$$ = {$CellContext`asymflg}, $CellContext`p$$ = -6 + \ $CellContext`x + 2 $CellContext`x^2, $CellContext`q$$ = -4 + $CellContext`x^2, \ $CellContext`ran$$ = {-10, 10}, $CellContext`zoom$$ = 2, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`p$$], -4 - 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$CellContext`x + \ $CellContext`x^2 -> TraditionalForm[-2 - $CellContext`x + $CellContext`x^2], -8 + \ $CellContext`x^3 -> TraditionalForm[-8 + $CellContext`x^3], (-3 + $CellContext`x) (-1 + \ $CellContext`x^2) -> TraditionalForm[(-3 + $CellContext`x) (-1 + $CellContext`x^2)], -4 - 3 $CellContext`x^2 + $CellContext`x^4 -> TraditionalForm[-4 - 3 $CellContext`x^2 + $CellContext`x^4], (-4 + $CellContext`x)^2 (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3) -> TraditionalForm[(-4 + $CellContext`x)^2 (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3)], 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4 -> TraditionalForm[ 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4]}}, {{ Hold[$CellContext`dom$$], {-5, 5}, Dynamic[ Tooltip[ Row[{"domain ", $CellContext`zoom$$ $CellContext`dom$$}], "Adjust the minimum and maximum values for \!\(\*\nStyleBox[\"x\",\n\ FontSlant->\"Italic\"]\)."]]}, {-20, 1}, {-1, 20}, {1, 1}}, {{ Hold[$CellContext`ran$$], {-10, 10}, Dynamic[ Tooltip[ Row[{"range ", $CellContext`zoom$$ $CellContext`ran$$}], "Adjust the minimum and maximum values for \!\(\*\nStyleBox[\"y\",\n\ FontSlant->\"Italic\"]\)."]]}, {-40, 1}, {-1, 40}, {1, 1}}, {{ Hold[$CellContext`zoom$$], 1, Dynamic[ Tooltip[ StringForm["zoom \[Times] `` ", $CellContext`zoom$$], "Adjust perspective."]]}, 1, 20, 1}, {{ Hold[$CellContext`optns$$], {}, Tooltip[ Row[{ Spacer[40], "graph options"}], "Display asymptotes and removable discontinuities, if any."]}, \ {$CellContext`asymflg -> " show asymptotes", $CellContext`holeflg -> " show holes"}}}, Typeset`size$$ = {360., {111., 114.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`p$15042$$ = False, $CellContext`q$15043$$ = False, $CellContext`dom$15044$$ = {0, 0}, $CellContext`ran$15045$$ = {0, 0}, $CellContext`zoom$15046$$ = 0, $CellContext`optns$15047$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`dom$$ = {-5, 5}, $CellContext`optns$$ = {}, $CellContext`p$$ = -4 - 3 $CellContext`x^2 + $CellContext`x^4, $CellContext`q$$ = -2 - \ $CellContext`x + $CellContext`x^2, $CellContext`ran$$ = {-10, 10}, $CellContext`zoom$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`p$$, $CellContext`p$15042$$, False], Hold[$CellContext`q$$, $CellContext`q$15043$$, False], Hold[$CellContext`dom$$, $CellContext`dom$15044$$, {0, 0}], Hold[$CellContext`ran$$, $CellContext`ran$15045$$, {0, 0}], Hold[$CellContext`zoom$$, $CellContext`zoom$15046$$, 0], Hold[$CellContext`optns$$, $CellContext`optns$15047$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> (Off[ MessageName[Infinity, "indet"], MessageName[Power, "infy"], MessageName[Graphics, "gptn"]]; If[ And[ PolynomialQ[$CellContext`p$$, $CellContext`x], PolynomialQ[$CellContext`q$$, $CellContext`x]], Show[$CellContext`r = 0.01; $CellContext`f1[ Pattern[$CellContext`x$, Blank[]]] := Expand[$CellContext`p$$]/Expand[$CellContext`q$$]; $CellContext`f2[ Pattern[$CellContext`x, Blank[]]] := Evaluate[ Together[ Cancel[ $CellContext`f1[$CellContext`x]]]]; $CellContext`verasm = Solve[ Reduce[Denominator[ $CellContext`f2[$CellContext`x]] == 0, Reals]]; $CellContext`removdis = Solve[ And[ Reduce[$CellContext`q$$ == 0, Reals], Denominator[ $CellContext`f2[$CellContext`x]] != 0], $CellContext`x]; Plot[ $CellContext`f2[$CellContext`x], {$CellContext`x, \ $CellContext`zoom$$ Part[$CellContext`dom$$, 1], $CellContext`zoom$$ Part[$CellContext`dom$$, 2]}, Exclusions -> {Denominator[ $CellContext`f2[$CellContext`x]] == 0}, PlotStyle -> RGBColor[0, 0, 0], PlotRange -> {$CellContext`zoom$$ Part[$CellContext`ran$$, 1], $CellContext`zoom$$ Part[$CellContext`ran$$, 2]}, ImageSize -> {400, 250}], If[ And[$CellContext`verasm != {}, MemberQ[$CellContext`optns$$, $CellContext`asymflg]], Graphics[ ReplaceAll[{ Hue[0.7], Dashing[{0.02, 0.02}], Line[{{$CellContext`x, $CellContext`zoom$$ Part[$CellContext`ran$$, 1]}, {$CellContext`x, $CellContext`zoom$$ Part[$CellContext`ran$$, 2]}}]}, $CellContext`verasm]], {}], If[ And[ MemberQ[$CellContext`optns$$, $CellContext`asymflg], Not[ PolynomialQ[ $CellContext`f2[$CellContext`x], $CellContext`x]]], Plot[ Evaluate[ PolynomialQuotient[$CellContext`p$$, $CellContext`q$$, \ $CellContext`x]], {$CellContext`x, $CellContext`zoom$$ Part[$CellContext`dom$$, 1], $CellContext`zoom$$ Part[$CellContext`dom$$, 2]}, PlotStyle -> { Hue[0.7], Dashing[{0.02, 0.02}]}], {}], If[ And[$CellContext`removdis != {}, MemberQ[$CellContext`optns$$, $CellContext`holeflg]], Epilog -> { RGBColor[1, 0, 0], ReplaceAll[{{ GrayLevel[1], Disk[{$CellContext`x, $CellContext`f2[$CellContext`x]}, Scaled[$CellContext`r {1, GoldenRatio}]]}, Circle[{$CellContext`x, $CellContext`f2[$CellContext`x]}, Scaled[$CellContext`r { 1, GoldenRatio}]]}, $CellContext`removdis]}, {}]], "N(x) and D(x) must both be polynomials in x.", {0, 0}]), "Specifications" :> {{{$CellContext`p$$, -4 - 3 $CellContext`x^2 + $CellContext`x^4, Style["N(x) = ", Small]}, { 1 -> 1, 3 $CellContext`x -> 3 $CellContext`x, -1 + $CellContext`x -> TraditionalForm[-1 + $CellContext`x], -6 + $CellContext`x + 2 $CellContext`x^2 -> TraditionalForm[-6 + $CellContext`x + 2 $CellContext`x^2], -8 + $CellContext`x^3 -> TraditionalForm[-8 + $CellContext`x^3], ( 1 + $CellContext`x) (-8 + $CellContext`x^3) -> TraditionalForm[(1 + $CellContext`x) (-8 + $CellContext`x^3)], -4 - 3 $CellContext`x^2 + $CellContext`x^4 -> TraditionalForm[-4 - 3 $CellContext`x^2 + $CellContext`x^4], (-4 + $CellContext`x)^2 \ (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3) -> TraditionalForm[(-4 + $CellContext`x)^2 (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3)], 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4 -> TraditionalForm[ 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4]}, ControlType -> PopupMenu}, {{$CellContext`q$$, -2 - $CellContext`x + \ $CellContext`x^2, Style["D(x) = ", Small]}, { 1 -> 1, $CellContext`x -> $CellContext`x, $CellContext`x^2 -> \ $CellContext`x^2, $CellContext`x (1 + $CellContext`x)^2 -> $CellContext`x ( 1 + $CellContext`x)^2, -4 + $CellContext`x^2 -> TraditionalForm[-4 + $CellContext`x^2], -2 - $CellContext`x + \ $CellContext`x^2 -> TraditionalForm[-2 - $CellContext`x + $CellContext`x^2], -8 + \ $CellContext`x^3 -> TraditionalForm[-8 + $CellContext`x^3], (-3 + $CellContext`x) (-1 + \ $CellContext`x^2) -> TraditionalForm[(-3 + $CellContext`x) (-1 + $CellContext`x^2)], -4 - 3 $CellContext`x^2 + $CellContext`x^4 -> TraditionalForm[-4 - 3 $CellContext`x^2 + $CellContext`x^4], (-4 + $CellContext`x)^2 \ (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3) -> TraditionalForm[(-4 + $CellContext`x)^2 (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3)], 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4 -> TraditionalForm[ 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4]}, ControlType -> PopupMenu}, {{$CellContext`dom$$, {-5, 5}, Dynamic[ Tooltip[ Row[{"domain ", $CellContext`zoom$$ $CellContext`dom$$}], "Adjust the minimum and maximum values for \!\(\*\n\ StyleBox[\"x\",\nFontSlant->\"Italic\"]\)."]]}, {-20, 1}, {-1, 20}, {1, 1}, ControlType -> Slider}, {{$CellContext`ran$$, {-10, 10}, Dynamic[ Tooltip[ Row[{"range ", $CellContext`zoom$$ $CellContext`ran$$}], "Adjust the minimum and maximum values for \!\(\*\n\ StyleBox[\"y\",\nFontSlant->\"Italic\"]\)."]]}, {-40, 1}, {-1, 40}, {1, 1}, ControlType -> Slider}, {{$CellContext`zoom$$, 1, Dynamic[ Tooltip[ StringForm["zoom \[Times] `` ", $CellContext`zoom$$], "Adjust perspective."]]}, 1, 20, 1, ControlType -> Slider}, {{$CellContext`optns$$, {}, Tooltip[ Row[{ Spacer[40], "graph options"}], "Display asymptotes and removable discontinuities, if any."]}, \ {$CellContext`asymflg -> " show asymptotes", $CellContext`holeflg -> " show holes"}, ControlType -> CheckboxBar, ControlPlacement -> Top}}, "Options" :> { TrackedSymbols -> {$CellContext`p$$, $CellContext`q$$, \ $CellContext`dom$$, $CellContext`ran$$, $CellContext`asymflg, \ $CellContext`holeflg, $CellContext`optns$$, $CellContext`zoom$$}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{405., {228., 233.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.5903319304984126`*^9}, CellID->253972386], Cell[TextData[{ StyleBox["\n\t\t\t\t", FontColor->GrayLevel[0]], "\n3.\tIf ", StyleBox["deg(N(x)) > deg(D(x))", FontColor->RGBColor[1, 0, 0]], ", then the numerator will increase faster than the denominator for large \ values of |x|, which causes y \[RightArrow] \[Infinity]. There is no \ horizontal asympotote in this case.\n\n\t", StyleBox["Example 3:", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" Consider the function h(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ SuperscriptBox["x", "3"], "-", " ", "8"}], "x"], TraditionalForm]], FontColor->GrayLevel[0]], ". ", StyleBox["Choose N(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "-", " ", "8"}], TraditionalForm]], FontWeight->"Plain", FontColor->GrayLevel[0]], " ", StyleBox["and D(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox["x", TraditionalForm]], FontColor->GrayLevel[0]], " ", StyleBox["from the drop-down\tmenu in the interactive frame below. Check \ the \"Show Asymptotes\" box. \n\t\n\t", FontColor->GrayLevel[0]], StyleBox["Question 3:", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\t(a) \tProvide a sketch the graph of h. \n\t\t\t\t\t(b) \tFind \ the equations of any vertical and horizontal asymptotes. (Use the domain, \ range and zoom sliders, if necessary.)\n\t\t\t\t\t(c)\tFind f(10) and f(100), \ writing their answers in fractional form. Fill in the blank:\n\t\t\t\t\t\t\t\ As x \[RightArrow] \[Infinity], y \[RightArrow] ____.\n\t\t\t\t\t(d)\tFind \ f(-10) and f(-100), writing their answers in fractional form. 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3 $CellContext`x^2 + $CellContext`x^4, Style["N(x) = ", Small]}, { 1 -> 1, 3 $CellContext`x -> 3 $CellContext`x, -1 + $CellContext`x -> TraditionalForm[-1 + $CellContext`x], -6 + $CellContext`x + 2 $CellContext`x^2 -> TraditionalForm[-6 + $CellContext`x + 2 $CellContext`x^2], -8 + $CellContext`x^3 -> TraditionalForm[-8 + $CellContext`x^3], ( 1 + $CellContext`x) (-8 + $CellContext`x^3) -> TraditionalForm[(1 + $CellContext`x) (-8 + $CellContext`x^3)], -4 - 3 $CellContext`x^2 + $CellContext`x^4 -> TraditionalForm[-4 - 3 $CellContext`x^2 + $CellContext`x^4], (-4 + $CellContext`x)^2 \ (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3) -> TraditionalForm[(-4 + $CellContext`x)^2 (-6 + 17 $CellContext`x - 14 $CellContext`x^2 + 3 $CellContext`x^3)], 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4 -> TraditionalForm[ 2 - $CellContext`x - 5 $CellContext`x^2 + $CellContext`x^3 + 3 $CellContext`x^4]}, ControlType -> PopupMenu}, {{$CellContext`q$$, -2 - 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This brings up a special case in #3 \ above. ", FontColor->RGBColor[0, 0.67, 0]], StyleBox["\n\n\t", FontColor->RGBColor[0, 0, 1]], StyleBox["Example 4:", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" In the panel below, choose the following from the dropboxes:\n\t\ \t\t\tN(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "3"], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["- 8\n\t\t\t\tD(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox["\t \n\t\n\tNotice that the ", FontColor->GrayLevel[0]], StyleBox["difference", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" of the degree of the numerator and the degree of the denominator \ is ", FontColor->GrayLevel[0]], StyleBox["1", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[".\n\tClick the \[OpenCurlyDoubleQuote]show asymptotes\ \[CloseCurlyDoubleQuote] box. Use the slider to zoom out. Notice that the \ graph follows a ", FontColor->GrayLevel[0]], StyleBox["linear", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" pattern for large values of |x|, approaching a ", FontColor->GrayLevel[0]], StyleBox["slant", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" (or ", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox["oblique", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[") ", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox["asymptote", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[".", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox["\n\n\tWhat is the equation of this slant asymptote? Using \ polynomial division, we can write f(x) in mixed-polynomial form:", FontColor->GrayLevel[0]], StyleBox["\n\t\t\t\t", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", FractionBox[ RowBox[{ SuperscriptBox["x", "3"], "-", "8"}], RowBox[{ SuperscriptBox["x", "2"], " "}]]}], TraditionalForm]], TextAlignment->Center], " = x - ", Cell[BoxData[ FormBox[ FractionBox["8", SuperscriptBox["x", "2"]], TraditionalForm]]], "\n\t\t\t\t ", StyleBox[" \[UpArrow] \[UpArrow]\n\t\t\t\t\t quotient \ remainder\n\t\t\t\t (polynomial) (rational function)", FontSize->13, FontColor->RGBColor[0, 0.67, 0]], "\n", StyleBox["\n\t", FontSize->16], StyleBox["So, for |x|\[RightArrow] \[Infinity], the remainder polynomial \ gets closer and closer to 0, so in general, f(x) \[RightArrow] x.\n\tThe \ slant asympote, therefore, has equation y = x.\n\t \n\t ", FontColor->GrayLevel[0]], StyleBox["Question 4", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[":", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" Consider the function f(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ StyleBox["3", FontColor->GrayLevel[0]], FormBox[ RowBox[{ SuperscriptBox["x", "4"], "+", SuperscriptBox["x", "3"], "-", RowBox[{"5", SuperscriptBox["x", "2"]}], "-", "x", "+", "2"}], TraditionalForm]}], RowBox[{ SuperscriptBox["x", "3"], "-", "8"}]], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[". Choose N(x) = 3", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "4"], "+", SuperscriptBox["x", "3"], "-", RowBox[{"5", SuperscriptBox["x", "2"]}], "-", "x", "+", "2"}], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["and D(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "-", "8"}], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[" from the drop-down menu in the interactive frame below. Check \ the \"Show \t\t\t\t \t\t\t\t\t\t\t Asymptotes\" box. \n\t \n\t \t(a) \t\ Begin with the zoom slider at 1. Zoom out using the slider and notice how \ the graph behaves for large values of |x|. Provide ", FontColor->GrayLevel[0]], StyleBox["three", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" sketches of f, at the \"zoom x 3\", \"zoom x 5\" and \"zoom x \ 10\" settings. \t\t\t\t\t", FontColor->GrayLevel[0]], "\n", StyleBox["\t\t\t\tFill in the blanks:\n\t\t\t\tThe difference of the degree \ of N(x) and the degree of D(x) is _____.\n\t\t\t\tThe graph of f(x) follows a \ ______pattern for large values of |x|.\n\t\t\t\t\n\t\t(b)\tUsing long \ division of polynomials, rewrite f(x) in mixed-polynomial form, filling in \ the blanks below (the first two blanks are numbers, the third is a \ polynomial):\n\t\t\n\t\t\t\tf(x) = ___x + ___ + ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox["_", RowBox[{ SuperscriptBox["x", "3"], "-", "8"}]], TraditionalForm]], FontColor->GrayLevel[0]], "\n\t\t\t", StyleBox["\t\n\t\t\tWhat is the pattern that f(x) follows for large values \ of |x|? Fill in the blank:\n\t\t\t\n\t\t\t\tAs |x| \[RightArrow] \ \[Infinity], f(x) \[RightArrow] ______. 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Choose the appropriate N(x) and D(x) functions from the drop-\t\ down menu below. Check the \[OpenCurlyDoubleQuote]Show Asymptotes\ \[CloseCurlyDoubleQuote] and \[OpenCurlyDoubleQuote]Show Holes\ \[CloseCurlyDoubleQuote] boxes.\n\t\n\t", FontColor->GrayLevel[0]], StyleBox["Question 5", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[":", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" \t(a)\tFactor the functions N(x) and D(x) completely. Fill in \ the blanks. (review factoring, if necessary)\n\t\n\t\t\t\t\t\t\tf(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"(", " ", ")"}], " ", RowBox[{"(", " ", ")"}]}], RowBox[{ RowBox[{"(", " ", ")"}], " ", RowBox[{"(", " ", ")"}]}]], TraditionalForm]], FontColor->GrayLevel[0]], "\n\t\t\t\t\n\t\t\t\t\t", StyleBox["(b) \tDo you notice a common factor in N(x) and D(x)? 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As seen in the examples above, \ writing rational functions in mixed-polynomial form sometimes allows us to \ better analyze the End Behavior of the function. \n\n\t", StyleBox["Example 6:", FontSlant->"Italic", FontVariations->{"Underline"->True}, FontColor->GrayLevel[0]], StyleBox[" ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" Consider the function f(x) = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ SuperscriptBox["x", "4"], "-", " ", RowBox[{"3", SuperscriptBox["x", "2"]}], " ", "-", " ", "4"}], RowBox[{ SuperscriptBox["x", "2"], " ", "-", " ", "x", " ", "-", " ", "2"}]], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[". Choose the appropriate N(x) and D(x) functions from the \ drop-down menu below. 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