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Students will be able to accomplish the \ following tasks.\ \>", "Text", CellChangeTimes->{{3.459501916812027*^9, 3.459501933671186*^9}, { 3.459502064888256*^9, 3.459502082872401*^9}}], Cell[TextData[StyleBox["Goals", FontWeight->"Bold", FontSlant->"Italic", FontVariations->{"Underline"->True}]], "Text", CellChangeTimes->{{3.456605020061042*^9, 3.4566050208640423`*^9}, 3.458497067826801*^9}], Cell[TextData[{ "Students will be able to graph initial functions in ", StyleBox["Mathematica", FontSlant->"Italic"], " and those same functions with various stretches applied." }], "Item", CellChangeTimes->{ 3.456609569251042*^9, {3.456609642519042*^9, 3.456609646513042*^9}, { 3.4572673825625916`*^9, 3.457267399419269*^9}, {3.4572677472854767`*^9, 3.4572678279288216`*^9}, 3.457267946144279*^9, {3.458496907014301*^9, 3.458496910858051*^9}, 3.458497067826801*^9, {3.4585139294485874`*^9, 3.458513944964113*^9}, {3.459270227907666*^9, 3.4592702285482874`*^9}, { 3.4595020943253794`*^9, 3.4595021125907707`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"parameters", "=", RowBox[{"{", RowBox[{"a", ",", "c", ",", "d", ",", "x"}], "}"}]}]], "Input", CellChangeTimes->{{3.457304966819664*^9, 3.457305015602542*^9}, { 3.457305065477542*^9, 3.457305138868167*^9}, {3.457305170305667*^9, 3.457305175961917*^9}, {3.457305495098973*^9, 3.4573055480823307`*^9}, 3.45734869762527*^9, {3.458496058076801*^9, 3.458496078108051*^9}, 3.458496549701801*^9, {3.4584970204674253`*^9, 3.458497024795551*^9}, { 3.458497067826801*^9, 3.4584971006861753`*^9}, 3.4584978789049253`*^9}, CellID->10998], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"a", ",", "c", ",", "d", ",", "x"}], "}"}], TraditionalForm]], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{ 3.458496081420551*^9, 3.458496549701801*^9, 3.458497026889301*^9, { 3.458497067826801*^9, 3.4584971006861753`*^9}, 3.4584978789049253`*^9}] }, {6}]], Cell["\<\ Students will be able to graph initial functions and their inversions and \ predict what happens at \"interesting\" points.\ \>", "Item", CellChangeTimes->{ 3.456609569251042*^9, {3.456609642519042*^9, 3.456609653702042*^9}, { 3.4572674029030867`*^9, 3.457267438038089*^9}, 3.458497067826801*^9, { 3.4585139516359453`*^9, 3.4585139535421834`*^9}, {3.458514067557079*^9, 3.4585140798851247`*^9}, {3.459502122965638*^9, 3.4595021483090897`*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Concept Exploration", "Section", CellChangeTimes->{{3.4462996971190157`*^9, 3.446299723414039*^9}, { 3.456614564946227*^9, 3.4566145737392273`*^9}}], Cell[CellGroupData[{ Cell["Stretching Graphs - revisited", "Subsubsection", CellFrameColor->RGBColor[ 0.6449835965514611, 0.758632791638056, 0.2516823071641108], CellChangeTimes->{{3.4572682917399263`*^9, 3.457268298197912*^9}, { 3.45726861307681*^9, 3.4572686272230935`*^9}, {3.458515362189418*^9, 3.458515368751876*^9}, {3.459502205277475*^9, 3.459502211043063*^9}}, FontColor->RGBColor[ 0.6449835965514611, 0.758632791638056, 0.2516823071641108]], Cell["Plot the following two graphs on the same set of axes. 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(The red function values are all twice the \ corresponding blue function' s values.)\ \>", "Answer", CellChangeTimes->{ 3.4595323664349117`*^9, {3.4595324045130367`*^9, 3.4595324185599117`*^9}, { 3.459602706419465*^9, 3.4596027092164116`*^9}, {3.4596027432329073`*^9, 3.4596027455142155`*^9}, {3.459605862748063*^9, 3.4596058687788877`*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Example 1", "Subsubsection", CellChangeTimes->{{3.4463006149870167`*^9, 3.4463006181425595`*^9}}], Cell["\<\ For the following 2 sets of functions, first predict what the graph will look \ like, then plot the functions on the same set of axes. For the plot, please \ make sure to...\ \>", "Text", CellChangeTimes->{{3.4595026155109243`*^9, 3.4595026541832943`*^9}}], Cell[CellGroupData[{ Cell["\<\ Choose an appropriate window. This is sometimes easier to determine after \ the plot is generated initially.\ \>", "Item", CellDingbat->"\[FilledSmallSquare]", CellChangeTimes->{{3.459502657683339*^9, 3.4595026745741806`*^9}, { 3.4595029090456715`*^9, 3.4595029093737984`*^9}, {3.459602836433325*^9, 3.459602836792638*^9}}], Cell["\<\ Make sure the two graphs are different colors of your choosing. 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Let's explore this below. 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Why \ is it that they only meet at this one point?\ \>", "Text", CellChangeTimes->{{3.4595052265355453`*^9, 3.4595052560667953`*^9}}], Cell["\<\ At y = 1. The reciprocal of 1 is one. But, also, this is the vertex of the parabola, and it is a minimum, so all other function values of \ the parabola are greater than 1. This means that all other function values \ of the reciprocal function are less than one. The minimum of a function is \ the maximum of its reciprocal.\ \>", "Answer", CellChangeTimes->{{3.4595052697855453`*^9, 3.4595052715824203`*^9}, { 3.4595061224644256`*^9, 3.459506193946971*^9}, {3.459604350793781*^9, 3.4596043626728086`*^9}, {3.459607153261155*^9, 3.45960716830649*^9}}], Cell["\<\ 2) When do the vertical asymptotes begin to appear? Why is that?\ \>", "Text", CellChangeTimes->{{3.4595052665355453`*^9, 3.4595052671605453`*^9}, { 3.4595052996292953`*^9, 3.4595053094886703`*^9}}], Cell["\<\ When the parabola touches the x - axis, thus having a function value of 0. The reciprocal of 0 is undefined, hence yielding a vertical \ asymptote.\ \>", "Answer", CellChangeTimes->{{3.4595053113480453`*^9, 3.4595053138792953`*^9}, { 3.4595061972281365`*^9, 3.4595062300711355`*^9}, {3.459604365767608*^9, 3.4596043693625774`*^9}, {3.459607157026395*^9, 3.4596071706343765`*^9}}], Cell["\<\ 3) As \"a\" gets larger, what happens to the graph of the reciprocal \ function? Why is that?\ \>", "Text", CellChangeTimes->{{3.4595053149730453`*^9, 3.4595053471136703`*^9}}], Cell["\<\ As the \"a\" moves farther from the x - axis, its value increases. And, the bigger a number is, the smaller the value of its reciprocal. \ Therefore, as \"a\" gets larger, the reciprocal values \"hug\" the x - axis \ closer, as those values are closer to y = 0.\ \>", "Answer", CellChangeTimes->{{3.4595053488480453`*^9, 3.4595053506605453`*^9}, { 3.4595062322117467`*^9, 3.459506290148876*^9}, {3.459604372691831*^9, 3.4596043815698414`*^9}, {3.45960716058853*^9, 3.4596071760713196`*^9}}] }, Open ]] }, Open ]] }, Open ]] }, AutoGeneratedPackage->None, ScreenStyleEnvironment->"SlideShow", WindowToolbars->"EditBar", WindowSize->{1272, 683}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"7.0 for Mac OS X x86 (32-bit) (July 20, 2009)", StyleDefinitions->Notebook[{ Cell[ StyleData[StyleDefinitions -> "Default.nb"]], Cell[ StyleData[StyleDefinitions -> "StyleMenuClear.nb"]], Cell[ CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[ StyleData[All, "Working"], DockedCells -> { Cell[ GraphicsData[ "Bitmap", 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