(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 11766, 404] NotebookOptionsPosition[ 10344, 354] NotebookOutlinePosition[ 10782, 371] CellTagsIndexPosition[ 10739, 368] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Polynomial Graphing Basics\n", StyleBox["(Factors with higher powers)", FontSize->36] }], "Title", CellChangeTimes->{{3.4935031198101873`*^9, 3.493503126939185*^9}, { 3.692556892785404*^9, 3.692556934883192*^9}}, TextAlignment->Center], Cell["\"Pass Through\" and \"Bounce Points\"", "Subtitle", CellChangeTimes->{{3.4935031314923477`*^9, 3.493503154498498*^9}, { 3.4935041172076693`*^9, 3.4935041244133368`*^9}}], Cell["\<\ by Ruth Dover Illinois Mathematics and Science Academy\ \>", "Subsubtitle", CellChangeTimes->{{3.6925571945308867`*^9, 3.692557206726015*^9}}], Cell["\<\ Throughout this notebook, all polynomials are assumed to have real number \ coefficients.\ \>", "Text", CellChangeTimes->{ 3.493503169554718*^9, {3.692557180108842*^9, 3.692557181480226*^9}}], Cell[TextData[{ "The exponent of a factor in a polynomial is called the ", StyleBox["multiplicity", FontWeight->"Bold"], " of that factor. Since each factor of (", StyleBox["x", FontSlant->"Italic"], " - ", StyleBox["r", FontSlant->"Italic"], ") corresponds to a zero (or root or ", StyleBox["x", FontSlant->"Italic"], "-intercept) at ", StyleBox["x", FontSlant->"Italic"], " = ", StyleBox["r", FontSlant->"Italic"], " , we define the multiplicity of the root at ", StyleBox["x", FontSlant->"Italic"], " = ", StyleBox["r", FontSlant->"Italic"], " to be the multiplicity of the factor (", StyleBox["x", FontSlant->"Italic"], " - ", StyleBox["r", FontSlant->"Italic"], ") .\n\nExample: Let ", StyleBox["y = (", FontSlant->"Italic"], "x", StyleBox["+ ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"7", ")"}], "4"], RowBox[{"(", StyleBox[ RowBox[{"x", "-", "4"}], FontSlant->"Italic"], StyleBox[")", FontSlant->"Italic"]}], SuperscriptBox[ StyleBox[ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], FontSlant->"Italic"], "3"]}], TraditionalForm]]], " . Then the factor (", StyleBox["x", FontSlant->"Italic"], " + 7) has multiplicity 4, the factor (", StyleBox["x", FontSlant->"Italic"], " - 4) has multiplicity 1, and the factor (", StyleBox["x", FontSlant->"Italic"], " + 2) has multiplicity 3. Similarly, the zero at ", StyleBox["x", FontSlant->"Italic"], " = -7 has multiplicity 4, the zero at ", StyleBox["x", FontSlant->"Italic"], " = 4 has multiplicity 1, and the zero at ", StyleBox["x", FontSlant->"Italic"], " = -2 has multiplicity 3." }], "Text", CellFrame->1.5, CellFrameColor->RGBColor[0, 0, 1], CellChangeTimes->{ 3.493503222669812*^9, {3.493503298523549*^9, 3.493503539749617*^9}, 3.493504103540021*^9}], Cell["\<\ We want to examine the role of the multiplicity of each factor and its effect \ on the graph of the polynomial. \ \>", "Text", CellChangeTimes->{3.4935035857613163`*^9}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[1, 0, 0]], StyleBox[" notes", FontColor->RGBColor[1, 0, 0]], ": \n(1) If you want/need to control the range, the syntax is shown on ", Cell[BoxData[ FormBox[ SubscriptBox["y", "3"], TraditionalForm]]], "below. You may copy and paste this syntax into any other commands and edit \ the values to find a good viewing window.\n\n(2) For functions 4, 5, and 6, \ it's your turn to do some work! Either:\n(a) Click below each ", Cell[BoxData[ FormBox[ SubscriptBox["y", "k"], TraditionalForm]]], " heading so that a horizontal bar appears. Then start typing (and ", StyleBox["Mathematica", FontSlant->"Italic"], " will make room for you), following the syntax used on the other functions\n\ OR\n(b) Copy the syntax from one of the functions that is given to you. \ Then move your cursor below the ", Cell[BoxData[ FormBox[ SubscriptBox["y", "k"], TraditionalForm]]], " and click so that a horizontal line appears. Then paste the command and \ edit it as necessary." }], "Text", CellChangeTimes->{{3.493557633234479*^9, 3.493557676158845*^9}, { 3.493557716901401*^9, 3.493557738300024*^9}, {3.493557805019475*^9, 3.493557838976616*^9}, {3.4935579001104803`*^9, 3.49355820640878*^9}}], Cell[CellGroupData[{ Cell["Graphs", "Section", CellChangeTimes->{{3.493503700545476*^9, 3.493503701143302*^9}}], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "1"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.4935038010693417`*^9, 3.4935038033890123`*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "3"}], ",", " ", "3.6"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.493503704089267*^9, 3.493503728146611*^9}, { 3.493503771041582*^9, 3.493503778074535*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "2"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.493503821472509*^9, 3.493503825771402*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "^", "2"}], RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "3"}], ",", " ", "3.6"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.493503952139802*^9, 3.4935039604127817`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "3"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.4935038396093616`*^9, 3.493503851617812*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "^", "2"}], RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], "^", "2"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "3"}], ",", " ", "3.6"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "50"}], ",", " ", "50"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.493503952139802*^9, 3.4935039604127817`*^9}, { 3.493557766982909*^9, 3.493557782788656*^9}, {3.49355787537889*^9, 3.493557880953261*^9}, {3.493558815588358*^9, 3.493558821449291*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Your turn. (See the instructions above.", "Subsection", CellChangeTimes->{{3.692557584133298*^9, 3.692557597025691*^9}}], Cell[TextData[{ "\tLet ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "4"], "="}], TraditionalForm]], FormatType->"TraditionalForm"], " ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", " ", "+", " ", "2"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{"x", " ", "-", " ", "1"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"x", " ", "-", "3"}], ")"}], "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "5"], "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "3"], RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}], "4"]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "6"], "=", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "5"], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], "3"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}], "2"], "."}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.692557601232939*^9, 3.6925576116961727`*^9}, { 3.692557679369626*^9, 3.692558138655632*^9}}] }, Open ]], Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "4"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.493503862301806*^9, 3.493503867114441*^9}}], Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "5"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.493503889891178*^9, 3.493503893048751*^9}}], Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "6"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.493503919467341*^9, 3.4935039228394413`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Questions", "Section", CellChangeTimes->{{3.692558790042418*^9, 3.692558791178219*^9}}], Cell[TextData[{ "Under what circumstances did you find a \[OpenCurlyDoubleQuote]pass-through\ \[CloseCurlyDoubleQuote] point? A \[OpenCurlyDoubleQuote]bounce\ \[CloseCurlyDoubleQuote] point? Explain (clearly) the relationship between \ the exponents on each of the factors in the polynomial functions and the \ behavior of the graphs at ", StyleBox["x", FontSlant->"Italic"], " = \[Dash]2, 1, and 3." }], "Text", CellChangeTimes->{{3.692558798075509*^9, 3.6925589101174097`*^9}}], Cell["\<\ Determine the degree of each polynomial. What do all of the graphs of odd \ degree have in common? How do these differ from the graphs with an even \ degree? (Your patterns should be consistent with the graphs in the previous \ notebook, PolyBasics1.nb.\ \>", "Text", CellChangeTimes->{{3.692558923370193*^9, 3.692559021282322*^9}}] }, Open ]] }, Open ]] }, WindowSize->{657, 707}, WindowMargins->{{167, Automatic}, {Automatic, 31}}, PrivateNotebookOptions->{"VersionedStylesheet"->{"Default.nb"[8.] -> False}}, FrontEndVersion->"11.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (September \ 21, 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 260, 7, 139, "Title"], Cell[830, 31, 179, 2, 51, "Subtitle"], Cell[1012, 35, 153, 4, 49, "Subsubtitle"], Cell[1168, 41, 204, 5, 49, "Text"], Cell[1375, 48, 1907, 77, 184, "Text"], Cell[3285, 127, 180, 4, 49, "Text"], Cell[3468, 133, 1307, 30, 264, "Text"], Cell[CellGroupData[{ Cell[4800, 167, 91, 1, 64, "Section"], Cell[CellGroupData[{ Cell[4916, 172, 176, 3, 45, "Subsection"], Cell[5095, 177, 469, 14, 32, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[5601, 196, 172, 3, 45, "Subsection"], Cell[5776, 201, 449, 14, 32, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[6262, 220, 174, 3, 45, "Subsection"], Cell[6439, 225, 752, 21, 32, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[7228, 251, 127, 1, 44, "Subsection"], Cell[7358, 254, 1454, 54, 51, "Text"] }, Open ]], Cell[8827, 311, 172, 3, 45, "Subsection"], Cell[9002, 316, 172, 3, 37, "Subsection"], Cell[9177, 321, 174, 3, 37, "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[9388, 329, 94, 1, 64, "Section"], Cell[9485, 332, 485, 10, 68, "Text"], Cell[9973, 344, 343, 6, 87, "Text"] }, Open ]] }, Open ]] } ] *)