(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 14379, 489] NotebookOptionsPosition[ 12751, 432] NotebookOutlinePosition[ 13189, 449] CellTagsIndexPosition[ 13146, 446] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Polynomial Graphing Basics\n", StyleBox["(Linear Factors)", FontSize->36] }], "Title", CellChangeTimes->{{3.493324736512966*^9, 3.493324737156581*^9}, { 3.692555709036313*^9, 3.69255571848533*^9}, {3.692556720272129*^9, 3.692556725805833*^9}}, TextAlignment->Center], Cell["\<\ by Ruth Dover Illinois Mathematics and Science Academy\ \>", "Subsubtitle", CellChangeTimes->{{3.4933248001225233`*^9, 3.4933248038493757`*^9}, { 3.692555730703878*^9, 3.692555742674326*^9}}], Cell[TextData[{ "Before this notebook, you have dealt with special types of polynomial \ functions, namely, quadratic functions and power functions of the form ", StyleBox["y", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "n"], TraditionalForm]], FormatType->"TraditionalForm"], ", where ", StyleBox["n", FontSlant->"Italic"], " is a positive integer. In this notebook, we will give a formal general \ definition of a polynomial function in one variable and examine some basic \ properties of these functions." }], "Text", CellChangeTimes->{ 3.493324830314372*^9, {3.692557034405141*^9, 3.692557094492289*^9}, { 3.692559114954172*^9, 3.692559153766704*^9}}], Cell[TextData[{ StyleBox["A ", FontSize->14], StyleBox["polynomial function", FontSize->14, FontWeight->"Bold"], StyleBox[" in the variable ", FontSize->14], StyleBox["x", FontSize->14, FontSlant->"Italic"], StyleBox[" is a function that may be written as", FontSize->14], "\n", StyleBox["p", FontSize->16, FontSlant->"Italic"], StyleBox["(", FontSize->16], StyleBox["x", FontSize->16, FontSlant->"Italic"], StyleBox[") = ", FontSize->16], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["a", "n"], "\[CenterDot]", SuperscriptBox["x", "n"]}], "+", RowBox[{ SubscriptBox["a", RowBox[{"n", "-", "1"}]], "\[CenterDot]", SuperscriptBox["x", RowBox[{"n", "-", "1"}]]}], "+", "\[CenterEllipsis]", "+", RowBox[{ SubscriptBox["a", "1"], "\[CenterDot]", SuperscriptBox["x", "1"]}], "+", SubscriptBox["a", "0"]}], TraditionalForm]], FontSize->16], "\n", StyleBox["where ", FontSize->14], StyleBox["n", FontSize->14, FontSlant->"Italic"], StyleBox[" is a non-negative integer and each coefficient ", FontSize->14], Cell[BoxData[ FormBox[ SubscriptBox["a", "i"], TraditionalForm]], FontSize->14], StyleBox["is a constant number.", FontSize->14], "\nThe largest value of ", StyleBox["n", FontSlant->"Italic"], " for which ", Cell[BoxData[ FormBox[ SubscriptBox["a", "n"], TraditionalForm]]], " \[NotEqual] 0 is called the ", StyleBox["degree", FontWeight->"Bold"], " of the polynomial. \nIf ", StyleBox["n", FontSlant->"Italic"], " = 0 or if ", StyleBox["p", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = 0, we say that the degree of ", StyleBox["p", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") is 0." }], "Text", CellFrame->1.5, CellFrameColor->RGBColor[0, 0, 1], CellChangeTimes->{{3.493324862182124*^9, 3.493324890653247*^9}, { 3.493324950960638*^9, 3.4933249602991667`*^9}, {3.493324993672702*^9, 3.493325022594961*^9}, {3.493325099160261*^9, 3.4933251237132597`*^9}, { 3.493325154568265*^9, 3.493325253968787*^9}, {3.493325307578992*^9, 3.493325514722865*^9}, {3.692555788047925*^9, 3.6925557938192368`*^9}, 3.692555937645213*^9, {3.692556051375868*^9, 3.6925560923189163`*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Question 1", "Section", CellChangeTimes->{{3.4934855742286043`*^9, 3.4934855761404676`*^9}}], Cell["\<\ We will start to explore polynomial functions with increasingly larger \ degrees. For each of the functions below (beyond the first example!), expand \ and note the degree and the zeroes of the function. \ \>", "Text", CellChangeTimes->{{3.4933256327292356`*^9, 3.4933256862091417`*^9}, { 3.493485564517666*^9, 3.493485565197318*^9}, {3.692556119003269*^9, 3.692556128980991*^9}}], Cell[TextData[{ "The following command (in boldface, below) will plot the graph of ", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["x", FontSlant->"Italic"], " - 1 for -6.5 \[LessEqual] ", StyleBox["x", FontSlant->"Italic"], " \[LessEqual] 6.5. To \"execute\" the command, put the cursor somewhere in \ the line below with \"Plot\" and click. Then press Shift-Enter. " }], "Text", CellChangeTimes->{{3.3944945470815783`*^9, 3.394494549304199*^9}, { 3.399815528576805*^9, 3.399815531910755*^9}, {3.493323986059205*^9, 3.493324000374734*^9}, {3.49332406974091*^9, 3.493324080663919*^9}, { 3.493325748863676*^9, 3.493325751253644*^9}}], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "1"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.4934852863905487`*^9, 3.493485295512478*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "6.5"}], ",", " ", "5.5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.399815542873377*^9, 3.399815647327448*^9}, { 3.493324084370346*^9, 3.493324086169952*^9}}], Cell["Fill in the information below:", "Text", CellChangeTimes->{{3.493325865577022*^9, 3.493325926244174*^9}}], Cell[TextData[{ "Degree:\n\nZeroes: ", StyleBox["x", FontSlant->"Italic"], " = " }], "Text", CellChangeTimes->{{3.493325934065152*^9, 3.493325942305312*^9}, { 3.4934846596388702`*^9, 3.4934846691798677`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "2"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.493485307030383*^9, 3.4934853101709557`*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "6.5"}], ",", " ", "6.5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.493325961433028*^9, 3.493325979623427*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Expand", "[", RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}]}], "]"}], "//", "TraditionalForm"}]], "Input", CellChangeTimes->{{3.49332598441008*^9, 3.493325996158104*^9}, { 3.493502182073421*^9, 3.493502185202903*^9}}], Cell[TextData[{ "Degree:\n\nZeroes: ", StyleBox["x", FontSlant->"Italic"], " = " }], "Text", CellChangeTimes->{{3.4933260080296593`*^9, 3.49332601407181*^9}, { 3.493484671763632*^9, 3.493484676355031*^9}}], Cell[TextData[{ StyleBox["Note: ", FontColor->RGBColor[1, 0, 0]], StyleBox["You may find it helpful to change the range as you go. ", FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" chooses a range by itself, but in some cases, it may be helpful \ to override this to see what happens around the ", FontColor->GrayLevel[0]], StyleBox["x", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["-axis more clearly. To do this, see the coding below for ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SubscriptBox["y", "3"], TraditionalForm]]], StyleBox[" and edit this as you continue. Be sure that you get a good sense \ of the entire graph. (If in doubt, ask your teacher!)", FontColor->GrayLevel[0]] }], "Text", CellChangeTimes->{{3.399815678576252*^9, 3.399815793725881*^9}, { 3.399815887817803*^9, 3.399815961725923*^9}, 3.399819100940125*^9, { 3.493324373877323*^9, 3.4933243742798433`*^9}, {3.4934847702740097`*^9, 3.493484774032857*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[ FormBox[ SubscriptBox["y", "3"], TraditionalForm]], "None"]], "Subsection", CellChangeTimes->{{3.493485322655527*^9, 3.493485325560855*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "6.5"}], ",", " ", "5.5"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "60"}], ",", " ", "60"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.399815542873377*^9, 3.399815647327448*^9}, { 3.3998158094339323`*^9, 3.399815846015758*^9}, {3.399819112229395*^9, 3.399819127971438*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Expand", "[", RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], "]"}], "//", "TraditionalForm"}]], "Input", CellChangeTimes->{{3.4934847884381037`*^9, 3.4934847996987762`*^9}, { 3.493502193485877*^9, 3.493502199318928*^9}}], Cell[TextData[{ "Degree: \n\nZeroes: ", StyleBox["x", FontSlant->"Italic"], " = " }], "Text", CellChangeTimes->{{3.493484804126987*^9, 3.493484816118102*^9}}], Cell[TextData[{ "Instructions for continuing:\nOK, so now it's your turn. First, move your \ cursor below this text (but above question 2) so that it turns into a \ horizontal I-bar. Click. A horizontal line should appear. Then copy the \ text for the entire Plot command above and paste it into the space below the \ horizontal line. (You can do this by selecting the bracket to the right of \ the \"cell\" and then doing copy/paste.) Edit the function to examine the \ graph of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "4"], "=", RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "4"}], ")"}]}]}], TraditionalForm]]], ". Then copy and edit the commands for Expand and then for the Degree and \ Zeroes. Then repeat this for ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["y", "5"], "=", RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "4"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "5"}], ")"}]}]}], TraditionalForm]]], ". " }], "Text", CellChangeTimes->{{3.493485075623464*^9, 3.4934851195000277`*^9}, { 3.493485172850601*^9, 3.493485271998884*^9}, {3.493485331757339*^9, 3.493485458880973*^9}, {3.4934858419285297`*^9, 3.493485890754232*^9}, { 3.4934875542778263`*^9, 3.493487636914105*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Question 2", "Section", CellChangeTimes->{{3.4934854726650476`*^9, 3.493485478096002*^9}}], Cell[TextData[{ "What is happening to the degree of the polynomial as you consider ", Cell[BoxData[ FormBox[ SubscriptBox["y", "1"], TraditionalForm]]], "through ", Cell[BoxData[ FormBox[ SubscriptBox["y", "5"], TraditionalForm]]], "?\n\n" }], "Text", CellChangeTimes->{{3.493485484303652*^9, 3.4934855470220623`*^9}, 3.493485778533907*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Question 3", "Section", CellChangeTimes->{{3.493485554356894*^9, 3.493485557445261*^9}}], Cell["\<\ Write a few sentences describing features of the graphs that you observed. \ Include features such as zeroes, maximum and minimum points, and anything \ else that you noticed. \ \>", "Text", CellChangeTimes->{{3.493485605438973*^9, 3.493485669513379*^9}, { 3.493485773435408*^9, 3.4934857756064796`*^9}, {3.4934860059371758`*^9, 3.493486008454247*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Question 4", "Section", CellChangeTimes->{{3.493485783989379*^9, 3.4934857865257263`*^9}}], Cell["\<\ How does the degree affect the graph? \ \>", "Text", CellChangeTimes->{{3.4934857930355377`*^9, 3.493485808084856*^9}}] }, Open ]] }, Open ]] }, WindowSize->{785, 687}, WindowMargins->{{272, Automatic}, {52, Automatic}}, 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, 294, 8, 139, "Title"], Cell[864, 32, 204, 5, 49, "Subsubtitle"], Cell[1071, 39, 708, 19, 88, "Text"], Cell[1782, 60, 2335, 89, 134, "Text"], Cell[CellGroupData[{ Cell[4142, 153, 99, 1, 64, "Section"], Cell[4244, 156, 396, 7, 49, "Text"], Cell[4643, 165, 663, 16, 49, "Text"], Cell[CellGroupData[{ Cell[5331, 185, 174, 3, 45, "Subsection"], Cell[5508, 190, 343, 9, 32, "Input"], Cell[5854, 201, 112, 1, 30, "Text"], Cell[5969, 204, 217, 7, 68, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[6223, 216, 174, 3, 45, "Subsection"], Cell[6400, 221, 366, 11, 32, "Input"], Cell[6769, 234, 335, 10, 32, "Input"], Cell[7107, 246, 213, 7, 68, "Text"], Cell[7323, 255, 1057, 26, 89, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[8417, 286, 172, 3, 45, "Subsection"], Cell[8592, 291, 655, 19, 32, "Input"], Cell[9250, 312, 396, 12, 32, "Input"], Cell[9649, 326, 166, 6, 68, "Text"], Cell[9818, 334, 1621, 43, 148, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[11488, 383, 97, 1, 64, "Section"], Cell[11588, 386, 360, 12, 70, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[11985, 403, 95, 1, 64, "Section"], Cell[12083, 406, 372, 9, 87, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[12492, 420, 97, 1, 64, "Section"], Cell[12592, 423, 131, 5, 68, "Text"] }, Open ]] }, Open ]] } ] *)