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Notebook[{
Cell[CellGroupData[{Cell[TextData["Calculus\nlesson 29"], "Title",
Evaluatable->False,
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Cell[TextData["Translations"], "Subtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"As we have seen with the graphs of the parabola, circle, ellipse and \
hyperbola if we replace x by x+ a constant it will shift the graph to the \
left or right. If we add a constant the graph will be shifted up or down \
that number of units. Try this with some different equations. Graph this \
function by highlighting the cell and pressing enter."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Plot[f[x]= Abs[x],{x,-4,4}];"], "Input",
AspectRatioFixed->True],
Cell[TextData[
"Change the following equation and see if you can shift the graph 2 units \
left and 2 units down."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Plot[f[x]= Abs[x],{x,-4,4}];"], "Input",
AspectRatioFixed->True],
Cell[TextData["Try this one"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Plot[f[x]=(1/2)x^2,{x,-4,4}];"], "Input",
AspectRatioFixed->True],
Cell[TextData[
"Then see if you can shift the graph 3 units right and 2 units up. Use the \
following equation."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Plot[f[x]=(1/2)x^2,{x,-4,4}];"], "Input",
AspectRatioFixed->True],
Cell[TextData[
"Start with the graph of this function then using the second equation see if \
you can shift it down 2 units and Pi/2 units to the left."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Clear[f,x]\nPlot[f[x]= Abs[4Sinx],{x,-Pi,4Pi}];"], "Input",
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Cell[TextData["Clear[f,x]\nPlot[f[x]= Abs[4Sinx],{x,-Pi,4Pi}];"], "Input",
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Cell[TextData[
"If a fraction has a polynomial for both numerator and denominator we say \
that the fraction is a rational polynomial expression. Rational polynomial \
expressions are called rational functions.\nIf we graph the rational function \
f[x]= 1/x we note that as x gets larger positively the value of f[x] gets \
smaller and smaller and as x gets smaller positively then f[x] gets larger \
and larger. As the |x| gets smaller f[x] increases negatively and as |x| \
increases f[x] gets smaller. Here is what the graph looks like. Highlight \
the following cell and press enter."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Clear[f,x]\nPlot[f[x_]=1/x,{x,-4,4},PlotStyle->RGBColor[1,0,0], \
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PageWidth->PaperWidth,
AspectRatioFixed->True],
Cell[TextData[
"We can see that the graph approaches the y axes but never touches. The y \
axes is called a vertical asymptote (a line that the graph approaches but \
never touches). It is important to note the graph goes up on one side of the \
asymptote and reappears from the down direction on the other side. This \
always happens when the expression in the denominator is a linear expression. \
When we replace the x in the previous equation by -x see what happens. \
Highlight the cell to the right of the equation and press the enter button."],
"Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
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AspectRatioFixed->True],
Cell[TextData[
"You can see the graph is flipped upside down about the x axes. What do you \
think will happen if we graph f[x]= 1/(x-3) +2? Try it:"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Clear[f,x]\nPlot[f[x]= 1/(x-3)+2,{x,-8,8},\nPlotStyle->RGBColor{0,0,1}];"],
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Cell[TextData[
"As you can see this shifted the graph to the right and up. The vertical \
asymptote is now the line x=3. If we place a negative in front of the \
fraction the graph should flip up side down about the line y=2. Try it to \
see if it actually happens. "], "Text",
Evaluatable->False,
AspectRatioFixed->True]}, Open]],
Cell[CellGroupData[{Cell[TextData["Rational Functions II\nLesson 39"], "Subtitle",
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Cell[CellGroupData[{Cell[TextData["Poles and Zeroes"], "Section",
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the function f[x] equals zero. These values of x are called the",
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StyleBox[" zeros ",
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FontWeight->"Bold"],
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function approaches but never touches a vertical asymptote. ",
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are poles of the function."], "Input",
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"We will begin by investigating graphs of rational functions by considering \
the special case of functions that are factored into linear factors that \
occur only once each and that have more factors in the denominator than in \
the numerator. This last stipulation will ensure that the x axes will be a \
horizontal asymptote.\nA rational function that is composed of unique \
nonrepeating linear factors changes signs at every zero of the numerator and \
denominator, this means the graph must cross the x axes at every zero and \
jump across the x axes at every pole.\nThe graphs can be sketched quickly if \
we draw vertical dashed lines at every pole and place dots at every zero. We \
then start at the right with a large positive value for x and substitute it \
in the equation and see if f[x] is positive or negative. Then we start above \
or below the x axes and follow the above rules.\nLets try it with the \
function f[x]=-[(x)(x-7)]/[(x+5)(x+2)(x-2)(x-5)]\nFirst we plot the zeros and \
vertical asymptotes. Highlight the cell to the right and press enter. "],
"Text",
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Cell[TextData[
"Clear[f,x]\nmy = Plot[f[x]=x,{x,-8,8},PlotStyle->RGBColor[1,1,1],\n\t\t\t\t\t\
DisplayFunction->Identity];\nShow[my,Graphics[{{Dashing[{0.05,0.05}],\n\t\t\t\
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DisplayFunction->$DisplayFunction];"], "Input",
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Cell[TextData[
"Then a large positive value for x yields a negative value for f[x] so we \
begin below the x axes and cross the x axes at zeros and jump across it at \
the poles. This is what it should look like. Highlight the cell and press \
enter."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Clear[f,x,p,l]\npl = Plot[f[x_]= \n \
-(x(x-7))/((x+5)(x+2)(x-2)(x-5)),{x,-8,8},\n\t\t\t\t\t \
PlotStyle->RGBColor[1,0,0],\n DisplayFunction->Identity];\n\
Show[pl,Graphics[{{Dashing[{0.05,0.05}],\n\t\t\t\t\
Line[{{5,8},{5,-8}}],Line[{{-5,8},{-5,-8}}],\n\t\t\t\t\
Line[{{2,8},{2,-8}}],Line[{{-2,8},{-2,-8}}]},\n\t\t\t\t\
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DisplayFunction->$DisplayFunction];"], "Input",
PageWidth->PaperWidth,
AspectRatioFixed->True],
Cell[TextData[
"Try this one: f[x]= (x+5)(x-3)(x-1)/(x-4)(x+2)(x+4)(x-1)\nWe can begin by \
canceling out the (x-1) factors but you must remember to put a hole in the \
graph at x=1."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Clear[f,x,p,l]\npl = Plot[f[x_]= ((x+5)(x-3))/((x-4)(x+2)(x+4)),\n\t\t\t\t\t\
{x,-8,8},PlotStyle->RGBColor[1,0,0],\n\t\t\t\t\tDisplayFunction->Identity];\n\
Show[pl,Graphics[{{Dashing[{0.05,0.05}],\n\t\t\t\t\
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Cell[CellGroupData[{Cell[TextData["Check your understanding"], "Subsection",
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Cell[TextData[
"In the following rational functions tell what the zeros would be, where the \
poles would be drawn and for a large value of x will you start graphing above \
or below the x axes."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"1. f[x]= (x-2)(x+3)/(x-5)(x+2)(x-3)\n\n2. f[x]= -X(x-3)/(x-1)(x+2)(x+3)"],
"Text",
Evaluatable->False,
AspectRatioFixed->True]}, Open]]}, Open]]}, Open]],
Cell[CellGroupData[{Cell[TextData["Lesson 44"], "Subtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["Factors of polynomial functions"], "Section",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"CalcAndMath",
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
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Cell[TextData[{
StyleBox[
"In lesson 39 we considered the graphs of rational polynomial functions. \
In this lesson we will investigate factors of polynomials and graphing \
second,third and fourth degree polynomial equations.\nIf all ",
Evaluatable->False,
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StyleBox["coefficients",
Evaluatable->False,
AspectRatioFixed->True,
FontVariations->{"Underline"->True}],
StyleBox[" of a polynomial are ",
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StyleBox["real numbers, the polynomial is called a real polynomial.",
Evaluatable->False,
AspectRatioFixed->True,
FontVariations->{"Underline"->True}],
StyleBox[
" The degree is the value of the greatest exponent. Carl Friedrich Gauss \
proved that every real polynomial of degree N has exactly N linear factors. \
Some factors may be complex linear factors but must occur in conjugate pairs \
(x+2i)(x-2i). A factor of the form ",
Evaluatable->False,
AspectRatioFixed->True],
StyleBox["(x\[Currency]+4)",
Evaluatable->False,
AspectRatioFixed->True,
FontVariations->{"Underline"->True}],
StyleBox[" is called and ",
Evaluatable->False,
AspectRatioFixed->True],
StyleBox[
"irreducible quadratic factor because it can't be factored into linear real \
factors.",
Evaluatable->False,
AspectRatioFixed->True,
FontVariations->{"Underline"->True}],
StyleBox[
" This factor can never equal zero for any real number value for x. Thus \
",
Evaluatable->False,
AspectRatioFixed->True],
StyleBox[
"irreducible quadratic factors never cause a polynomial to equal zero. ",
Evaluatable->False,
AspectRatioFixed->True,
FontVariations->{"Underline"->True}],
StyleBox[
"\nOnly linear real factors can cause a polynomial to equal zero. ",
Evaluatable->False,
AspectRatioFixed->True],
StyleBox[
"The graph will cross the x axes at a zero if the factor occurs an odd \
number of times and will touch but not cross the x axes if it occurs an even \
number of times. \n",
Evaluatable->False,
AspectRatioFixed->True,
FontWeight->"Bold"],
StyleBox[
"If you have the function f[x]= \
(x+4)(x+2)\:203a(x\[Currency]+3)(x-5)\:2039(x-7)\[Currency] at what values of \
x will the graph touch the x axes and at what values of x will it cross the x \
axes? After answering the question check it out by highlighting the \
following cell and press enter.",
Evaluatable->False,
AspectRatioFixed->True]
}], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Clear[f,x]\nPlot[f[x_]= (x+4)(x+2)^4(x^2+3)(x-5)^3(x-7)^2,\n\t\t\t\t\
{x,8,-8}];"], "Input",
AspectRatioFixed->True],
Cell[TextData[
"The turning point theorem tells us that the graph of a polynomial function \
has fewer turning points than the degree of the polynomial."], "Text",
CellFrame->True,
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[
"Thus the graph of a third degree polynomial function has at most two turning \
points; the graph of a fourth-degree polynomial function has at most three \
turning points;etc.\nThe term of highest degree in a polynomial is the \
dominant term because, for large absolute values of x, the value of the \
highest degree term will be greater than the absolute value of the sum of all \
the other terms in the equation. \nWe can tell by the dominant term what the \
ends of the graph of a polynomial function will do. The graph will increase \
or decrease according to the sign of the highest degree term evaluated when x \
is a large negative and a large positive number. If the value of the term is \
negative it will decrease if positive it will increase. \nExample: f[x]=x\
\:203a-3x\:2039+2x-9 For positive values of x, x\:203a would be a positive \
number. For negative values of x, x\:203a would be a positive number. So \
both ends of the graph will increase. Take a look."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Clear[f,x]\nPlot[f[x_]= x^4-3x^3+2x-9,{x,-8,8}];"], "Input",
AspectRatioFixed->True],
Cell[TextData[
"What do you think would happen to the ends of this polynomial function:\n\
f[x]=x\:2039+2x\[Currency]-x+4 Will the negative side increase or decrease? \
How about the positive side? Try it and see."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Clear[f,x]\nPlot[f[x_]=x^3+2x^2-x+4,{x,-8,8}];"], "Input",
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Cell[TextData[
"Just as is the case with linear equations the constant shifts the graph up \
or down. The same is true of a polynomial function. The general form of an \
n-th degree polynomial function is f[x]= \
ax\:02dc+bx\:02dc\[LongDash]\:2215+cx\:02dc\[LongDash]\[Currency]+...+k\nThe \
sum of all real and complex roots is -b/a and the product of all real and \
complex roots is k/a if the degree of the polynomial is even and -k/a if the \
degree is odd. The average value of all roots is"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData[" Average value of all roots =-b/na"], "Text",
CellFrame->True,
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FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False}],
Cell[TextData[
"This will give us a good idea of the x value of the center of the graph. \
Using this fact find the x coordinate of the center of the following \
polynomial function. f[x]= -x\:2039+3x\[Currency]-2x+3\nWhat does the graph \
do to the right of this center point and to the left? Try it and see if you \
were correct."], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Clear[f,x]\nPlot[f[x]=-3x^3+3x^2-2x+3,{x,-4,4}];"], "Input",
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Cell[TextData[
"Try another polynomial function and do the same as the one above then check. \
f[x]= x\:203a-3x\[Currency]+2x+5"], "Text",
Evaluatable->False,
AspectRatioFixed->True],
Cell[TextData["Clear[f,x]\nPlot[f[x]=x^4-3x^2+2x+5,{x,-4,4}];"], "Input",
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