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Amplitude, Period and Phase Shifts for Trigonometric Functions\
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  ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-6.63644, -3.16076, 0.046247,
   0.0357277}}],

Cell["\<\
Put the cursor anywhere in this yellow box and press shift-return to check \
your answer:\
\>", "Text"],

Cell[BoxData[
    \(Print["\<The amplitude is 3.\>"]\)], "Input",
  FontColor->RGBColor[1, 1, 0],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "\tLet's see if we can generalize what the amplitude is for any function of \
the form ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["A sin(x) ",
    FontWeight->"Bold"],
  "or",
  StyleBox[" f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["A cos(x) ",
    FontWeight->"Bold"],
  "by looking at several graphs.  Enter the input in the yellow box below to \
see the graphs of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " ",
  "= ",
  StyleBox["A sin(x)",
    FontWeight->"Bold"],
  " for several different values of ",
  StyleBox["A",
    FontWeight->"Bold"],
  ".  "
}], "Text"],

Cell[BoxData[
    \(Clear[f]; \n
    Do[Print["\<The graph of f(x) = \>", n, "\< sin(x)\>"]; 
      Plot[n\ Sin[x], {x, \(-4\)\ Pi, 4\ Pi}, PlotRange -> {\(-5\), 5}, 
        PlotStyle -> {Hue[n/10], Thickness[0.008]}, 
        AspectRatio -> Automatic], {n, \(-5\), 5, 2}]\)], "Input",
  AnimationDisplayTime->3.03785,
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "What do you notice about the amplitudes of the functions when ",
  StyleBox["A ",
    FontWeight->"Bold"],
  "is a positive number? a negative number?  How can we generalize these \
observations to a single statement about the amplitude of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["A sin(x) ",
    FontWeight->"Bold"],
  "that holds true for both positive and negative numbers? \n\tNow see if the \
statement you just made holds true for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["A cos(x).  ",
    FontWeight->"Bold"],
  "Input the program below to see the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " ",
  "= ",
  StyleBox["A cos(x)",
    FontWeight->"Bold"],
  " for different values of ",
  StyleBox["A",
    FontWeight->"Bold"],
  ". "
}], "Text"],

Cell[BoxData[
    \(Clear[g]; \n
    Do[Print["\<The graph of f(x) = \>", n, "\< cos(x)\>"]; 
      Plot[n\ Cos[x], {x, \(-4\)\ Pi, 4\ Pi}, PlotRange -> {\(-5\), 5}, 
        PlotStyle -> {Hue[n/10], Thickness[0.008]}, 
        AspectRatio -> Automatic], {n, \(-4\), 4, 2}]\)], "Input",
  AnimationDisplayTime->5.11859,
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Cell[TextData[{
  "Can you make the same conclusions for the amplitude of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["A cos(x)",
    FontWeight->"Bold"],
  " as you could for sine?  \n\t To get an even better intuitive \
understanding of the effect that changing the constant ",
  StyleBox["A ",
    FontWeight->"Bold"],
  "has on the amplitude, you can animate the collection of graphs above.  \
That is, you can put the collection of graphs \"in motion\" so that each of \
them is displayed successively on the same set of axes.  To do this, click \
once on the blue cell bar (to the right) that brackets all the graphs and \
nothing else (this should be the bar in the second column from the left).  \
When it is highlighted (in black), hold down the",
  StyleBox[" ",
    FontWeight->"Bold"],
  "\"Control\" key and the \"y\" key at the same time.  You can control the \
speed of the animation using the arrow keys that will appear in the bottom \
left corner of the window.  To stop the animation, click anywhere in the \
notebook.   \n\tWe can conclude from our observations that for any \
trigonometric function of the form ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["A sin(x)  ",
    FontWeight->"Bold"],
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  StyleBox["A cos(x) ",
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  "has a graph with amplitude |",
  StyleBox["A|",
    FontWeight->"Bold"],
  ".  Notice the absolute value sign makes this statement true for both \
positive and negative values of ",
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  "= ",
  StyleBox["(3/4) sin(x)",
    FontWeight->"Bold"],
  ".  When you have finished, enter the input in the yellow box below and \
check your answer.  Make sure you try it yourself first since the computer \
makes graphing look easier than it is."
}], "Text"],

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Cell[TextData[{
  "You should have marked on your graph at least the same points that are \
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simple as graphing the original sine function and then changing the height of \
the graph to ",
  StyleBox["|A|",
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    FontWeight->"Bold"],
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    FontWeight->"Bold"],
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  StyleBox["Graphing Techniques",
    FontSlant->"Italic"],
  " notebook to refresh your memory.  When you have completed your graph, \
input the following program to check your answer."
}], "Text"],

Cell[BoxData[
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      PlotStyle -> Thickness[0.008], 
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Cell[TextData[{
  "Your procedure for graphing this function should have been to graph the \
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  StyleBox["x",
    FontSlant->"Italic"],
  "-axis.  Note: You could have reflected the original cosine function first \
and then changed the height.  \n\tWhat do you notice about the ranges for the \
functions above?  Write down the range in each case and then see if you can \
write down a generalized statement about the range of ",
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}], "Text"],

Cell[BoxData[
    \(Print["\<The range for f(x) = A sin(x) is [-|A|,|A|].\>"]; \n
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Cell[TextData[{
  "\tFrom our conclusions about the range of ",
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  StyleBox["(x) ",
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  StyleBox["(x)",
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    FontWeight->"Bold"],
  ".\nEssentially this property is just another way of stating the range for \
these functions. \n\tNow that we understand what the amplitude is for the \
functions ",
  StyleBox["f(x)",
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  " = A sin",
  StyleBox["(x)",
    FontWeight->"Bold"],
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  " = A cos",
  StyleBox["(x)",
    FontWeight->"Bold"],
  ", let's relate our findings to the world we are living in.  We know that \
sound can be thought of as a wave.  Suppose a certain instrument made a sound \
whose wave was the sine wave, ",
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  StyleBox["(x).  ",
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press shift-return."
}], "Text"],

Cell[BoxData[
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sound.\>"]\)], "Input",
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Cell[TextData[{
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    FontWeight->"Bold"],
  " Based on this information, compare the sounds represented by the \
following sound wave functions:\na) f(x) = ",
  Cell[BoxData[
      \(TraditionalForm\`\(\ -\+3\%1\)\)]],
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f(x) = ",
  Cell[BoxData[
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}], "Text"],

Cell[TextData[{
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    FontWeight->"Plain"]
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\>"],
  ImageRangeCache->{{{0, 286.813}, {176.875, 0}} -> {-5.49081, -37.5945, 
  0.0486484, 0.426251}}],

Cell["\<\
Note: The vertical asymptotes displayed in this graph are not actually part \
of the function, but they do serve as good indicators of where a period \
starts and ends. \
\>", "Text"],

Cell[TextData[{
  "\tFrom these graphs, you should have concluded that for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(x)",
    FontWeight->"Bold"],
  " and",
  StyleBox[" f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["cos(x)",
    FontWeight->"Bold"],
  " the period is ",
  StyleBox["2\[Pi]",
    FontWeight->"Bold"],
  ", and for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["tan(x)",
    FontWeight->"Bold"],
  " the period is ",
  StyleBox["\[Pi]",
    FontWeight->"Bold"],
  ".  Now let's see what happens to the period when we multiply the variable \
",
  StyleBox["x",
    FontWeight->"Bold"],
  " by a constant ",
  StyleBox["c",
    FontWeight->"Bold"],
  " before applying the function ",
  StyleBox["f",
    FontWeight->"Bold"],
  ".  The following program will plot the graphs of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " for various values of ",
  StyleBox["c",
    FontWeight->"Bold"],
  ".  See if you can figure out what the period is in each case.  "
}], "Text"],

Cell[BoxData[
    \(Clear[f]; \n
    Do[Print["\<The graph of f(x) = sin(\>", n, "\<x)\>"]; 
      \(Plot[Sin[n\ x], {x, \(-2\)\ Pi, 2  Pi}, PlotRange -> {\(-2\), 2}, 
        Ticks -> {{\(-4\) Pi\ /\((2\ n)\), \(-2\)\ \ Pi, \(-\ Pi\), 0, Pi, 
              2\ Pi, 4\ Pi/\((2\ n)\)}, {\(-2\), \(-1\), 1, 2}}, 
        PlotStyle -> {Thickness[0.008], Hue[2\ n/10]}]; \), {n, 1, 5}]\)], 
  "Input",
  AnimationDisplayTime->0.815731,
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Compare the period of each function (use the graph) with the formula for \
the function.  Do you notice any relation between the period of the function \
and the constant in the function?  How does the period of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " relate to the period of the original function ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(x)",
    FontWeight->"Bold"],
  "?  See if you can write a formula relating these values.  \n\tNow look at \
the graphs for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " ",
  "= ",
  StyleBox["cos(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " for different values of ",
  StyleBox["c",
    FontWeight->"Bold"],
  ".  Again, try to figure out the period in each case.  "
}], "Text"],

Cell[BoxData[
    \(Clear[f]; \n
    Do[Print["\<The graph of f(x) = cos(\>", n, "\<x)\>"]; 
      \(Plot[Cos[n\ x], {x, \(-2\)\ Pi, 2  Pi}, PlotRange -> {\(-2\), 2}, 
        Ticks -> {{\(-4\) Pi\ /\((2\ n)\), \(-2\)\ \ Pi, \(-\ Pi\), 0, Pi, 
              2\ Pi, 4\ Pi/\((2\ n)\)}, {\(-2\), \(-1\), 1, 2}}, 
        PlotStyle -> {Thickness[0.008], Hue[2\ n/10]}]; \), {n, 1, 5}]\)], 
  "Input",
  AnimationDisplayTime->0.815731,
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Again, try to find a formula that relates the period of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["cos(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " to the period of the original function ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["cos(x)",
    FontWeight->"Bold"],
  ". How does this formula compare to the formula you got for the sine \
functions?  \n\tNow look at the graphs of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " ",
  "= ",
  StyleBox["tan(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " for different values of ",
  StyleBox["c",
    FontWeight->"Bold"],
  " and try to find the period of each graph. "
}], "Text"],

Cell[BoxData[
    \(Clear[f]; \n
    Do[Print["\<The graph of f(x) = tan(\>", n, "\<x)\>"]; 
      \(Plot[Tan[n\ x], {x, \(-2\)\ Pi, 2  Pi}, PlotRange -> {\(-2\), 2}, 
        Ticks -> {{\(-Pi\)\ /\((2\ n)\), \(-2\)\ \ Pi, \(-\ Pi\), 0, Pi, 
              2\ Pi, \ Pi/\((2\ n)\)}, {\(-2\), \(-1\), 1, 2}}, 
        PlotStyle -> Hue[2\ n/10]]; \), {n, 1, 5}]\)], "Input",
  AnimationDisplayTime->0.815731,
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Do you notice a relation between the period of the function and the \
constant in the function?  Can you write a formula that relates the period of \
",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["tan(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " to the original function ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["tan(x)",
    FontWeight->"Bold"],
  "?  How does this formula compare to those for the sine and cosine \
functions above?  \n\t From our observations, we can conclude that if ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " is a trigonometric function, then the period of ",
  StyleBox["f(cx) ",
    FontWeight->"Bold"],
  "is ",
  StyleBox["p/c",
    FontWeight->"Bold"],
  " where ",
  StyleBox["p",
    FontWeight->"Bold"],
  " is the period of the basic trigonometric function ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  ".  So for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(cx) ",
    FontWeight->"Bold"],
  "and ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["cos(cx)",
    FontWeight->"Bold"],
  " the period is ",
  StyleBox["2\[Pi]/c",
    FontWeight->"Bold"],
  " since the period of both sine and cosine is ",
  StyleBox["2\[Pi]",
    FontWeight->"Bold"],
  "; for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " ",
  "= ",
  StyleBox["tan(cx)",
    FontWeight->"Bold"],
  " the period is ",
  StyleBox["\[Pi]/c",
    FontWeight->"Bold"],
  " since the period of tangent is ",
  StyleBox["\[Pi]",
    FontWeight->"Bold"],
  ". "
}], "Text"],

Cell["\<\
\tAgain, we can get even better understanding of how constants change the \
period of a function by animating the collection of graphs above. To do this, \
click on the blue bar on the right that brackets all the graphs and then \
depress the \"Ctrl\" and \"y\" keys simultaneously.  Use the arrow keys in \
the bottom left corner to slow down the animation, and imagine that someone \
is pushing on the \"ends\" of the function, compressing the function so that \
more waves or periods appear in the window at once.  Do this for each set of \
graphs above. \
\>", "Text"],

Cell[TextData[{
  "\tIn all of the examples above, the constant ",
  StyleBox["c",
    FontWeight->"Bold"],
  " was greater than ",
  StyleBox["1",
    FontWeight->"Bold"],
  ".  The periods in each example became shorter and shorter as ",
  StyleBox["c",
    FontWeight->"Bold"],
  " increased; in other words, we were able to see more and more periods in \
the same space for higher and higher values of ",
  StyleBox["c",
    FontWeight->"Bold"],
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  "?  We know that multiplying the variable ",
  StyleBox["x",
    FontWeight->"Bold"],
  " by ",
  StyleBox["c",
    FontWeight->"Bold"],
  " should horizontally stretch the graph of the original function.  But what \
does this do to the period of the function?   Let's experiment with the graph \
of ",
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  "= ",
  StyleBox["sin(x)",
    FontWeight->"Bold"],
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  StyleBox["c",
    FontWeight->"Bold"],
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  StyleBox["0 \[LessEqual] c \[LessEqual] 1",
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}], "Text"],

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          RowBox[{\(Print["\<The graph of f((1/\>", n, "\<)x)\>"]\), ";", 
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                \(Ticks -> {{\(-8\) Pi, \(-6\) Pi, \(-4\) Pi, \(-2\) Pi, 
                      2  Pi, 4  Pi, 6  Pi, 8  Pi}, Automatic}\), ",", 
                \(PlotRange -> {\(-5\), 5}\), ",", 
                \(PlotStyle -> {Hue[2\ n/10], Thickness[0.008]}\), ",", 
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          \({n, 1, 5}\)}], "]"}]}]], "Input",
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  FontColor->GrayLevel[0.666667],
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Cell[TextData[{
  "Was the result what you expected?  How do the periods here relate to the \
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  StyleBox["sin(x)",
    FontWeight->"Bold"],
  "?  Does our formula ",
  StyleBox["new period",
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  " = ",
  StyleBox["original period / c",
    FontWeight->"Bold"],
  "  still hold?  What about for ",
  StyleBox["f(x)",
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  StyleBox["cos(c\[CenterDot]x)",
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  StyleBox[" f(x)",
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  StyleBox["tan(c\[CenterDot]x)",
    FontWeight->"Bold"],
  " where ",
  StyleBox["0 \[LessSlantEqual] c \[LessSlantEqual] 1",
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",
  StyleBox["blue",
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  StyleBox["Sin",
    FontColor->RGBColor[0, 0, 1]],
  ".  Do the periods for these functions change the way you expected them to? \
 Does the formula ",
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  " ",
  "= ",
  StyleBox["original period / c",
    FontWeight->"Bold"],
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  StyleBox["c",
    FontWeight->"Bold"],
  " when ",
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Cell[TextData[{
  "\tFrom the examples above, you should have observed that the periods \
become longer and longer or more and more \"stretched out\" as the constant ",
  
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period stays the same for ",
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  ".  "
}], "Text"],

Cell[TextData[{
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  "?  We know from the ",
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  StyleBox["f(x)",
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  StyleBox["cos(-x)",
    FontWeight->"Bold"],
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    FontWeight->"Bold"],
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Cell[TextData[{
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    FontWeight->"Bold"],
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Cell[BoxData[
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Cell["\<\
\tJust as we related the concept of amplitude to the volume of a sound, we \
can relate the concept of frequency to another one of sound's physical \
characteristics.  Can you guess what increases as the period decreases, i.e., \
as the number of periods we see in one window increases?  If you don't \
already know the answer, put the cursor in the yellow box below and press \
shift-return.\
\>", "Text"],

Cell[BoxData[
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higher in 'pitch' or 'frequency'.\>"]\)], "Input",
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Cell[TextData[{
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  Cell[BoxData[
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Cell[TextData[{
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Cell[TextData[{
  "Exercise 2.4: ",
  StyleBox["Graph two periods of f(x)",
    FontWeight->"Plain"],
  " ",
  "= ",
  StyleBox["sin(5x).  Be sure to label the significant ",
    FontWeight->"Plain"],
  StyleBox["x",
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox["-values (there are eight of them).  What is the ",
    FontWeight->"Plain"],
  "period of the function",
  StyleBox["?",
    FontWeight->"Plain"]
}], "Text",
  FontWeight->"Bold"],

Cell[TextData[{
  "Exercise 2.5: ",
  StyleBox["Graph f(x)",
    FontWeight->"Plain"],
  " ",
  "= ",
  StyleBox["cos(",
    FontWeight->"Plain"],
  Cell[BoxData[
      FormBox[
        StyleBox[\(-\+3\%2\),
          FontWeight->"Plain"], TraditionalForm]]],
  StyleBox["x) making sure to label all significant ",
    FontWeight->"Plain"],
  StyleBox["x",
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox["-values.  What is the ",
    FontWeight->"Plain"],
  "period of",
  StyleBox[" this function?",
    FontWeight->"Plain"]
}], "Text",
  FontWeight->"Bold"],

Cell[TextData[{
  "Exercise 2.6: ",
  StyleBox["Graph at least two periods of f(x)",
    FontWeight->"Plain"],
  " ",
  "= ",
  StyleBox["tan(2x).  Be sure to label all ",
    FontWeight->"Plain"],
  StyleBox["x",
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox[
  "-intercepts and all vertical asymptotes.  What is the period?  What is the \
domain of this function?    ",
    FontWeight->"Plain"]
}], "Text",
  FontWeight->"Bold"]
}, Closed]],

Cell[CellGroupData[{

Cell["Phase Shifts", "Section",
  FontColor->RGBColor[1, 1, 0],
  Background->RGBColor[0, 0, 1]],

Cell[TextData[{
  "\tWe know from our previous work with functions that adding a constant ",
  StyleBox["c",
    FontWeight->"Bold"],
  " to the variable ",
  StyleBox["x",
    FontWeight->"Bold"],
  " before applying the function ",
  StyleBox["f",
    FontWeight->"Bold"],
  " causes a horizontal shift in the graph of the function.  The same thing \
occurs when ",
  StyleBox["f",
    FontWeight->"Bold"],
  " is a trigonometric function such as sine, cosine, or tangent.  Recall \
that the shift is to the left if ",
  StyleBox["c",
    FontWeight->"Bold"],
  " > ",
  StyleBox["0",
    FontWeight->"Bold"],
  " and to the right if ",
  StyleBox["c",
    FontWeight->"Bold"],
  " < ",
  StyleBox["0",
    FontWeight->"Bold"],
  ".  Enter the commands below for a demonstration of this phenomenon with \
the graphs of  ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(x)",
    FontWeight->"Bold"],
  " and ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(x+2\[Pi]/3)",
    FontWeight->"Bold"],
  ".  (Note: f(x) = sin(x+0\[Pi]/1) simplifies to f(x) ",
  "= ",
  "sin(x).)  When you are finished, to watch the full effect of the shift, \
animate the graphs by highlighting the blue cell bar containing them and \
simultaneously depressing Ctrl-Y.  (Use the arrow keys in the bottom left \
corner to reduce to the optimal speed for viewing the functions.)"
}], "Text"],

Cell[BoxData[
    \(Do[Print["\<The graph of f(x) = sin(x+(\>", n, "\<\[Pi]/\>", n + 1, 
        "\<))\>"]; 
      Plot[Sin[x + n\ Pi/\((n + 1)\)], {x, \(-2\)\ Pi, 2\ Pi}, 
        PlotRange -> {\(-3\), 3}, 
        Ticks -> {{\(-2\)\ Pi, \(-5\) Pi/3, \ \(-Pi\), \ \(-2\) Pi/3, Pi/3, 
              Pi, 4  Pi/3, 2  Pi}, Automatic}, 
        PlotStyle -> {Thickness[0.008], Hue[2  n/10]}], {n, 0, 2, 2}]\)], 
  "Input",
  AnimationDisplayTime->2.42,
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Notice that the second graph (",
  StyleBox["green",
    FontColor->RGBColor[0, 1, 0]],
  ") is the first graph (",
  StyleBox["red",
    FontColor->RGBColor[1, 0, 0]],
  ") shifted to the ",
  StyleBox["left",
    FontSlant->"Italic"],
  " by ",
  StyleBox["2\[Pi]/3",
    FontWeight->"Bold"],
  ".  This is what we should have expected since ",
  StyleBox["c",
    FontWeight->"Bold"],
  " = ",
  StyleBox["2\[Pi]/3",
    FontWeight->"Bold"],
  " > ",
  StyleBox["0",
    FontWeight->"Bold"],
  ".  See is you can predict the graph for ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(x - 3\[Pi]/4)",
    FontWeight->"Bold"],
  " by first drawing the original graph ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(x)",
    FontWeight->"Bold"],
  " and then drawing the shifted graph.  What amount should you shift by?  In \
what direction?  Check your answer by entering the input below.  "
}], "Text"],

Cell[BoxData[
    \(Do[Print["\<The graph of f(x) = sin(x-(\>", n, "\<\[Pi]/\>", n + 1, 
        "\<))\>"]; 
      Plot[Sin[x - n\ Pi/\((n + 1)\)], {x, \(-2\)\ Pi, 2\ Pi}, 
        PlotRange -> {\(-3\), 3}, 
        Ticks -> {{\(-2\)\ Pi, \(-5\) Pi/4, \ \(-Pi\), \(-Pi\)/4, 3  Pi/4, \ 
              Pi, 6  Pi/4, 2\ Pi}, Automatic}, 
        PlotStyle -> {Thickness[0.008], Hue[n/10]}], {n, 0, 3, 3}]\)], "Input",\

  AnimationDisplayTime->1.08728,
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Notice in this case the second graph (",
  StyleBox["green",
    FontColor->RGBColor[0, 1, 0]],
  ") is the first graph (",
  StyleBox["red",
    FontColor->RGBColor[1, 0, 0]],
  ") shifted to the ",
  StyleBox["right",
    FontSlant->"Italic"],
  " by 3\[Pi]/4."
}], "Text"],

Cell[TextData[{
  "\t",
  StyleBox["CAUTION",
    FontWeight->"Bold"],
  ": We recall from our earlier work on graphing functions that the graph of \
the function ",
  StyleBox["f(cx+b)",
    FontWeight->"Bold"],
  " is NOT obtained by horizontally compressing the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " by a factor of ",
  StyleBox["c",
    FontWeight->"Bold"],
  " and then shifting it by ",
  StyleBox["b",
    FontWeight->"Bold"],
  ".  Rather, we must first factor out the ",
  StyleBox["c",
    FontWeight->"Bold"],
  " and rewrite the function as ",
  StyleBox["f(c(x+b/c))",
    FontWeight->"Bold"],
  ".   This makes us realize that the procedure for graphing this function is \
to horizontally compress the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " by a factor of ",
  StyleBox["c",
    FontWeight->"Bold"],
  " and then shift it horizontally by ",
  StyleBox["b/c",
    FontWeight->"Bold"],
  " instead of just ",
  StyleBox["b",
    FontWeight->"Bold"],
  ".  "
}], "Text"],

Cell[TextData[{
  "\tWe must heed the same caution with the trigonometric functions sine, \
cosine, and tangent.  For example, consider the function ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(2x-\[Pi]/2)",
    FontWeight->"Bold"],
  ".   The graph of this function is NOT the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " = ",
  StyleBox["sin(2x)  ",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  "shifted horizonally by \[Pi]/2.  To see this, compare the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " = ",
  StyleBox["sin(2x - \[Pi]/2) ",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  "and the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " = ",
  StyleBox["sin(2x)  ",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  "shifted horizonally by \[Pi]/2 by executing the command box below.  "
}], "Text"],

Cell[BoxData[
    \(Plot[{Sin[2 \((x - Pi/2)\)], Sin[2\ x - \ Pi/2]}, {x, \(-\ \ Pi\), \ 
        Pi}, PlotRange -> {\(-1\), 1}, 
      Ticks -> {{\(-Pi\), \(-3\) Pi/4, \(-Pi\)/2, \(-Pi\)/4, Pi/4, Pi/2, 
            3  Pi/4, Pi}, {\(-1\), 1}}, \ 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1], Thickness[0.008]}]; 
    Print["\<The blue graph is the graph of f(x) = sin(2x-\[Pi]/2).\>", 
      "\<The red graph is the graph of f(x) = sin(2x) shifted by \[Pi]/2.\>"]
      \)], "Input",
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Notice that if the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " = ",
  StyleBox["sin(2x - \[Pi]/2)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " was the same as the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " = ",
  StyleBox["sin(2x)",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " shifted horizontally by ",
  StyleBox["\[Pi]/2",
    FontWeight->"Bold"],
  ", then we would not be able to distinguish the graphs.  But the two graph \
are distinct, so ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " = ",
  StyleBox["sin(2x - \[Pi]/2)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " must be ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " = ",
  StyleBox["sin(2x)",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " shifted by something other than ",
  StyleBox["\[Pi]/2",
    FontWeight->"Bold"],
  ".  In fact, we can see that the horizontal shift is ",
  StyleBox["\[Pi]/4",
    FontWeight->"Bold"],
  ".  This is the shift we expect when we rewrite ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " = ",
  StyleBox["sin(2x - \[Pi]/2)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " as ",
  StyleBox["f(x)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " = ",
  StyleBox["sin[2(x - \[Pi]/4)]",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  ".\n\tUse the next program to plot ",
  StyleBox["f(cx+b)",
    FontWeight->"Bold"],
  " where ",
  StyleBox["f",
    FontWeight->"Bold"],
  " is one of the trigonometric functions sine, cosine or tangent, and where \
",
  StyleBox["b ",
    FontWeight->"Bold"],
  "and",
  StyleBox[" c ",
    FontWeight->"Bold"],
  "are constants.  Experiment with different values for ",
  StyleBox["b",
    FontWeight->"Bold"],
  " and ",
  StyleBox["c",
    FontWeight->"Bold"],
  " and with different trigonometric functions by highlighting (ONLY) the \
parts of the program in ",
  StyleBox["blue",
    FontColor->RGBColor[0, 0, 1]],
  " and typing in the changes you want.  Remember to type in Sin, Cos, and \
Tan with capitals, \[Pi] as Pi.  Try to graph the function you chose first \
and then check your answer by executing the program.  Continue experimenting \
until you feel confident in graphing any trigonometric function of the form ",
  
  StyleBox["f(cx+b)",
    FontWeight->"Bold"],
  ".    "
}], "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{"c", " ", "=", " ", 
        StyleBox["3",
          FontColor->RGBColor[0, 0, 1]]}], 
      StyleBox[";",
        FontColor->RGBColor[0, 0, 1]], "\n", 
      RowBox[{"b", " ", "=", " ", 
        StyleBox[\(Pi/4\),
          FontColor->RGBColor[0, 0, 1]]}], ";", "\n", 
      RowBox[{\(f[x_]\), ":=", " ", 
        RowBox[{
          StyleBox["Cos",
            FontColor->RGBColor[0, 0, 1]], "[", "x", "]"}]}], ";", "\n", 
      \(Plot[{f[c\ x], f[c\ x + b]}, {x, \(-2\) Pi, 2  Pi}, 
        PlotStyle -> {{Hue[0.4]}, {Hue[0.6]}}, 
        Ticks -> {{\(-2\) Pi, \(-3\) Pi/2, \(-Pi\), \(-Pi\)/2, Pi/2, Pi, 
              3  Pi/2, 2  Pi, \(-2\) Pi/c, 2  Pi/c}, Automatic}]\), ";", 
      "\n", \(Print["\<The graph of \>", f[c\ x], 
        "\< (green) is shifted by \>", b/c, "\< to give the graph of \>", 
        f[c\ x + b], "\< (blue).\>"]\)}]], "Input",
  FontColor->GrayLevel[0.666667],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "\tBy now it should be clear to you that trigonometric functions of the \
form ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(bx+c)",
    FontWeight->"Bold"],
  ", ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["cos(bx+c)",
    FontWeight->"Bold"],
  " and ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " ",
  "= ",
  StyleBox["tan(bx+c)",
    FontWeight->"Bold"],
  " have a horizontal phase shift of ",
  StyleBox["c/b",
    FontWeight->"Bold"],
  ", NOT ",
  StyleBox["c",
    FontWeight->"Bold"],
  ".  "
}], "Text"],

Cell[TextData[{
  "Exercise 3.1: ",
  StyleBox["Graph f(x)",
    FontWeight->"Plain"],
  " = ",
  StyleBox["sin(x-\[Pi]/3) by first graphing f(x)",
    FontWeight->"Plain"],
  " ",
  "= ",
  StyleBox[
  "sin(x) and then shifting it.  What amount and direction should you shift \
the original graph?  What should the new labels on the ",
    FontWeight->"Plain"],
  StyleBox["x",
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox["-axis be? ",
    FontWeight->"Plain"]
}], "Text",
  FontWeight->"Bold"],

Cell[TextData[{
  "Exercise 3.2: ",
  StyleBox["Graph f(x)",
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