RIGHTEOUS RELATIONS  
AND 
FRIENDLY FUNCTIONS

All the  sciences using mathematics  employ it essentially to study relationships.  Physicists, engineers, chemists, biologists, economists, psychologists, sociologists, and other social scientists, seek to find connections among the various elements of their chosen fields and to arrive at a clearer understanding of why  these elements behave as they do.  In this lesson, you will examine one of the central ideas in the mathematical study of relationships.

Needs["Graphics`Colors`"]; Off[General::"spell1"]; Off[ParametricPlot::"ppcom"];

Great DOMAIN  and Home on the RANGE

RANGE AND DOMAIN

In your science class, you will investigate the way in which the growth of young turkeys is related to their age.  (YOU LUCKY INVESTIGATOR!)   You will weigh a turkey each month from the time of hatching until it has reached almost full size.  The data below is an example of such records.  The weights are to the nearest tenth of a kilogram.

Age in Months        0 (Just Hatched)     1    2    3    4    5    6     7     8     9
Weight in Kilograms         0.1           0.6   2.1  4.0  6.2  8.4  10.6  12.7  14.6  14.8

This set of data specifies a set of ordered pairs.  The first number in each pair represents the age of the turkey and the second number represents the weight of the turkey (Age, Weight).

Any set of ordered pairs is called a relation.  The set of all first coordinates of the ordered pair is the domain of the relation; the set of all second coordinates of the ordered pair is the range of the relation.   Therefore, the data listed above defines a relation
T whose domain is {0,1,2,3,4,5,6,7,8,9} and whose range is {0.1,.6,2.1,4.0,6.2,8.4,10.6,12.7,14.6,14.8}.

The relation consists of ordered pairs of real numbers.   Therefore, in the coordinate plane you can regard T as a set of 10 points represented by the graph below.

Clear[x, y]ListPlot[{{0,  .1}, {1,  .6}, {2, 2.1}, {3, 4.0}, {4, 6.2}, {5, 8.4}, {
          6, 10.6}, {7, 12.7}, {8, 14.6}, {9, 14.8}}, 
      PlotStyle -> {PointSize[ .02], Red}, 
      AxesLabel -> {"Age", "Weight"}]

The graph of a relation is the set of points whose coordinates are the ordered pairs belonging to the relation, in this case 
(Age, Weight).

EXAMPLE 1:  If  F is the relation whose graph is shown below, give a roster of the ordered pairs of (a)  F;
                                (b)  the domain of F;  
                                																															(c)  the range of F;

Clear[x, y]ListPlot[{{-2, 0}, {0, 0}, {2, 0}, {0, 3}, {0, -3}}, 
      PlotStyle -> {PointSize[ .02], Red}, 
      AxesLabel -> {"x", "y"}];

F = {
domain of F = {
range of F = {

SOLUTION:  F = {(-2,0),(0,0),(2,0),(0,3),(0,-3)}
           Domain of F = {-2,0,2}
           Range of F = {-3,0,3}

Some relations can be defined by a rule enabling you to determine the member or members of the range which are paired with each element of the domain.   The solution for the equation x - y = 4 is a relation L that can be specified as follows:

L = {(x,x-4)}

Clear[l, x]l[x_] = x - 4; Plot[l[x], {x, -4, 10}, 
      PlotStyle -> {{Red, Thickness[0.005]}}, 
      AxesLabel -> {"x", "y"}, 
      PlotLabel -> "             The relation x - y = 4."];

What is domain and range of L?

domain of L = {
range of L = {

SOLUTION:

The domain for x - y = 4 is the set of real numbers and the range for x - y = 4 is the set of real numbers.

EXAMPLE 2:  Determine the (a) domain and
                         (b) the range of:

X = {(x,y): y =  é(ÿ1ÿ-ÿxÿ¤ÿ) }

domain = {
range = {

SOLUTION:

(a) THE DOMAIN:   Because each value of y is to be a real number, each value of (1 - x¤) must be a nonnegative number.

What happens when x¤ > 1?  Try some values for x which make x¤ greater than 1.  To do that, select the next cell and press ENTER.

Clear[f, x, x0]f[x_] = 1 - (x^2);

While[x0 =!= Null, 
      x0 = Input["Type in a value x, or Return to quit: "]; 
      If[x0 =!= Null, Print["f[", x0, "] = ", f[x0]], 
        Print["Done."]]]

What happens when x¤ < 1?  Try some values for x which make x¤ less than or equal to 1.   To do that, select the next cell and press ENTER.

Clear[f, x]f[x_] = 1 - (x^2);

While[x0 =!= Null, 
      x0 = Input["Type in a value x, or Return to quit: "]; 
      If[x0 =!= Null, Print["f[", x0, "] = ", f[x0]], 
        Print["Done."]]]

What happens when x¤ = 1?  Try some values for x which make x¤ equal to 1.  What values of x make x¤ = 1?  To do that, select the next cell and press ENTER.

Clear[f, x]f[x_] = 1 - (x^2);

While[x0 =!= Null, 
      x0 = Input["Type in a value x, or Return to quit: "]; 
      If[x0 =!= Null, Print["f[", x0, "] = ", f[x0]], 
        Print["Done."]]]

We can look at three intervals for this function.  What do you think they are?

TYPE YOUR SOLUTION FOR THE INTERVALS HERE.

SOLUTION:   Are your choices for these three intervals
x æ 1
x … -1
and
-1 … x … 1 ?

WHY?

TYPE YOUR EXPLANATION HERE.

Let's look at the graphs of these intervals.

What does the graph of ( 1 - x¤) look like if x æ 1?

Clear[f, x]f[x_] = 1 - (x^2); Plot[f[x], {x, 1, 10}, PlotStyle -> {{Red, Thickness[0.01]}}, 
      Epilog -> Text[f[x], {8, -20}]];

Is (1 - x¤) a nonnegative number? EXPLAIN.

TYPE YOUR EXPLANATION HERE.

What does the graph of  (1 - x¤) look like if x … -1?

Clear[f, x]f[x_] = 1 - (x^2); Plot[f[x], {x, -1, -10}, 
      PlotStyle -> {{Red, Thickness[0.01]}}, 
      Epilog -> Text[f[x], {-6, -20}]];

Is (1 - x¤) a nonnegative number? EXPLAIN.

TYPE YOUR EXPLANATION HERE.

What does the graph of  (1 - x¤)  look like  if
   -1 … x … 1?

Clear[f, x]f[x_] = 1 - (x^2); Plot[f[x], {x, -1, 1}, 
      PlotStyle -> {{Red, Thickness[0.01]}}, 
      Epilog -> Text[f[x], {0.5,  .4}]];

Is (1 - x¤) a nonnegative number?  EXPLAIN.

TYPE YOUR EXPLANATION HERE.

(b)  THE DOMAIN:  The domain (-1 … x … 1 ) for the function 
y =  é(ÿ1ÿ-ÿxÿ¤ÿ)  , tells you that x¤ … 1.

Let's graph the function y =  é(ÿ1ÿ-ÿxÿ¤ÿ)  using the interval of the domain.

WHY?

TYPE YOUR EXPLANATION HERE.

Let's look at the graph of  y =  é(ÿ1ÿ-ÿxÿ¤ÿ)  .

Clear[f, x]f[x_] = (Sqrt[1 - (x^2)]); Plot[f[x], {x, -1, 1}, 
      PlotStyle -> {{Red, Thickness[0.01]}}, 
      Epilog -> Text[f[x], {8, -20}]];

What is the range of the function y =  é(ÿ1ÿ-ÿxÿ¤ÿ) ?

TYPE YOUR SOLUTION HERE.

SOLUTION:  0 … y … 1

FOR YOUR ENLIGHTMENT AND  CERTAIN PLEASURE---

Please do the following excellent excercises.

State the domain and the range for each of the following relations.

1.  {(5,1),(-3,0),(2,0)
    domain =
    range =

SOLUTION
domain = {5,-3,2}
range = {1,0}

2.  {(2,4),(3,9),(é7 ,7)}
    domain =
    range =

domain = {2,3,é7 }
range = {4,9,7}

3.  {(1,3),(5,3),(-8,3)}
    domain =
    range =

domain = {1,5,-8}
range = {3}

4.  {(25,-5),(8,-2é2)}
    domain =
    range =

domain = {25,8}
range = {-5,-2é2 }

5.   { ( ™í∕ , §íŸ ), (  ™í∕ ,  §íﬁŸ), (  ™í∕ ,  §í‡Ÿ)}
    domain = 
    range =

domain = {™í∕} 
range = {§íŸ , §íﬁŸ, §í‡Ÿ}

Tell the domain and range of the following relations.

 6.  {(x,y): y¤ = x¤}
				domain =
				range =

domain = all real numbers
range = all real numbers

7.  {(x,u): u =  éxÿ ÿ-ÿ ÿ2 }
    domain =
    range =    			

domain => x æ 2
range => u æ 0

Graph the following relations.

8.  {(x,y): y = x - 7}

Clear[f, x]f[x_] = x - 7; Plot[f[x], {x, -4, 10}, 
      Ticks -> {{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 
            8, 9, 10}, {-7, -6, -5, -4, -3, -2, 
            -1, 0, 1, 2, 3, 4, 5, 6, 7, 8}}, 
      AspectRatio -> Automatic, 
      PlotStyle -> {{Red, Thickness[0.005]}}, 
      AxesLabel -> {"x", "y"}];

9.  {(a,b): b = a¤}

Clear[f, a]f[a_] = (a^2); Plot[f[a], {a, -4, 4}, 
      PlotStyle -> {{Red, Thickness[0.005]}}, 
      AxesLabel -> {"a", "b"}];

10.  {(x,y): y = 2x}        

LINEAR natured RELATIONS

LINEAR RELATIONS

WELCOME TO A LINEAR RELATIONS WORLD.
Any relation which is an open sentence is called a linear relation.  The graph of a linear relation is either a straight line or a region of a plane bounded by a straight line.

The following graph shows the line y = 3x -2, in red, in the plane xy.   The line is solid because every point on the line satisfies the equation y = 3x -2.  This line separates the plane into two regions called half-planes.  There is a half-plane above the line and one below the line.

Clear[f, x]f[x_] = 3 x - 2; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Plot[f[x], {x, xmin, xmax}, 
        PlotStyle -> {{Red, Thickness[0.005]}}, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; all = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        DisplayFunction -> $DisplayFunction];

Look at the relation y æ 3x -2.  The line is still solid because all points on it still satisfy the equation y æ 3x -2, however now, the top part of the plane is shaded a perfect yellow because all the points above the line also satisfy the equation.

Clear[f, x]f[x_] = 3 x - 2; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 0.996, 0.328], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Plot[f[x], {x, xmin, xmax}, 
        PlotStyle -> {{Red, Thickness[0.005]}}, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; all = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        DisplayFunction -> $DisplayFunction];

O, interested student, examine the relation y … 3x -2.  The line is still solid because all points on it still satisfy the equation y æ 3x -2, however now, the bottom part of the plane is shaded true blue because all the points below the line also satisfy the equation.

Clear[f, x]f[x_] = 3 x - 2; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[0.319, 0.543, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Plot[f[x], {x, xmin, xmax}, 
        PlotStyle -> {{Red, Thickness[0.005]}}, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; all = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        DisplayFunction -> $DisplayFunction];

Now, math loving one, let's examine the inequality y > 3x -2.  Look at the graph below.  You will notice that the line is not solid now.  It is a dotted or dashed line.  The top half of the plane is shaded yellow.   Beautiful, is it not??

Clear[f, x]f[x_] = 3 x - 2; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 0.996, 0.328], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Graphics[{Thickness[0.005], Dashing[{ .05,  .05}], Red, 
          Line[{{xmin, ymin}, {xmax, ymax}}]}]; all1 = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> $DisplayFunction];

WHY IS THE LINE DASHED?
 WHY IS THE TOP HALF-PLANE SHADED?

TYPE YOUR EXPLANATION HERE, O EXCELLENT ONE.

SOLUTION:  There will be no points contained on the line which will make the inequality y > 3x -2 true, they will all be above the dashed line in the top half-plane.

Try some values {x,y} and see where they appear on the above graph.  (Click on the box to the right and press ENTER.)

Clear[x0]f[x_] = 3 x - 2; eq[{x_, y_}] = y > f[x]; pointlist = {}; While[x0 =!= Null, 
      x0 = Input[
          "Type in a coordinate of the form {x,y},            or Return to \
quit: "]; 
      If[x0 =!= Null && eq[x0], 
        AppendTo[pointlist, 
          Graphics[{PointSize[ .03], RGBColor[Random[], Random[], Random[]], 
              Point[{x0[[1]], 
                  x0[[2]]}]}]]; 
        Show[all1, pointlist]; For[i = 0, i < 1000, i++, Null]; ]];

Classy student, let's examine the inequality y < 3x -2.  Look at the graph below.  You will notice that the line is not solid now.  It is a dotted or dashed line.  The bottom half of the plane is shaded a quite lovely blue.

Clear[f, x]f[x_] = 3 x - 2; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[0.319, 0.543, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Graphics[{Thickness[0.005], Dashing[{ .05,  .05}], Red, 
          Line[{{xmin, ymin}, {xmax, ymax}}]}]; all2 = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> $DisplayFunction];

WHY IS THE LINE DASHED?
 WHY IS THE BOTTOM HALF-PLANE SHADED?

TYPE YOUR EXPLANATION HERE, O EXCELLENT ONE.

SOLUTION:  There will be no points contained on the line which will make the inequality y < 3x -2 true, they will all be below the dashed line in the bottom half-plane.

Try some values {x,y} and see where they appear on the above graph.  (Click on the next cell and press ENTER.)

Clear[x0]f[x_] = 3 x - 2; eq[{x_, y_}] = y < f[x]; pointlist = {}; While[x0 =!= Null, 
      x0 = Input[
          "Type in a coordinate of the form {x,y},            or Return to \
quit: "]; 
      If[x0 =!= Null && eq[x0], 
        AppendTo[pointlist, 
          Graphics[{PointSize[ .03], RGBColor[Random[], Random[], Random[]], 
              Point[{x0[[1]], 
                  x0[[2]]}]}]]; 
        Show[all2, pointlist]; For[i = 0, i < 1000, i++, Null]; ]];

FOR YOUR ENLIGHTMENT AND CERTAIN PLEASURE---

HERE ARE MORE HIGHLY MOTIVATING EXERCISES FOR YOU.
Sketch the graph of the relation defined by the given open sentence.

1.  y < x

Clear[f, x]f[x_] = x; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 0.556, 0.342], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Graphics[{Thickness[0.005], Dashing[{ .05,  .05}], Red, 
          Line[{{xmin, ymin}, {xmax, ymax}}]}]; all1 = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> $DisplayFunction];

2.  y æ x + 1

Clear[f, x]f[x_] = x + 1; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[0.549, 1.000, 0.329], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Plot[f[x], {x, xmin, xmax}, 
        PlotStyle -> {{Red, Thickness[0.005]}}, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> Identity]; all = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        DisplayFunction -> $DisplayFunction];

3.  y - 2x > 1

Clear[f, x]f[x_] = 2 x + 1; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 0.330, 0.330], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Graphics[{Thickness[0.005], Dashing[{ .05,  .05}], Red, 
          Line[{{xmin, ymin}, {xmax, ymax}}]}]; all1 = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> $DisplayFunction];

4.  y < 3x -4

Clear[f, x]f[x_] = 3 x - 4; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[0.771, 0.331, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Graphics[{Thickness[0.005], Dashing[{ .05,  .05}], Red, 
          Line[{{xmin, ymin}, {xmax, ymax}}]}]; all1 = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> $DisplayFunction];

5.  y … -1 - x

Clear[f, x]f[x_] = -1 - x; xmin = -10; xmax = 10; ymin = f[xmin]; ymax = f[xmax]; top = {RGBColor[0.106, 0.397, 1.000], 
        Polygon[{{xmin, ymin}, {xmin, ymax}, {xmax, ymax}}]}; bot = {RGBColor[1.000, 1.000, 1.000], 
        Polygon[{{xmin, ymin}, {xmax, ymin}, {xmax, ymax}}]}; axes = {Graphics[Line[{{xmin, 0}, {xmax, 0}}]], 
        Graphics[Line[{{0, ymin}, {0, ymax}}]]}; fplot = 
      Graphics[{Thickness[0.005], Dashing[{ .05,  .05}], Red, 
          Line[{{xmin, ymin}, {xmax, ymax}}]}]; all1 = 
      Show[Graphics[top], Graphics[bot], fplot, axes, Frame -> True, 
        AxesLabel -> {"x", "y"}, 
        DisplayFunction -> $DisplayFunction];

Funky FUNCTIONS

FUNCTIONS

When  you cast your eyes upon the graph of the relation {(-2,0),(1,-1),(1,3)} --- you find that two of the points of this wonderful graph are on the same vertical line.  The number 1 occurs as the first coordinate in two of the ordered pairs, (1,-1) and (1,3).  WHAT A DEAL -- TWO FOR ONE.  OKAY.  OGLE  the graph for this set of ordered pairs {(-2,0),(1,-1),(2,3),(3,2)} --- Look closely, my friend.  As you clearly apprehend, no two points of the graph lie on the same VERTICAL line.  Do you know WHY?

Listen Up!  When none of the ordered pairs of a relation have the same first (x) coordinate, that relation is a FUNCTION.   WOWEE!!

FOR YOUR ENLIGHTMENT AND CERTAIN PLEASURE---

State whether the relation indicated by each graph is a function.  If it is not a function, explain why.

1.

Clear[x, y]ListPlot[{{-2, -2}, {-1, -2}, {0, -2}, {1, 
          -2}, {2, 2}, {2, -2}, {3, -2}, {4, -2}}, 
      PlotStyle -> {PointSize[ .02], RGBColor[0.383, 0.112, 1.000]}, 
      AxesLabel -> {"x", "y"}]

2.

Clear[x, y]ListPlot[{{1, 4}, {0, 3}, {-1, 2}, {0, 1}, {1, 0}, {0, -1}, {
          -1, -2}, {0, -3}, {1, -4}}, 
      PlotStyle -> {PointSize[ .02], RGBColor[0.084, 1.000, 0.109]}, 
      AxesLabel -> {"x", "y"}]

3.

Plot[{x + 3, -x + 1, x - 1, -x - 3}, {x, -1, 1}, 
      AxesLabel -> {"x", "y"}, 
      PlotStyle -> {{RGBColor[0.689, 0.109, 1.000], Thickness[0.01]}}, 
      PlotRange -> {{-3, 3}, {-4, 4}}, 
      AspectRatio -> Automatic]

4.

Plot[{Sin[x]}, {x, 0, 2 π}, AxesLabel -> {"x", "y"}, 
      PlotStyle -> {{RGBColor[0.108, 0.113, 1.000], Thickness[0.01]}}, 
      AspectRatio -> Automatic]

5.

Plot[{x + 4, -x - 2, x, -x + 2, x - 4}, {x, -4, 4}, 
      AxesLabel -> {"x", "y"}, 
      PlotRange -> {{-4, 4}, {-1, 1}}, 
      Ticks -> {Automatic, {-1, 1}}, 
      PlotStyle -> {{RGBColor[1.000, 0.320, 0.320], Thickness[0.01]}}, 
      AspectRatio -> Automatic]

6.

Show[
      Graphics[{RGBColor[0.080, 0.990, 1.000], Thickness[0.01], 
          Circle[{2, 2}, 4]}, AspectRatio -> Automatic]]

7.

Show[
      Graphics[{RGBColor[0.108, 0.113, 1.000], Thickness[0.01], 
          Circle[{2, 2}, {9, 4}]}, AspectRatio -> Automatic]]

8.

Plot[ChebyshevT[7, x], {x, -1, 1}, 
      PlotStyle -> {{RGBColor[1.000, 0.320, 0.320], Thickness[0.01]}}, 
      AspectRatio -> Automatic]

BARRETTE BUCK TIME:

TO START THE ANIMATION:
 HIGHLIGHT THE ENTIRE CELL BELOW
  CHOOSE GRAPH FROM ABOVE MENU
   THEN CHOOSE:
    ANIMATE SELECTED GRAPHICS.

FOR ONE BARRETTE BUCK:
WHAT CURVE IS BEING GENERATED BY THE ROLLING CIRCLE?

PREVIEW OF COMING ATTRACTIONS

ΡΕΣΕΑΡΧΗ-Α-ΡΑΜΑ

Visit the wonderful world of mathematicians.
Your trip includes:
4 weeks visiting the WONDERS of the LIBRARY;
AND
an ADDITIONAL 2 weeks
visiting the
LAND OF WRITE.
JOIN us for this EXCITING and somewhat PERILOUS ADVENTURE.