by Phil Owens: powens@austincc.edu
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Feodor Chaliapin as Mephistopheles |
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| [Mephistopheles] Equations, written with pencil or pen, Must be visible in space, and when Difficulties in construction arise, We need only define it otherwise. |
For, what is formed after laws arithmetic Must also yield some delight geometric. |
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| Kurd Lasswitz, Der Faust-Tragodie | ||
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Prologue
You can skip this if you want, and go directly to the fun part,
The Transformation Lab
.
For the bold of heart, read on!
Mathematics has traditionally been divided into two main areas: Algebra and Geometry.
Algebra: A train leaving Detroit . . .
Historically, algebra has been the study of equations, and how to solve them. The ancient algebraists (for example, al-Khowarizmi, c. 825, who gave algebra its name) studied formal ways to solve certain problems that arose in everyday life (like dividing up a certain amount of money according to certain specifications). This was done by describing equations and solving them (how they were represented is another whole story, not to be told here). This tradition continues to the present time in high-school and college algebra books, and is exemplified by word problems of the type "A train leaves Detroit traveling at 70 mph. Two hours later … ".
In the 20th century, algebra split into two major mathematical areas, Analysis and Abstract Algebra, and both have become a much more abstract in their approach to the subject matter.
Geometry: The sum of the interior angles of a polygon is . . .
Geometry's stock-in-trade consists, not of equations, but of pictorialized objects: points, lines, angles, polygons, parabolas. Like algebra, it also has its roots in practical problems; for example, redrawing property boundaries after the annual Nile floods had receded, obliterating all boundary markers during the course of its at once destructive and rejuvenating flow. Of course, the seminal thinker in the study of geometry using axioms and deductive proof was Euclid (c. 300 B. C.).
In modern times, geometry has evolved into the modern subject known by mathematicians as Topology.
Analytic Geometry: putting algebra
and geometry together
| It is impossible not to feel stirred at the thought of
the emotions of men at certain historic moments of adventure and discovery
- Columbus when he first saw the Western shore, . . . Galileo when he first
turned his telescope to the heavens. Such moments are also granted to
students in the abstract regions of thought, and high among them must be
placed the morning when Descartes lay in bed and invented the method of co-ordinate
geometry.
Alfred North Whitehead |
Mathematicians have devised a clever and useful connection between Algebra and Geometry. Rene Descartes (1596-1650) was the genius who established the science of Analytic Geometry, which ties these two seemingly unrelated branches of mathematics together.
Rene Descartes
Figure 1 is the graph (geometrical representation) of the standard equation y = x2 :
Figure 1
But the story does not end here. Since a graph is a geometrical object, one can imagine moving it to another location on a coordinate, system, by shifting it around. But if one shifts a graph to another location, presumably that new graph will represent a correspondingly different equation . What will that new equation be? That is one of the questions we will be studying.
This concept is illustrated by Figure 2, in which the graph of the reference equation y = x2 (depicted in black) has been shifted to the right by one unit, and up by one unit, resulting in a graph (depicted in blue) whose equation (the study equation) is y = (x - 1)2 + 1.
Figure 2
Question: from this simple example, can you theorize a general rule that, given:
Shifts are not the only kind of transformation that one can perform on a graph. There are also stretching/shrinking transformations (as though the graph were on a rubber sheet which is being stretched in either a vertical or horizontal direction), and reflection transformations, by which a new graph is obtained by reflecting the old one across either the x-axis or y-axis. These descriptions will become more vivid as you experiment with the lesson to follow. After all, one-half of what we are studying is visual, and a picture is worth a thousand words! But first, a short description of the frames you will be experimenting with.
Back to main menu ?
The Equation Frame adopts the view of an equation as an algebraic object, with certain numerical parameters that can be supplied by the viewer (you). Five different kinds of equations are provided by the lesson to help you in your study:
quad: the quadratic polynomial (whose graph is a parabola). The reference equation is y = x2. The study equation (for which you provide the a, b, c, and d parameters) is y = a(bx + c) 2 +d.
abs: reference equation y = | x |. Study equation: y = a | bx + c | + d.
sin, cos, tan: a selection of three reference equations: y =
sin(x), y= cos(x) y = tan(x). Study equations:
y = a sin(bx + c) + d, and similarly for cos and tan
.
You will make your selection of a study equation from a frame like that shown in Figure 3.
Figure 3
You will be choosing a, b, c, and d. You choices will have an effect on the graph of the resulting study equation. You will see what the resulting graph looks like by clicking the "Graph" button. This will take you to the Graph Frame, where you will first see the reference equation. By clicking "Graph" again, you will see the graph of the study equation as it develops from the reference equation.
Back to main menu ?
The Transformation Frame adopts the view of an equation as a geometrical object, as shown in Figure 4.
Figure 4
As for the Equation Frame, you start by selecting a reference equation style (qua, abs, sin. cos, or tan). Then, rather than describing a study equation, you select directly the transformations that you wish to perform on the graph of the reference equation; for example, "shift right by 1", "reflect in the x-axis". You will then be able to see the study equation that effects those transformations, and as before, see the study equation (and its graph) develop by moving to the "Graph" screen.
Back to main menu ?
The "Graph" frame (which is depicted again here as Figure 5) is where the algebraic and geometric representations come together.
Figure 5
By clicking on "Graph", you can see the study graph evolve from the reference graph. By clicking on "Go Back", you can return to the previous frame to study a new equation and graph.
Back to main menu ?
Here is your "Graph Transformation Laboratory". It has been preconditioned with the study equation
y = (x - 1)2 + 1
for your initial mathematical romp. Click on the "Equation" button, experiment with this model, then change some things, and see what develops. Have fun!
Back to main menu ?
After playing with the lesson for a while, you might want to try the following experiments.
Experiment 1:
You will note that for study equation y = (2x + 4)2 (b=2, c=4), the horizontal shift is not "four units to the left", as might be expected. The amount of horizontal shift is determined by some combination of parameters b and c. Can you (experimentally) determine the formula for horizontal shift from the values of b and c? Can you derive it algebraically?
Back to the lesson ? Back to the main menu ?
Experiment 2:
For the quadratic and absolute value equations, the vertical and horizontal stretch/shift transformations are not very dramatic. They become more so for the trigonometric functions. Experiment with various values of these parameters and observe their effects on the graphs. In particular, for horizontal shifts, you may want to use multiples and fractions of pi (1.57, 6.28, etc.), as the x-axis is scaled in radian units (360 degrees = 2*pi radians).
In an electronics context (as well as other applied physics contexts), where one studies "sine waves", these parameters have different names. Relate our terminology, horizontal shift, horizontal stretch/shrink and vertical stretch/shrink to the more familiar electronics terminology: phase shift, frequency, period, wave length, and amplitude.