## The Genius of René Descartes – Part 2 (The Parabola)

In my previous blog, I discussed how René Descartes (1596 – 1650) discovered a way to synthesize geometry and algebra, which resulted in a revolution in mathematics. This synthesis is the reason Descartes is credited as the father of analytic of geometry. Because of Descartes’s discovery, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on a conic curve.

In this blog, I will discuss how teachers can use modern computer graphing technology to help students gain a better understanding of the definition of a parabola and the equation of a parabola. In a future blog, I will discuss some of the magical properties and applications of the parabola. This discussion is intended to provide teachers with a general approach to teach the parabola. The specific approach is left to the discretion of the individual teacher. Teachers may want to provide a handout that students complete as the lesson progresses. Students should be required to use their calculators to make various calculations and verify specific facts during the lesson. When graphs are projected on a screen, presentations can be dynamic, especially with a moving trace mark and mouse drawn line segments.

Now for a quick review of the parabola. All parabolas are defined in terms of a fixed point, the focus of the parabola, and a fixed line, the directrix of the parabola. Point (x, y) is on the parabola if and only if the distance from (x, y) to the focus point equals the distance from (x, y) to the directrix line. All parabolas have an axis of symmetry and the directrix of the parabola is perpendicular to the axis of symmetry. The vertex and focus are on the axis of symmetry, and the vertex point is equidistant from the focus and directrix. Study the basic parabolic graph below.

Teaching Points: (Depending on the class, teachers need to give appropriate coaching.)

• Similar to the graph above, show the graph of a parabola and its directrix, in a handout and or on a projection screen.
• Students should be told something about a focus point and the directrix line but not the definition of a parabola. The definition of a parabola and the derivation of the equation will come at a later time.
• For at least four points (x, y) on the parabola, have students use the theorem of Pythagoras to calculate the distance from (x, y) to the focus and from (x, y) to the directrix line.
• Hopefully, most students will realize an amazing property of the parabola. For every point (x, y) on the parabola, the distance from (x, y) to the focus always equals the distance from (x, y) to the directrix. The class can experiment with other points on the parabola.
• Repeat the experiment with the equation y = 0.75x2. Does this new parabola have the same amazing property?
• Consider the parabola with equation y = kx2. How does changing k change the focus point? In general, how does changing k affect the shape of the graph? If we know the focus point of the parabola, can we find the equation of the parabola? Teachers and students can answer these questions by experimenting and just fiddling around with a computer graphing program.
• In another lesson, teachers can show students how the equation y = kx2 comes about. Since all parabolas are geometrically similar figures, the equation of most parabolas can be found by applying the standard equation transformations rules.
• Special equation transformation rotation rules:
Rotate graph 90 degrees counterclockwise about (0, 0): Replace x with –y and y with x.
Rotate graph 90 degrees clockwise about (0, 0): Replace x with y and y with –x.
Rotate graph 180 degrees about (0, 0): Replace x with -x and y with –y.
• Homework practice problems and exam questions.
Given the vertex and focus of a parabola, have students find the equation of the parabola and sketch the graph of the parabola and the directrix.
Given the graph of a parabola with key points, find the equation of the parabola, find the x-y coordinates of the focus and find the equation of the directrix line.

Teachers might consider allowing students to have some fun by having them do a project in which they are told to experiment and fiddle around with a computer graphing program to see what interesting relation graphs they can come up with. I can guarantee that an inquisitive student will come up with a graph that no human in the history of mankind as ever seen. Other than myself and some of my students, no human has ever seen the graph of the relation 2xSin(3x) + 2y = 3yCos(x + 2y) + 1.

The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered like any equation in a text book. Example: The inequality 2x – 10Sin3(3x) + 4y2 ≤ 25 is entered as 2x -10Sin(3x)^3 + 4y^2 ≤ 25. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results. The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives users a quick summary of all of the magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

## The Genius of René Descartes – Part 1

René Descartes (1596 – 1650), a French philosopher, mathematician and writer, discovered a way to synthesize geometry and algebra that resulted in a revolution in mathematics and science. Without Descartes’s brilliant insight, it would not have been possible to develop differential calculus, integral calculus, and many other branches of mathematics. What was revolutionary to Descartes’s contemporaries, now seems natural and almost intuitively obvious, a part of our culture. (Before Isaac Newton, the concept of gravity was unknown, and now all adults and most children know something about gravity.)

So what was Descartes’s world changing discovery all about? He first invented a right angle based coordinate system in which every point in the Euclidean plane is assigned a unique ordered pair of numbers, which represents the point’s location, denoted by (x, y) where both x and y are real numbers. He then demonstrated how to create algebraic equations or formulas to calculate the distance between two points, midpoint of a line segment, and the slope of a line. With these basics established, he showed how to find an x-y variable equation that describes the relationship between the x-coordinate and y-coordinate for every point on a curve and only those points on the curve. Once the equation of a curve is known, the equation can be algebraically manipulated to reveal important properties of the curve and solve a wide variety of application problems.

The diagrams below illustrates how Descartes’s great discovery is used to calculate the distance between points A and B, and the slope of the line that contains points A and B. The distance calculation is, of course, a direct application of the theorem of Pythagoras. If we let AB equal the distance from point A to point B and let m equal the slope of the line that contains points A and B, then AB = √( 82 + (-6)2 ) = √(100) = 10 units, and m = Δy / Δx = -6/8 = -3/4 or -0.75.

From the definition of a conic section and the theorem of Pythagoras, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on the curve. Study the graph of the circle and its equation. If you listen carefully, you will hear Pythagoras whisper from his grave, “x squared plus y squared equals 4 squared for every point (x, y) on the circle.” The graphs below are the graphs of various conic curves and a line. All equations are special cases of the general conic equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are real number constants.

The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered as indicated to the right of the graphic. Example: The inequality x2 + 4y2 ≤ 64 is entered as x^2 + 4y^2 ≤ 64. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results.

The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives a quick summary of all the magical mouse control clicks. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.