(11)   Graphs and Ellipses  

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• Learn or re-acquire the use of graphs in cartesian coordinates.

• Become acquainted with linear graphs, the parabola and the rectangular hyperbola. Also learn to prepare tables of paired values as preparation for plotting a line.

• Learn about graphs defined implicitly, without isolating y (or x). The circle is used as an example, also demonstrating a multiply-valued graph.

• Learn about the cartesian equation of an ellipse, with a worked example.

• Learn about the historical definition of the ellipse as the collection of points whose distances from 2 given points (the foci) has the same sum.

Stories and extras: The focusing property of an ellipsoid, in particular the focusing of whispers in the old chamber of the US House of Representatives. Also the painting of that chamber by Samuel Morse, inventor of the telegraph.

Start this lesson by explaining that the most useful and most common use of cartesian coordinates is to create graphs. On the board:

"A graph is a graphical representation of a mathematical relationship."

It is the bridge connecting shapes of lines, as seen by our eyes, with mathematical relationships and formulas.

Before getting into graphs (with which many of you here are already familiar, maybe all of you), let us first review what we know about Cartesian coordinates:

Start the discussion of graphs by a review of coordinates

-- What are "systems of coordinates"?

Methods of labeling points in space by a set of numbers, called their "coordinates."

The point's distances from two straight axes--the "x axis" usually drawn horizontally, and the "y axis" perpendicular to it

These are two numbers which give its position:

  x is the distance measured parallel to the x axis. It is measured from the y axis--to the right it is positive, to the left, negative.

  y is the distance measured parallel to the y axis. It is measured from the x axis--up is positive, down is negative.

They are (0,0), that is (zero,zero)

Yes

Polar coordinates also have an "origin" O as reference point, but instead of using (x,y) to label the position of a point P, they use the distance r from O to P, and the angle φ ("phi"--Greek f) between the line OP ("radius"--hence the letter "r") and some reference line.

Graphs--the material of section (11a).

[Note to the teacher: It is easier for the student to start with concrete examples than with abstract formulas, which need mental translation]

  A graph is a way of using coordinates to present visually the relationship between quantities. The relationship can be something observed--for instance, stock market prices (for example, as given by the "Dow Jones Index") against time, or the temperature of a patient in a hospital against time, etc. When either of these graphed quantities goes up or down, the graph will instantly show it, also telling how steep and how big the change is.

  You should be familiar with graphs, they are widely used (if the students use graphing calculators, bring that up). Graphs are even more useful for mathematically defined variations, and can be used to represent many kinds of shapes--including ellipses.

Then present section (11a), using the questions below in the presentation and/or for review.

A line drawn in a system of coordinates, on which all points (x,y) satisfy some relation between x and y.

Yes; the relationship between x and y need not be a mathematical one. However, this graph will not include the origin, because we have no data before x = 1776.

A straight line.

[That is the example in the lesson. Draw the line on the blackboard, but don't label the axes, only the origin. Then as answers come in (below) label also the intersections with the axes with their values of y and x].

   For any value of x, plug it in to the right side, calculate the value of the expression and get the paired value of y.
  The collection of all such pairs describes a line, which turns out to be straight.

On the y-axis, x=0. Put this in the equation and get y=2.

On the x-axis, y=0, so    –(2/3)x + 2 = 0.
Add (2/3)x to both sides:    2 = (2/3)x
Divide by 2:             1 = (1/3)x
Multiply by 3:            x=3.

They are if the relation between x and y has the form y = ax + b, where a and b are two numbers of either sign.

They are not with other relations, e.g. y = 3x² which is a parabola, or y = 3/x which is a hyperbola [also, if you replace "3" by any other number, positive or negative]

You bet. It is a straight line parallel to the x axis and passing the y-axis at y=2. For any value of x, y equals 2.

  [One may add a comment on the word "linear" in mathematics. The equation of a graph giving straight line may also be written "ax + by = c", and mathematicians call this a "linear" expression.
  This has been generalized to more variables, for instance 3x + 5y -2z is said to be linear, even though the points in 3-dimensional (x,y,z) space which satisfy (say) 3x + 5y -2z = 11 do not form a line but a flat plane.]

A parabola. Describe on the blackboard.

[The example below is given for illustration. It should not be on any test, and is optional material].

A hyperbola. [Describe on the blackboard, tabulate and sketch]

x =	1	2	3	4	6	8	12
y =	12	6	4	3	2	1.5	1

and

x =	-1	-2	-3	-4	-6	-8	-12
y =	-12	-6	-4	-3	-2	-1.5	-1

Point out that at x = 0, y is not defined--it is + infinity if we approach from the right, - infinity if we come from the left, in either case the point cannot be drawn.

[The next question is best left for the teacher to answer]

--Does the equation of a line always have the form y = f(x) , where f(x) is "some expression involving x"?

[No. This form is however the one almost always used, because with it, finding points on the line is very easy. You just choose your value of x, plug it into the formula and immediately get the corresponding y.

   The "expression involving x" is called "a function of x" which is why the shorthand for it is f(x).

   However, any equation connecting x and y can be used. In such cases, if we choose x, we may need some extra work to get y.

[The next example shows one of them.]

Draw a circle on the board, mark its center with O, put a system of Cartesian axes through the center, select a point P on the circle, draw its radius R and its projection A on the x axis. Mark AP as y, OA as x.

x² + y² = 25

Because for any such point, the triangle OAP is a right-angle triangle, and the above relation follows from the theorem of Pythagoras.

On the x axis, y = 0, so x² = 25, giving two solutions, x=5 or x= –5

On the y axis, y = 0, so y² = 25, giving two solutions, y=5 or y= –5

It is the same circle as before--we only need to multiply both sides by 25 to show this is another form of the same equation as before.

It is an ellipse

[Optional: Where does it cut the axes?
Optional: Let us derive 2 additional points on the ellipse. (calculator needed--see text of the lesson). Students may do part or all of the derivation..]

Its greatest length and width.

16 and 8 units--not 8 and 4. Each half of the major axis is 8.

An ellipse is the collection of all points for which the sum of the distances R₁ + R₂ has a constant value

Foci--each one, a focus.

Because Kepler's first law places the Sun at a focus--not at the center of the ellipse, which is the origin of our axes.

[Tell story of old chamber of the House of Representatives in the US Capitol. First however ask the class if anyone had been to the capitol, and ask those who have, if they remember something special about the big room where statues were collected. Let a student tell it, later, if necessary, fill in more details.]