Department of Mathematics Education

An Exploration with Algebraic Relations


James W. Wilson

This exploration with graphs of mathematical relations grew out of some class discussions with prospective teachers as we examined the use of the computer for graphing mathematical relations. The use of the computer allows us to explore and conjecture and develop ideas with respect to the mathematical relations and their graphs.

The particular relation was

Cursory discussion led to agreement that the curve would cross the y-axis at three points -- -1, 0, and 1 -- and likewise the curve would cross the x-axis at three points -- -2, 0, and 2. Also, for large x, y is large, etc. The computer provides a tool to draw the graph, look closely at particular regions of the graph, and so forth. It was clear that the interesting part of the graph would be near the origin. Using Algebra Xpresser (Hoffer, 1989) the graph was obtained as follows.

A point of discussion was to explore changes in the graph that resulted from changes in the relation. For example, the following resulted when the 4 on the right hand side was change to 2. The curve as a somewhat similar shape but is more compressed, crossing the x-axis at points closer to the origin.

Why not go further and see what happens when the constant on the right hand side is 1, producing a relation symmetric with respect to x and y? Surely the symmetry will add some demand to the graph. The result is the following.

The graph now looks like the composite of two graphs -- a line and an ellipse. It is a single relation, but some algebra -- either by hand or by symbol manipulator -- can lead to a factored form of the equation,

and one factor corresponds to the line and the other corresponds to the ellipse.

Building Some Curves

Why not reverse this process? Let's begin with two mathematical relations and consider their product to build a graph. Then coefficients or constant terms in the product can be changed to build related curves.

Take the ellipse, express its equation, and graph it. For example,

is an ellipse with major axis along the line y = -x and contains the points (0,2),
(0,-2), (2,0), and (-2,0). A graph is in Figure 1.

Next, consider any line with a graph passing through this ellipse. A simple case would be y = x. But, rather than graphing the ellipse and then the line as an overlay, consider the following relation

The graph of this relation will be the ellipse and the line through it. Figure 2 is a graph of the new relation.

Using some algebra, this latter relation is transformed into

A family of curves can be generated by considering

where a and b are real numbers. In fact, if a = b and both are positive, then all that changes from Figure 2 to a new graph is scaling. For example, the three graphs

are shown in Figure 3.

Much more interesting, however, are the curves where a and b are not equal. For example, in Figure 4, a = 3 and b = 4.

In Figure 5, a = -4 and b = 4.

In Figure 6, a = 0 and b = 4.

In Figure 7, graphs for a = .5, 1, 2, and 3 are given for b = 4 in each case. In Figure 8, the values of a and b are reversed.

Figure 7. Graphs for a = .5, 1, 2, 3; b = 4

Figure 8. Graphs for b = .5, 1, 2, 3; a = 4

Perhaps the interesting cases are where a and b are nearly equal. Figure 9 is for a = 3.9 and Figure 10 is for a = 4.1, with b = 4 in each case.

Figures 11 and 12 each show a set of curves that can be produced by varying different coefficients. Each is a set of 21 graphs where a coefficient has been stepped over 2 units in steps of 0.1. Figure 11 varies the coefficient of the term from 0 to 2; Figure 12 varies the coefficient of x from 3 to 5.

Figure 11. Varying the coefficient of

Figure 12. Varying the coefficient of x

If we rewrite the equation as

the three-dimensional graph would be a surface -- much like an undulating blanket looping through the origin (0,0,0)

Now, to examine cross-sections where z = -3, -2, -1, 0, 1, 2, 3 gives the
following graphs (contour lines?):

There is more to the investigation of these curves stimulated by the look at this equation, but let us turn our attention to applying the technique of multiplying relations to build composite curves that can in turn be modified to build a family of curves. Consider a very simple case, beginning with the equations for two lines:

y - x = 0
y + x = 0.

The product of the two relations is

(y - x)(y + x) = 0

and the graph is

If the product is set equal to some constant other than 0, the graph is a hyperbola from the family of hyperbolas having these two lines as asymptotes. The following graph shows the graphs for three hyperbolas in this family. Similarly, if any other lines were selected, the technique could generate a family of hyperbolas.

(y - x)(y + x) = - 1
(y - x)(y + x) = - 10
(y - x)(y + x) = - 40

The technique can be used to build families of curves for which new conjectures and problems can be formed. For example, if the component curves are two overlapping ellipses, then the entire family of curves may be bounded. It is illustrative, but not necessary, to examine inequalities rather than equations. Let the product be

and multiply the factors in the left hand side to get

If the constant term is replaced by 15 or by 17, the following graphs are found. It is illustrative, but not necessary, to examine inequalities rather than equations.

15 17

As the constant term is replaced by numbers lower than 15 the region in the center gets smaller and the shape approaches a square with the corners rounded. When will the center region reduce to a point? Why?

As the constant term is replaced by numbers larger than 17 the four small regions get smaller. When will they reduce to four points? What are the points? Why?

Finally, one more problem . . .What is the equation for the following graph?

Does this help?

I used

for the large circle and

for the horizontal parabola. Their product and expansion are

Now, replace the second coefficient of 49 with 47 to separate the intersections points in the graph and multiply this expression by an espression for the small circle representing the eye.

Experiment with other replacements of this coefficient, or others in the expression, to see variations in the result.