EMT 668
Final Project #1

Considering the graphs of the equation
xy = ax + by + c

by
Brian Seitz

The best way to examine an equation such as xy = ax + by + c is to construct graphs using Algebra Xpresser. I will attempt to construct various graphs to show tendencies of the equation. The first graph sets the value of a=2, b=4 and varies the value of c.

The next graph is a graph making a=-2 and b=4 while again adjusting the c value.

From the first two graphs, I can decipher much material and ideas about our general equation xy = ax + by + c. Keeping a, b, and c positive (in the first graph) gives hyperbolas predominantly in quadrant I and partly in quadrants II and IV. The points that the hyperbolas cross the x-axis are equal to -1/2 the value of the c value. For example, the green hyperbola crosses the x-axis at (0,-3) and the red one crosses at (0, -5). As c gets larger, the hyperbolas go farther away from the found origin (which I will soon prove is the intersection of the two asymptopes) and the curves seem to get more narrow as they go farther away.

In the second graph, the values of a and b are negative and positive, respectively. C is positive for both the green and red graphs, but they are clearly unique and in separate quadrants, but why? The c value seems to be altering the location of the hyperbola.

Also graphing x - b = 0 and y - a = 0 produces the following graph with x = 4 and y = -2. These two lines seem to be the asymptopes of the equation xy = -2x + 4y. The two hyperbolas will never cross the asymptopes. The intersection of these two lines (4,-2) is the "origin" for the hyperbolas constructed from the equation xy=-2x + 4y + c.

My hypothesis is that if c < 8 (4 x 2), than the hyperbola is similar to the green one below and if c > 8, the hyperbola is similar to the red one below. By similar, I am saying that the hyperbola is in the same quadrants (with respect to the origin of the equation) and of similar shape.

This graph above is simply a combination of the past two graphs, proving that the hyperbolas never cross the asymptopes.

Constructing a couple more graphs while altering the value of c proves my stated hypothesis, that the c value alters the location of the hyperbola and the value of c defines where the hyperbola will form.

My hypothesis seems to be valid for when c= -10 (less than 8) the hyperbola is formed similar to when c = 6 or c=0. When c = 15, (greater than 8) the hyperbola is similar to c = 10.

This graph is once again showing my belief that if c = a x b, then the graph is the asymptopes. If c< a x b then all hyperbolas will be similar (same quadrants and shapes). If c > a x b, than all hyperbolas are opposite of the ones when c < a x b. The blue graph (c=8) is farther from the intersection of the asymptopes (-5,-2) than the green graph (c=-5). The dark green graph has c=15 and one can clearly see it is opposite the blue and light green graph where c < 10.

This final graph, again states the previous findings, only the equations are different.