Final Project #1

xy = ax + by + c

by

Brian Seitz

The best way to examine an equation such as

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.

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