Final Project I

by

Beth Richichi

Consider graphs of the equation xy=ax+by+c.

The values of a and b are held fixed at 1. The value of c is greater
than or equal to zero. Here we have a set of hyperbolas with the horizontal
asymptote at y=1 and the vertical asymptote at x=1.

Here the values of c are less than or equal to zero. When the value of
c is negative, the hyperbolas "reverse" in direction. The asymptotes
remain the same as those of the previous graph. Notice that when c=-1 (the
red curve) we see asymptotes intead of hyperbolas which intersect the axes
at x=1 and y=1. Notice that the larger value of c moves the curve further
away from each asymptote.

An examination of the above curves illustrates the "job" of the c value. The c value determines the location of the hyperbola in terms of quadrant placement and distance from the asymptotes.

This time let's hold the values of a and b fixed, yet let a and b be different values.

Let's examine the equations of x-b=0 and y-a=0 with an a value of 2 and a b value of 3. The equations x-3=0 and y-2=0 yield asymptotes at x=3 and y=2. Multiplying these two equations,

- (x-3)(y-2)=0
- xy-2x-3y+6=0
- xy=2x+3y-6

Thus, from the equation (x-a)(y-b)=0, we can see that we will find the horizontal asymptote at y=a and the vertical asymptote at x=b. When (x-a)(y-b) is not equal to 0, we have the equation (x-a)(y-b)=k for some value of k not equal to 0. As demonstrated above, this equation will result in a the graph of a family of curves, or more specifically, a family of hyperbolas.

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