Part One: Graphing Calculator Exploration of xy=ax+by+c

By: Mary Wisniewsky

We are going to explore various graphs of the equation xy=ax+by+c.

This is a rational equation and therefore will most likely contain asymptotes.

1.) First lets take a look at the graph when a and b are fixed and c is varied.

When c is the only coefficient being varied, all the graphs for each equation contain the asymptotes of y=1 and x=1 (except for when c=-1 but we will take a look at this).

Why are x=1 and y=1 asymptotes for these equations?

Well when c=-2,

xy=x+y-2

(1)(1)=1+1-2

1=0 is not a true statement.

In order for the equation to intersect these two lines, the right side of the equation needs to also equal xy=(1)(1)=1.

Thus, when c=-1 we have for the right hand side of the equation:

x+y-1=1+1-1=2-1=1.

This is why x=1 and y=1 is not an asymptote for this case.

What will our asymptotes be for other cases?

2.) Let's observe the graph of the equation when b and c are fixed and a is varied

Similar to Case 1, we do have an asymptote for x=1, but y=1 is not the only asymptote.

The lime green graph is similar to our first case therefore it has similar asymptotes.

However, take a look at the cases when a=-2, 0, 2, -5.

These graphs all contain the asymptote x=1, but when a=-2, 0, 2, & -5 they also have the asymptote

y=-2, 0, 2, -5.

Could the value of a correspond to graph having an asymptote of y-a=0?

3) Now let's take a look at the graph of the equation when a and c are fixed and b is varied

Similarly to the above cases, when one of the coefficients in front of x or y are fixed, most all of the graphs have one of the same asymptotes.

Since a is fixed, most of the graphs contain the asymptote y-a=0, which in this case is y-1=0 or y=1.

In this case, b is being varied. Take a look at when b=-2, 0, 2, -5.

For these values of b, the graphs have an asymptote of x=-2, 0, 2, -5 respectively.

Therefore when b is varied the asymptote for the graph in most cases will be x-b=0.

So to sum up, the graph of xy=ax+by+c, may have asymptotes of:

x-b=0 and y-a=0.

However, we did see cases where depending on the value of c, the graph did not contain those asymptotes.

Let's take a further look into these asymptotes when b=-5 and a=2

and what the graph of (x-b)(y-b) is set equal to a constant looks like

Here in this graph, we can see that also the asymptotes for (x+5)(y-2)=1 are x+5=0 and y-2=0

which is in the format of x-b=0 and y-a=0.

The graph also resembles the graphs of the various equations we graphed before.

What happens when we expand the equation (x+5)(y-2)=1?

When we expand the equation and rearrange it so that xy is on the left-hand side, we have come to our original equation.

Compare the two graphs:

These two equations have identical graphs!

So when expanding (x-b)(y-a)=d, such that d is a constant, you have:

where (d-ab)=c