Graphs of the equation:

xy=ax+by+c

What would one think the graphs of the equation **xy=ax+by+c** would
look like, for different values of **a**, **b**, and **c**? The
easiest way to see how **a**, **b**, and **c** affect the graphs
of the equation is to keep two of them constant and vary the third. To start,
**a** and **b **will remain constant while **c** varies. **Figure
1** shows multiple graphs of the equation for **a=b=1** and **c=.25, 1,
2, 3**. The
colors of the values of **c** just mentioned correspond to the same color
in the figure.

Just from looking at **Figure 1**, it seems as though there are two
asymtotes. They look as though they are at **x=1** and **y=1**. **Figure
2** shows the same graphs as **Figure 1** along with the lines for
**x=1** and **y=1**.

The lines of **x=1** and **y=1** are indeed the asymtotes of the graphs drawn in **Figure
1**. This can be seen by plugging **1** into the equation that we have
been dealing with for the values of **x** and **y** and keeping **a**,
**b**, and **c** the same as they were in the development of **Figure
1**. When all the values are plugged in, the left side of the equation
has a value of **1** and the right side of the equation has a value other
than **1**. This tells us that the asymtotes must be **x=1** and **y=1**.

When **a** and **c** are held constant and **b** is varied or
**b** and **c** are held constant and **a** is varied, the graphs
only have one asymtote in common. If **a** and **c** are held constant,
the horizontal asymtote remains the same however the vertical asymtote changes
for every different value of **b**. Also, if **b** and **c** are
held constant, the vertical asymtote remains the same however the horizontal
asymtote changes for every different value of **a**. This is shown in
**Figure 3** below. The graph on the left corresponds to when **a**
and **c** are held constant and the graph on the right corresponds to
when **b** and **c** are held constant.

Notice how all the graphs on the left have a common horizontal asymtote
of **y=1** but do not have a common vertical asymtote and all the graphs
on the right have a common vertical asymtote of **x=1** but do not have
a common horizontal asymtote. This can be derived once again by plugging
in values for **x** and **y** into the equations for each graph.

What would you expect to happen if **a=2**, **b=1**, and **c**
varied as before? What would the graphs look like? Can one make any connections
between these graphs and the ones already described? **Figure 4** demonstrates
the graphs when **a=2**, **b=1**, and **c** varied as before.

The colors of the graphs in **Figure 4** correspond to **c=.25, 1,
2, 3** as was
the case in **Figure 1**. Look where the asymtotes are located now. They
are at **x=1** and **y=2**.
This is different from before. In **Figure 1**, both **a** and **b**
where equal to 1 and the asymtotes were at **x=1** and **y=1** and
now in **Figure 4**, **a=2** and **b=1** and the asymtotes are
at **x=1** and **y=2**. Could there be a correlation between the value
of **a** and **b** and the asymtotes? Let us look at a couple more
examples before making statements about this.

**Figure 5** displays graphs of **a=1**, **b=2**, and **c**
varied as before.

Notice that the asymtotes have changed again. The horizontal asymtote
is now **y=1** and the vertical asymtote is **x=2**. Now what would
you expect if **a=b=2** and** c** varied as before? In **Figure 6**,
**a=b=2** and **c** varies as before.

Now the asymtotes are located at **x=2** and **y=2**.

From what was learned from the figures presented, one can determine the
horizontal and vertical asymtotes. The horizontal asymtote is determined
by the value of **a** and the vertical asymtote is determined the value
of **b**. When **a** or **b** is negative, the method of determining
asymtotes still holds even though it was not shown here. The reader can
prove this or take it to be true.

When **c** is negative, nothing that has been discussed already changes.
What this does is makes the graphs go to the asymtotes faster. This can
be seen in **Figure 7**.

Another variation worth mentioning is when the left side of the equation
equals the right side of the equation. How would think the graphs would
look if the left and right side of the equation were equal? **Figure 8**
has the answer.

The green graph above is when the left and right sides of the equation
are equal. The other three graphs are just like they have always been but
the green graph becomes two straight lines of which one is an asymtote to
the other graphs. By choosing different values for **a**, **b**, and
**c**, one can get both sides of the equation to be equal with part of
the graph being a vertical asymtote. This is left up to the reader to experiment
further into.