GRAPHS OF THE EQUATION
xy = ax + by + c


The purpose of this investigation is to explore various real values for a, b, and c given the equation xy = ax + by + c. In particular, we wish to hold two of the variables constant while varying the third. The first order of business is to establish the general shape of the "parent" graph by allowing a = 1, b = 1, and c = 0.

Notice that the graph appears to be a hyperbola which is asymptotic to the line y = 1 (drawn in green). We shall now try different values of c in order to see how this parent graph is altered.

Notice that the parent graph xy = x + y is shown in red, xy = x + y + 1 is shown in green, and xy = x + y + 3 is shown in blue. We should note that all three graphs are asymptotic to the line y = 1 as well as the line x = 1. It is easily seen from the graph that xy = x + y crosses the y-axis at 0, xy = x + y + 1 crosses the y-axis at -1, and xy = x + y + 3 crosses the y-axis at -3. We are also able to examine the equations xy = x + y - 3 (shown in purple) and xy = x + y - 4 (shown in aqua). This time, we see that xy = x + y - 3 crosses the x-axis at 3 and it crosses the y-axis at 3. Furthermore, xy = x + y - 4 crosses the x-axis at 4 and it crosses the y-axis at 4. These graphs, as the ones before, are also asymptotic to the lines y = 1 and x = 1. At this point, it is possible to make several conjectures:

1) the graph of the equation xy = x + y + c is a hyperbola asymptotic to y = 1 and x = 1;

2) If c > 0, the graph of xy = x + y + c crosses the y-axis at (-c);

3) If c < 0, the graph of xy = x + y + c crosses both the x-axis and y-axis at c.

Let's consider the same equation, but allow a = b = 2. It follows that our equation would be xy = 2x + 2y and we must compare this graph with the graph of the equation xy = x + y.

The graph of xy = x + y is shown in red, whereas the graph of xy = 2x + 2y is shown in green. The two graphs are similar in that they both cross the y-axis at 0; however, they are different in that xy = x + y is asymptotic to the line y = 1 and xy = 2x + 2y is asymptotic to the line y = 2 (shown above in blue). Now, how does a constant transform the graph?

Let's treat each graph individually. The graph xy = 2x + 2y is shown in green, is asymptotic to the graph graph y = 2 and x =2, and crosses the y-axis at 0. The graph of xy = 2x + 2y + 1 is shown in red and crosses the y-axis at -.5--which is one-half of the constant. The graph xy = 2x + 2y + 3 is shown in blue and crosses the y-axis at -1.5, and this is one-half the constant. The graph of xy = 2x + 2y - 1 is shown in gold and crosses the y-axis at .5. The graph of the equation xy = 2x + 2y - 3 crosses the y-axis at 1.5. Once again, we may generalize:

1) The graph of xy = 2x + 2y is a hyperbola asymptotic to y = 2 and x = 2;

2) If the equation xy = 2x + 2y + c, the graph crosses the y-axis at (-c/2).


We will continue this investigation along the same lines by graphing xy = 3x + 3y + c.

Here, we see that the hyperbolas are all asymptotic to the lines y = 3 and x = 3. The red graph, green graph, and blue graph represent the equation xy = 3x + 3y + c where c = 0 , c = 1, and c = 3, respectively. Similarly, the gold graph and the purple graph represent the xy = 3x + 3y + c where c = -1 and c = -3, respectively. As before, these graphs cross the y-axis at (-c/3).

So far, we have only discussed positive values of a and b, let's try a negative value--such as a = b = 4. Hence, the equation is xy = -4x - 4y + c.

 

We have viewed enough graphs now to make several generalizations. Consider the general form xy = ax + by + c, where a=b, and a, b, and c are real numbers.

The equation will always graph as a hyperbola. The hyperbola is always asymptotic to the lines x = a (or b) and y = a (or

b).

The y-intercepts of the graph may be obtained by the formula .

If you were to translate the origin to the point of intersection of the asymptotes and consider the four quadrants, the

hyperbolas lie in the first and third quadrants. EXCEPTION: if a = b = 1 or if a = b = -1 and xy = ax + by + c and c < 0,

the hyperbola will lie in the second and fourth quadrants.

COULD THIS LESSON BE EXTENDED?
 

One possibility for extending this notion is to explore the conic section definition of a hyperbola. By definition, a hyperbola is the set of all points P in the plane such that the difference of the distances from P to two fixed points is a given constant. Each of the fixed points is a focus of the hyperbola and we shall refer to these as F1 and F2. In addition, the center of the hyperbola is the midpoint of the line segment joining its foci. Indeed, we may use the locus of points to trace-out a hyperbola. Click here to see the "envelope" of lines created by this formal definition.


Another possibility would be to explore the equation xy = ax + by + c, where a and b are different. We have already tested the case where a = b and made several observations; however, let's just look at a few graphs. Consider


and

It appears as though our asymptotes will no longer be vertical and horizontal because the graph seems to be "tilted," or "rotated." Furthermore, students could be asked to explore the equations for the asymptotes, make observations about x-intercepts and y-intercepts and then generalize, and explore other values of a and b.


A third option would be to have students research other conic sections and their definitions, take the equation xy=ax+by+c and note what changes would have to be made to the equation in order to transform it into a circle, or a parabola, or an ellipse. In addition, students could be asked to explore the relation

and determine under what conditions the relation is a hyperbola, or a parabola, etc.. They could use the fact that the discriminant aids in making this determination:

Students could also explore several equations such as

and their graphs using a program such as Algebra Xpresser.


Problem Number 2 of Final Exam