Here we will investigate the graphs involving and . We will start with .

This is probably no surprise if you remember that an equation of the form defines a circle with center at the origin and radius = r.

Now, add xy to the expression, so we have .

Now lets vary the coefficient of the xy term, we'll call it a,

a=.5

a=1

a=1.5

a=1.75

a=2

a=2.5

Note that it appears that as a increases from 0 (the circle equation) to a=2, it first appears to be an ellipse then "opens up" to look like two lines around a=2. For a>2, it appears to be two hyperboli.

Now lets look at a<0.

a=-.5

a=-1

a=-1.5

a=-2

a=-2.5

For a<0, we see a similar pattern. The circle opens up to look like an ellipse, then two parallel lines at a=-2, then two hyperboli.

Click here to see an animation of this as a varies between -4 and 4.