In this write-up, I examine the graphs of the equation

for different values of the coefficient *a*. The different
types of curves generated for various values of *a* will
be discussed, as well as the values of *a* that mark the
boundaries between these curves. The behavior of the curves as
they approach the boundaries will also be examined.

When a = 0 the equation simplifies to the following:

This equation results in the graph of a circle centered at the origin with radius of length 3, as shown below.

For some values of *a* in the original equation:

the resulting graph has an elliptical shape. This oval shaped
curve exists when the absoluate value of *a* is greater than
0 and less than 2. When *a* is a positive number, the elliptical
curve is oriented so that the major axis is the line y = -x. The
graph below shows the curves generated for *a* = .3 (purple
graph), *a* = 1 (red graph), *a* = 1.5 (blue graph),
and *a* = 1.8 (green graph).

As seen in the graph, as the value of *a* increases from
0, the curve is elongated along the major axis. The curves generated
when *a* is a negative number are oriented so that the major
axis is the line y = x. The graph below shows the curves generated
for *a* = -.2 (purple graph), *a* = -1 (red graph),
*a* = -1.4 (blue graph), and *a* = 1.8 (green graph).

When comparing two curves where the absolute value of *a*
is equal, the curve with the oppositve coefficient can be found
by rotating the original curve ninety degrees about the origin.
As the absolute value of *a* approaches 2, the graph approaches
the boundary where the curve is no longer elliptical. The closer
the coefficient gets to the boundary, the more the graph is stretched
along the major axis.

When the absolute value of the coefficient of a equals two, the following two equations result:

The graph of each equation becomes the set of two parallel lines. The graph of the first equation is shown in red and the graph of the second equation is shown in blue.

We can find the equations for all four of these lines by factoring the original equations to yield:

Taking the square root of both sides of the equation results in the following:

, which becomes or

, which becomes or .

This gives us the equations for the four lines:

, shown as the upper blue line in the graph above and parallel to

, shown as the lower blue line in the graph above

, shown as the upper red line in the graph above and parallel to

, shown as the lower red line in the graph above

As mentioned above, the blue lines are parallel to one another
and the red lines are parallel to one another. We can see by the
slopes that the blue lines are perpendicular to the red lines.
Another interesting obervation is that the lines generated by
the coefficient *a* = -2 are parallel to y = x, which was
the major axis of symmetry for the ellipses formed with negative
values of *a*. Likewise, the lines generated by the coefficient
*a* = 2 are parallel to y = -x, which was the major axis
of symmetry for the ellipses formed with positive values of *a*.

When the absolute value of *a* is greater than 2, the
resulting graph has the shape of a hyperbola. For values of *a*
closer to the boundary, there is only a slight curve, approaching
a linear graph with a slope of 1 or -1. As the magnitude of *a*
grows larger, the hyperbolic shape is more pronounced as the curve
approaches both the x-axis and the y-axis.

For positive values of *a*, the axis of symmetry for the
curve is y = -x. The following graph shows the curves for *a*
= 2.1 (green graph), *a* = 8 (blue graph), *a* = 16
(purple graph), and *a* = 100 (red graph).

For negative values of *a*, the axis of symmetry for the
curve is y = x. The graph below shows the curves for *a*
= -2.1 (green graph), *a* = -5 (blue graph), *a* = -10
(purple graph), and *a* = -100 (red graph).

Similar to the elliptical shaped curves, when comparing two
curves where the absolute value of *a* is equal, the curve
with the oppositve coefficient can be found by rotating the original
curve ninety degrees about the origin. As the absolute value of
*a* approaches infinity, the graph approaches the graph of
the union of x = 0 and y = 0.

As the coefficient of the middle term in the second degree polynomial changes, the characteristics of its graph change to various conic sections. When the absolute value of the coefficient is zero, the graph is a circle. As the magnitude of the coefficient increases, the curve becomes elliptical and approaches two parallel lines. As the magnitude increases further beyond the boundary of the parallel lines, the graph is hyperbolic. For the equation

the boundary between the elliptical and hyperbolic curves occurs
when the coefficient *a* has a magnitude of 2. The graph
below shows the intersection of the curves for *a* of magnitude
0, 1, 1.5, 2, 3, and 5.