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.
