How to determine the shape of a conic graph based on its equation

This is relatively easy to do. The shape of the graph is determined entirely by the coeffiecients and the squared terms.

If there is an x² term and a y² term (on the same side of the equation) and the two coeffiecients are equal, then the graph is a circle.

If there is an x² term and a y² term (on the same side of the equation) and the two coeffiecients are not equal, then the graph is an ellipse.

If there is only an x² term or a y² regardless of the coeffiecients, the graph is a parabola.

If there is an x² and a y² term on the same side of the equation and one term is negative, the graph is a hyperbola.