What information do you need to get algebraically, to graph a conic section?
There are additional questions asked about the graphs and the equations, but to get a good sketch of the graph:
You need to know whether the axes have been rotated. (You'll need trigonometry to get the graph if the have been.)
You need to identify the type or kind of conic section.
You need to put the equation in standard form for its type.
(Well, you don't "need" this to graph something like
Depending on the type of conic, you'll need other information depending on how detailed you want your graph:
Circle : center and radius
Ellipse : center and either the lengths or the endpoints of the major and minor axes
(Sometimes we are also interested in the coordinates of the foci.)
Parabola : vertex, direction it opens, perhaps 2 more points
(Sometimes we are also interested in the parameter
Hyperbola : center, directions of opening,
(Sometimes we are also interested in the foci.)