How do you graph conic sections algebraically?
It depends on what type of conic section it is. Graphing a parabola is different from graphing an ellipse, which is different from graphing a hyperbola. And the circle is another conic section.
That's just too much to try to handle in one post. (My old calculus book had a section for each of them, plus sections for translation and rotation of axes).
You may be able to find individual answers to "How do you graph a parabola algebraically? and so on.