# Graphing Conic Sections Algebraically

## Key Questions

• 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 $y = {x}^{2} - x$, if you'll settle for a sketch based on it being an upward opening parabola with $x$-intercepts $0$ and $1$)

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 $p$, the focus, and the directrix.)

Hyperbola : center, directions of opening, $a$ and $b$ to find the asymptotes
(Sometimes we are also interested in the foci.)

• A conic section in general form has the equation

$A {x}^{2} + B x + C {y}^{2} + D y + E = 0$

when either $A$ or $C$ is missing (i.e $A = 0$ or $C = 0$), you're dealing with a parabola.

when $A$ and $C$ has the same signs, you're dealing with either an ellipse or a circle. If $A$ and $C$ have the same value, you're dealing with a circle. Otherwise, you're dealing with an ellipse

when $A$ and $C$ have different signs, you're dealing with a hyperbola

• 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.