Conic concepts

So far, we have talked about how to graph two shapes: lines, and parabolas. This unit will discuss parabolas in more depth. It will also discuss circles, ellipses, and hyperbolas. These shapes make up the group called the conic sections : all the shapes that can be created by intersecting a plane with a double cone .

On the left is a double cone. If you intersect the double cone with a horizontal plane, you get a circle. If you tilt the plane a bit, you get an ellipse (as in the bad clip art picture on the right).If you tilt the plane more, so it never hits the other side of the cone, you get a parabola. If the plane is vertical, so it hits both cones, you get a hyperbola.

We are going to discuss each of these shapes in some detail. Specifically, for each shape, we are going to provide...

• A formal definition of the shape, and
• The formula for graphing the shape

These two things—the definition, and the formula—may in many cases seem unrelated. But you will be doing work in the text exercises to show, for each shape, how the definition leads to the formula.