Parabolas, Ellipses, Circles & Hyperbolas

Why are these called Conic Sections? It's about "cones" and "cuts" ("sect" is Latin for "cut"). Still unclear? To delve, obtain two traffic cones and set one on top of the other, upside-down so they're tip-to-tip. Next, steal a high-power laser from a top-secret government lab and use it to slash through the stacked cones in one swipe. (Think Darth Vader's light saber and Luke's arm.) Voila: the molten edge of the remaining cones will make a circle, ellipse, parabola or hyperbola, depending on the angle of the cut! I assume that clears things up for you.
PARABOLA CONFUSION: This chapter only covers the non-function version of parabolas, where there's only x's and y's and no 'f(x)'. If in class you're currently learning about factoring, max/min problems, the Quadratic Formula, etc., then you should instead check out Graphing Quadratics or Solving Quadratics.

Intro to Conics

In this video I'll teach you to how to just look at an equation and know if it's a circle, ellipse, hyperbola or parabola, and we'll look at what they have in common. I'll also emphasize the most common mistakes students make with the conic formulas, as well as explaining the differences between the two common parabola equations you'll see.


In this video we'll see how to graph circles from their equations, as well as how to get a circle's equation from its graph. We'll also check out the distance and midpoint formulas, and use them for a circle word problem. Finally, we'll use completing the square to put a random circle equation in standard form.


In this rather long video we'll hit all the crazy details of the stretched-out circles we call ellipses: vertices, co-vertices, co-co-vertices (I made that one up), foci (that one's real), and the "constant sum". To find the foci of an ellipse, we'll use a mutant Pythagorean theorem unique to ellipses: b2+c2=a2.


These hyper parabolas take conic craziness to another level, combining all the craziest stuff we've seen in graphing: asymptotes, foci, vertices, weird dashed-line boxes. Even a minus sign! Plus, you've got to just look at the equation and figure which way it opens. To find the foci, we're back to the usual Pythagorean theorem: a2+b2=c2.

Parabolas: Directrix & Focus

I saved parabolas for last because even though you probably think you know something about parabolas from past chapters, there are a couple new details, like focus and directrix, that are very similar to hyperbolas and ellipses.