A parabola is represented by a quadratic equation. Remember those things? Well, it turns out they were holding out on us. They were hiding some of their secret properties. Now we're going to grill them hard, because knowing parabolas inside and out will help us understand all of the conics. And we want all the help we can get.
Before, when we wanted to talk about parabolas, we would focus on the vertex and the x- and y-intercepts. You can tell that the vertex, the tippy-tip of the bump of the parabola, is still important, because we bolded it. The intercepts, though? They're old news. Ancient history. More forgotten than what's-his-face.
We've got some new hotness to define parabolas with now. They are the directrix, that line beneath the parabola, and the focus, the point inside of it. Every point, P, on a parabola is the same (perpendicular) distance from the directrix as it is from the focus. They're like two bratty siblings who can't stand it if the other one gets a single drop of ice cream more than they do.
We already knew that parabolas are symmetric around their vertex. But now we can see that the line of symmetry passes through the focus, too, and it is perpendicular to the directrix.
That Looks Painful
The focus is always going to be inside the curve of a parabola. It feels safe inside the parabola's comforting arms. The graph will always bend away from the directrix, though. And the vertex will be sitting on the parabola, right in the middle.
This is true no matter which way the parabola points: up or down, left or right.
Wait, what?
Yep, parabolas have been going to yoga twice a week, and they can bend themselves into a completely new orientation. We're going to have to deal with them like that, even if we have to turn our heads completely sideways to do it. At least we can always find the directrix, vertex, focus, and line of symmetry for each type of graph; they always follow the same order.