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Once again, when you start, don’t tell them we’re doing parabolas! Tell them we’re going to create another club. This time the requirement for membership is: you must be exactly the same distance from the point (0,3) that you are from the line $y=-3$ . For instance, the point (3,3) is not part of our club—it is 3 units away from (0,3) and six units away from $y=-3$ .
Now, let them work in groups on “All the Points Equidistant from a Point and a Line” to see if they can find the shape from just that. If they need a hint, tell them there is one extremely obvious point, and two somewhat obvious points. After that they have to dink around.
When all or most groups have it, go through it on the blackboard, something like this. The extremely obvious point is the origin. The “somewhat” obvious points are (-6,3) and (6,3). Show why all those work.
Now, can any point below the x-axis work? Clearly not. Any point below the x-axis is “obviously” (meaning, after you show them for a minute) closer to the line, than to the point.
So, let’s start working up from the origin. The origin was in the club. As we move up, we are getting closer to the point, and farther away from the line. So how can we maintain equality? The only way is to move farther away from the point, by moving out. In this way, you sketch in the parabola.
Now, you introduce the terminology. We’re already old friends with the vertex of a parabola. This point up here is called the focus. This line down here is the directrix. The focus and directrix are kind of like the center of a circle, in the sense that they are central to the definition of what a parabola is, but they are not themselves part of the parabola. The vertex, on the other hand, is a part of the parabola, but is not a part of the definition.
The directrix, of course, is a horizontal line: but what if it isn’t? What is the directrix is vertical? Then we have a horizontal parabola. Of course it isn’t a function, but it’s still a shape we can graph and talk about, and we have seen them a few times before. If you have time, work through $x=4{y}^{2}\u20138y$ .
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