Skip to 0 minutes and 11 secondsAll right, so let's have a look at the basic parabola in a Cartesian framework. Here is y equals x squared. Very familiar object to school kids. We can plot it very easily by making some points. There's the point 1,1, there's the point 2, 4. Over here is the point minus 2, 4. Minus 1, 1. And of course, this point here, very important-- the origin, 0, 0, also on the parabola. And in each case, the y-coordinate is in fact the square of the x-coordinate. Of course, it goes up in this direction indefinitely. OK, so Apollonius would have looked at a parabola like this in a number of ways.

Skip to 1 minute and 0 secondsSo one way is as a slice of a cone, like we've talked about. But Apollonius knew that there was another way of thinking about a parabola like this, in terms of sort of a metrical definition. So this is a construction that involves an auxiliary point, f, called a focus. So f is called focus. And then auxiliary line, it's called l, in green, which is called the directrix.

Skip to 1 minute and 29 secondsOK, so how do these get involved? Well, Apollonius would have said, OK, we take an arbitrary point, p, on the parabola, and then this distance from the focus, f, to this point, p always equals the vertical distance from the point p to our line, l. So in other words, this distance and this distance are equal no matter where p is on the parabola. So we can write in terms of distances that the distance from f to p over the distance from l to p equals 1. So here we connect with this ancient Greek way of thinking about parabolas, going back to Apollonius, with this more modern algebraic version, introduced by Descartes in this Cartesian setting.

Skip to 2 minutes and 34 secondsThis is gonna be a major theme for us, being able to connect the geometry and the algebra. So we'll want to extend this understanding to more general parabolas, not this particularly simple kind. For that we need to get our algebra involved. So we'll have plenty of opportunity to investigate that in the steps in this activity.

# The parabola

The parabola has some lovely properties, going back to Apollonius. In particular it can be defined by means of a special point called the focus, and a special line called the directrix.

In this introductory video we look at the simplest quadratic relation \(\normalsize{y=x^2}\) which defines a parabola. There is a metrical definition, in terms of a point \(\normalsize{F}\) called the focus and a line \(\normalsize{l}\) called the directrix. The definition is often stated as follows: that if \(\normalsize{X}\) is a point on the parabola then the distance from \(\normalsize{X}\) to the focus \(\normalsize{F}\) exactly equals the (perpendicular) distance from \(\normalsize{X}\) to the directrix \(\normalsize{l}\).

When we work in the Cartesian plane, we like to avoid the irrationalities contained in the square roots involved with distances, so it is often more convenient to restate this condition in terms of quadrances.

With the Cartesian coordinate system of Descartes, we can reasonably easily convert this kind of geometrical condition into an algebraic one. This gives us considerable power in understanding not only the basic parabola \(\normalsize{y=x^2}\) but also more general ones, obtained by scaling and translating.

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