 Any conic section is defined by the ratio of the distance between a point and the focus and the point and the directrix. If the ratio is exactly equal to one, the conic section is a parabola. In that case, there is a point midway between the focus and the directrix. This will be the point on the curve closest to the directrix and focus. This point is the vertex. Suppose the directrix is horizontal and the focus at some point f. By a horizontal and vertical translation, we can move the vertex to the origin. And at that point, the focus will be at some location 0p, that's p units above the vertex, and the directrix will be the line y equals negative p, that's p units below. So if x, y is a point on the parabola, the distance between the point and the focus will be, and the distance between the point and the directrix will be, and since the ratio of these two distances is equal to one, this allows us to set up an equation, and we'll do a bunch of algebraic simplification to find, and this proves the first part of the following result. The equation of a parabola with focus 0p and directrix y equals negative p is 4py equals x squared. And if our directrix happens to run vertically, we can do a very similar set of manipulations to find the equation of a parabola with focus p0 and directrix x equals negative p for px equals y squared. So let's see if we can find the focus of the parabola. Now the key strategy here is that we know the equation for a parabola with focus 0p and directrix y equals minus p, so suppose we start with that one, and our equation will be 4py equals x squared. Now our goal is to find a sequence of transformations that will give us this equation 8y plus 2 equals x minus 3 squared. So one of these things is not like the other. The first thing we might observe here is that in our parabola we have x squared, but we really want x minus 3 squared. And we can get an x minus 3 if we shift horizontally to the right by 3 units. And if we do that, our equation becomes we also want a y plus 2 instead of a y, and so that means we're going to shift vertically downward by 2 units. Which we'll make our equation into, and we want coefficient 8 instead of coefficient 4p, and so that means that we'll get the same equation if p is equal to 2. And what this means is we can start with a parabola 4 times 2y equals x squared, which has directrix y equals minus 2 and focus at 0,2. We can then apply the same transformations. We shift this parabola horizontally to the right by 3 units. Then shift it vertically downward by 2 units. And this will move our focus to 3,0 and the directrix to y equals negative 4. Or let's try to write the equation of a parabola with focus 4,2 and directrix y equals 8. So let's start by graphing this information. So to begin with the vertex is midway between the focus and the directrix, and so this means it will be at 4,5. Now if we move the vertex back to the origin, we have to move it left 4 and down 5, and this moves both the focus and the vertex. The focus will move to 0, negative 3, and the directrix will move to y equals 3. And so that tells us that p is equal to negative 3, and so our equation becomes... Now let's put everything back to where it was. We'll need to move the graph right 4 units, and so our equation becomes then up 5 units, which gives us our final equation. Or suppose we know where the vertex and the focus are. So graphing label. So remember the vertex is midway between the focus and the directrix, and given the way the points are located, that means our directrix must be vertical. With equation x equal to 1. And it also means our parabola is horizontal. And again we can move the vertex to the origin by translating left 3 and down 8. And we do that the focus will be at 2,0, and the directrix will be at x equals negative 2. So p is equal to 2. And since the directrix is vertical, we begin with a graph of 4px equals y squared. We found p equal to 2, and we'll shift everything back into place, right 3, then up 8. And that gives us our final equation.