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Master Writing the equation of a parabola given the vertex, focus or directrix
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Welcome, ladies and gentlemen. So what I'd like to do is show you how to write the equation of a parabola given certain pieces of information, such as what is the vertex, what is the focus, and what is the directrix. However, I only give you two. We always kind of leave out one. And in reality we don't need all three pieces of information, because to write the equation of a parabola all we really need to know is h and k, which is a vertex.
So we have to figure out the vertex, and then we have to figure out what the value p is. Now remember, p is the distance from the vertex to the focus, or opposite of p is the distance from the vertex to the directrix. So as long as we have one of those, we can figure out what the value of p is. My best practice for writing the equation of a parabola is to sketch the information. So remember, the vertex and the focus are points, and the directrix is a line.
So in reality, all we're simply doing is plotting points and a line, whatever's given. So my first example, I'm just going to plot my vertex, which is at 2, negative 3. And I'm going to label it so I don't forget what I did. And then I'm going to plot the focus, which is at 2, comma, negative five-- so over 2, down to negative 5. And then I'm going to plot that.
Now again, I think this is important not only to see the information, but it also just helps you visualize. Remember that a parabola either opens up up and down, like we did when we learned quadratic equations, or it opens up left or right. If it's opening up or down, x is squared, and that's our equation. If it's opening left or right, y is squared, and this is the question we're going to use.
So since our vertex always opens up towards our focus, you guys can see that my parabola is going to open down. Therefore, here is going to be the equation that I'm going to use. Remember, h and k represent the coordinates of your vertex, where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex. So we can see here that 2, negative 3, is our h and our k. The only other piece of information we need to figure out is what exactly is p.
Well, remember, p is the distance from the vertex to the focus. Well, since I graphed it, I can actually just count 1, 2. But since it's going 2 and then down, it means it's going in the negative direction. So I'm just going to write p equals negative 2 above. Now, all I'm simply going to is plug them into my equation, because I know that the parabola is opening down. So I'm just going to type in x minus h, which is 2 squared, equals 4 times p, which is negative 2, times y minus k, which is negative 3.
Now all I need to do is just go ahead and simplify. So x minus 2 is just going to lay leave like that. 2 squared equals negative 8, and then y minus the negative 3 is going to be y plus 3. Now, sometimes your teacher might ask you to also put this in standard form. So to do that, this would be in conic section form, which is preferred. But to write in standard form, what we basically need to do is solve for our linear variable. So we need to solve for y.
So to do that, in best practice it's going to be to expand on my binomial squared, which isn't that bad because it just takes you to a perfect square trinomial, which is x squared, minus 4x, plus 4. And then distribute my negative 8, which equals negative 8y, minus 24. Now to solve for y, all I simply need do is add 24, and I get-- let's go and switch these around. So negative 8y equals x squared minus 4x, plus 28. And then we just divide by negative 8.
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