Subscribe! http://www.freemathvideos.com Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classrooms and allow more time for teaching within the classroom. Please feel free to share my resources with those in need or let me know if you need any additional help with your math studies. I am a true educator and here to help you out. Welcome ladies and gentlemen. So what I'd like to do is show you how to write the equation of a parabola when the vertex is at 0, 0 or at the origin, given either the focus or the directrix. So the main important thing when we want to write equation, we need to understand, what information do we need.
So I wrote two forms of our conic section. Here is going to be when we have a vertical parabola, either opening up or down, and here's where we're going to have a horizontal parabola either opening up to the right or to the left. And if you look at those two equations, the information that we need, we know that x and y represent the infinite set of points that make up the parabola. But h and k represents the vertex, which fortunately in this problem we know is 0, 0. So we already know the values for h and k.
The only other value we don't know is p. But we know what p means. p represents the distance from the vertex to the focus, whereas the opposite of p represents the distance from the vertex to the directrix. So we basically for these problems need to figure out the value of p, but not only figure out the value of p, determine if our parabola opens up or down or opens up left or right.
So the easiest way to be able to do that is plot the information you're given. And I tell my students whenever we're writing the equation of conic sections to plot the information you're given.
So first thing is we know we have a vertex at 0, 0. So I'm going to plot the information and I'm going to label it v. The next thing is we have a point negative 2, 0. And that's going to be the focus. Now what's important about the focus is remember, that's going to be the point that the parabola opens up towards.
So we have four options. The parabola either opens up, down, left, right. Well if the focus is to the right of the vertex, then we know that the parabola has to open up to the right. That's very helpful because now what that tells us is, this is going to be the equation that we are going to use.
So I'll simply just recreate that equation. And then all we have to do is plug in the information that we are given. Well we know that the vertex is at 0, 0. So I'm going to plug in 0 in for x and 0 in for k. The only other piece information we don't know is what p is. Well remember, p is that distance from the vertex to the focus. Sense that distance is in the negative direction, p is equal to negative 2. So it's important that direction matters for our focus. Yes, it's going a value of 2, but it's going to be negative 2.
So I'll just plug in this information, y minus 0 squared equals 4 times negative 2 times x minus 0. Now I can just go ahead and simplify. That gives me y squared equals negative 8 times x.
Now that's in the conic section form. You might see your answer though or your teacher might want it in standard form. So when that's the case, what we basically want to do is solve for our linear variable, which in this case is x. So to do that, I'm just going to divide by a negative 8 on both sides. And what I obtain is x equals negative 1/8 y squared.
In the next example, again, before I even have to do anything, let's just plot the information. A focus is a point, and I know the vertex is at 0, 0. Point 0, 1/3, that's going to be right there. So my focus is really close to my vertex. Therefore I already know, just on this information, very little thinking I need to do. The parabola opens up.
That's very important though because now I know rather than using this equation, I need to use this equation. To kind of skip a step, I'm not going to rewrite the information. I'm just going to start plugging in what we know.
We know the vertex is at 0, 0. So again, h and k are 0, 0. However now the distance from the vertex to the focus-- because here's at 0, 0, and now this one 0, 1/3. So that means I traveled 1/3 up. Since that's positive, p is equal to 1/3. So now I write my equation x minus 0 squared equals 4 times 1/3 times y minus 0. Now we just go ahead and simplify. So this gives us x squared equals 4/3 y.
So in conic section form, that's perfectly good. If we want to rewrite this in the standard form, I need to isolate the y, solve for y basically. So what I'll do is I'll multiply by the reciprocal on both sides. And then when I obtain is y equals 3/4 x squared. So you could have in this equation or in that equation, kind of whatever works.
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