>> When we first look at this equation, it might seem that things were quite a bit jumbled up, but just realize that that standard form has X squared isolated on the left side of the equation and the other information on the right. And we might start from that point to rearrange our equation. Just put this term over here, and that term over here. Alright. Now, let's go from there. We want to isolate X squared so we would divide on both sides by negative 2, or we could think of multiplying on both sides by negative one-half. Over here, the negative 2 would go out, and over here we have negative one-half Y. Now, the trick is to force the factorization of 4 out of this coefficient. That is, 4 times what, you see, will give negative one-half, and really, if, if there's any dilemma about identifying the value that should go in here, and there, there should be probably a dilemma in this problem. It's, it's not self-evident as it was before, but what we can do is this. We can set up a little equation to help us. We can say, well, I'd like to know what 4 can be multiplied times, and I'll call that unknown value P, to be equal to negative one-half. You see, 4 times what is negative one-half. That's what we're trying to find out for this other factor. Now, dividing on both sides by 4 or multiplying by one-fourth, we find P to be negative one-half. Down here's the parenthesis. It's, I'm multiplying on both sides by one-fourth. You see, multiply over here by one-fourth, and the 4 disappears. And multiply over here by one-fourth, and we get a P value of negative one-eighth. Alright. So if P is negative one-eighth, then this factor is negative one-eighth, and now we have the directed distance from vertex to focus, alright, to construct the graph, we would think this way. Well, let's see. From vertex to focus is negative one-eighth. That means we go down one-eighth. So it's pretty tight here on the, on the origin, and the directrix [phonetic], if we want identified, is one-eighth up, one-eighth in the other direction, you see. So it's kind of like this, and the graph, then, will have this kind of an appearance. Once again, as the focus gets really close to the, to the vertex, we have this wraparound quality, and the graph tends to look a bit thin, and as it grows farther and farther away, it looks fatter and fatter. Alright. We're not restricted in our study here to parabolas that open up and down. We can talk about parabolic curves that open to the side. We're not talking about functions here, you see. We are, we're talking just about curves, geometric figures. OK. So, we know at this point that if the graph opens up or down, then the equation takes this form. X squared is 4PY. It turns out that when the graph opens to the side, either to the right or to the left, then the equation takes the form Y squared equals 4PX.