Hey, everyone. So let's see if we can solve this problem. In this problem, we're asked to identify the vertex, focus, and directrix of each of these parabolas. Now, to solve these problems, what I'm going to do is use the standard equation for any parabola, which is 4p⁡(y−k)=(x−h)2. Now in a lot of parabolas, you're going to see that this k and this h are not actually in the equation because that means that the parabola is at the origin. So if we didn't have this h and we didn't have this k, our equation would be 4p⁡y=x2. So these are the 2 equations that we can use when solving problems associated with parabolas.

So let's see what we can do with these examples. And we'll start with example a. Now, one of the things that I notice is that, for example, a, we don't see a k or an h in this equation because the Y is by itself, and the X2 is by itself. So that means we're going to use this version over here. So what we can do is recognize that 4py=x2 is the standard form of this equation, and we already know because this is an example where the parabola is at the origin that the vertex is going to be at the point (0,0). So we've already figured that out. Now what we can do from here is figure out the focus and directrix by using the standard equation, and I could set 4p=16. I can divide 4 on both sides, cancelling the fours, giving us that p = 4. So this is our p value, and notice that our p value came out positive. That means that our parabola is going to open upward. So the focus is going to be somewhere up a certain amount of units, and then the directrix is going to be the same amount of units down. So what we can do is find the absolute value of P, which is just going to be positive 4, and what this means is for our vertex if it's at (0,0), the focus is going to be up 4 units. So our focus is going to be at the position (0, 4), and then our directrix is going to be down 4 units since it's opposite the direction that the parabola opens, meaning our directrix is going to be at y = -4, and this would be the focus and directrix for this first parabola.

But now let's take a look at parabola b. For parabola B, we need to find the focus, vertex, and directrix here. And something that I notice is that once again, the y and the x2 are by themselves. So because of this, we can use this version of the equation again. And we also know once again that the vertex would be at (0,0) whenever we have a parabola that's at the origin. Now the next thing that we need to do is find the focus, and to find the focus, well, notice we need to find the p value so we set 4p equal to everything that's in front of the y, which we can see is 1d, and to solve for p we can go ahead and take this 4 and move it to the other side by multiplying both sides by 14. This is the same as dividing 4 on both sides, in case you're wondering. And what we can do is cancel the fourth here, giving us that p = d times 14. Now I can multiply the 3 and the 4, which is one over 12. So that means that p is going to be 112. Now because p is positive, we know that this parabola opens up, and if I take the absolute value of p, we're going to get positive one twelfth again. So that means that for our focus and directrix, we need to go up a certain amount of units, which is p, and we need to go up to one-twelfth. So our focus is going to be at (0, 1/12) and then our directrix is going to be at the line y = -1/12, so it's going to be in the opposite direction. So that's how you can find the focus as well as the directrix for parabola b.

But now, let's take a look at parabola c. For parabola c, what I noticed about this equation is th