Hey, everyone, and welcome back. So continuing on our journey through conic sections, we've looked at the circle and the ellipse shape. We're now going to take a look at a familiar shape, which is the parabola. The way that you can get a parabola in conic sections is by taking a three-dimensional cone and slicing it with a heavily tilted 2-dimensional plane, giving you this parabola shape. Now, when it comes to parabolas and conic sections, the problem-solving is a bit different because you're going to need to know how to find something called the focus and the directrix. If this all sounds overwhelming, don't sweat it, because we're actually going to learn the focus is just a point on your graph, whereas the directrix is just a line. And the point and the line for the focus and directrix turn out to actually be the same distance from the vertex of the parabola. They may not be in the same direction, but they are the same distance. So let's get right into things and see how we can solve these types of problems.

Now when it comes to finding the focus point, if your parabola opens up, the focus point is going to be somewhere up on the graph a certain amount of units. Whereas if the parabola opens down, the focus point is going to be down a certain amount of units. So if we take a look at this graph, for example, since we can see our parabola opens up, the focus point is going to be somewhere up here. Now when finding the directrix line, if the parabola opens up, the directrix line is actually going to be somewhere down on your graph. Whereas if the parabola opens down, the directrix line will be up a certain amount of units. And since again, our parabola opens up, the directrix line is going to be somewhere down here.

Now to actually figure out where we can directly find our focus and directrix, we need to take a look at the equation for parabolas. We've seen this equation in previous videos, which is the standard form for any parabola. When it comes to conic sections, the equation looks like this. Notice we do have some similarities between the two equations, but there's one main difference, which is that we have p4 rather than a. The reason for this is because this p variable is actually going to help us to graph our parabola, and it allows us to find both the focus point and the directrix line.

Now if you have a parabola with the vertex at the origin, the equation is going to look like this, basically just the h and k go away. And if we wanted to find the p value for this parabola specifically, well, what we can do is look at what the equation looks like. Notice that we have a 4 in front of this y, and in the general form of the equation, we have 4p in front of y. So we can set 4p equal to 4. And to solve for p, we just divide 4 on both sides of the equation. That will get the fours to cancel, giving us that p=1. So our p value for this parabola is 1. And to find the focus and directrix, that means that for our focus, we need to go up 1 unit to find the focus point at 1 or excuse me, 1. And to find the directrix line, we need to go down 1 unit, which is going to give us a line right about here. And whenever drawing the directrix is going to be this type of dash line, and we can see that our dash line is right here at y=-1.

Now something else I want to mention is that if you get a positive p value, which is what we have up here, then the parabola is going to open up like we can see happen in this example. Whereas if you get a negative p value, the parabola is actually going to open down. Now to make sure we can put this all together and understand the general problem-solving, let's actually try an example. And in this example, we are given the equation of a parabola and asked to graph it over here. So let's see what we can do.

Our first step should be to find the vertex, and to find the vertex, I can see that this h corresponds with this 1. So our horizontal position is 1, whereas I can see this k corresponds with 2. So the vertical position is going to be 2. That means that the center or excuse me, not the center, the vertex of our parabola is going to be horizontally 1 unit to the right and up 2 units, giving us the vertex right about there. Now our next step is going to be to find the p value, and to find the p value, I can see that our 4p, which is going to be in front of the y−k, corresponds with this 8 in front of the y−2. So we can recognize that 4p=8, and to find p, we just divide 4 on both sides, giving us that p=2. Now our third step is going to be to find the focus, and to find the focus we need to go a certain amount of p units, the absolute value of p, from the vertex. And I can see here that our p value is positive, meaning our parabola is going to open up, and since our parabola opens up, we need to go up p amount of units. So if I start at our vertex, I can go up 1, 2 units since our p value is 2, and that's going to give us our focus at the point 1 comma 4. So that's the third step.

Now our fourth step is going to be starting from the focus go left and right 2 p units, and this will tell us the width of our parabola. Now 2 times the absolute value of p is the same thing as 2 times the absolute value of 2, which is just 4. So if I start at our focus, I can go to the left 1, 2, 3, 4 units, and I can go to the right 1, 2, 3, 4 units. Giving me points at negative 3 comma 4, and another point at 5 comma 4. Now our fifth step is going to be to connect to the outside points with a smooth curve. So if I connect these outside points, that's going to give us a parabola, which looks like this. So here we have our parabola shape, and our last step is going to be to find the directrix line. And recall that our directrix goes in the opposite direction our parabola opens. Since our parabola opens up, our directrix is going to be down the absolute value of p amount of units from the vertex. And since our p value is 2, we need to go down 1, 2 units, which will give us a Directrix line right along the x-axis. And since we're on the x-axis, that means our Directrix line is at a y value of 0.

So this is how you can find the focus, the directrix, and the shape of the parabola on a graph. So this is how you can solve parabolas in conic sections. Hope you found this video helpful. Thanks for watching.