Subscribe! http://www.freemathvideos.com Want more math video lessons? Welcome, ladies and gentlemen, so what I'd like to do now is show you how to find the focus and the directrix, as well as the vertex, when we have a parabola in conic section form.
So just like we did in the last video where we wanted to determine the opening, does a graph open up, down, left, or right, that's going to be the first thing we're going to want to do in these problems as well.
So, and again, that's all going to hinge on what variable is being squared. So you can see we have two conic sections [INAUDIBLE] here-- one, where the x is squared, and then one where the y squared. Well, when the x is squared, just like quadratics, the graph is going to open up or down. And for the y squared that means the graph is going to open up to the right or to the left. So, and the first thing we want to do for all of these videos, I want to look at which variable is going to be squared.
So obviously, if I expanded x minus 2 squared and I expand it out, I would have x as squared. So that means this graph is going to be opening up or it's going to be opening down. OK?
Now, in the previous examples, we didn't have an h and a k and we just had do, you know the vertex was at 0, 0. And that was fine, and that kind of made it helpful and easy to find the vertex-- or the focus and the directrix. But now, we have some x plus and x minus, y plus and y minus. And what that represents is our h and our k, which is going to represent the vertex of our parabola. Remember the vertex of the parabola is going to be your absolute min or max of the point. It's where it kind of that line of symmetry goes through your vertex.
So that's going to be the first thing we're going to want to identify. So remember, notice in the equation, kind of like when we first started doing vertex form of quadratics. It's x minus h, x minus k. And a lot of times, this might be helpful to think of these as in parentheses on their own because you're doing x minus the value h, y minus the value k.
So why that's important is because if I was going to put these in parenthesis here, you can see that h is equal to 2. h is not negative 2. h is 2. So when I'm trying to find my vertex, I'm going to have 2.
And then here, well, the easiest way to write this is y minus a negative 1 because y minus a negative 1, that's y minus k. Well, minus a negative 1 is the same thing as plus, so that's why it's always going to be kind of we think about it as the opposite. So my vertex though ends up being 2 comma negative 1.
Now to identify the vertex, so I focus [INAUDIBLE] see my previous video. If you remember, I plotted them. And I think plotting them is so important to just visualize and understand where exactly everything means, how it's orientated, and just so you can actually find the points as well. So also if you have a problem where you actually have to graph this, just my rough sketch should be able to help you out as well.
OK, so the first thing I'm going to do is since now I have that vertex information, I'm going to plot it. So that's at positive 2, negative 1, and I label it vertex so I don't forget. Now remember, the graph either opens up or down. That's why I wrote that up there so I don't forget it.
The next thing we need to do is figure out the value p. Remember p represents the distance from the vertex to the focus. If p is positive, then the graph either opens up, or opens up to the right. If p is negative, the graph either opens down or opens up the left depending on the orientation of the parabola.
So to find p, remember p is going to be your coefficient of your linear term. So my linear term here is why. My linear term over here is x.
So whatever that is 4p is equal to it. So in this case, you can see my value is 12, so I'm going to write for p is equal to 12. Solve for p. p is equal to 3.
Now, again, my graph either opens up or down. So since the value of p is 3, from my vertex I'm going to go up 1, 2, 3. And I'm going to label that my focus. So now I know my graph opens up towards its focus, so it's going to open up to the right.
But remember, the opposite of p is going to be the distance from the vertex to the directrix. So that's going to be down 1, 2, 3. OK, so all we need to do for these problems is label the focus, so my focus is a coordinate point.
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