Welcome back, everyone. So up to this point, we've been talking about parabolas in recent videos. And most of the parabolas that we've dealt with throughout this entire course have looked something kind of like this, where we have this vertical parabola that opens either up or down. Now the question becomes, what happens if I were to take this parabola and tilt it so it's on its side? How much would the equation really change, and how would the problems differ? Well, there are going to be some changes to the equation and graph, but don't sweat it because in this video, I'm going to show you how the equation is going to change, and I think you're going to find that the concept is pretty straightforward. So let's get right into things.
So when it comes to horizontal parabolas, they are just like the vertical ones, except the x's and y's are going to be switched. So basically, all you're dealing with is the inverse of vertical parabolas when you have a horizontal one. Now the directrix that you see, and we talked about that in the previous video, is always going to be perpendicular to the axis of symmetry. So if we look at an example of a vertical parabola, like we have down here, the ones we typically see, the axis of symmetry would be right down the middle here. And notice how the directrix is perpendicular or horizontal. So when dealing with vertical parabolas, the directrix is always going to be horizontal. But now let's say you have a horizontal parabola. Well, notice that the axis of symmetry is now going to be a horizontal line, which means the directrix is going to be vertical. So the directrix is vertical for the horizontal parabola and horizontal for the vertical parabola.
Now notice when looking at a parabola at the origin, the equations for the vertical and horizontal parabola look very similar. The only real difference is the y's and x's are switched, so we have: 4 p y = x 2 here, and we have: 4 p x = y 2 there. If you look at the specific equation for these, we have: 2 y = x 2 and: 2 x = y 2 . So you're literally just switching the two variables from the vertical parabola to get to a horizontal one. Now a way that you can remember this is to notice that we actually have a y squared. We don't typically see the y variable squared when dealing with these types of equations.
Now I also want to mention, it's important to keep track of the sign of the p value that you get. Because if your p value is positive, then the parabola is going to open up if you have a vertical parabola. However, if the p value is negative, the parabola is going to open down. Now as for a horizontal parabola, if the p value is positive, the parabola is going to open to the right. Whereas if the p value is negative, the parabola will open to the left. And I think this makes sense since the negative numbers that we see on a graph are typically going to be down and to the left on your graph, whereas the positive numbers that we see are typically up and to the right on a graph.
Now to make sure we understand this concept, let's see if we can actually apply this to an example, and we'll go through the steps here. So here we're asked to graph the parabola given this equation. Now I noticed that we have a y², which means our parabola is going to open sideways. And since I see that our p value, whatever we have in front here is positive, that means the parabola is going to open to the right. The first step we're going to take in graphing this is finding the vertex or center of our parabola. But I noticed that we don't have any h value, and we don't have any k value either. We just have 8x = y². So that means our h and our k are both going to be 0, so this parabola is going to be centered at the origin.
Our next step will be to calculate the p value. To calculate the p value, I can recognize that this 8 is going to be equal to 4p. So if: 4 p = 8 , I can solve for p by dividing both sides of the equation by 4. This gives me that p is equal to 2. Our next step is to find the focus. And we discussed in the last video that the focus is always going to be in the direction that the parabola opens. Since we discussed before that this parabola will open to the right, that means our focus is going to be to the right. So if we start here at our origin, we're going to go p units to the right, and p equals 2, so we're going to go 1, 2 units to the right. And this here would be our focus at (2,0).
Our next step is going to be to find the width of the parabola. And since we have a parabola that's going to open to the right, the width is going to be up and down. So we need to go up and down 2p units to find our width. Now we need to start from the focus point, which we determined was right here. So if I go up 2p units, well, 2p is the same thing as 2 times 2, which is 4. So I can go up 4 units to get to this point, which is at (2,4), and then I can go down 4 units from our focus point, which will get me right here to (2, -4). Now our last step is going to be to connect these points with a smooth curve. All I need to really do from here is draw what the parabola is going to look like, and it should look something like this. And our last step is going to be to find the directrix. And recall that the directrix is always in the opposite direction that the parabola opens. So since the parabola opens to the right, the directrix is going to be here to the left. Specifically, it's going to be to the left, p units. So if I go ahead and look at our p value, which is 2, we need to go 1, 2 units to the left, so we'll have that x is equal to -2. So our directrix is the line x = -2, and our focus is the point (2,0). This is the graph of our shape, and that is how you can solve horizontal parabolas. Hope you found this video helpful. Thanks for watching, and let's move on.