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 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 a d*yy2=x2 here, and we have d*xx2=y2 there. If you look at the specific equation for these, we have 2y=x2 and 2x=y2. 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. And we also know that we don't usually see parabolas opening up sideways, so we can recognize that the uncommon y squared is associated with the uncommon sideways parabola.

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 y squared, which means our parabola is going to open sideways. And since I see that our p value is positive, that means the parabola is going to open to the right.

Now 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 an 8 x and then equals y squared. 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.

Now our next step will be to calculate the p value. So to calculate the p value, I can recognize that this 8 is going to be equal to 4p. So if 4p is equal to 8, I can solve for p by dividing both sides of the equation by 4. This gives me that p is equal to 2. So our p value is 2.

Now our third step is to find the focus. And we discussed in the la