Parabolas - Online Tutor, Practice Problems & Exam Prep

Parabolas can be derived from slicing a cone with a tilted plane, resulting in a focus point and a directrix line. The focus is a point on the graph, while the directrix is a line, both equidistant from the vertex. The equation for a vertical parabola is $(y-k)=4p(x-h)$. For horizontal parabolas, the equation switches to $(x-h)=4p(y-k)$. The sign of p determines the direction of the opening.

Parabolas as Conic Sections

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.

Graph the parabola $-4\left(y+1\right)=\left(x+1\right)^2$, and find the focus point and directrix line.

• A.
• B.
• C.
• D.

If a parabola has the focus at $\left(0,-1\right)$ and a directrix line $y=1$, find the standard equation for the parabola.

• A. $4y=x^2$
• B. $4\left(y-1\right)=x^2$
• C. $-4y=x^2$
• D. $-4\left(y+1\right)=x^2$

Parabolas as Conic Sections Example 1

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 x squared is by itself. So that means we're going to use this version over here. So what we can do is recognize that 4p⋅y=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 by 4 on both sides, cancelling the fours, giving us that p is equal to 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 somwhere 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, 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 x squared 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=13, and to solve for p I 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 I can do is cancel the 4 here, giving us that p is equal to 1/12. So that means that p is going to be 1/12. Now because p is positive, we know that this parabola opens up, and if I take the absolute value

Horizontal Parabolas

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.

Graph the parabola $8\left(x+1\right)=\left(y-2\right)^2$ , and find the focus point and directrix line.

• A.
• B.
• C.
• D.

If a parabola has the focus at $\left(2,4\right)$ and a directrix line $x=-4$ , find the standard equation for the parabola.

• A. $12\left(x+1\right)=\left(y-4\right)^2$
• B. $-\left(x+1\right)=\left(y-4\right)^2$
• C. $12x=y^2$
• D. $4\left(x-1\right)=\left(y+4\right)^2$

Horizontal Parabolas Example 1

Hey everyone, and welcome back. So let's give this example a try. Here we're asked to identify the vertex, focus, and directrix for each of these parabolas below, and we'll start with parabola a. I notice that this parabola has a y squared, which means we're dealing with a horizontal parabola. The equation for a horizontal parabola is 4p (x-h)=(y-k)². Using this equation as a reference, I see nothing is subtracted from the x and y, indicating that our h and k are 0. The vertex of this parabola is at the origin (0,0).

The focus is calculated using the p value. Setting the terms in front of the x equal, we find 4p=4, leading to p = 1. With a positive p value, the horizontal parabola opens to the right. The focus, moving 1 unit to the right from the vertex, is at the coordinate (1,0) and the directrix is the line x = -1.

For parabola b, the y term is subtracted by 2, indicating the vertex is at (0,2). Setting 4p=9, we find p = 9/4. The parabola opens to the right; the focus is at (9/4, 2) and the directrix is x = -9/4.

Looking at parabola c, with x and y terms subtracted by 4 and adds 2 respectively, this implies k = -2, so the vertex is at (4, -2). Setting 4p=16 yields p = 4. The focus from this vertex moves 4 units right to (8, -2) and the directrix is x = 0.

Finally, for parabola d, with the vertex affected by subtracted x by 1, setting 4p=−2 gives p = -1/2. With a negative p, the parabola opens left; the focus located at (1/2, 0) moving left by 1/2 unit, and the directrix is x = 3/2.

This methodology allows one to solve these types of problems with horizontal parabolas. Hope you found this video helpful, and thanks for watching.

Here’s what students ask on this topic:

What is the standard form equation of a parabola?

The standard form equation of a parabola depends on its orientation. For a vertical parabola, the equation is $(y-k)=4p(x-h)$. For a horizontal parabola, the equation is $(x-h)=4p(y-k)$. Here, (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus or directrix.

How do you find the focus and directrix of a parabola?

To find the focus and directrix of a parabola, you need the value of p from the standard form equation. For a vertical parabola $(y-k)=4p(x-h)$, the focus is at (h, k + p) and the directrix is the line y = k - p. For a horizontal parabola $(x-h)=4p(y-k)$, the focus is at (h + p, k) and the directrix is the line x = h - p.

What is the difference between vertical and horizontal parabolas?

The main difference between vertical and horizontal parabolas lies in their orientation and equations. A vertical parabola opens either up or down and has the equation $(y-k)=4p(x-h)$. A horizontal parabola opens either left or right and has the equation $(x-h)=4p(y-k)$. The sign of p determines the direction of the opening: positive p means up or right, and negative p means down or left.

How do you graph a parabola given its equation?

To graph a parabola given its equation, follow these steps: 1) Identify the vertex (h, k) from the equation. 2) Calculate the value of p from the coefficient of the linear term. 3) Determine the focus and directrix using the value of p. 4) Plot the vertex, focus, and directrix on the graph. 5) Draw the parabola by connecting points that are equidistant from the focus and directrix. For vertical parabolas, use $(y-k)=4p(x-h)$, and for horizontal parabolas, use $(x-h)=4p(y-k)$.

What is the significance of the p value in a parabola's equation?

The p value in a parabola's equation is crucial as it determines the distance between the vertex and the focus, as well as the vertex and the directrix. For a vertical parabola $(y-k)=4p(x-h)$, a positive p value means the parabola opens upwards, while a negative p value means it opens downwards. For a horizontal parabola $(x-h)=4p(y-k)$, a positive p value means the parabola opens to the right, while a negative p value means it opens to the left.