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 standard form of a parabola's equation is y^2 = 4px for horizontal parabolas and x^2 = 4py for vertical ones. The sign of p determines the direction of opening, influencing the graph's shape and properties.

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. And 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. Now, 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. And 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 4p 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 the 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 a 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. They'll get the fours to cancel, giving us that p is 4 over 4, which is 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 equals negative one.

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, well, I can see that this h corresponds with this 1. So our horizontal position is 1, as I can see this k corresponds with 2. So the vertical position is going to be 2. So 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, well, I can see that our 4p, which is going to be in front of the y minus k, corresponds with this 8 in front of the y minus 2. So we can recognize that 4p is equal to 8, and to find p, we just divide 4 on both sides. Giving us that p is 8 over 4, which is 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 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 X2 is by itself. So that means we're going to use this version over here. So what we can do is recognize that 4py=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 4 on both sides, cancelling the fours, giving us that p = 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 somewhere 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 (0,0), 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 x2 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 equal to everything that's in front of the y, which we can see is 1d, and to solve for p we 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 we can do is cancel the fourth here, giving us that p = d times 14. Now I can multiply the 3 and the 4, which is one over 12. So that means that p is going to be 112. Now because p is positive, we know that this parabola opens up, and if I take the absolute value of p, we're going to get positive one twelfth again. So that means that for our focus and directrix, we need to go up a certain amount of units, which is p, and we need to go up to one-twelfth. So our focus is going to be at (0, 1/12) and then our directrix is going to be at the line y = -1/12, so it's going to be in the opposite direction. So that's how you can find the focus as well as the directrix for parabola b.

But now, let's take a look at parabola c. For parabola c, what I noticed about this equation is th

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 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

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 about this parabola is that we have a y², and because we have a y² that means we're dealing with a horizontal parabola. The equation for a horizontal parabola is 4p⁢⁡⁢⁡⁢⁡⁢⁡x−h⁡=y−k2. So what we can do is use this equation as a reference for each of these examples. Looking at this first example, I notice nothing is subtracted from the x, and nothing is subtracted from the y. So in this case, our h would be 0 and our k would be 0. The vertex of this parabola is going to be at the origin (0,0). The vertex is pretty straightforward for this parabola. Now, we need to also find the focus, and to find the focus, we can use this p value. So what I can do is set everything that's in front of the x, the x minus h here, equal to everything in front of the x there. So, that means that 4p⁢⁡⁢⁡⁢⁡=4, and what I can do is divide 4 on both sides to get p by itself. Now we're going to get that p is 4 divided by 4, which is 1. Now notice how our p value came out to positive 1. Since we have a positive p value, that means that our horizontal parabola is going to open to the right. Since we're opening to the right, that means our focus is going to be somewhere in there, and our directrix is going to be somewhere back here, just for example. So, what we can do is take a look here and see how things are going to behave based on this p value. Now the absolute value of positive one is also just 1, and so looking at our vertex, if we were to go 1 unit to the right, our focus would be at the position 10, because horizontally we'd be at a position of 1 for our focus, and our directrix line which is going to be x is equal to the line would be one position back from 0 which is at negative one. So, this would give us the vertex, the focus, and the directrix for this first parabola. But now, let's take a look at our next parabola, parabola b. In this equation, I notice that some number is being subtracted from the y. Because of this that means we are not going to be dealing with a parabola at the origin. So, what I can do is take a look at what we have in front of x, and I see that nothing is being subtracted from the x. So, that means h is 0. And I can see that 2 is being subtracted from the y, so k is 2. Our vertex is going to be at the horizontal position of 0 and the vertical position of 2, and that's our vertex. Now, from here I need to find the focus point, and to find the focus point I need to first find the &#p value. So I'll set 4p⁢⁡⁢⁡⁢⁡ in front of everything that's in front of the x, which is going to be 9. If I divide 4 on both sides that'll get the force to cancel, giving me that &#p is equal to 9 over 4. Now notice that once again we got a positive &#p value, and because we got a positive &#p value, our parabola is going to open to the right. And that means our focus will be somewhere there, for example, and our directrix is going to be behind the graph. So, what we can do here is use this &#p value and go to the right to find our focus, and to the left to find our directrix. So our focus is going to be to the right horizontally we're at 0, and going to the right we'd be at 9 4ths by this &#p value 2, and by the way, we're doing this by the absolute value of &#&#&#p. So the absolute value of 9 4ths is just 9 4ths, but I need to go to the right by that many units, so that's why we ended up with this for our focus, and our directrix is going to be at x equals negative 9 fourth since we need to go to the left by 9 fourths which we're starting from 0. So that would be our directrix. So this would be the vertex focus and the directrix for parabola b. But now let's take a look at parabola c. Well, I can see once again that we have numbers that are being subtracted from the x and y, so that means we're not at the origin. Now looking at this equation, I can see that h corresponds with this 4. I can see that k corresponds with 2, but notice that we have a plus 2, and the equation says that we have y minus k. So because we have an opposite sign here, y plus 2² is the same thing as y minus negative two squared. So that means that our k is actually negative 2, so we need to put a negative number in 4 k. So our vertex is going to be at the position 4 negative 2. Now from here, I can find the focus and directrix and to find the focus I first need to find the &#p value. So we'll set 4p⁢⁡⁢⁡⁢⁡ equal to everything in front of the x which is 16 and If I go ahead and divide 4 on both sides that'll get the 4 to cancel giving me that &#p is 16 over 4 which is 4. Now notice our &#&#&#p value came out positive So that means that our parabola is going to open to the right since it's a horizontal parabola. Now to find the focus based on where our vertex is at, we're going to the right 4 units. And horizontally, we're at a position of 4. So going to the right 4 more units would put us at a position of 8. So we're going to be at 8 negative 2 for the focus. And as for the directrix, we need to go back 4 units, and if you take 4 and subtract out 4 we're going to get an x value of 0. Now, keep in mind that this is all again assuming the absolute value of &#&#&#&#&#&#&#&#&#&#&#&#&#&#p which is 4, but since I saw it was positive that's just gonna be the same number. So this is how you can find the vertex, the focus, and the directrix for parabola c. But now let's move on to our last parabola which is parabola d. What I can do is I can look at this equation and I can first try to find the vertex. Now our 'h' is going to be whatever is being subtracted from the x, which in this case is 1. And our k is going to be whatever subtracted from the y, but since I see nothing subtracted from the y, our k is 0. So that means that the vertex is going to be at 10. Now before I go any farther, what I'm actually going to do is draw a graph of what this parabola is going to look like. Because I think that and and we don't have to draw a graph for this, but I think that by looking at a graph, it's going to actually help us to really visualize where the focus and directrix end up. So I can see that our vertex is at the position 10, which would be let's just say right here on our graph. Let's say that's an x value of 1. So, this is where the vertex is going to be, and our parabola is either going to open to the right or to the left. And to figure that out, we need to find this &#p value, which will also tell us the distance to the focus and directrix. So to find &#p, well we've been doing this for that previous 3 problems, we just set 4p⁢⁡⁢⁡⁢⁡ equal to everything in front of x. So in this case our 4p⁢⁡ is going to be equal to negative 2, and I can go ahead and divide 4 on both sides of this equation that'll get the force to cancel there, giving me that &#&#&#p is equal to negative 2 over 4, which reduces to negative one half. But notice how the &#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#&#p value came out negative this time. And because the &#p value came out negative, that means our parabola is actually going to open to the left in this example. So we have a parabola that opens to the left. And because this parabola opens to the left, the focus is going to be somewhere to the left this time, and our directrix is going to be somewhere to the right. So it's going to be opposite of the other examples that we've had because now it opens the opposite direction. Let's see if we can find the focus point and directrix line. To find the focus point, well, I need to go to the left by a &#p amount of units. And notice that we start at a position of 1, and our focus point is going to be 1 half units to the left, but I need to find the absolute value of &#p first and the absolute value of negative one half is positive one half. So if I go to the left, one half units, well, one minus one half is one half. So that's going to be one half and 0 giving us our focus point onetwo comma 0. And to find our directrix line, I need to go to the right onetwo units. And to the right, well, we would have onetwo plus our position here which is 1. So we'd have 1 plus 1 half on the x axis which would put us at 3 half. So our directrix line is going to be x equals 3 halves, And that is how you can find the vertex, focus, and directrix for parabola d. So this is how you can solve these types of problems with horizontal parab...

Here’s what students ask on this topic:

What is the standard form of a parabola's equation?

The standard form of a parabola's equation depends on its orientation. For a vertical parabola, the equation is $x^2=4py$. For a horizontal parabola, the equation is $y^2=4px$. Here, $p$ is a constant that determines the distance from the vertex to the focus and the directrix. The sign of $p$ indicates the direction in which the parabola opens.

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

To find the focus and directrix of a parabola, you first need the standard form of the equation. For a vertical parabola $x^2=4py$, the focus is at $(0,p)$ and the directrix is the line $y=-p$. For a horizontal parabola $y^2=4px$, the focus is at $(p,0)$ and the directrix is the line $x=-p$. The value of $p$ is found by comparing the given equation to the standard form.

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 standard form $x^2=4py$. A horizontal parabola opens either left or right and has the standard form $y^2=4px$. The sign of $p$ determines the direction of the opening: positive $p$ means up or right, while 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 from the equation. 2) Calculate the value of $p$ by comparing the equation to the standard form. 3) Determine the focus by moving $p$ units from the vertex in the direction the parabola opens. 4) Find the width by moving $2p$ units perpendicular to the axis of symmetry from the focus. 5) Draw a smooth curve through these points. 6) Identify the directrix, which is $p$ units in the opposite direction from the vertex.

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 from the vertex to the focus and the directrix. For a vertical parabola $x^2=4py$, a positive $p$ means the parabola opens upwards, while a negative $p$ means it opens downwards. For a horizontal parabola $y^2=4px$, a positive $p$ means it opens to the right, while a negative $p$ means it opens to the left.