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 two dimensional plane giving you this parabola shape. Now, when it comes to problems 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 directs. 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 direct is just a line and the point and the line for the focus and directs 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 gonna be somewhere up on the graph, a certain amount of units. Whereas if the Parabola opens down the focus point is gonna 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 gonna be somewhere up here. Now when finding the direct line, if the Parabola opens up, the direct line is actually gonna be somewhere down on your graph. Whereas if the parable opens down the direct line will be up a certain amount of units. And since again, our Parabola opens up, the direct line is gonna be somewhere down here to actually figure out where we can directly find our focus and directs. 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 similar between the two equations. But there's one main difference which is that we have four P 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 directors 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 want 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 four in front of this Y. And in the general form of the equation, we have a four P in front of Y. So we can set four P equal to four. And the sulfur P, we just divide four on both sides of the equation. They'll get the four to cancel, giving us that P is 4/4, which is one. So our P value for this Parabola is one and to find the focus in directs, that means that for our focus, we need to go up one unit to find the focus point at one or excuse me, 01. And to find the direct line, we need to go down one unit which is going to give us a line right about here. And whenever drawing the directs 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 that 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 asks to 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 one. So our horizontal positions one, whereas I can see this K corresponds with two. So the vertical position is going to be two. So that means that the center or excuse me, not the center, the vertex of our parabola is going to be horizontally one unit to the right and up two 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 four P, which is going to be in front of the Y minus K corresponds with this eight in front of the Y minus two. So we can recognize that four P is equal to eight. And to find P, we just divide four on both sides giving us that P is 8/4, which is two. 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 12 units since our P value is two, and that's going to give us our focus at the 0.1 comma four. So that's the third step. Now, our fourth step is going to be starting from the focus, go left and right to P units. And this will tell us the width of our parabola. Now two times the absolute value of P is the same thing as two times the absolute value of two, which is just four. So if I start at our focus, I can go to the left 1234 units and I can go to the right 1234 units giving me points at negative three, comma four and another point at five comma four. Now, our fifth step is going to be to connect to the outside points with a smooth curve. So if I connect to 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 direct line and recall that our direct goes in the opposite direction. Our Parabola opens since our Parabola opens up, our director is going to be down the absolute value of amount of units from the vertex. And since our P value is two, we need to go down 12 units which will give us a direct line right along the x axis. It says we're on the x axis. That means our direct line is at a Y value of zero. So this is how you can find the focus, the directors 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.

2

Problem

Problem

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

3

Problem

Problem

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

$-4\left(y+1\right)=x^2$

4

example

Parabolas as Conic Sections Example 1

Video duration:

7m

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Hey, everyone. So let's see if we can solve this problem in this problem, we're asked to identify the vertex focus and direct tricks 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 four P times Y minus K is equal to X minus H squared. 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 four P times Y is equal to X squared. So these are the two equations that we can use when solving problems associated with Parabolas. So let's see if what we can do with these examples. And we'll start with example, a. Now, one of the things that I noticed 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 four py equals X squared is the standard form of this equation. And we already know because this is an example where the parable is at the origin that the vertex is going to be at the 0.00. So we've already figured that out. Now, what we can do from here is figure out the focus and directs by using the standard equation and I can set four P equal to everything that's in front of the Y which is 16. So or P is equal to 16, I can divide four on both sides, canceling the fours giving us that P is equal to 16/4, which is four. 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 directors 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 four. And what this means is for our vertex, if it's at 00, the focus is going to be up four units. So our focus is going to be at the position zero positive four. And then our direct is going to be down four units since it's opposite the direction that the Parabola opens meaning our directs is going to be at Y equals negative four. And this would be the focus and directs 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 directives 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 00 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 four P equals everything that's in front of the Y which we can see is one third. And to solve for P, I can go ahead and take this four and move it to the other side by multiple, both sides by 1/4. This is the same as dividing four on both sides in case you're wondering, and what I can do is cancel the fours here giving us that P is equal to one third times 1/4. Now, I can multiply the three and the four which is 1/12. So that means that P is going to be 1/12. Now, because P is positive, we know that this problem opens up. And if I take the absolute value of P, we're going to get positive 112 again. So that means that for our focus and directs, we need to go up a certain amount of units which is P and we need to go up to 112. So our focus is going to be at zero and then zero plus 112 is 112. And then our direct is going to be at the line of Y equals negative 112 since it's going to be in the opposite direction. So that's how you can find the focus as well as the direct for Parabola B. But now let's take a look at Parabolas C for parabolic C. What I noticed for this equation is that we're not at the origin this time because notice that we have an A K value and we have an H value. So that means that we're going to be at some new location for the H value. This corresponds with two. So the vertex is going to be at two and then the K value is one. So it's gonna be at two comma one. So that's gonna be the vertex of R Parabola. Now, next, I need to find the P value and find the P value I can use this version of the equation. So I set four P equal to everything that's in front of the Y minus K which is eight. So four P is equal to eight and then I can divide four on both sides of this equation. They'll give us that P is equal to 8/4, which is two and the absolute value of P is going to be two as well. And I notice that the original P value is positive. So the Parabola still opens up, which means our focus is going to be up and our directors would be down. Now looking at this vertex, this H value doesn't change because if you have some kind of parable atoms going to write a small graph over here, if you have a parable, that's at the position 21, which would be right there for the vertex and it opens up, then the focus is gonna be somewhere up here P amount of units notice how we're still going to be in the same horizontal position, but vertically we're now shifted up. So what this means is for our focus, our focus is gonna be at the position two because that stays the same. And then we're going to have the absolute value P plus one which is three. So it's gonna be at position +23 and then the directs is going to be two units below this and one minus the one minus two is comes out to negative one. So our focus is going to be at position two comma three and our directs is the line Y equals negative one. That's paraly C. Now let's move on to our last parable, which is para D. And for this one, we have that negative 12 times Y is equal to X plus one squared. And I see that we do have a value here. So that means that we are not going to be at the origin because we do have a, an H there. Now, I noticed that we don't have a K. So because we don't have a KK is equal to zero because the Y is just by itself. But for this H value, we have one. But notice that we have a plus sign rather than a minus sign. And X plus one squared is the same thing as X minus negative one squared. So because of this, we can recognize that H must be negative one. So our H value, our horizontal position is gonna be at negative one. So our vertex is at the position negative +10. So this is our vertex. Now, next, what we want to do is find the P value and I can use this version of the equation where I set four P equal to everything that's in front of the Y. And so we're going to have that four P is equal to negative 12. And if I divide four on both sides, we can cancel the force here giving us that P is equal to negative 12/4, which is negative three. But this is interesting because notice we got a negative P value this time. So what does this mean? Well, what this means is our Parabola is now going to open down instead of opening up. And that means likewise, our focus is also going to be somewhere down instead of up. I'll take the absolute value of P which will get rid of the negative sign giving us that the absolute value of P is three. Now, if I go ahead and find the focus and director, it's based on the vertex, I recognize that the problem is gonna open down. So the vert or the focus is gonna be somewhere down here. So to subtract three from zero, which means our focus is going to be at the position negative one negative three and our directs is going to be the line Y is equal to zero plus three because the directs is up this time, which is at positive three. If you're having trouble visualizing this, just imagine this, we have our graph which looks something like that. And we figured out that our vertex is at the position negative 10, our Parabola opens down. So that's why we need to subtract three because the focus is somewhere down there. Whereas direct is gonna be somewhere up here since it's opposite the direction the Parabola opens. So this is why we added three to get our directors and we subtracted three to get our focus. So this is gonna be the focus point and the direct line for Parabola D. So that is how you can solve this type of problem. If you're not necessarily given a graph or asked to graph the situation, you can just try to understand where the focus and directs are going to be and you can use the equation to solve from there. So hope you found this video helpful. Thanks for watching and let me know if you have any questions.

5

concept

Horizontal Parabolas

Video duration:

5m

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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 Xs and Ys 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 direct tricks 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 directs is perpendicular or horizontal. So when dealing with vertical parabolas, the direct 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 direct is going to be vertical. So the directs 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 pre look very similar. The only real difference is the Ys and X's are switched. So we have a four py equals X squared here and we have a four PX equals Y squared there. If you look at the specific equation for these, we have two Y equals X squared and two X equals Y squared. 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 para 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 whatever we have in front here is positive, that means the parable 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 eight X and then equals Y squared. So that means our H and RK are both going to be zero. 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 eight is going to be equal to four P. So if four P is equal to eight, I can solve for P by dividing both sides of the equation by four, this give me that P is equal to 8/4, which is two. So our P value is two. Now our third 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 paradela 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 is equal to two. So we're gonna go 12 units to the right. And this here would be our focus at 20. Now, our fourth 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 two P 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 two P units, well, two P is the same thing as two times two, which is four. So I can go up one, 234 units to get to this point, which is at 24 and then I can go down 1234 units from our focus point, which would get me right here to two negative four. Now, our fifth 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 direct tricks. And we're called that the direct is always in the opposite direction that the Parabola opens. So since the Parabola opens to the right, the directors is going to be here to the left. And specifically, it's going to be to the left P units. So if I go ahead and look at our P value, which is two, we need to go 12 units to the left. So we'll have that X is equal to negative two. So our direct tricks is the line X equals negative two and our focus is the 0.20. This is the graph of our shape and that is how you can solve horizontal parabola. So hope you found this video helpful. Thanks for watching and let's move on.

6

Problem

Problem

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

7

Problem

Problem

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

$4\left(x-1\right)=\left(y+4\right)^2$

8

example

Horizontal Parabolas Example 1

Video duration:

7m

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Hey everyone and welcome back. So let's give this example a try. So here we're asked to identify the vertex focus and directors for each of these Parabolas below. And we'll start with Parabola A. Now what I noticed about this Parabola is that we have a Y squared and because we have a Y squared, that means we're dealing with a horizontal parabola. Now, the equation for a horizontal parabola is four P times X minus H is equal to Y minus K squared. So what we can do is use this equation as a reference for each of these examples. Now looking at this first example, I noticed nothing is subtracted from the X and nothing is subtracted from the Y. So that means that in this case, our H would be zero and our K would be zero. So the vertex of this parabola is going to be at the origin 00. So 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 say 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 four P is going to equal four. And what I can do is divide four on both sides to get P by itself. And we're going to get that P is four divided by four, which is one. Now notice how our P value came out to positive one, since we a positive P value, that means that our horizontal Parabola is going to open to the right. And so since we're opening to the right, that means our focus is going to be somewhere in there and our directors 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 one. And so looking at our vertex, if we were to go one unit to the right, our focus would be at the position 10 because horizontally we would be at a position of one for our focus. And our direct line which is going to be X is equal to the line would be one position back from zero, which is at negative one. So this would give us the vertex, the focus and the direct for this first Parabola. But now let's take a look at our next Parabola para B. And in this equation, I noticed 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 zero and I can see that two is being subtracted from the Y. So K is two. So our vertex is going to be at the horizontal position of zero and the vertical position of two. 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 four P in front of everything that's in front of the X which is going to be no. And if I divide four on both sides, that'll get the force to cancel, giving me that P is equal to 9/4. Now notice that once again, we got a positive P value and because we've 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 directors 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 into the left to find our directs. So our focus is going to be to the right horizontally, we're at zero and going to the right, we would be at nine fourths by this P value you two. And by the way, we're doing this by the absolute value of P. So the absolute value of nine fourths is just nine fourths, 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 directors is going to be at X equals negative nine fourths since we would need to go to the left by nine fourths, which we're starting from zero. So that would be our direct. So this would be the vertex focus and directs 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 four and I can see that K corresponds with two. But notice that we have a plus two and the equation says that we have Y minus K. So because we have an opposite sign here, Y plus two squared is the same thing as Y minus negative two squared. So that means that our K is actually negative two. So we need to put a negative number in four K. So our vertex is going to be at the position four negative two. Now, from here, I can find the focus and directors and to find the focus, I first need to find the P value. So we'll set four P equal to everything in front of the X which is 16. And if I go ahead and divide four on both sides, that'll get the force to cancel. Giving me that P is 16/4 which is four. 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 four units and horizontally we're at a position of four. So going to the right four more units would put us at a position of eight. So we're going to be at eight negative two for the focus. And as for the directors, we need to go back four units. And if you take four and subtract that four, we're going to get an X value of zero. Now keep in mind that this is all again, assuming the absolute value of P which is four, 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 directs for Parabola C. But now let's move on to our last Parabola which is Parabola D. Now 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 one and our K is going to be whatever subtracted from the Y. But since I see nothing subtracted from the Yrk is zero, so that means that the vertex is going to be at 10. Now, before I go any further, what I'm actually going to do is draw a graph of what this problem is going to look like because I think that 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 directors 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 one. So this is where the vertex is going to be. And our problem 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 directors. So to find P well, we've been doing this for the previous three problems, we just set four P equal to everything in front of X. So in this case, our four P is going to be equal to negative two. And I can go ahead and divide four on both sides of this equation, they'll get the force to cancel there giving me that P is equal to negative 2/4, which are 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 parable is actually going to open to the left in this example. So we have a Parabola that opens to the left. And because this parable opens to the left, the focus is actually going to be somewhere to the left this time and our directors 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. So let's see if we can find the focus point and direct 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 one and our focus point is going to be one 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 it's going to be one half and zero, giving us our focus 0.1 half comma zero. And to find our directors, I need to go to the right, one half units and to the right. Well, we would have one half plus our position here, which is one. So we'd have one plus one half on the X axis, which would put us at three halves. So our direct line is going to be X equals three halves. And that is how you can find the vertex focus and direct for Parabola D. So this is how you can solve these types of problems with horizontal problems. Hope you found this video helpful and thanks for watching.