04 - Graphing Parabolas - Vertex and Axis of Symmetry - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . I'm Jason with math and | |
| 00:02 | science dot com . Today we're going to jump into | |
| 00:05 | the concept of graphing parabolas . Specifically , we're going | |
| 00:08 | to talk about the vertex of the parabola and the | |
| 00:10 | axis of symmetry . I want you to have an | |
| 00:12 | idea of the overall roadmap of where we're going this | |
| 00:14 | lesson . We'll talk about the vertex in the axis | |
| 00:16 | of symmetry . In the next few lessons , we'll | |
| 00:18 | start working on how to shift these parabolas around the | |
| 00:22 | xy plane . In other words , how to write | |
| 00:24 | down the equation of a parabola and instead of it | |
| 00:26 | being centered in the center of the xy plane , | |
| 00:29 | how it can be moved left and moved right and | |
| 00:31 | so on . But before we get to that point | |
| 00:33 | we have to understand some ideas and concepts in in | |
| 00:35 | this case we're gonna be talking about the vertex in | |
| 00:37 | the axis of symmetry . So what I need you | |
| 00:40 | to do is to understand really burn it in your | |
| 00:44 | mind . The basic parabola . Uh the reason I'm | |
| 00:51 | talking about the basic problem is because in your mind | |
| 00:53 | you need to have a general idea or or a | |
| 00:56 | very um I say a general idea , but what | |
| 00:57 | I mean is a , you have to burn this | |
| 00:59 | image in your mind of what a parabola looks like | |
| 01:02 | and where it is centered . I'm gonna call that | |
| 01:04 | the basic parabola . The reason is because when we | |
| 01:06 | start shifting the parabola around , then all we're gonna | |
| 01:09 | do is take the original equation of the basic parabola | |
| 01:13 | and change it very slightly in order to move it | |
| 01:15 | around . So we have to have an idea of | |
| 01:17 | what the basic one looks like . So you've already | |
| 01:20 | encountered this , we've talked about it many times before | |
| 01:22 | . The basic problem is very simple . F of | |
| 01:24 | X is equal to x square . This is the | |
| 01:26 | most basic parabola that you can get . And if | |
| 01:29 | you don't want to think about functions , you can | |
| 01:30 | write it instead of F of X . You can | |
| 01:32 | say that Y is equal to X squared . Usually | |
| 01:34 | when you start algebra you look at it in terms | |
| 01:37 | of why is equal to X squared . And then | |
| 01:39 | later on we understand the concept of a function and | |
| 01:42 | so you replace the why with ffx . But functionally | |
| 01:44 | these two things , these two representations are saying , | |
| 01:47 | the same thing . What they're saying is that we | |
| 01:49 | stick numbers into this side of the equation and we | |
| 01:52 | square them . And then the result gets applied to | |
| 01:55 | . In this case it's a variable why ? In | |
| 01:56 | this case it's a notation which is a little more | |
| 01:59 | clear saying where that that this function is a function | |
| 02:01 | of X because X is what we're changing . And | |
| 02:04 | then the results kinda get spit out there and we | |
| 02:07 | talked about the idea of what a function is in | |
| 02:08 | the past . So uh we're not gonna do this | |
| 02:12 | for every single problem but for this one , because | |
| 02:14 | the basic problem is so important , I want to | |
| 02:16 | write down a few points so we have X as | |
| 02:18 | an input and then we have , I'm gonna use | |
| 02:20 | the Y notation . Uh why is able to X | |
| 02:23 | squared ? So we'll just make a quick little table | |
| 02:25 | and I know that we've actually done this before but | |
| 02:27 | I just want to do it here because we're going | |
| 02:29 | to graphic uh as best we can and then we're | |
| 02:31 | going to play around with it and and kind of | |
| 02:34 | like so in the case of that we need to | |
| 02:36 | have an idea of the basic problem in our minds | |
| 02:38 | . So we need to pick some point . So | |
| 02:39 | let's go from negative three , negative two , negative | |
| 02:41 | 10123 Of course you could go all the way to | |
| 02:45 | negative five or negative 10 or positive five or whatever | |
| 02:48 | . But in this case I'm just gonna stick to | |
| 02:50 | negative three to positive three . So when we take | |
| 02:52 | a negative three and we square it negative three times | |
| 02:55 | negative three just gives us a nine negative two . | |
| 02:58 | We square it . We get a positive four . | |
| 03:00 | Same thing here , we get a positive 10 squared | |
| 03:02 | is a 01 squared is a 12 squared is 43 | |
| 03:07 | squared is nine . So right away before we even | |
| 03:09 | graph anything , you can already see some symmetry in | |
| 03:12 | the in the shape of the problem . The center | |
| 03:15 | of the thing is basically here , if we go | |
| 03:17 | to positive values of X like this , then we | |
| 03:20 | get larger and larger outputs which is the Y values | |
| 03:24 | . But if we go to the negative values of | |
| 03:26 | X , we get the exact same outputs as as | |
| 03:30 | in the other side , the one matches with the | |
| 03:31 | one and so on . So I can kind of | |
| 03:33 | like just kind of like make it super obvious . | |
| 03:36 | This one goes with this one , this one goes | |
| 03:38 | with this one and the nine goes with this one | |
| 03:41 | . And so we say that the graph of the | |
| 03:43 | parietal is symmetric and we'll see it a little more | |
| 03:45 | clearly when we draw it . But basically whether you | |
| 03:48 | go to negative X values or positive X values , | |
| 03:51 | the outputs of the function give you exactly the same | |
| 03:54 | thing . So it's like a mirror image , right | |
| 03:57 | ? Um So let's go ahead and draw this graph | |
| 03:59 | . It doesn't need to be perfect . I'm not | |
| 04:00 | gonna pull out the crazy graph paper or anything like | |
| 04:04 | that . We're just going to uh in general right | |
| 04:07 | down what we need to burn the image in our | |
| 04:10 | mind of what this problem is . So here we | |
| 04:11 | have X . And here we have I can put | |
| 04:13 | F of X . Here . I'll just put why | |
| 04:15 | Since we're doing everything in terms of why . And | |
| 04:17 | then we need to have some kind of tick marks | |
| 04:19 | . Right ? So again , I went from in | |
| 04:21 | my graph here negative three to positive three . So | |
| 04:23 | let me go 123 So this is three . This | |
| 04:27 | is to this is one , this is zero . | |
| 04:30 | This is negative one , negative two negative three . | |
| 04:32 | So I'll put negative three negative two negative one . | |
| 04:34 | Right ? And then I when I graft this guy | |
| 04:37 | , I ended up getting a large value of nine | |
| 04:39 | . So I'm gonna go ahead and put nine Tech | |
| 04:40 | marks 123456789 I could go on of course , but | |
| 04:46 | I don't have any points that are beyond that place | |
| 04:50 | . So let's go and plot this thing . We | |
| 04:52 | have X comma Y which is zero comma zero . | |
| 04:54 | Which means we have a point right here we have | |
| 04:57 | one common one , which means we have one comma | |
| 04:59 | one . Right here we have two comma four , | |
| 05:03 | which means we have two comma 1234 So that one | |
| 05:06 | is actually somewhere right around there and then we have | |
| 05:09 | three common nine , which means we have three . | |
| 05:11 | And if I've counted correctly and we stand in front | |
| 05:13 | to try to get it to look as good as | |
| 05:15 | I can get it , it's not perfect . But | |
| 05:17 | let's make it put a little bit higher Somewhere right | |
| 05:20 | around here . So 123456789 is where that point is | |
| 05:25 | . Now let's go the other direction negative one comma | |
| 05:27 | one means negative one comma one is right here , | |
| 05:30 | negative two comma four means negative two , comma four | |
| 05:33 | is right here and negative three comma nine means let | |
| 05:37 | me stand in front so I can try to line | |
| 05:38 | it up as good as I can . That's something | |
| 05:41 | like that . It's not exactly right . So you | |
| 05:43 | see this is not a straight line . I mean | |
| 05:45 | if you try to draw a straight line through these | |
| 05:47 | two points right here , then you don't hit this | |
| 05:49 | one and you definitely don't hit this one so you | |
| 05:51 | can see it's a curved kind of thing . So | |
| 05:53 | I'm gonna do my best to sketch it but just | |
| 05:55 | keep in mind that it's not going to be perfect | |
| 05:57 | . So let's go down here through these two points | |
| 05:59 | . We bend over , we go down here , | |
| 06:02 | we go up through these points like this of course | |
| 06:06 | I could go off to the computer . But the | |
| 06:07 | point is when you're on your , when you're on | |
| 06:09 | your paper , when you're on your exams , you're | |
| 06:10 | not gonna have a computer , right ? So of | |
| 06:12 | course I could show you a computer image and I | |
| 06:13 | have shown you already computer images of what a problem | |
| 06:16 | it really looks like . It's a beautiful curve that | |
| 06:18 | goes down like a smiley face kind of and then | |
| 06:20 | it goes up for the for the one that opens | |
| 06:22 | upward anyway . But for this , I want you | |
| 06:25 | to burn this image in your mind . This is | |
| 06:27 | what we call the basic parabola . It just is | |
| 06:30 | F of X is equal to X squared , or | |
| 06:32 | why is equal to X squared . And we plot | |
| 06:34 | the points like this to show you the general shape | |
| 06:37 | of the thing . But just keep in mind between | |
| 06:39 | all of these points , Their infinite points in between | |
| 06:41 | all of these points which I could of course fill | |
| 06:43 | out in the table there that would trace out the | |
| 06:45 | shape perfectly . And of course I could go on | |
| 06:47 | and on forever writing points down . But the purpose | |
| 06:50 | of this is not to be exact and it's not | |
| 06:53 | to put a million points on the purpose of it | |
| 06:55 | is so that you can see the basic shape . | |
| 06:56 | What can we learn from this ? Right ? This | |
| 06:59 | Parabola has a lowest point here , Right . And | |
| 07:03 | this point here is called has a special name . | |
| 07:06 | It's called the vertex . Mhm . So any time | |
| 07:09 | when I say or a book says or a teacher | |
| 07:12 | says , the vertex of the Parabola is located at | |
| 07:15 | blah blah blah . All it means is that's the | |
| 07:17 | lowest point of the parabola . Okay , of course | |
| 07:20 | , I've drawn a problem that opens up like this | |
| 07:23 | . So the lowest point is the vertex and this | |
| 07:25 | vertex is located at zero comma zero . Now , | |
| 07:28 | later on , we're going to grab this parabola and | |
| 07:31 | we're gonna move it all around . So I can | |
| 07:32 | have the problem over here or I can have the | |
| 07:34 | problem over here , I can have the problem over | |
| 07:36 | here . The vertex of the problem will just be | |
| 07:39 | the point at the lowest part of that parabola . | |
| 07:42 | If it opens up like this , we'll see in | |
| 07:44 | a minute . The problem can actually go upside down | |
| 07:46 | , in which case the vertex will be the tippy | |
| 07:48 | top . So it's either going to be the maximum | |
| 07:50 | value of the problem or the minimum value of the | |
| 07:52 | problem like it is here . But either way it's | |
| 07:54 | called a vertex and it has a coordinate point in | |
| 07:57 | this case zero comma zero . So for the basic | |
| 07:59 | problem , the vertex is located at zero comma zero | |
| 08:02 | . And as I've already hinted , we can of | |
| 08:05 | course have parable is john all over this plane and | |
| 08:08 | they can open up or they can open down , | |
| 08:10 | let's say how to problem that went and opened upside | |
| 08:12 | down like this . In this case the vertex would | |
| 08:15 | be whatever the maximum value was here . We're gonna | |
| 08:18 | call that the vertex Uh here as well and I | |
| 08:21 | give it some coordinate points or whatever . I'll do | |
| 08:24 | some more detailed example later . But this coordinate might | |
| 08:26 | be like 5:06 or something . They're So the vertex | |
| 08:30 | is basically the maximum value . Now , the other | |
| 08:33 | thing that I want to talk to you about is | |
| 08:35 | what we call the axis of symmetry . It has | |
| 08:38 | a very complicated sounding name , but it's actually very | |
| 08:40 | , very simple . Okay . This thing , another | |
| 08:43 | way to say it instead of calling it an axis | |
| 08:45 | of symmetry is you can think of it as its | |
| 08:47 | mirror image . Right ? This piece of paper is | |
| 08:49 | a rectangle , right ? This piece of paper is | |
| 08:52 | symmetric , it has a symmetry to it , right | |
| 08:54 | ? Because why I can take this piece of paper | |
| 08:57 | and I can fold it in half . And so | |
| 08:59 | this piece paper has an axis of symmetry along the | |
| 09:02 | long direction . If I could draw a dotted line | |
| 09:04 | like through this piece of paper I have an axis | |
| 09:07 | of symmetry of this thing because I can fold it | |
| 09:08 | into mirror image . Also if I turn the piece | |
| 09:10 | of paper sideways , I have an axis of symmetry | |
| 09:13 | going like this because I can fold it across now | |
| 09:16 | . Of course not every line is an axis of | |
| 09:19 | symmetry . For instance if I try to draw an | |
| 09:21 | axis of symmetry through the corners like that , it | |
| 09:24 | doesn't quite work because if I try to fold it | |
| 09:26 | from corner to corner , you see it doesn't really | |
| 09:28 | line up and the thing has to line up exactly | |
| 09:31 | for it to be a mirror image or an axis | |
| 09:33 | of symmetry . So this parable has a very special | |
| 09:35 | axis of symmetry . This one does as well . | |
| 09:37 | And you can see that that axis of symmetry is | |
| 09:40 | a line like this that goes vertically and cut goes | |
| 09:46 | right through the vertex . It always has to go | |
| 09:48 | through the vertex which is the center and this is | |
| 09:50 | what we call the axis of symmetry . What it | |
| 09:57 | means is it's the dotted line that I can fold | |
| 09:59 | that terrible on itself and have it perfect . Have | |
| 10:02 | it perfectly lined up . There's only one access that | |
| 10:04 | works and it goes right through the vertex . The | |
| 10:07 | axis of symmetry of this problem is the y axis | |
| 10:09 | right here because the thing is centered there , I | |
| 10:11 | can fold this thing on itself . Now . Before | |
| 10:14 | we close the lesson out , I've introduced the concept | |
| 10:16 | of what a vertex is . I've introduced the concept | |
| 10:19 | of what an axis of symmetry is . Now . | |
| 10:20 | I just want to write down and sketch a couple | |
| 10:22 | of quick examples to show you numerically what these axes | |
| 10:25 | are because in your problems , your your problem might | |
| 10:28 | say here's a parabola , tell me what the vertex | |
| 10:30 | is and what the axis of symmetry is and you'll | |
| 10:32 | have to know how to do that . So let's | |
| 10:35 | go over here and do that . Um What if | |
| 10:37 | I had a parabola that looks like this ? I'll | |
| 10:39 | go ahead and draw the axis , draw the problem | |
| 10:41 | . We'll talk about what it is . Um Obviously | |
| 10:45 | let's take a look at this problem . It's going | |
| 10:47 | to be that nice standard , beautiful parable of shape | |
| 10:49 | like this . Of course , This is X . | |
| 10:51 | And this is why Okay , this is kind of | |
| 10:54 | a gimme because we already drew it on the board | |
| 10:56 | here . But what is the vertex of this problem | |
| 10:58 | ? The vertex or the location of the vertex is | |
| 11:02 | just the lowest point of the problem . And it's | |
| 11:03 | right here at the origin . So the vertex just | |
| 11:06 | as we said before is zero comma zero . The | |
| 11:08 | axis of symmetry . I'm gonna write that as axis | |
| 11:12 | of symmetry . You have to express that as a | |
| 11:15 | line because an axis is a line , right ? | |
| 11:19 | But the line that is the kind of the mirror | |
| 11:21 | image center point here is the line right here at | |
| 11:25 | X is equal to zero . Remember vertical lines in | |
| 11:28 | algebra , when you plot vertical lines , they always | |
| 11:31 | have the same form X equals something . Right ? | |
| 11:34 | If I have a line at X equals five , | |
| 11:37 | that means I have a vertical line over here . | |
| 11:39 | X is equal to five because the X values of | |
| 11:41 | every point on this line over here , is that | |
| 11:43 | five units of X ? Right ? In this case | |
| 11:45 | it's X is equal to zero because the line is | |
| 11:47 | going right through X is equal to zero . We | |
| 11:49 | talked about vertical lines in the past , you should | |
| 11:51 | know that , but I'm just refreshing your memory . | |
| 11:53 | So the vertex is a point . The axis of | |
| 11:56 | symmetry is a line . So this is not just | |
| 11:58 | an X value . This is a line of points | |
| 12:01 | that defines this guy . Now , let's take this | |
| 12:04 | basic parabola and let's move it around a little bit | |
| 12:07 | . Let's go over here and draw another problem which | |
| 12:12 | is over here . Let's draw it over here like | |
| 12:17 | this . So it looks exactly the same . It's | |
| 12:19 | just shifted over . This is one , this is | |
| 12:23 | two . This is three . All right . So | |
| 12:26 | what would be the vertex in the axis of symmetry | |
| 12:28 | here ? The vertex what would be the vertex ? | |
| 12:32 | Well , that's the lowest point . The point here | |
| 12:34 | . I've just shifted it from this point over here | |
| 12:36 | . The point over here is located at two comma | |
| 12:38 | zero . X is to y is zero . So | |
| 12:41 | the vertex is now at a different location than the | |
| 12:43 | vertex before . And the axis of symmetry . Yeah | |
| 12:49 | . Is what ? It has to be a vertical | |
| 12:51 | line , right ? It has to be a vertical | |
| 12:52 | line that bisects or cuts this parable in half . | |
| 12:56 | It has to go right through the vertex . So | |
| 12:58 | the access of cemetery is X is equal to two | |
| 13:00 | because it's a vertical line located at X is equal | |
| 13:03 | to to all the set of points that go through | |
| 13:05 | there . That's the axis of symmetry . So , | |
| 13:09 | here , we just basically took this basic parabola and | |
| 13:11 | we just shifted it to the right and we can | |
| 13:13 | very easily see how to handle the vertex and so | |
| 13:15 | on . Now let's take and shift this Parabola in | |
| 13:18 | a different location entirely . Right ? So instead of | |
| 13:21 | shifting it over here , let's draw the parabola . | |
| 13:24 | Let's draw the problem way up here in space , | |
| 13:27 | something like this . So it's not on any of | |
| 13:29 | these access . What would the vertex in the axis | |
| 13:31 | of symmetry be ? In that case ? Um Well | |
| 13:35 | , you could say Uh 1 , 2 , 3 | |
| 13:40 | , one . Yeah , Kind of set up 1 | |
| 13:44 | , 2 . That's what I'm trying to go for | |
| 13:45 | here like this . If I had drawn it like | |
| 13:48 | this , then I could kind of died a line | |
| 13:51 | over here and down the line over here and this | |
| 13:53 | is X is equal to three and y is equal | |
| 13:55 | to two . Right , then what would the vertex | |
| 13:58 | B ? The vertex It's just the location of this | |
| 14:01 | minimum 0.3 comma two . Basically you just read off | |
| 14:05 | the coordinates of where the lowest part of that parabola | |
| 14:07 | is . And then what would the axis of symmetry | |
| 14:09 | ? B . Yes . Uh huh . It has | |
| 14:14 | to be the vertical dotted line that cuts this thing | |
| 14:18 | in half and it goes right down . You can | |
| 14:20 | see it goes right through three . Right ? So | |
| 14:22 | this axis of symmetry is X is equal to three | |
| 14:25 | . So you can kind of see the pattern here | |
| 14:27 | . The axis of symmetry always goes through the first | |
| 14:30 | part of the vertex because that's the X coordinate of | |
| 14:33 | where the vertex is , X equals zero . Goes | |
| 14:35 | through this point X is equal to to the line | |
| 14:38 | X is equal to , goes through the point to | |
| 14:41 | the vertical line , X is equal to three goes | |
| 14:43 | through this so they all match . So the axis | |
| 14:45 | of symmetry is always going to be the first number | |
| 14:47 | in the coordinate of the vertex . Alright now , | |
| 14:51 | so far all of these parables except for this one | |
| 14:55 | that I drew have opened up and we're going to | |
| 14:58 | find out an easy way to figure out if a | |
| 15:00 | parabola opens up or opens down later . But for | |
| 15:03 | now let's focus on what would the vertex in the | |
| 15:06 | axis of symmetry look like for a problem that opens | |
| 15:09 | kind of upside down . So instead of a smiley | |
| 15:12 | face , like all of these problems , what if | |
| 15:13 | it opens as a frowny face it goes upside down | |
| 15:16 | . So if the problem goes upside down you might | |
| 15:18 | have something like this . So let's take a look | |
| 15:20 | at a quick sketch of one of these guys and | |
| 15:23 | by the way it's not rocket science , it's not | |
| 15:25 | gonna be any harder . Right ? So let's say | |
| 15:27 | the problem instead of opening up , which is the | |
| 15:29 | standard problems shape , It actually opens and comes down | |
| 15:33 | like this . What is the vertex in the axis | |
| 15:35 | of symmetry here ? It's not a big surprise . | |
| 15:37 | The vertex is located at the highest point of this | |
| 15:41 | guy , which is right at the origin and the | |
| 15:44 | axis of symmetry is that x is equal to zero | |
| 15:47 | . Same thing . It's just the first coordinate here | |
| 15:49 | , because the line that bisects a is X is | |
| 15:51 | equal to zero here . This is the X axis | |
| 15:53 | , this is the y axis , so X is | |
| 15:55 | equal to zero here . It goes right through there | |
| 15:57 | . That's the vertex in the axis of symmetry . | |
| 16:00 | And then we'll do one more before we close it | |
| 16:01 | down . What if we had a problem over here | |
| 16:05 | that let's try to be a little more precise , | |
| 16:09 | let's say this is negative one , this is this | |
| 16:11 | is 123 And let's say that our vertex is gonna | |
| 16:15 | end up being there . So I kind of gave | |
| 16:16 | it away a little bit . But the point is | |
| 16:18 | is that this Parabola goes something like this like this | |
| 16:26 | , It opens upside down . What's the vertex in | |
| 16:28 | the axis of symmetry ? What you can see that | |
| 16:30 | the vertex is the is the kind of where these | |
| 16:32 | kind of X and Y values there , the coordinates | |
| 16:35 | of this top point here . So the vertex is | |
| 16:39 | negative one comma three , that's X comma Y . | |
| 16:42 | And the axis is going to be this line , | |
| 16:45 | whatever this line is , that by sexist thing which | |
| 16:47 | has to be X is equal to negative one . | |
| 16:49 | Right here , it's the one that goes through there | |
| 16:52 | . So this lesson was all about introducing concepts , | |
| 16:55 | we're going to be taking these parabolas that we've been | |
| 16:57 | learning about and we've kind of talked about problems being | |
| 17:00 | quadratic functions , right ? And we have graft some | |
| 17:02 | of them of course we already know how to graph | |
| 17:04 | things . But what we have not done is done | |
| 17:07 | terminology . You have to know what the vertex is | |
| 17:09 | because why later on down the road we're going to | |
| 17:11 | be writing these problems . Are these quadratic equations in | |
| 17:15 | terms of their vertex . In other words , we're | |
| 17:17 | gonna , you know when we talked about equations of | |
| 17:19 | lines , you know , we learned three different ways | |
| 17:22 | to write the equation of a line or at least | |
| 17:23 | a couple of different ways . We learned the Mx | |
| 17:26 | plus B . And then we learned the point slope | |
| 17:28 | form and we learned a couple of different ways to | |
| 17:30 | write the equations of lines . Well , there's more | |
| 17:32 | than one way to write the equation of a parabola | |
| 17:35 | . And so we're gonna learn about those . But | |
| 17:37 | in order for us to get there , we have | |
| 17:39 | to understand what the vertex and the axis of symmetry | |
| 17:41 | of a problem is . So I want , you | |
| 17:43 | know , mostly burn this in your mind , burn | |
| 17:46 | the shape of the problem in where the lowest point | |
| 17:48 | is for the basic problem , X squared which is | |
| 17:51 | vertex is at 00 and X axis of symmetry being | |
| 17:53 | X is equal to zero as a vertical line . | |
| 17:57 | And then we did some other accommodations . We moved | |
| 17:59 | it around and look at where the vertex moved . | |
| 18:01 | Because in the next few lessons , we're gonna start | |
| 18:03 | shifting these problems , we're gonna write equations that move | |
| 18:06 | these problems . And I didn't show you what the | |
| 18:08 | equation of this problem is that moves it over here | |
| 18:10 | or the equation that moves it over here , we're | |
| 18:12 | gonna write those equations down . It's gonna be much | |
| 18:14 | easier if you understand the concept , the concepts in | |
| 18:17 | this lesson . So make sure you understand this . | |
| 18:19 | Follow me on to the next lesson will start shifting | |
| 18:21 | parabolas around in the xy plane . |
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