01 - Shifting Ellipses and Hyperbolas in the XY Plane - Part 1 (Central Conics) - By Math and Science
Transcript
| 00:00 | Hello , welcome back to algebra . We're continuing to | |
| 00:02 | study the concept of comic section . Specifically here we're | |
| 00:05 | going to talk about when we shift ellipses in hyperbole | |
| 00:08 | is in the xy plane . So up until now | |
| 00:11 | we have a great length discussed all of the comic | |
| 00:14 | section circles ellipses , parabolas and hyperbole . But as | |
| 00:17 | you might now know , the circles and the parabolas | |
| 00:21 | are a little easier to graph and a little easier | |
| 00:23 | to understand . The ellipses and the hyperbole are a | |
| 00:25 | little bit more involved . Specifically the hyperbole is take | |
| 00:28 | a lot more effort to graph and sketch them . | |
| 00:30 | But if you remember back all of the lessons , | |
| 00:32 | which by the way , if you haven't watched those | |
| 00:34 | lessons , please stop now and go watch those those | |
| 00:36 | original lessons . The last lesson we've done on ellipses | |
| 00:39 | and hyperbole as you'll need that information for what we're | |
| 00:41 | talking about here . But in those previous lessons we | |
| 00:44 | have considered ellipses and high purple is only centered at | |
| 00:48 | the origin . So here is the xy plane and | |
| 00:51 | that ellipses centered right there . It's either horizontal or | |
| 00:54 | vertical , but its center is on the origin . | |
| 00:56 | And also the hyperbole was centered either horizontally or vertically | |
| 01:00 | , but the center of it was on the origin | |
| 01:02 | . Now we're going to talk about how do we | |
| 01:04 | take those ellipses in those hyperbole and shift them around | |
| 01:07 | to different locations in the xy plane ? Now , | |
| 01:10 | if you remember back , we have learned how to | |
| 01:13 | shift functions and shift graphs already in this class . | |
| 01:16 | When you think about the simple case of the line | |
| 01:18 | mx plus B , that's just uh denotes what the | |
| 01:21 | line is . Well , if you change the y | |
| 01:23 | intercept or if you change the equation of the line | |
| 01:26 | , you can put that line anywhere you want in | |
| 01:28 | the xy plane . You can what we call shift | |
| 01:31 | that line . Later we talked about parabolas and we | |
| 01:34 | also talked about circles and we talked about how do | |
| 01:36 | we shift Parabolas ? And circles around in the Xy | |
| 01:38 | plane ? We've already done that material . But since | |
| 01:42 | it lifts isn't hyperbole is worth so much more complicated | |
| 01:44 | . I'm saving the shifting part until right now . | |
| 01:47 | So before we get there , I want to talk | |
| 01:49 | to you about something you might see in your textbook | |
| 01:51 | called The Concept of a Central Comic . I know | |
| 01:53 | the first time somebody told me it was a central | |
| 01:55 | comic , I didn't understand what that means . A | |
| 01:57 | Central Connick is very simple . It's just the Konik | |
| 02:00 | sections that have a center to them . It turns | |
| 02:03 | out that if you think about it only a circle | |
| 02:05 | has a center and the lips has a center and | |
| 02:08 | the hyperbole has a center . We call those the | |
| 02:10 | Central comics . The other comic section called the Parabola | |
| 02:13 | , it actually doesn't really have a center . So | |
| 02:15 | we don't really call it a Central Connick . So | |
| 02:17 | if you're in your book and you talk about the | |
| 02:19 | concept of a Central Connick , you're only talking about | |
| 02:23 | the circles the ellipses or the hyperbole is . So | |
| 02:25 | here we have the Central Connick now , they're all | |
| 02:29 | comic sections , even the Parabola as a comic section | |
| 02:32 | . But when you think about it , a circle | |
| 02:35 | has a centre that is the equal distant point from | |
| 02:38 | from all of the boundaries of what we uh call | |
| 02:42 | a circle . So we have the concept of a | |
| 02:43 | circle . That is a central concept because it has | |
| 02:46 | a center , we can't have any lips and I'm | |
| 02:48 | terrible at drawing ellipses . I know that's really bad | |
| 02:51 | , but if you think about it , it has | |
| 02:52 | a central center . So , you know , the | |
| 02:54 | ellipses and the lips has a central uh center also | |
| 02:58 | . So it's called a central Central Connick . Also | |
| 03:01 | , the hyperbole , whether or not is horizontal like | |
| 03:04 | this or vertical , you might say , well , | |
| 03:05 | where is the center of this thing ? In fact | |
| 03:07 | , that's kind of a bad drawing . Um it | |
| 03:09 | should come in and be a little more , I'm | |
| 03:11 | really bad at freehand drawing these things . It should | |
| 03:13 | come in and be more kink like that . Yeah | |
| 03:15 | , something like that . So you might say where | |
| 03:17 | is the center of this thing ? Well , yes | |
| 03:19 | , the curves are kind of weird , but the | |
| 03:20 | center of it is still well defined . The center | |
| 03:23 | of this thing is right in the middle . It's | |
| 03:24 | equal distance from all the other little parts of this | |
| 03:27 | thing that we call the comic section . So the | |
| 03:29 | circle of the ellipse and the hyperbole to are all | |
| 03:34 | what we call Central Comics uh there . So we | |
| 03:37 | can you can draw a little line here like this | |
| 03:41 | and a little line around this to show you that | |
| 03:42 | these are the Central Comics now . What about The | |
| 03:44 | Lonely Parabola out here ? So Parabola can either look | |
| 03:47 | like this . Mhm . Or it can have of | |
| 03:53 | course the mirror images upside down . Where is the | |
| 03:56 | center of this thing ? Well , there is no | |
| 03:58 | center up here is not equal distant to all the | |
| 04:00 | places on it down here isn't . So there is | |
| 04:02 | no real center to , it tipped down here isn't | |
| 04:05 | the center , It's not equal distance for all the | |
| 04:07 | points . Now , it is true that the Parable | |
| 04:09 | has a focus . The problem has a focus somewhere | |
| 04:11 | right around there , but that's not equal distant to | |
| 04:13 | all the points on the Parabola . So this is | |
| 04:15 | not a central colic . All four of these are | |
| 04:22 | what we call the comic sections , because you can | |
| 04:24 | take a cone and slice them and you can get | |
| 04:26 | all of those shapes , but you do not have | |
| 04:28 | a center of the parabola . So it's not called | |
| 04:30 | a central Connick . Now , why do we care | |
| 04:32 | about that ? Well , you can shift these guys | |
| 04:34 | around . Of course we've taken in graph parabolas shifted | |
| 04:37 | in the xy plane all day long , but we | |
| 04:40 | talk about the Central Comics because these are the ones | |
| 04:42 | we're going to focus on shifting here . Now , | |
| 04:44 | it turns out that we've already shifted the circles before | |
| 04:46 | we're gonna talk in a minute about the equation of | |
| 04:48 | a circle . Have already done many lessons on shifting | |
| 04:50 | circles around because they're so easy . But here we're | |
| 04:53 | going to focus only on the ellipse and only on | |
| 04:55 | the hyperbole . So let's go down a trip down | |
| 04:59 | memory lane before we talk about the election hyperbole . | |
| 05:01 | And let's talk about the circle , which we have | |
| 05:05 | discussed at great length . The equation of the circle | |
| 05:07 | is X squared . The basic equation plus why square | |
| 05:11 | Is equal to the Radius Square ? So you might | |
| 05:13 | have a radius of 20 radius squared of 25 on | |
| 05:16 | the right square root of that is five . So | |
| 05:18 | the radius would then be five . So what this | |
| 05:21 | equation of a circle means is it means the radius | |
| 05:28 | is are the square root of what's on the right | |
| 05:31 | hand side , and the center was located at zero | |
| 05:35 | comma zero . Right . So this was the basic | |
| 05:38 | equation of a circle , X squared plus , Y | |
| 05:40 | squared is equal to some number squared . And then | |
| 05:43 | the number of the radius just defines how big the | |
| 05:45 | circle is in the center of it is in the | |
| 05:46 | xy plane . How did we shift circles around long | |
| 05:50 | ago . All we did is we did a replacement | |
| 05:52 | . We replaced the X variable with something . We'll | |
| 05:55 | talk about the second , we replace the Y variable | |
| 05:57 | with something . And whatever we replace it by is | |
| 05:59 | the numbers that shift the graph of that circle around | |
| 06:03 | the xy plane . So for example , if instead | |
| 06:07 | of this which is centered at 00 you have the | |
| 06:10 | equation uh X minus two , quantity squared Plus Why | |
| 06:17 | -4 , quantity squared is equal to some radius squared | |
| 06:20 | . I don't care what the radius is right now | |
| 06:22 | let's just say it's the same radius is a circle | |
| 06:24 | above . Then what this equation means is that this | |
| 06:28 | guy has a center in a new location given by | |
| 06:31 | these numbers here . Now this is a minus to | |
| 06:33 | any -4 . So that means basically this guy is | |
| 06:36 | going to have the same size the same radius are | |
| 06:41 | , but the center Is going to be located at | |
| 06:44 | 2:04 . Now the one thing you do have to | |
| 06:48 | remember , we talked about this before is when you | |
| 06:50 | have a minus sign , when you basically replace the | |
| 06:53 | X variable with x minus two and you replace the | |
| 06:56 | y variable with y minus four . The minus sign | |
| 06:59 | means you shift it to the right in the positive | |
| 07:02 | X direction . And the minus sign for the why | |
| 07:04 | means you shift in the positive Y direction , that's | |
| 07:07 | very backwards from what you might think . You probably | |
| 07:09 | think that a minus sign will shift it to the | |
| 07:11 | other to the left , toward the negative , but | |
| 07:13 | it doesn't shift it to the right . And we | |
| 07:14 | talked about this at great length but I'm going to | |
| 07:16 | review it really quickly for you here . Now let's | |
| 07:19 | say this circle has a radius of five . This | |
| 07:22 | graph means it's centered at the origin . That means | |
| 07:24 | the circle boundary are the X and Y values . | |
| 07:27 | So when you put the X and Y . Values | |
| 07:29 | of that that are on that circle in here and | |
| 07:32 | you square the x and Y values , you're gonna | |
| 07:34 | get five square 25 on the right hand side . | |
| 07:36 | That is what defines the boundary of the circle . | |
| 07:39 | Now , when you look at this thing , you've | |
| 07:41 | replaced X with something called x minus two And you've | |
| 07:44 | replaced why with Y -4 ? What that means is | |
| 07:47 | for these numbers to be squared and still equal , | |
| 07:50 | let's say it's the same radius 2025 5 squared on | |
| 07:53 | the right hand side . Then in this equation I | |
| 07:55 | must put values of X . M two units bigger | |
| 07:58 | than I did here . And I must put why | |
| 08:00 | values in four units bigger than I did up here | |
| 08:03 | . Why ? Because I'm subtracting off the number four | |
| 08:06 | . So if I I'm subtracting off the number two | |
| 08:08 | . So if I know the circle at the origin | |
| 08:11 | has to equal let's say 25 on the right hand | |
| 08:13 | side . Then in order for this equation to also | |
| 08:15 | equal 25 , I need to put units two units | |
| 08:19 | bigger than whatever the numbers were over here , because | |
| 08:22 | then I'll subtract off the two and I'll square it | |
| 08:24 | and then also tracked off before in our square it | |
| 08:26 | . So because I need to put X values two | |
| 08:29 | units bigger to make it equal the same thing as | |
| 08:31 | it did up here , I've shifted the X coordinates | |
| 08:34 | of every point on the circle to the right along | |
| 08:37 | the positive X direction . Two units . Right ? | |
| 08:40 | Because I have to put why values of the circle | |
| 08:43 | and four units bigger than I do up here ? | |
| 08:45 | It means that all of the y values of the | |
| 08:48 | coordinates of the circle are all shifted up . So | |
| 08:50 | that's why the minus signs shift to the right and | |
| 08:53 | up . Okay , if you were to have plus | |
| 08:55 | signs here it would be exactly opposite . You would | |
| 08:57 | shift them down or to the left . All right | |
| 09:01 | . So the way that we pull off the shifting | |
| 09:03 | business in general for pretty much for any function is | |
| 09:07 | we take the X variable and we put either X | |
| 09:09 | plus or minus the number , we take the Y | |
| 09:12 | value and we do plus or minus the number . | |
| 09:14 | So in order to do the shifting Mhm . What | |
| 09:17 | we do is we take the X . Variable and | |
| 09:19 | we replace it with something called x minus H . | |
| 09:22 | This H is just a number in this case I | |
| 09:23 | picked two but in your books you're going to see | |
| 09:26 | h it shifted h units to the right and we | |
| 09:29 | replace the y variable with y minus . Usually you | |
| 09:33 | see the letter K in the text books in this | |
| 09:35 | case I picked the number four for an example . | |
| 09:37 | But here all you have to know is that to | |
| 09:39 | shift any circle any number of units you just replace | |
| 09:42 | the X variable with x minus some number that you're | |
| 09:44 | going to shift and why minus some number for the | |
| 09:47 | Y shift ? If you want to shift , it | |
| 09:48 | shifted in the opposite direction , the minus has become | |
| 09:51 | a pluses plus for the same sort of arguments . | |
| 09:54 | Okay . The reason I'm bringing up all this is | |
| 09:56 | because we've graphs circles before and we've shifted circles all | |
| 09:59 | over the xy plane , so you know how to | |
| 10:00 | do it now . It turns out that for ellipses | |
| 10:03 | and for hyperbole is the exact sort of same thing | |
| 10:06 | happens . All you have to do is find the | |
| 10:08 | original equation of the ellipse and replace the X values | |
| 10:12 | with this x minus H , which is a shift | |
| 10:14 | X minus however many units you want to shift it | |
| 10:16 | and the y value gets replaced with a Y with | |
| 10:20 | a wise shift . And so when you go back | |
| 10:22 | to the equation of the ellipse that we've used for | |
| 10:24 | many , many , many lessons , you just take | |
| 10:26 | the X variable . You replace it with x minus | |
| 10:28 | some number or X plus some number and you take | |
| 10:31 | the Y variable and you replace it with at Y | |
| 10:34 | plus or minus something that's going to shift in the | |
| 10:36 | y direction . Same thing for the ellipse . So | |
| 10:39 | it's not so difficult once you know how to sketch | |
| 10:41 | ellipses and high purple is at the origin , shifting | |
| 10:43 | them around is really not that difficult . So that's | |
| 10:47 | the general idea of what we're doing . And so | |
| 10:49 | here we're going to revisit all of the equations that | |
| 10:53 | we have already come to know and love in terms | |
| 10:56 | of their shifted versions . I've did a summary of | |
| 10:59 | this when we had the lips is at the origin | |
| 11:02 | and when we had high purple is at the origin | |
| 11:03 | , all I have done here is take the exact | |
| 11:05 | same equations and put a shift in there and then | |
| 11:08 | I shift the graph around . So for instance , | |
| 11:10 | if you remember , the equation of an ellipse can | |
| 11:12 | be horizontal or vertical . If it's a horizontal ellipse | |
| 11:15 | , the original equation was just X squared over a | |
| 11:18 | squared plus Y squared over B squared is one . | |
| 11:21 | Here . All we've done is we've introduced a shift | |
| 11:23 | in X and a shift and why it's exactly the | |
| 11:25 | same idea as the circle . This c square is | |
| 11:28 | a squared minus b squared , is exactly what we've | |
| 11:30 | already learned before . Nothing has changed the some of | |
| 11:33 | the focal radi i is exactly what we've already learned | |
| 11:35 | before . Nothing has changed . What has happened though | |
| 11:38 | is that this ellipse is no longer centered at the | |
| 11:40 | origin . We've shifted it to the right H . | |
| 11:42 | Units and we've shifted it up K units . If | |
| 11:46 | you have minus signs , you go to the right | |
| 11:48 | and up in the positive X . And Y direction | |
| 11:50 | . If you have plus signs up here , then | |
| 11:51 | you go the opposite directions . Just like we do | |
| 11:53 | for circles . Now , when you if you remember | |
| 11:57 | back in your mind when we did ellipses , we | |
| 11:59 | said that the X intercept in the Y intercept where | |
| 12:02 | the ellipse crosses come , comes from what's on the | |
| 12:05 | bottom here , Right ? So if you think about | |
| 12:07 | an ellipse centered at the origin , it's going to | |
| 12:09 | look something like this is gonna be like this and | |
| 12:11 | the X intercept is going to be A and the | |
| 12:14 | Y intercept is going to be be . But since | |
| 12:16 | now the ellipses now shifted , it's really not the | |
| 12:18 | intercept , it's just the distance from the center to | |
| 12:21 | the edges . What we call A . That's the | |
| 12:23 | same thing as if the ellipse was centered at the | |
| 12:25 | origin , it would be a unit . It's just | |
| 12:27 | , this is the distance we call it A . | |
| 12:29 | And then the distance in the Y direction from the | |
| 12:31 | center to the edge of the ellipse is called be | |
| 12:33 | . So before those were the intercept locations . But | |
| 12:38 | the lips may not have an intercept with X and | |
| 12:40 | Y . Like , like here if you shift it | |
| 12:42 | far enough away , so A and B . Are | |
| 12:44 | just the length of the ellipse in the X direction | |
| 12:47 | , Well , half of half of its length and | |
| 12:50 | half of its length in the Y direction . Right | |
| 12:52 | now , when this equation c squared is a squared | |
| 12:54 | minus B squared C . Is always in the distance | |
| 12:56 | from the center of the ellipse to a focus . | |
| 12:58 | So that's how I have it drawn here . So | |
| 13:01 | these definitions and the center here is that hk these | |
| 13:04 | definitions don't change , They're not different . It's just | |
| 13:07 | that when it's at the center we usually say oh | |
| 13:08 | the X intercept and the Y intercept . But now | |
| 13:10 | that you know , it may not be at the | |
| 13:12 | center , we just say A . And B . | |
| 13:13 | Are the length in the X . And Y . | |
| 13:15 | Direction of the ellipse . Same exact song and dance | |
| 13:18 | for this . Here we know this one's vertical because | |
| 13:22 | notice the equation is exactly the same . But here | |
| 13:25 | we have A . And B flipped the B . | |
| 13:26 | Number is always the smaller number for the ellipse . | |
| 13:28 | So since A . Is always bigger , that means | |
| 13:31 | that's horizontal . And since A . Is bigger here | |
| 13:34 | , that means it's going to be stretched more in | |
| 13:36 | the Y . Direction . But the numbers mean the | |
| 13:39 | same exact thing A . Represents how long is in | |
| 13:42 | the Y direction . The long direction B . Represents | |
| 13:45 | how wide it is in the short direction , the | |
| 13:48 | center is here . And then A . And B | |
| 13:51 | . Uh I've already told you X . And Y | |
| 13:52 | . Direction . And then see is just the distance | |
| 13:54 | from the center to the focus . So it's the | |
| 13:56 | same meanings as what we have above . So , | |
| 13:58 | if you think about that too , what we learned | |
| 14:00 | in regular lips is A . B . And C | |
| 14:02 | . Mean exactly the same thing as they do there | |
| 14:04 | . The only thing different is that we have now | |
| 14:06 | shifted this thing over . So when you're gonna sketch | |
| 14:09 | this first , you have to figure out where the | |
| 14:11 | center is and then you sketch from there . All | |
| 14:14 | right now we talk about our friend , the hyperbole | |
| 14:18 | , same exact thing . The original equation of a | |
| 14:20 | hyperbole at the origin is just X squared over A | |
| 14:23 | squared minus Y squared over B squared is one here | |
| 14:27 | . All we've done is introduced a shift in X | |
| 14:29 | and Y . This equation C squared is a squared | |
| 14:33 | plus B , squared is exactly the same as it | |
| 14:34 | was before . So the equation is the same . | |
| 14:37 | We've just introduced a shift . Now this one is | |
| 14:40 | what we call horizontal horizontal here because the X term | |
| 14:45 | is positive . The X term is the first term | |
| 14:48 | here . It's the positive term . So it's in | |
| 14:50 | the X direction , horizontal . If you look at | |
| 14:52 | the vertical version , A and B are in the | |
| 14:54 | same locations . A and B . For hyperbole is | |
| 14:56 | don't flip around , but they always stays in the | |
| 14:59 | front in the first term . But in this case | |
| 15:01 | the wise term is squared and is positive . So | |
| 15:04 | that means it's vertical . That's exactly the same rules | |
| 15:06 | we learned in hyperbole as I just told you to | |
| 15:08 | figure out if the hyperbole with horizontal or vertical , | |
| 15:10 | you just look and see what term is positive . | |
| 15:12 | It's exactly the same thing here . All we have | |
| 15:14 | done is introduce a shift in the X and Y | |
| 15:17 | directions , which shifts the center of the hyperbole off | |
| 15:20 | in those directions . This equation is exactly the same | |
| 15:23 | . The difference of focal radio for hyper polices called | |
| 15:26 | the difference of focal radio is to a that definition | |
| 15:29 | has not changed either . So really it's not that | |
| 15:32 | big of a deal to deal with ellipses and hyperbole | |
| 15:35 | is that are shifted . But there's just a couple | |
| 15:37 | things you got to remember number one . The rules | |
| 15:39 | don't really change . The definitions of A . B | |
| 15:41 | and C Are not different , but you do have | |
| 15:43 | to be a little bit careful because for instance , | |
| 15:46 | when we get to graphing these hyperbole as you're going | |
| 15:48 | to have to sketch the assam tops in order to | |
| 15:51 | actually sketch the graph . And we've done that many | |
| 15:53 | many times when it's at the origin , it's very | |
| 15:56 | easy . But whenever it's shifted , you got to | |
| 15:58 | be careful because the way in which you draw the | |
| 16:01 | the asem top lines here , everything is relative to | |
| 16:05 | the new center of this guy here . So it's | |
| 16:08 | just like in the ellipse case , you know , | |
| 16:12 | A . And B is the length and width of | |
| 16:14 | the ellipse , but that's all reference to the center | |
| 16:16 | of the ellipse , right ? And so the same | |
| 16:18 | thing is happening here . Whenever we get to graphing | |
| 16:20 | these and we have to graph these Assumpta these lines | |
| 16:23 | in order to sketch it , we're going to have | |
| 16:25 | to reference everything we do when we sketch these things | |
| 16:29 | relative to the new center of just like we were | |
| 16:32 | doing before . We sketched everything before . We always | |
| 16:35 | referenced it from the center . It's just the center | |
| 16:37 | was always the center of the xy plane . So | |
| 16:40 | here we have to be a little more careful when | |
| 16:41 | we're shifted over to graph all of our lines and | |
| 16:44 | all of our sometimes and everything relative to the new | |
| 16:46 | center , which is not at the origin anymore . | |
| 16:50 | So that's a general overview of what a shifted Connick | |
| 16:53 | section is and what a central Connick is . I | |
| 16:56 | do want to do a couple of quick problems . | |
| 16:57 | Um they're not hard problems , but I want to | |
| 17:01 | get our feet wet here by doing a couple of | |
| 17:03 | quick things and we're going to start that process right | |
| 17:07 | now . So let's right , we want to write | |
| 17:10 | the equation of this comic section with the new center | |
| 17:13 | that I'm gonna give you here . So for instance | |
| 17:15 | , if if you give you a comic section X | |
| 17:17 | squared over 25 plus why squared over four ? And | |
| 17:24 | that's equal to one . And I want the new | |
| 17:26 | center of this thing to be located at zero comma | |
| 17:28 | negative five . I want to write the equation of | |
| 17:31 | this guy , first of all , what kind of | |
| 17:34 | iconic is it ? Well , if you look at | |
| 17:36 | this it's got a plus sign . So you know | |
| 17:38 | right away and it's got different numbers on the bottom | |
| 17:40 | . So you know right away . It's in the | |
| 17:41 | lips . All right . So what do we do | |
| 17:44 | to take this ellipse which is at the origin ? | |
| 17:46 | And shift it over here to the location where the | |
| 17:49 | center is at zero comma negative five . All you | |
| 17:51 | do is take the shifted values and stick them in | |
| 17:53 | here . So the X . has shifted zero units | |
| 17:58 | . So really you don't need to do anything at | |
| 18:00 | all . But I'm going to put it as a | |
| 18:01 | shift of zero . The y numbers are shifted by | |
| 18:05 | negative five units . Now you can put a plus | |
| 18:07 | on here that's fine . But I'm just gonna show | |
| 18:08 | you it's like shifting a negative of negative five units | |
| 18:11 | . So every time you do it you have a | |
| 18:13 | minus sign and whatever you shift is going to either | |
| 18:15 | be plus or minus . So you'll see in a | |
| 18:17 | second that's going to turn into a plus sign . | |
| 18:20 | All right . So you can just put the plus | |
| 18:22 | right away . That's fine with me . But I'm | |
| 18:24 | gonna show you it's shifting shifts always go with a | |
| 18:27 | minus sign and then whether you shift left or right | |
| 18:29 | , you just put however many units you want to | |
| 18:30 | shift . That's why it becomes a plus sign at | |
| 18:34 | the end of the day anyway . All right . | |
| 18:36 | So what you're gonna have here , this is going | |
| 18:37 | to become x squared over 25 . This is going | |
| 18:41 | to become y plus five squared over four and this | |
| 18:46 | is going to equal one . And this is the | |
| 18:48 | new equation of this ellipse which is no longer centered | |
| 18:51 | at the origin , its center is at zero comma | |
| 18:53 | negative five . Okay . And you know it's any | |
| 18:58 | lips because it has the form of an ellipse with | |
| 19:02 | a plus sign . And also is this a horizontal | |
| 19:05 | or a vertically oriented lips ? Well , since the | |
| 19:07 | X squared has a 25 under it and the Y | |
| 19:09 | . Term has a four under it . The bigger | |
| 19:11 | number here is 25 . So it's going to be | |
| 19:13 | stretched more in the X direction . So this is | |
| 19:16 | a horizontal you live . So it might look something | |
| 19:19 | like this . Who knows where it is in the | |
| 19:21 | ex wife . Well we know exactly where I could | |
| 19:23 | draw an X . Y . Plane and I can | |
| 19:24 | put the center right here . Whatever I just I | |
| 19:26 | want you to know that it's a horizontal lips shifted | |
| 19:29 | where the center is at 00 comma negative five . | |
| 19:32 | So it's right along on the Y axis shifted down | |
| 19:35 | essentially . And that's all I want you to know | |
| 19:38 | . I'm not gonna graph a sketch of every one | |
| 19:39 | of these things . It's gonna slow us down . | |
| 19:42 | Mhm . Alright , next problem , what if I | |
| 19:44 | tell you , take this comic section and put write | |
| 19:48 | an equation where the center is , where I give | |
| 19:49 | it to here . So it's X squared minus y | |
| 19:52 | squared is 49 . And the new center of whatever | |
| 19:55 | this thing is , I wanted to be a negative | |
| 19:57 | four comma three and I ask you to draw that | |
| 19:59 | for me . Okay well first of all what kind | |
| 20:02 | of comic section is it ? You might not be | |
| 20:04 | 100% sure at first but we want to one on | |
| 20:06 | the right hand side . So we'll take the X | |
| 20:08 | squared minus two Y squared . We'll divide the left | |
| 20:11 | hand side by 49 . We'll take the right hand | |
| 20:14 | side and also divide by 49 . So on the | |
| 20:18 | left will we break this up ? Which we've done | |
| 20:20 | many times will be X squared over 49 . The | |
| 20:23 | minus sign drops down . Why squared over 49 ? | |
| 20:27 | And that's going to equal to one . So it | |
| 20:29 | looks exactly like a hyperbole right ? Because it has | |
| 20:32 | an X squared term of Y squared term numbers on | |
| 20:34 | the bottom and a minus sign . Now this is | |
| 20:36 | a high purple is centered at the origin because of | |
| 20:38 | the X . Shift and the Y shift or non | |
| 20:40 | existent . So it's already centered at the origin . | |
| 20:43 | So in order to shift it into this location I'm | |
| 20:47 | going to shift the X . Direction . All shifts | |
| 20:49 | go with the minus sign , I'm shifting negative four | |
| 20:51 | units , I'm gonna write it like this And then | |
| 20:55 | over 49 Minour sign comes down because it's a hyperbole | |
| 20:58 | and this guy is going to be why minus all | |
| 21:01 | shifts have a minus sign and then three units And | |
| 21:05 | then 49 and that's going to equal one . Now | |
| 21:08 | I said all shifts have a minus side . Well | |
| 21:10 | what happens is if you shift a negative unit a | |
| 21:13 | negative direction this becomes a plus sign . So you | |
| 21:15 | have x plus four quantity squared over 49 minus y | |
| 21:21 | minus three quantity squared over 49 Equals one . And | |
| 21:27 | that's what you have . Let me double check myself | |
| 21:30 | . That's correct . And then you ask yourself well | |
| 21:31 | , what is this thing ? Well , you know | |
| 21:33 | , it's a hyper Bella right ? Is it a | |
| 21:39 | vertical or horizontal hyperbole ? You do not look at | |
| 21:42 | the bottom numbers to figure out if hyperbole is a | |
| 21:44 | vertical or horizontal for hyperbole . As you just check | |
| 21:46 | and see which term is positive , The X term | |
| 21:48 | is positive . The y term is negative because the | |
| 21:51 | X term is positive . This is a horizontal hyperbole | |
| 21:56 | . Okay , horizontal hyperbole . All right , So | |
| 21:59 | you see what you have to do to shift any | |
| 22:02 | kind of section to a new location . You're just | |
| 22:04 | simply substituting in what the shifted version of X and | |
| 22:07 | Y are now I ran out of space over there | |
| 22:10 | . So what we're gonna do is our final problem | |
| 22:12 | over here and I think we have enough room . | |
| 22:14 | What if I give you the context ? Section four | |
| 22:16 | , X squared plus Y squared is equal to 16 | |
| 22:22 | . And I say , well let's first put it | |
| 22:23 | into the standard form of a comic section . We're | |
| 22:25 | gonna divide by the 16 . So we're gonna have | |
| 22:27 | the four X squared over 16 plus the Y squared | |
| 22:31 | over . The 16 is equal to the 16 over | |
| 22:34 | the 16 . And then in the next step , | |
| 22:36 | I can simplify this because the four and the 16 | |
| 22:38 | 4 divided by four is one and 16 divided by | |
| 22:41 | four is four . So this would just be x | |
| 22:43 | squared over four plus Y squared over 16 . And | |
| 22:48 | on the right hand side is just going to equal | |
| 22:49 | one . So this is any lips actually that is | |
| 22:53 | centered at the origin and the four and the 16 | |
| 22:56 | govern how the thing is uh kind of shaped what's | |
| 22:59 | what's the what side is bigger , but now what | |
| 23:01 | we want to do , I forgot to actually even | |
| 23:02 | write it down in the problem statement . I want | |
| 23:04 | this ellipse to be at the location of one common | |
| 23:07 | negative four . I want the center of it to | |
| 23:09 | be at one common negative four . So all I | |
| 23:11 | really have to do is take this version of the | |
| 23:13 | equation I have and substitute that shift in . So | |
| 23:17 | the X term becomes X shift by one unit Quantity | |
| 23:23 | squared over the four on the bottom . Then I | |
| 23:27 | have the plus sign and then the Y is shifted | |
| 23:29 | by a negative for units Squared . And on the | |
| 23:34 | bottom of that is going to be the 16 And | |
| 23:37 | that's going to equal to one . So when you | |
| 23:40 | figure all that out , it's going to be X | |
| 23:42 | -1 , quantity squared over four plus y plus four | |
| 23:48 | quantity squared over 16 is equal to one . Double | |
| 23:52 | Check myself , X minus one squared over four . | |
| 23:54 | Y plus four squared over 16 is equal to one | |
| 23:56 | . That's all correct . And you can tell by | |
| 23:58 | looking at this because of the plus sign . It's | |
| 23:59 | in your lips What kind of ellipses is a horizontal | |
| 24:04 | or vertical for ellipses ? Since there's no plus or | |
| 24:07 | negative terms , you just look at what's on the | |
| 24:08 | bottom . The X term has a four and the | |
| 24:11 | white term has a 16 . That means the Y | |
| 24:13 | direction is stretched more . So this guy is a | |
| 24:16 | vertical . If I can spell vertical , your lips | |
| 24:20 | , sorry , I ran out of space there . | |
| 24:21 | So it's in the lips that looks something like this | |
| 24:23 | . And of course it shifted where the center of | |
| 24:25 | it . Is that one uh over to the right | |
| 24:27 | one unit and negative four down . So this is | |
| 24:30 | the equation there . It's a vertical ellipse . All | |
| 24:33 | right , so , we've done quite a bit in | |
| 24:35 | this lesson . We have reviewed the concept of shifting | |
| 24:38 | in general . We've taken a look at circles and | |
| 24:41 | reminded ourselves how we shift circles around . We've done | |
| 24:43 | that before and we're using that concept to shift comic | |
| 24:46 | sections . In fact , this concept of shifting functions | |
| 24:49 | around by just replacing the X and Y variable can | |
| 24:52 | be used for any function or any graph . If | |
| 24:54 | you want to shift it , X number of units | |
| 24:56 | , you just put the shift in as we've done | |
| 24:58 | here for the X . And the Y variable . | |
| 24:59 | And that's going to shift the whole graph wherever you | |
| 25:01 | want it to be in the xy plane . But | |
| 25:03 | we're not done with shifted hyperbole as an ellipses . | |
| 25:06 | We need to get some practice graphing and do some | |
| 25:08 | other things . So follow me on to the next | |
| 25:10 | lesson , we're gonna conquer that stuff right now . |
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01 - Shifting Ellipses and Hyperbolas in the XY Plane - Part 1 (Central Conics) is a free educational video by Math and Science.
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