07 - Graphing Parabolas in Vertex Form & Shifting Horizontally - By Math and Science
Transcript
| 00:00 | Hello , Welcome back to algebra . We're now going | |
| 00:02 | to focus our attention on taking our basic shape of | |
| 00:05 | a parabola and shifting it left and right . That | |
| 00:08 | means we're only going to shift the parabola along the | |
| 00:11 | X axis . In the previous lesson , we talked | |
| 00:13 | about shifting the parabola in the Y axis . So | |
| 00:16 | if you haven't watched that please stop and go do | |
| 00:18 | that now . Now we'll be talking about shifting parabolas | |
| 00:20 | in the X direction . And then in the next | |
| 00:22 | lesson will shift it on both directions at the same | |
| 00:25 | time . So we're kind of inching our way there | |
| 00:27 | . So to get started , I want you to | |
| 00:29 | re acquaint yourself with what we have left this on | |
| 00:31 | the board in the last several lessons because it's the | |
| 00:33 | basic shape of a parabola . So we have the | |
| 00:35 | basic problem shape vertex is at the origin . This | |
| 00:39 | is a table of values for this parabola . Why | |
| 00:41 | is equal to X squared ? Now you all know | |
| 00:43 | by now that the general idea of the general form | |
| 00:45 | of a parabola uh for how it opens and closes | |
| 00:49 | as far as being steep or shallow is a times | |
| 00:51 | X squared the number in front here A governs how | |
| 00:55 | steeply or how shallow the parabola opens up . We | |
| 00:58 | talked about that before . So this table of values | |
| 01:00 | is the basic shape where we have a equals one | |
| 01:03 | . So it's basically why is equal to one X | |
| 01:05 | . Squared . And here is the table of values | |
| 01:07 | . Basic problem vertex , which is the minimum value | |
| 01:10 | right at the origin . And then in the last | |
| 01:14 | lesson , we talked about uh shifting horizontally . So | |
| 01:18 | we're gonna say , I'm going to relabel this , | |
| 01:19 | I'm gonna say this is a summary of horizontal , | |
| 01:21 | I'm sorry , vertical , which is what we did | |
| 01:23 | in the last lesson , vertical shift . And I'm | |
| 01:27 | gonna mark up this board a little bit because I | |
| 01:29 | want to inch you . They're one step at a | |
| 01:32 | time . And by the way at the end of | |
| 01:33 | the lesson we'll have another computer demo . So please | |
| 01:35 | stick around for that because the computer can really help | |
| 01:38 | solidify things after I've done the lecturing . Uh This | |
| 01:41 | was a summary of a vertical shift , so here's | |
| 01:43 | your basic parabola . No shift when it's just a | |
| 01:45 | X . Squared . When you add one to the | |
| 01:47 | side . Another way of thinking about it is pulling | |
| 01:50 | that one over next to the Y . And that | |
| 01:52 | is shifting up one unit . Uh Similarly shifting three | |
| 01:56 | units would look like why minus one equals X . | |
| 01:59 | Squared . And so shifting down is when you have | |
| 02:01 | an opposite sign . So I told you in the | |
| 02:03 | last lesson to in your brain when you have the | |
| 02:06 | shift written next to the variable on the left hand | |
| 02:08 | side of the equal sign when we're shifting up and | |
| 02:10 | down , that if it was a minus sign , | |
| 02:13 | your shifting up in the positive direction , and if | |
| 02:15 | it was a plus sign , your shifting down going | |
| 02:17 | kind of in the negative direction . So it's kind | |
| 02:19 | of exactly the opposite of what you would expect . | |
| 02:21 | And we went to great lengths to explain why that | |
| 02:24 | is the case in the last lesson . So if | |
| 02:26 | you haven't watched that , please do it now . | |
| 02:28 | But this is the results of that . Now . | |
| 02:29 | What I wanna do , you wanna make too little | |
| 02:32 | changes to this and I want to do their changes | |
| 02:35 | don't don't matter . But they will help you understand | |
| 02:36 | today's topic Here . I have the Y -1 in | |
| 02:39 | this example written on the left , I'm just gonna | |
| 02:41 | put parentheses around it . You would agree that parentheses | |
| 02:44 | don't change anything , it doesn't make it a different | |
| 02:46 | expression , I'm just kind of grouping it together and | |
| 02:48 | you'll understand why I'm doing that in just a second | |
| 02:51 | . I'm gonna put Princess around that because I'm gonna | |
| 02:53 | say this kind of goes together . I'm gonna put | |
| 02:55 | princes around that . Which kind of says that this | |
| 02:57 | goes together and you already know this because we talked | |
| 03:00 | about the shifting so much . But when we shift | |
| 03:02 | up one unit , for instance , let me use | |
| 03:04 | this color when we shift up one unit , which | |
| 03:06 | is this equation as an example . Then in general | |
| 03:10 | what I'm talking about is I don't have that much | |
| 03:12 | space to write it . But basically if this is | |
| 03:14 | your Xy playing then instead of the problem coming down | |
| 03:18 | like this , it shifted up one unit , which | |
| 03:20 | is what I'm trying to represent here . This is | |
| 03:22 | one unit shifted in the up direction , so a | |
| 03:24 | minus sign means we shift up one unit and there's | |
| 03:27 | an example in the other direction . I'll draw a | |
| 03:29 | little quick little access down here . I should have | |
| 03:31 | drawn it kind of a little more favorably sorry about | |
| 03:34 | that but minus two means we're gonna have to units | |
| 03:37 | down and this parabola is going to be , that's | |
| 03:39 | a very bad joining , I'm sorry about that but | |
| 03:41 | it's supposed to be kind of like centered right there | |
| 03:43 | . The vertex is two units down , so downward | |
| 03:46 | shift , upward shift . And now I've drawn some | |
| 03:48 | parentheses to group them here . So what I'm gonna | |
| 03:50 | do is now I'm just gonna write on the board | |
| 03:53 | um what it looks like when you shift a parabola | |
| 03:56 | left or right and then after I write it down | |
| 03:59 | you're going to scratch your head and say that makes | |
| 04:01 | no sense at all . But then you're gonna have | |
| 04:03 | to listen to me talk for a minute so you | |
| 04:04 | understand why I could just tell you to memorize it | |
| 04:07 | . But that's no fun . Plus you're just a | |
| 04:09 | monkey and you're just not doing any thinking . I | |
| 04:11 | don't want you to be a monkey . And do | |
| 04:12 | I think I want you to understand ? So I'm | |
| 04:14 | gonna write down how it looks to shift left and | |
| 04:16 | right , which is a little bit weird looking . | |
| 04:18 | And then you'll fully understand it when you listen to | |
| 04:20 | me babble on for a few minutes about it . | |
| 04:23 | All right ? So I'm gonna do it on this | |
| 04:24 | board . So we're gonna talk about the horizontal shift | |
| 04:29 | . And the reason I present the horizontal shift after | |
| 04:33 | the vertical shift is it's a little bit harder to | |
| 04:34 | understand , but I will make it clear horizontal shift | |
| 04:39 | of a problem . So what I wanna do , | |
| 04:43 | a horizontal shift looks like this horizontal shift looks like | |
| 04:47 | this . Uh The basic equation is why is equal | |
| 04:51 | to X squared . So if you shift horizontally , | |
| 04:55 | it's gonna look like this parentheses X -1 squared . | |
| 05:00 | Right ? This is a shift , right ? Let | |
| 05:05 | me erase this . Actually . We're gonna shift the | |
| 05:07 | equation . Let's do it like this . We're gonna | |
| 05:09 | shift equation right ? One unit , one unit . | |
| 05:15 | And I'm gonna do a quick little sketch . It's | |
| 05:16 | not gonna be a graph in all of its glory | |
| 05:19 | right here . I want to get as much on | |
| 05:21 | the board as I can . So I kind of | |
| 05:22 | have to squish things in a little bit sometimes . | |
| 05:26 | So if the basic traveler goes and touches down the | |
| 05:28 | center like that . So when we shift one unit | |
| 05:31 | to the right , this is one unit to the | |
| 05:32 | right . The problem is no longer centered in the | |
| 05:35 | origin . It goes like this . That is what | |
| 05:37 | we say . When we say one unit shifted to | |
| 05:39 | the right now , before we go any farther . | |
| 05:42 | Because I'm gonna give a lot more examples and I'm | |
| 05:44 | going to talk to you a lot about why it's | |
| 05:45 | happening before we go any further . I want you | |
| 05:48 | to go back to what you already understand . A | |
| 05:50 | vertical shift up , which is this Whenever you have | |
| 05:53 | the plus one on the right hand side can be | |
| 05:55 | written like this with the minus one on the left | |
| 05:58 | hand side , a minus sign shifts it kind of | |
| 06:01 | in the positive Y direction , a minus sign . | |
| 06:04 | And why here shifts it in the positive y direction | |
| 06:07 | . I'm telling you without any proof yet that a | |
| 06:09 | minus sign here shifts it in the positive X direction | |
| 06:13 | . So really what you need to know is that | |
| 06:15 | when you write the shifts next to the variables , | |
| 06:18 | you just go in the opposite direction of the sign | |
| 06:21 | . If it's a minus sign , it shifted to | |
| 06:22 | the positive direction . So if it's a minus and | |
| 06:25 | why it shifted positive , why if it's a minus | |
| 06:28 | and X . It shifted positive X . Okay . | |
| 06:30 | And that's why I put these parentheses around and that's | |
| 06:33 | why you know we kind of do one before the | |
| 06:34 | other . So this is a single unit shift to | |
| 06:38 | the right . What do you think would happen if | |
| 06:40 | we wanted to shift three units ? three units to | |
| 06:43 | the right , I'm sorry , three units to the | |
| 06:44 | left would be X instead of plus instead of minus | |
| 06:48 | , it would be plus . Okay notice it's still | |
| 06:51 | square . That makes sure . Makes allows you to | |
| 06:54 | have a parable of shape . This is a shift | |
| 06:57 | left three units , three units . Now , what | |
| 07:03 | do you think that looks like a shift left of | |
| 07:06 | three units looks like this . Here's negative one , | |
| 07:09 | negative two , negative three . Have to extend my | |
| 07:11 | graph a little bit . So this is negative three | |
| 07:13 | . And that means that graph no longer goes and | |
| 07:15 | it's centered uh touching the origin , it goes down | |
| 07:19 | and touches the negative three . So the vertex is | |
| 07:21 | right here at negative three . That's what we mean | |
| 07:23 | . When we say it shifted left three units along | |
| 07:26 | the X . Direction . So of course this is | |
| 07:28 | X . And this is why . So we've kind | |
| 07:31 | of established that a negative science shift to the right | |
| 07:33 | a positive signs shifts to the left . But just | |
| 07:35 | for giggles , we'll do one more example Where we | |
| 07:38 | say , what happens if we have the equation X | |
| 07:41 | -4 quantity squared ? This means we shift because it's | |
| 07:46 | a minus sign , it goes opposite so to the | |
| 07:48 | right four units . And what does that look like | |
| 07:55 | ? I don't really think I need to draw it | |
| 07:56 | , but I will anyway , so we have 1234 | |
| 08:00 | units to the right . The problem is now centered | |
| 08:03 | there and of course my shape isn't exactly right . | |
| 08:05 | You gotta realize it's going to have that perfect parabola | |
| 08:07 | shape when I'm drawing it freehand freehand , I'm not | |
| 08:10 | gonna be able to do that . All right , | |
| 08:13 | So that is the bottom line . That's what you're | |
| 08:15 | going to see in books . You're going to see | |
| 08:17 | uh an equation that says , hey , if you | |
| 08:19 | want to shift the problem to the right , then | |
| 08:21 | you put parentheses around the X square term . Remember | |
| 08:24 | the basic the basic equation for all of these is | |
| 08:27 | just X square . What you're doing is you're replacing | |
| 08:29 | the variable with the parentheses and you're putting a shift | |
| 08:32 | inside . And that whole thing is square , right | |
| 08:35 | ? And so it's like X square . But you | |
| 08:37 | have now a shift in there and that whole thing | |
| 08:40 | has to be squared . And I'm telling you kind | |
| 08:42 | of without proof that when it's a minus sign , | |
| 08:44 | it shifts that many units that way . When it's | |
| 08:46 | a plus sign it's that many units shifted that way | |
| 08:49 | . But what we want to do now is discuss | |
| 08:51 | why that's the case and that takes a few minutes | |
| 08:54 | to explain . But it's not it's not crazy hard | |
| 08:56 | but you need to kind of open your mind because | |
| 08:58 | it's uh the first time I learned that it was | |
| 09:01 | very hard for me to understand , I'm gonna try | |
| 09:02 | to short circuit all of the difficulty and try to | |
| 09:05 | cut down to the to the bottom line here . | |
| 09:07 | I think the easiest way to do it . Uh | |
| 09:10 | First of all is do a table of values . | |
| 09:13 | So let's take let me think , I want to | |
| 09:16 | do the table of values here . I want to | |
| 09:17 | go to the next board . I want to go | |
| 09:19 | to the next port . I'm gonna take one of | |
| 09:20 | these examples . I'm gonna take this first example x | |
| 09:23 | minus one quantity squared . This is the one we're | |
| 09:25 | gonna work with . We're gonna do a quick table | |
| 09:27 | of values on that and we're gonna examine it in | |
| 09:30 | detail . So if you have x and y equals | |
| 09:34 | x minus one quantity squared . And we're just gonna | |
| 09:37 | do a table of values . This is exactly what | |
| 09:39 | we did when we did the previous shifting . And | |
| 09:41 | we're gonna go negative three negative two , negative 1012 | |
| 09:46 | and three . And I need you to kind of | |
| 09:48 | bear with me because we do need to plug these | |
| 09:50 | values into here and I want you to see what | |
| 09:52 | the outputs really look like . So let's start with | |
| 09:56 | zero . That's actually easier . If you put zero | |
| 09:58 | in here Then what you're gonna get is 0 -1 | |
| 10:01 | quantity squared . That's what you would have if you | |
| 10:03 | put it in there and of course you get negative | |
| 10:05 | one quantity squared , you're just gonna get a one | |
| 10:08 | out of that . Now , if you put a | |
| 10:10 | one in here , one minus one is going to | |
| 10:12 | give , you will do it like this one minus | |
| 10:14 | one squared is going to be zero squared , Which | |
| 10:19 | is going to be zero . So when you put | |
| 10:21 | a one in , you actually get a zero out | |
| 10:23 | . When you put a to win , what do | |
| 10:24 | you get ? You get 2 -1 quantity square which | |
| 10:27 | is one squared , which is one . Now I | |
| 10:30 | want you to stop for just a second . I | |
| 10:32 | haven't filled the rest of the table out , but | |
| 10:34 | I want you to see that you can already see | |
| 10:36 | some symmetry forming in here . Notice that there's always | |
| 10:39 | symmetry in that first example , the symmetry was about | |
| 10:42 | this point . All of the numbers on both sides | |
| 10:45 | of zero were symmetric . That's why the graph is | |
| 10:47 | symmetric about this zero point right here . Right . | |
| 10:50 | But already , before we even finish any other part | |
| 10:53 | of the table , uh we already see that we | |
| 10:55 | have a center point kind of with some symmetry on | |
| 10:58 | both sides , but it's not located in the middle | |
| 11:01 | , it's shifted so that the symmetry , the symmetric | |
| 11:03 | point is at X is equal to one . What | |
| 11:06 | did I tell you ? I said that when it's | |
| 11:08 | minus one is going to be shifted to the right | |
| 11:10 | . That means that the center point of the problem | |
| 11:12 | should be shifted over one unit . And we're already | |
| 11:15 | seeing that the center point of the probable is actually | |
| 11:17 | shifted over one unit before we even finished the table | |
| 11:20 | . But let's go ahead and finish the table because | |
| 11:22 | it's helpful Three plugged in here . The 3 -1 | |
| 11:26 | quantity square , which is two squared , which is | |
| 11:29 | four . All right . So we have these values | |
| 11:31 | . Let's go to the negative territory . Right ? | |
| 11:33 | So we're putting negative one in here negative one minus | |
| 11:35 | one , quantity squared . That's negative too , Quantity | |
| 11:39 | squared , which is four . Let's pause for a | |
| 11:41 | second . See how we have a center point . | |
| 11:43 | We have symmetry here and we have a four and | |
| 11:45 | a four . We have symmetry building here . Now | |
| 11:49 | let's put it to in here . Um negative two | |
| 11:52 | . I'm sorry , minus one quantity square . This | |
| 11:54 | will be negative three , quantity squared , which is | |
| 11:57 | going to give you nine . And the last one | |
| 11:59 | will do . Is this 1 -3 -1 , quantity | |
| 12:02 | squared is negative four , quantity squared . I can | |
| 12:06 | barely fit in here to be 16 . Now , | |
| 12:08 | I didn't continue my chart this direction . I could | |
| 12:10 | do that . In fact , it might be a | |
| 12:13 | good idea to do that . Let's go up one | |
| 12:14 | more unit 24 And let's put it in here 4 | |
| 12:18 | -1 . I didn't do this on my paper , | |
| 12:21 | but let's do it here . Three squared is nine | |
| 12:24 | . Now the nine you see matches the nine up | |
| 12:26 | here and just for absolute completeness will go to 55 | |
| 12:30 | minus one quantity squared is four squared is 16 . | |
| 12:34 | So the 16 matches the 16 , the nine matches | |
| 12:36 | the nine . The four matches the four of the | |
| 12:38 | one matches the 10 is now in the center . | |
| 12:41 | This parable is centered here , this is the center | |
| 12:46 | . However , the center of this Parabola is no | |
| 12:48 | longer at X is equal to zero . The center | |
| 12:50 | of this parabola is now shifted one unit over to | |
| 12:53 | when X is equal to one , right ? And | |
| 12:55 | that's the bottom line . So this is a graph | |
| 12:57 | of a parabola that looks something like this . If | |
| 13:01 | you were to graphic , I'm not gonna do it | |
| 13:03 | rigorously . But if you do this guy x minus | |
| 13:06 | one quantity squared , the basic problem would go in | |
| 13:09 | the center like this , but this has shifted one | |
| 13:11 | unit over like this and it's just too sharp here | |
| 13:15 | , it's more flattened out in the real shape of | |
| 13:17 | a problem . But I'm doing things freehand . So | |
| 13:19 | the bottom line is when you have a minus sign | |
| 13:23 | here , it shifted over . So that's kind of | |
| 13:25 | like the first veneer layer of of explaining to you | |
| 13:28 | why shifts work like this . Of course the table | |
| 13:31 | of values helps you understand . But I want to | |
| 13:33 | probe just a little bit deeper . Talk a little | |
| 13:36 | bit more so you'll understand . Why is it really | |
| 13:38 | shifted to the right ? Let's write this equation down | |
| 13:41 | in large letters , right ? Why equals X -1 | |
| 13:47 | quantity squared ? Why am I writing it in big | |
| 13:49 | letters ? Because I want you to focus on this | |
| 13:51 | thing right here . This is a parabola . What | |
| 13:54 | it means is that whatever is inside and I want | |
| 13:58 | you listen carefully here because these following words are the | |
| 14:00 | most important in the whole lesson , right ? The | |
| 14:02 | worthy whatever is inside of this thing , inside of | |
| 14:07 | the princes , the whole enchilada inside of here , | |
| 14:09 | whatever is in here is what is squared . So | |
| 14:12 | if I stick a three as what is the whole | |
| 14:15 | thing inside here and squared ? I'm gonna get that | |
| 14:17 | nine . If I stick to the whole thing , | |
| 14:20 | the entire thing evaluates to being a two in the | |
| 14:22 | in the inside and then I get the four out | |
| 14:24 | . If I Sticking whatever I put in here on | |
| 14:27 | the inside of those princes . If I end up | |
| 14:29 | with a five in there after I've done everything . | |
| 14:30 | If I get a five inside the parentheses , like | |
| 14:33 | the whole thing , then I squared . I get | |
| 14:34 | the 25 . The shape of the Parabola basically comes | |
| 14:38 | about because whatever I have in here is squared . | |
| 14:41 | And so then I build my table of values . | |
| 14:43 | But notice that whenever I I have something like X | |
| 14:49 | -1 squared , then what ends up happening is I | |
| 14:53 | have to have uh the easiest way to think about | |
| 14:56 | it is to do the following , sorry , to | |
| 14:58 | switch gears on you . But let me go back | |
| 14:59 | to this . The basic parameters , shape is F | |
| 15:02 | X is equal to x square . When I put | |
| 15:04 | a zero into this , I square it and I | |
| 15:07 | get a zero out . That is why the center | |
| 15:09 | of the problem is right here . So let's focus | |
| 15:11 | on this this point . This one right here , | |
| 15:13 | let's focus on this point right here , zero comma | |
| 15:15 | zero . I put a zero in . I get | |
| 15:16 | a zero out . If I put a zero and | |
| 15:19 | and end up squaring it , then I get a | |
| 15:21 | zero out . Okay , But what happens in this | |
| 15:24 | equation is the only way that I can get a | |
| 15:26 | zero in here to square it is I have to | |
| 15:28 | have an X value , one unit bigger than I | |
| 15:31 | do for the base equation so that I can have | |
| 15:34 | a zero in there . Why ? Because I have | |
| 15:37 | to have a one here for X to make one | |
| 15:39 | minus one to give me a zero here to square | |
| 15:42 | it . So in other words , the bottom of | |
| 15:44 | this parabola is always going to happen when whatever I'm | |
| 15:47 | sticking in here is zero and then I square it | |
| 15:49 | and then I get zero , that's the bottom of | |
| 15:50 | the parabola . But the only way to get a | |
| 15:52 | zero in here is to have a to feed it | |
| 15:55 | an X value , one unit bigger than the basic | |
| 15:58 | equation . The base equation being y equals x squared | |
| 16:01 | , Stick at zero in , get a zero out | |
| 16:03 | . But here , if I stick at zero in | |
| 16:05 | , I just have negative one squared and I have | |
| 16:07 | one that's not the bottom of the parabola . The | |
| 16:11 | only way to shift that zero point to the right | |
| 16:14 | is to feed an X . Value in here one | |
| 16:17 | unit bigger than I'm originally feeding for the base equation | |
| 16:21 | . That's why it shifted to the right . You | |
| 16:23 | see that happens for every single point , right ? | |
| 16:25 | So if I put a a one in here , | |
| 16:28 | which is this point right here , one minus 10 | |
| 16:31 | square it . That's the very bottom of the problem | |
| 16:33 | . And it happens for every other point here . | |
| 16:36 | So that's why the center of the problem is now | |
| 16:38 | at this point when X is equal to one . | |
| 16:40 | If in an alternate universe , I had the equation | |
| 16:44 | X -4 quantity squared . This equation , this Parabola | |
| 16:50 | is a basic problem , shape shifted four units to | |
| 16:52 | the right . Four units . 1234 Why ? Because | |
| 16:57 | the only way that I can put a zero in | |
| 16:59 | here to square it to get zero as the bottom | |
| 17:02 | of the problem is to put an X value and | |
| 17:04 | that's an X value along the X axis . Four | |
| 17:06 | units bigger than usual , so that I have four | |
| 17:09 | minus four , that gives me zero and I square | |
| 17:11 | it , you see . And so that's how you | |
| 17:13 | need to think about horizontal shifts . You need to | |
| 17:15 | think of it . This x minus business here means | |
| 17:18 | I need to be feeding values into the function four | |
| 17:21 | units bigger to get to the same bottom point of | |
| 17:23 | that Parabola that I do in the basic function , | |
| 17:25 | that's about as clear as I can make it . | |
| 17:27 | I want you to roll that around your brain a | |
| 17:29 | little bit um and make sure you kind of understand | |
| 17:32 | and the same thing happens in the other direction . | |
| 17:34 | So we didn't do a table of values , but | |
| 17:37 | let's do it for for I've already told you that | |
| 17:40 | shifting to the left , I said , hey , | |
| 17:41 | if you have , I can just look at this | |
| 17:43 | one here . If you have the equation , uh | |
| 17:45 | X minus four square . That's that's a shift , | |
| 17:47 | I'm sorry X plus three square . That's a shift | |
| 17:49 | to the left three units . Why does this thing | |
| 17:52 | shifted to the left three units ? Because the only | |
| 17:56 | way that I can get a zero in here which | |
| 17:59 | is going to be squared to give me a zero | |
| 18:01 | for why ? Which will be the bottom of this | |
| 18:02 | parabola . The only way that can happen is if | |
| 18:05 | I stick a value of X in here , three | |
| 18:07 | units less than usual , negative three plus three is | |
| 18:11 | zero , which is squared . That's why it shifted | |
| 18:13 | to the left . And that process happens for every | |
| 18:15 | point on the parabola . So when you think of | |
| 18:17 | the problem as a curve traced out by all of | |
| 18:20 | these infinite points , right ? X comma Y they | |
| 18:24 | all end up being carbon copied and moved over because | |
| 18:27 | of that shift there . The x minus thing . | |
| 18:29 | Because as I feed numbers in here to have the | |
| 18:32 | same uh based , it's easier to think in terms | |
| 18:36 | of the origin zero comma zero point , you need | |
| 18:38 | to have it in this case three units smaller than | |
| 18:41 | zero in order to get uh negative three plus three | |
| 18:46 | giving you zero . Which will be the bottom of | |
| 18:47 | the problem . And of course the same thing happens | |
| 18:49 | for all the other points . So that's a lot | |
| 18:52 | of talking . Most books are just going to tell | |
| 18:54 | you if it looks like this , it shifted left | |
| 18:56 | if it looks like this shifted right but I want | |
| 18:58 | to give you a little bit more than that . | |
| 18:59 | So now I think we're ready to write down my | |
| 19:02 | little summary which is something I did . Um Before | |
| 19:08 | where did I run a space here ? Let me | |
| 19:10 | put it over here I guess . Um So I'm | |
| 19:13 | gonna do a little summary of horizontal shifting to a | |
| 19:16 | summary here . Now in all of these examples I | |
| 19:24 | chose I said well it's uh like x minus one | |
| 19:28 | quantity square . But you all know that there's another | |
| 19:30 | little part of problems . You can have a number | |
| 19:32 | in front of the X . Right ? It's a | |
| 19:34 | X squared . So in general , when the shape | |
| 19:37 | is a little more general than what I have here | |
| 19:39 | , for instance , the equation why is equal to | |
| 19:41 | A Parentheses , X -1 Quantity Squared . You shift | |
| 19:47 | the equation a X squared , right ? Mhm . | |
| 19:54 | One unit . See in these boards are just eliminated | |
| 19:58 | because I'm trying to make it easy for you now | |
| 20:00 | , I'm opening things up and I'm saying these problems | |
| 20:02 | can have another value in front of the X , | |
| 20:04 | which determines how narrow or wide the parabola is . | |
| 20:07 | But whatever this number is , you're just shifting the | |
| 20:09 | whole thing to the right one unit . Right , | |
| 20:12 | let's pick another one . Uh If you have a | |
| 20:18 | X -6 Quantity Squared , you're going to shift the | |
| 20:23 | equation , the base equation A . X . Squared | |
| 20:28 | . Right . one unit . Right . six units | |
| 20:34 | by six units . Because of course there's a six | |
| 20:36 | right there . And then lastly we're gonna go the | |
| 20:38 | other direction . We're gonna say why is equal to | |
| 20:40 | a X plus seven Quantity squared ? We're going to | |
| 20:45 | shift the curve A . X . Where left , | |
| 20:51 | How many units ? seven years , right . Same | |
| 20:54 | sort of thing . I need a negative seven in | |
| 20:56 | here just to get a zero to square . That | |
| 20:58 | means that the center of the problem is going to | |
| 21:00 | be shifted left seven units . So in general , | |
| 21:03 | the most general thing that you might see in a | |
| 21:06 | book is you might see something like this . Let | |
| 21:08 | me write it right under here . I have a | |
| 21:10 | little more room here in general . Yeah . Right | |
| 21:15 | . In general you're gonna see something like this . | |
| 21:17 | Why is equal to a X minus H quantity squared | |
| 21:22 | ? Right ? And then what you see usually under | |
| 21:25 | that is if A . Is greater than zero , | |
| 21:28 | not a . Sorry if H is greater than zero | |
| 21:32 | . Yes , H is greater than zero . You | |
| 21:34 | shift to the right . eight units shift right . | |
| 21:42 | H . Units . And exactly the same thing if | |
| 21:48 | H is less than zero shift left H units because | |
| 21:55 | if H is less than zero , H . Is | |
| 21:57 | negative . So if H is negative it's x minus | |
| 22:00 | and negative , which means it's X plus something . | |
| 22:02 | So when you see an X plus something here , | |
| 22:04 | you shifted to the left . When you see an | |
| 22:05 | X minus something , you shifted to the right . | |
| 22:07 | This is what you'll see in a book for shifting | |
| 22:11 | horizontally , right ? But usually they won't explain it | |
| 22:14 | or even tell you why it works or whatever . | |
| 22:16 | They'll just tell you memorize it . But now you | |
| 22:18 | can kind of see the cemetery of what's going on | |
| 22:20 | when you're shifting in the Y direction . You put | |
| 22:23 | your shift kind of in parentheses near the Y . | |
| 22:25 | Variable . If it's a minus sign , you're shifting | |
| 22:27 | the problem up . If it's a positive sign , | |
| 22:30 | your shifting that problem down for a for a horizontal | |
| 22:36 | shift , you write that shift next to the X | |
| 22:38 | . Variable . Again you have parentheses , it's got | |
| 22:40 | to be squared because it's a parabola , X . | |
| 22:42 | Has got to be squared . And then uh if | |
| 22:45 | this is bigger than zero , you're shifting to the | |
| 22:47 | right . If it's less than zero , you're shifting | |
| 22:48 | to the left . So if it's like a minus | |
| 22:50 | one , you shift to the right . If it's | |
| 22:51 | a like a uh minus four again you're shifting to | |
| 22:54 | the right , if there's a plus you're shifting to | |
| 22:56 | the left . And I tried to explain why that | |
| 22:59 | works by just thinking about the table of values . | |
| 23:01 | So what I want to do now is follow me | |
| 23:03 | onto the computer dima where I can play around with | |
| 23:05 | it even more and show you how the shifting really | |
| 23:07 | works . Okay , welcome back here , we have | |
| 23:11 | shifting quadratic equations horizontally , so right now I want | |
| 23:15 | you to focus on this side , I have the | |
| 23:17 | parabola , X squared y is equal to export graft | |
| 23:20 | . This is the same graph for the basic problem | |
| 23:22 | that we always uh do and then what's above here | |
| 23:26 | , ignore the one for right now it's X minus | |
| 23:29 | in this case zero , there's no shift at all | |
| 23:31 | . That's why the parable is sitting right here at | |
| 23:33 | the origin . Now if we make it X -1 | |
| 23:38 | , what happens ? Okay , first , ignore , | |
| 23:41 | ignore this for a second and let's take a look | |
| 23:43 | at the table of what's going on here . What's | |
| 23:46 | happening is uh Let me go back for a second | |
| 23:49 | and you can see a little clear right here this | |
| 23:51 | value is 20 . Let's go to the one value | |
| 23:53 | or the zero value . Let's go here , it's | |
| 23:55 | 000 . But when I move it over You can | |
| 23:58 | see that that zero point here I've highlighted in red | |
| 24:01 | has now shifted shifted down on the table . So | |
| 24:03 | now it's 1:00 . And if I go over It's | |
| 24:08 | 2:00 and if I go over again , you know | |
| 24:11 | four comma zero and so on . As I change | |
| 24:14 | that guy , you can see the problem dancing left | |
| 24:16 | and right and you can see this red zero shifting | |
| 24:19 | around and that's the zero point of the parabola . | |
| 24:22 | So when you have something like X plus two squared | |
| 24:25 | , it's a left hand shift . And now the | |
| 24:27 | zero point of the problem , the zero point of | |
| 24:29 | the problem is here , right ? Why ? Because | |
| 24:32 | I need to stick a negative two in here for | |
| 24:34 | X in order to give me a zero in order | |
| 24:36 | to give me a minimum of the proble way down | |
| 24:38 | over here . And so as you go deeper , | |
| 24:40 | negative same thing , I need a negative three in | |
| 24:42 | here to give me a zero to give me a | |
| 24:44 | square to give me a minimum of the proble . | |
| 24:46 | And so as I'm shifting the problem around , if | |
| 24:49 | you look at this red zero here , you'll see | |
| 24:51 | that minimum point that parabola dancing around . And that | |
| 24:55 | is why the shifting really happens . Um you can | |
| 24:57 | pick any other point on here . You want to | |
| 24:59 | let's go to x minus one squared and take a | |
| 25:02 | look at the value right above it . When you | |
| 25:03 | put a zero in , you get negative one square | |
| 25:06 | giving you positive one . And all of these are | |
| 25:09 | basically just plugging in the table of values directly into | |
| 25:11 | this equation as we said . So this whole shifting | |
| 25:15 | business happens for every one of these points . That's | |
| 25:17 | why the whole entire shape of the curve is what | |
| 25:19 | is transferred . Because the shifting doesn't happen for anyone | |
| 25:22 | point . It happens for any value of X . | |
| 25:24 | We stick in here . All right now one more | |
| 25:27 | thing I wanna talk to you about Notice over here | |
| 25:30 | , I have something like this . So I have | |
| 25:32 | X -1 quantity square . Where did I get this | |
| 25:34 | from ? Remember ? You already know how to take | |
| 25:38 | X minus one quantity squared and how to do foil | |
| 25:41 | on it . That's x minus one times X minus | |
| 25:43 | one . So F . O . I L . | |
| 25:45 | If you do the first Outside Inside last multiplication and | |
| 25:49 | collect terms , this is what you're gonna get . | |
| 25:51 | So these two things , these two equations are exactly | |
| 25:55 | the same thing , why is equal to x squared | |
| 25:57 | minus two , X plus one is exactly the same | |
| 26:00 | thing as why is equal to x minus one , | |
| 26:03 | quantity square . Because if I do the foil on | |
| 26:05 | this , this is what I get as an output | |
| 26:07 | here and you can see that that's happening . I've | |
| 26:10 | done the equation two ways because I want you to | |
| 26:12 | understand that basically when you see any parabola like this | |
| 26:16 | , if it has a square term like this right | |
| 26:18 | here , then you know , it's a parabola . | |
| 26:20 | But looking at it like this doesn't really give you | |
| 26:23 | any idea which way it shifted . I mean x | |
| 26:25 | squared minus four X plus four . It doesn't tell | |
| 26:27 | you at all that it shifted to units to the | |
| 26:29 | right . You basically can't tell by looking at this | |
| 26:32 | where this thing is shifted . It's very hard to | |
| 26:34 | tell , but you can tell very easily by looking | |
| 26:37 | at this version over here . Right ? So this | |
| 26:40 | version over here , when we're talking about shifting problems | |
| 26:43 | is usually what we focus on . But just keep | |
| 26:46 | in mind , you can do F . O . | |
| 26:47 | I L . And this is a totally equivalent way | |
| 26:49 | of writing the problem . It's just you can't see | |
| 26:51 | very easily how the thing is shifted . So we | |
| 26:53 | can go over and look at another one like X | |
| 26:54 | plus three or X plus . Uh Yeah let's do | |
| 26:57 | X plus three quantity squared . So you can see | |
| 26:59 | very easily that this has shifted three units to the | |
| 27:02 | left . But when you blow it out with the | |
| 27:03 | multiplication , this is what you get . And you | |
| 27:05 | can't tell which way this thing is shifted by looking | |
| 27:07 | at it , but just keep in mind , there's | |
| 27:09 | always two ways to write parable is you can write | |
| 27:11 | them all blown out and they're full blown multiplication . | |
| 27:13 | Uh And then you can also compact if I factor | |
| 27:16 | it and then this can tell you more readily which | |
| 27:18 | way the thing is shifted . The last thing I | |
| 27:21 | want to leave you with is also very important . | |
| 27:23 | This whole shifting business . Uh for this equation I | |
| 27:25 | have a one out here because remember the general form | |
| 27:28 | is a X squared . Now let's change it to | |
| 27:31 | make it a two , or let's change it to | |
| 27:33 | make it a three C . It makes it more | |
| 27:34 | and more narrow as I increase this guy . So | |
| 27:37 | let's land on four quantity X minus zero squares . | |
| 27:40 | Which means it's basically for X square . This is | |
| 27:42 | the table of values , but whenever I go to | |
| 27:46 | the right , that shape is what is shifted to | |
| 27:48 | the right . So the shape of the curve is | |
| 27:51 | really governed by the four that's out in front . | |
| 27:54 | The shifting is governed completely by is what is inside | |
| 27:57 | of the parentheses here . And if you take this | |
| 27:59 | X plus one quantity squared and do foil on it | |
| 28:02 | and then multiply by the four on the outside . | |
| 28:04 | This is the full blown equation that you get again | |
| 28:07 | , you can't tell which way the thing is shifted | |
| 28:09 | by looking at that you kind of have to use | |
| 28:11 | these other forms and that's why we're learning them or | |
| 28:13 | one of the reasons why we're learning them . But | |
| 28:15 | no matter what shape you pick , the shifting still | |
| 28:17 | happens exactly as you would expect and notice that zero | |
| 28:20 | point going up and down . In fact , even | |
| 28:22 | if I turn it upside down and make it like | |
| 28:24 | negative two X squared the minus two means that shape | |
| 28:28 | is shifted to the right and the mind and the | |
| 28:30 | you know the plus three or plus four means is | |
| 28:32 | shifted to the left . So all you have to | |
| 28:34 | do to worry about the shifting is figure out what's | |
| 28:36 | inside the parentheses . That tells you how it shifted | |
| 28:38 | the coefficient out in front tells you if it opens | |
| 28:40 | up or opens down and how narrow the thing is | |
| 28:43 | because you can see that it gets more narrow as | |
| 28:46 | we make as we change that coefficient . So now | |
| 28:48 | follow me onto the board , we're going to complete | |
| 28:50 | and conclude this lesson . Hello , welcome back . | |
| 28:54 | I hope you've enjoyed the computer demo . I like | |
| 28:56 | putting them together because it really can help solidify some | |
| 28:59 | difficult kind of concepts ahead of time . It's very | |
| 29:02 | very powerful I think for you to see how things | |
| 29:04 | move around by just dragging some sliders and you can | |
| 29:06 | see a bunch of equations that once there was one | |
| 29:08 | thing that I showed in the video demo or the | |
| 29:11 | computer demo that I want to point out here notice | |
| 29:14 | I want to show you one more thing that I | |
| 29:17 | did talk about in the computer demo and that is | |
| 29:19 | that if we're talking about these parables being shifted left | |
| 29:22 | and right , this is the best form to use | |
| 29:24 | because it's very easy to see how many units . | |
| 29:26 | The thing is shifted , but keep in mind that | |
| 29:28 | this is still a binomial square , this is a | |
| 29:31 | binomial squared and you can expand and multiply these guys | |
| 29:34 | out As follows . For instance , this would be | |
| 29:38 | X -4 times x minus four . And you know | |
| 29:41 | how to do this , multiplication , Just multiply it | |
| 29:43 | out and you get x squared minus this is four | |
| 29:46 | X . Outside terms to give you minus four X | |
| 29:50 | . And then I'm gonna four times negative four is | |
| 29:52 | positive 16 . And so , what you'll get is | |
| 29:54 | X squared minus eight X plus 16 . So , | |
| 29:58 | the equation why equals x squared minus eight X plus | |
| 30:01 | 16 . If you make a table of values and | |
| 30:03 | plot it , it is exactly the same thing as | |
| 30:06 | this . Why ? Because this is just a factor | |
| 30:08 | form of this . That's all right . But it | |
| 30:11 | is very hard to look at this thing . This | |
| 30:13 | X squared minus eight X plus 16 is basically impossible | |
| 30:16 | to look at it and off top your head . | |
| 30:17 | No , where that parable is shifted . But by | |
| 30:20 | having it in this form , it's very easy because | |
| 30:23 | when you factor it and get it into this exactly | |
| 30:26 | this . Easy to understand form . We already know | |
| 30:28 | that any time it's X minus something quantity squared , | |
| 30:31 | we just shift the basic shape to the right , | |
| 30:33 | right ? So sometimes you'll see parable is written like | |
| 30:36 | this and sometimes you'll see Paraiba was written like this | |
| 30:38 | . And I remember the first time I learned this | |
| 30:40 | stuff , I was like , why do we have | |
| 30:41 | different ways of writing stuff ? Well , the truth | |
| 30:43 | is they both represent the same thing , but writing | |
| 30:45 | it in this form , which is what we're focusing | |
| 30:47 | on in these lessons , it is much , much | |
| 30:49 | easier for figuring out where the thing is moved to | |
| 30:52 | . That's why we have different ways of writing things | |
| 30:54 | . So make sure you understand this , We have | |
| 30:56 | now covered the concept of shifting a parable horizontally . | |
| 30:59 | And in the previous lesson , we've covered the concept | |
| 31:01 | of shifting a parable of vertically . Now , in | |
| 31:04 | the next lesson , we're going to combine those two | |
| 31:05 | and shift a parabola anywhere in the xy plane . | |
| 31:08 | Using a vertical shift , combined with a horizontal shift | |
| 00:0-1 | . |
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07 - Graphing Parabolas in Vertex Form & Shifting Horizontally is a free educational video by Math and Science.
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