06 - Graphing Parabolas - Shifting Vertically (Quadratic Functions) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra and Jason with math | |
| 00:02 | and science dot com . Today , what we're gonna | |
| 00:04 | do is learn about graphing parabolas . Specifically we're gonna | |
| 00:07 | focus on shifting a parabola up and down . So | |
| 00:10 | I'm gonna give you the big picture and then dive | |
| 00:12 | into shifting these things up and down . So you | |
| 00:13 | understand what we're doing . We have the basic shape | |
| 00:16 | of a parabola , We talked about it in the | |
| 00:17 | last lesson and in fact for a very long time | |
| 00:19 | in algebra And this parabola is centered on the origin | |
| 00:24 | right here where the vertex of it is touching the | |
| 00:26 | 00 point in the origin . Well with a very | |
| 00:29 | slight modification to the equation of this parabola , we | |
| 00:32 | can move the parabola . It will still retain the | |
| 00:35 | complete shape . Everything is exactly the way it is | |
| 00:37 | . But we with a very simple change to the | |
| 00:40 | equation , we can move that parabola anywhere we want | |
| 00:43 | . It's almost like grabbing it with your hands , | |
| 00:45 | you can move the whole thing around uh in anywhere | |
| 00:49 | in the xy plane . Now , usually most books | |
| 00:51 | will give you the equation of a parabola in order | |
| 00:53 | to do that , where you can shift it along | |
| 00:55 | the X . Direction and then shift it along the | |
| 00:58 | Y . Direction and by shifting it in X . | |
| 01:00 | And y . You can move that parabola anywhere you | |
| 01:02 | want . But what I'm gonna do is break it | |
| 01:04 | up into two pieces . First we're gonna talk about | |
| 01:06 | just shifting this parabola up and down . What does | |
| 01:09 | it look like when we change the equation ? Just | |
| 01:12 | to move the problem up and down . Why do | |
| 01:14 | I start with that ? Because it's really easy to | |
| 01:15 | understand ? Okay then after that , in the next | |
| 01:18 | lesson we're gonna shift this parabola left and right , | |
| 01:20 | we're gonna grab it and just shift it along the | |
| 01:22 | X . Axis and you'll understand how that is accomplished | |
| 01:25 | . Then we're gonna put both of the two things | |
| 01:27 | together , shifting it along why and shifting it along | |
| 01:30 | X . And that's going to allow us to pull | |
| 01:32 | this parabola anywhere in the xy plane . And by | |
| 01:35 | the way this process applies to any function , right | |
| 01:38 | ? Any anything at all um cubic X . Cubed | |
| 01:42 | things . Later on we're gonna apply it to circles | |
| 01:44 | . We're gonna be graphing circles in the xy plane | |
| 01:46 | . We're going to use a similar idea to move | |
| 01:49 | that circle anywhere around the xy plane . So what | |
| 01:51 | you're learning here is not something just for parable as | |
| 01:54 | it's really used for any function . All right so | |
| 01:56 | let's go dive back into it . We're gonna be | |
| 01:58 | learning how to shift this problem only up and down | |
| 02:01 | in the y axis along the Y axis . So | |
| 02:04 | here we have , we talked about in the last | |
| 02:06 | lesson I last couple of lessons , I left it | |
| 02:08 | up on the board . We have a parabola , | |
| 02:09 | why is equal to X square ? We have a | |
| 02:11 | table of values and we plotted some points to get | |
| 02:13 | the general shape . All right . But don't forget | |
| 02:16 | that Parabolas are uh in general here I have X | |
| 02:19 | squared . This is the problem we've graphed here , | |
| 02:21 | but in general general , right . In general the | |
| 02:26 | actual equation of a parabola , the most basic form | |
| 02:29 | of this guy is really a times X squared . | |
| 02:32 | Remember we talked about this in the last lesson , | |
| 02:34 | the number A is just a number . It could | |
| 02:37 | be 1234 so on it could be negative one negative | |
| 02:40 | two negative three . It can be fractions as well | |
| 02:42 | , but this number controls the steepness of the problem | |
| 02:46 | . If A is a bigger number , this parabola | |
| 02:48 | closes in and becomes very narrow . If A is | |
| 02:51 | a low number , like a fraction like one half | |
| 02:54 | , then this parabola opens up like a flower . | |
| 02:57 | Okay ? And then if A becomes negative then the | |
| 02:59 | problem flips upside down . And as a as a | |
| 03:02 | frowny face down here . And as it gets deeper | |
| 03:04 | negative like negative four , negative five , negative six | |
| 03:07 | . Then the problem is very very closed off down | |
| 03:10 | downstairs . Now we've talked about that before , that's | |
| 03:12 | more of a review . We've done that in the | |
| 03:14 | past . This is the general equation of a problem | |
| 03:17 | . Now what we have graft here Is this equation | |
| 03:20 | where we have basically chosen that a . Is equal | |
| 03:23 | to one because there's an invisible A here that's not | |
| 03:26 | there because in this case is just one so you | |
| 03:28 | have to write it . So this is the case | |
| 03:30 | of that . So we're going to focus on learning | |
| 03:32 | how to do this for this . Very simple case | |
| 03:34 | when Y , f of X or y is equal | |
| 03:37 | to X squared . So here's the table of values | |
| 03:39 | running from negative three to positive three . And we | |
| 03:41 | have the corresponding outputs . I want you to remember | |
| 03:43 | these numbers 149 and then 149 In this direction . | |
| 03:48 | And the problem of course is centered at 0:00 . | |
| 03:51 | Now what we're gonna do is we're gonna make a | |
| 03:52 | very small change to this equation to shift this parable | |
| 03:56 | up in the Y direction and then in just a | |
| 03:58 | few minutes we'll be shifting it down . That change | |
| 04:01 | is so easy to understand . That's why I'm incorporating | |
| 04:04 | it first . Let's take a look , let's change | |
| 04:07 | this equation . This triangle means change . You're gonna | |
| 04:10 | learn that in calculus . So it's my little shorthand | |
| 04:13 | . Let's change this , Change this equation to the | |
| 04:17 | following . Instead of F . F X is equal | |
| 04:19 | to x square , which is what we graphed over | |
| 04:21 | there . Let's change it to y . Is equal | |
| 04:24 | to X square . This is the basic shape of | |
| 04:27 | what we've graph . But we're gonna make one small | |
| 04:28 | change , we're gonna add one to it and that's | |
| 04:31 | gonna end up shifting this graph , right ? So | |
| 04:34 | let's see how that's gonna happen . Let's draw another | |
| 04:36 | table of values . So we have to have inputs | |
| 04:39 | on the X . And outputs on the Y . | |
| 04:41 | Like this and I want to use the same values | |
| 04:44 | here . So I'm gonna go negative three , negative | |
| 04:46 | two , negative 1012 and three . And let's see | |
| 04:51 | what's gonna happen . The equation is now X squared | |
| 04:55 | plus one . So you see what's really going on | |
| 04:57 | here is we're taking essentially the basic curve that we've | |
| 05:00 | already graphed the X square curve and whatever values we | |
| 05:03 | calculate for why as we stick these numbers and we're | |
| 05:06 | going to calculate the y . Value . But before | |
| 05:08 | we assign it we're gonna just add one . So | |
| 05:11 | what's gonna end up happening is it's gonna be a | |
| 05:13 | carbon copy of this entire table . It's just that | |
| 05:16 | every one of these output values for why has won | |
| 05:19 | added to it because we're just adding one on the | |
| 05:21 | outside . So let's go and and do it here | |
| 05:24 | . If we have uh negative three square , that's | |
| 05:26 | gonna be 99 plus one is 10 . So 10 | |
| 05:29 | goes here actually , I'm gonna um use a different | |
| 05:32 | colour . 10 goes here negative two squared , that's | |
| 05:35 | four plus one is five , negative one squared . | |
| 05:38 | That's one plus one is 20 squared plus one is | |
| 05:41 | one . And then one squared plus was one . | |
| 05:44 | Then you add one to it , you get to | |
| 05:46 | and you can see the pattern . You're gonna have | |
| 05:47 | a five and a 10 here because if you put | |
| 05:49 | that three in here , it's nine plus one is | |
| 05:51 | 10 . So the numbers that you get for this | |
| 05:53 | equation uh is no longer the same numbers that we | |
| 05:57 | started with over there , we have a one kind | |
| 06:00 | of this is a mirror image kind of . You | |
| 06:01 | can see the symmetry here because we have these guys | |
| 06:05 | are symmetric with one another . These guys are symmetric | |
| 06:08 | with one another , and these guys are symmetric with | |
| 06:10 | one another . Notice we have the same symmetry here | |
| 06:13 | . So these were cemetery symmetric , symmetric , symmetric | |
| 06:16 | centered about this kind of this mirror image kind of | |
| 06:19 | the center point here in the center , we still | |
| 06:21 | have a mirror image , but the numbers are actually | |
| 06:23 | just different now . So when we plot this thing | |
| 06:26 | , what are we going to get ? Let me | |
| 06:29 | go ahead and just do a sketch of it right | |
| 06:30 | here . What do you think we're gonna get if | |
| 06:33 | you notice what's going on , all that's happened because | |
| 06:36 | we added to one is every one of these output | |
| 06:38 | values just has a one added to it . So | |
| 06:41 | if you can visualize your basic curve , the one | |
| 06:43 | that we graft a long time ago , every one | |
| 06:45 | of these points is exactly kind of relatively where it | |
| 06:48 | should be . It's just there's a one attitude . | |
| 06:50 | So for instance , this point Is going to be | |
| 06:52 | shifted one up because instead of zero comma zero , | |
| 06:57 | it's now zero comma one . So the bottom of | |
| 07:00 | the Parabola actually falls one unit higher right there and | |
| 07:04 | everything happens to all the other points . These points | |
| 07:06 | are shifted up , these points are shifted up by | |
| 07:09 | one unit and these points are shifted up by one | |
| 07:11 | unit . So what ends up happening is if I | |
| 07:13 | am not going to sit here and plot every little | |
| 07:15 | point , I mostly want to sketch things for you | |
| 07:17 | . But this parabola is going to have the same | |
| 07:19 | exact shape as the one on the left hand board | |
| 07:22 | . It's gonna except it's going to be shifted up | |
| 07:24 | exactly one unit because this is one unit up , | |
| 07:29 | so it's gonna be shifted up one unit . So | |
| 07:36 | the bottom line is when you take a quote unquote | |
| 07:39 | basic curve , I always told you I want you | |
| 07:41 | to think of the X . Square . Why is | |
| 07:43 | equal to X square ? That parabola is the basic | |
| 07:46 | shape of a problem . It's the most basic one | |
| 07:48 | you can have . Burn it in your mind . | |
| 07:50 | I told you remember this is why because that basic | |
| 07:52 | shape instead of looking at this equation of some crazy | |
| 07:55 | equation , this is how I want you to think | |
| 07:56 | about it . This is the basic shape of the | |
| 07:58 | thing and I'm just adding one to every single output | |
| 08:02 | of this function here really ? The Y values , | |
| 08:04 | I'm adding one to him . So I take everything | |
| 08:06 | in this table , I add one and that because | |
| 08:09 | of that shifts the whole graph up because every one | |
| 08:11 | of those points are in the same locations are just | |
| 08:14 | shifted up one unit . All right now , what | |
| 08:18 | do you think is going to happen ? Yes . | |
| 08:20 | When we shift it by some other number other than | |
| 08:24 | one ? Right , let's go and do that . | |
| 08:28 | And let's change this equation again . Let's make it | |
| 08:32 | . Why is equal to X squared plus drum roll | |
| 08:35 | , please let's make it a plus three . What | |
| 08:37 | do you think is gonna happen ? Well what's gonna | |
| 08:40 | happen is we're gonna run it through a table . | |
| 08:43 | We're going to basically get these base values for the | |
| 08:45 | X . Squared but instead of adding one to it | |
| 08:48 | we're gonna add three to it . And so what's | |
| 08:50 | gonna end up happening is that entire Parabola is going | |
| 08:52 | to be shifted up three units . So the bottom | |
| 08:55 | line is to shift really any curve . But we're | |
| 08:58 | talking now about Parabolas up along the Y axis , | |
| 09:01 | you just add however many units you want on the | |
| 09:03 | right hand side and that's going to shift all the | |
| 09:05 | values up an equal amount . So just because this | |
| 09:08 | is probably the first time you're you're seeing this , | |
| 09:11 | I want to make sure everybody's on the same page | |
| 09:13 | . So let's go X . And Y . Is | |
| 09:15 | equal to X squared plus three . And we're gonna | |
| 09:18 | just do it real quick because it only takes a | |
| 09:19 | second . So negative three , negative two , negative | |
| 09:22 | 10123 And now we're gonna go a little faster because | |
| 09:25 | you understand what's really going on every one of these | |
| 09:28 | points . When we put in a negative three we | |
| 09:31 | get negative three square we've got a nine . But | |
| 09:34 | now we're gonna be adding three to that . Nine | |
| 09:36 | plus three is 12 . So what's gonna happen is | |
| 09:38 | you're gonna have a 12 a value of 12 right | |
| 09:40 | here . And this is gonna be a seven and | |
| 09:42 | this is gonna be a 4 to 3 of four | |
| 09:46 | . We have a three or four . This is | |
| 09:48 | gonna be a mirror image here with seven and 12 | |
| 09:50 | . So you see it's the same sort of thing | |
| 09:52 | . We still have the symmetry . This one goes | |
| 09:55 | with this one , this one matches with this one | |
| 09:57 | . So the shape of the curve remains intact and | |
| 10:00 | it's the same sort of thing when you put a | |
| 10:01 | two in , we're squaring it , that's four plus | |
| 10:03 | three is seven . one goes in that square , | |
| 10:06 | that's one plus the three here gives me the four | |
| 10:09 | and so on . And so when you graph this | |
| 10:11 | guy , what you think is gonna happen 123 you're | |
| 10:15 | shifting up three units up like this . And then | |
| 10:18 | the graph of it is basically going to be the | |
| 10:20 | exact same shape . You're not adjusting the shape of | |
| 10:23 | the parabola . It's the same shape as the basic | |
| 10:26 | parabola that you have here , but it's now shifted | |
| 10:29 | up three units . All right . And so I | |
| 10:32 | want to do one in the opposite direction . So | |
| 10:34 | you understand what if you have the equation why is | |
| 10:37 | equal to x squared minus two ? So here we're | |
| 10:41 | not shifting up any number of units . When you | |
| 10:44 | have a minus sign over there , you're shifting them | |
| 10:46 | down . Why is that the case ? Because the | |
| 10:48 | basic curve is X squared , which are these values | |
| 10:51 | When you subtract two ? I'm taking every output and | |
| 10:54 | I'm moving them down . So in other words , | |
| 10:56 | this center point instead of being at zero will be | |
| 10:59 | shifted down two units . These points will be shifted | |
| 11:02 | down two units . These points will be shifted down | |
| 11:04 | to you and so on . So the curve will | |
| 11:06 | look exactly the same but it will be shifted down | |
| 11:09 | those number of units . So just to make sure | |
| 11:12 | we're all on the same page , why is equal | |
| 11:13 | to x squared -2 ? What values do we get | |
| 11:18 | ? All right , well we have let's go negative | |
| 11:21 | three negative two negative 1012 and three . And then | |
| 11:26 | we have what let's just do the first one . | |
| 11:28 | So this will be 93 negative three scores 99 minus | |
| 11:31 | two is a seven . And if you go through | |
| 11:34 | it you're gonna get a to hear a negative one | |
| 11:36 | here , a negative to here , A negative one | |
| 11:40 | here , A two and a seven . Now , | |
| 11:44 | make sure you understand what's going on here ? You | |
| 11:46 | might say , well where's the center point ? Where | |
| 11:48 | is the cemetery going on here first ? Let's make | |
| 11:49 | sure the numbers are correct . When you put a | |
| 11:51 | zero in here , that zero square , zero minus | |
| 11:53 | two gives you negative to a one goes in , | |
| 11:56 | that's going to give you one minus two , gives | |
| 11:58 | you a negative one . A two goes in two | |
| 12:01 | squared is 44 minus two is two . So all | |
| 12:03 | the numbers are correct , three squared is nine minus | |
| 12:05 | two is seven . So where's the symmetry in this | |
| 12:08 | case ? The symmetry here is going to be these | |
| 12:13 | numbers are going to be symmetric . These numbers are | |
| 12:15 | gonna be symmetric and these numbers are gonna be symmetric | |
| 12:17 | . So you can see you still have symmetry . | |
| 12:19 | But the whole thing is shifted down because all the | |
| 12:21 | numbers are shifted down from before . So whenever you | |
| 12:25 | draw this guy , you have an X . Uh | |
| 12:30 | y . If I can draw X and Y correctly | |
| 12:33 | and you're gonna have 12 units down the Y axis | |
| 12:38 | , same exact shape . Nothing is different other than | |
| 12:43 | the fact that it's pulled down like that . So | |
| 12:46 | so far we've taken the basic shape of the problem | |
| 12:48 | which is X . Squared . And we can shift | |
| 12:50 | it up by adding numbers to the right hand side | |
| 12:53 | of the equation . The shift all of the values | |
| 12:55 | equally up . We can subtract values that shift all | |
| 12:58 | of the output . Why values equally down . Which | |
| 13:01 | means the thing is only gonna go up and down | |
| 13:02 | in the Y . Direction . Okay now I need | |
| 13:05 | to do a summary here but when we do the | |
| 13:07 | summary after the summary we're gonna go off to the | |
| 13:10 | computer and I'm gonna show you a little demo that's | |
| 13:12 | gonna make it even more rock solid in your mind | |
| 13:16 | . But let me just make sure we're on the | |
| 13:18 | same page . This a shift up three units shift | |
| 13:26 | down . That's down two units . All right . | |
| 13:32 | So let's summarize summary . Right now we have to | |
| 13:39 | generalize things a little bit . If you have the | |
| 13:41 | equation why is equal to a X . Squared C | |
| 13:47 | . In this example we had just set equal to | |
| 13:49 | one to make it the most basic probability that you | |
| 13:51 | have . But it applies to any problem of any | |
| 13:54 | shape if you have , why is equal to three | |
| 13:58 | X squared ? The only difference will be the problem | |
| 14:00 | will be very steep because remember what is in front | |
| 14:02 | of the X . Squared controls the steepness or the | |
| 14:05 | how closed off the parabola is . So if you | |
| 14:07 | start with any parabola with any number out in front | |
| 14:10 | of the X squared , then basically you have no | |
| 14:13 | shift this problem will be centered on the origin , | |
| 14:17 | the vertex go down and touch the origin . But | |
| 14:19 | if you have the problem , why is equal to | |
| 14:21 | a X squared Plus 1 ? You're just adding one | |
| 14:25 | to whatever . Probably you started with and you shift | |
| 14:29 | up one unit . Now , sometimes oftentimes this has | |
| 14:38 | represented a different way . I introduced it to you | |
| 14:41 | this way with the plus one on the other side | |
| 14:42 | because it makes it very easy for you to understand | |
| 14:44 | . But oftentimes and books , it's written like this | |
| 14:47 | . If you take this one and you subtract it | |
| 14:50 | , moving it over to the other side , what | |
| 14:52 | you get is why minus one equals X squared . | |
| 14:56 | This equation is exactly the same as this one . | |
| 14:59 | So in your book , if you see The -1 | |
| 15:03 | on the left , a lot of times people are | |
| 15:04 | like , what does that mean ? Well it's written | |
| 15:07 | that way for a reason . I will get to | |
| 15:08 | in just a second . But mentally in your mind | |
| 15:11 | you can imagine that one going to the other side | |
| 15:13 | meaning you're going to shift the thing up , let | |
| 15:14 | me get the rest of the board in place and | |
| 15:16 | I'll go back and talk a little bit about this | |
| 15:18 | minus sign for now . Just know you can move | |
| 15:20 | the one over and ride it like that . If | |
| 15:23 | you have y is equal to X squared some parabola | |
| 15:26 | plus some other number like a three . Then you | |
| 15:29 | shift shift sorry up three units . And again this | |
| 15:39 | can be written instead of writing the three on the | |
| 15:41 | right hand side if you like , you can move | |
| 15:43 | it to the left hand side , which a lot | |
| 15:44 | of books do . So I want I'm gonna get | |
| 15:46 | you used to this . All I've done is take | |
| 15:47 | that three , move it over . This is exactly | |
| 15:50 | representing the same thing as this . It is a | |
| 15:51 | parabola shifted up three units . Yeah . And finally | |
| 15:56 | let's go in the other direction . What if you | |
| 15:58 | have why is equal to X squared -2 ? This | |
| 16:01 | is a downshift , this is shifting the whole problem | |
| 16:04 | down shift down two units . All right . And | |
| 16:12 | then you can take that too and you can move | |
| 16:15 | it over by adding to to both sides . And | |
| 16:17 | then it'll be y plus two is equal to a | |
| 16:20 | X squared . So you might see the problem with | |
| 16:22 | the shift instead of written on the right hand side | |
| 16:24 | , it might be written on the left hand side | |
| 16:26 | with a plus . So when you have this guy | |
| 16:27 | has shifted down two units . So I need to | |
| 16:30 | talk a little bit about this before we leave the | |
| 16:32 | topic . All right . Yeah . Why do we | |
| 16:35 | move and write the shift over here next to the | |
| 16:38 | Y . Right . I've tried to introduce you with | |
| 16:40 | tables a table of values to show you that when | |
| 16:43 | you have a shift here , you're just adding numbers | |
| 16:45 | to the to the Y values . And it's very | |
| 16:47 | easy if you add one , you shift the whole | |
| 16:49 | thing up . If you subtract two , you shift | |
| 16:51 | the whole thing down . But usually or I shouldn't | |
| 16:55 | say . Usually oftentimes we write equations like this where | |
| 16:58 | the shift is written on the other side of the | |
| 17:00 | equal sign , but it gets confusing because this is | |
| 17:03 | why -1 , but yet it means you're shifting up | |
| 17:07 | one unit . So when people see why -1 , | |
| 17:09 | it appears that you're shifting a thing down because it | |
| 17:13 | has a minus sign there . But actually it means | |
| 17:15 | you're shifting it up , so it's kind of opposite | |
| 17:18 | . So you might say well why do I even | |
| 17:19 | write it like this ? Why am I putting it | |
| 17:21 | over here ? That's confusing . It's because later on | |
| 17:23 | when we get to the equation of a circle can't | |
| 17:26 | get too much ahead of myself . But when we | |
| 17:28 | get to the equation of a circle , this is | |
| 17:29 | going to be the only way that is easy for | |
| 17:31 | us to write the equation of a circle down . | |
| 17:33 | So the reason we put the numbers over here is | |
| 17:35 | because this number it's written next to the Y . | |
| 17:38 | And it reminds us a shift in the Y direction | |
| 17:41 | . This number is written next to the Y . | |
| 17:43 | So it reminds us is three units shift in the | |
| 17:45 | Y direction . This number is written next to the | |
| 17:48 | Y . So it reminds us that it's a two | |
| 17:49 | unit shift in the Y . Direction . But when | |
| 17:52 | you see the shifts written down next to the variable | |
| 17:54 | , you have to remember that it shifted in the | |
| 17:57 | opposite direction . Why minus means shift up ? Three | |
| 18:01 | minus means shift up . Why ? Plus two means | |
| 18:04 | shift down . So in your mind you have to | |
| 18:06 | think if the shift is written next to the variable | |
| 18:09 | on the left hand side of the equal sign , | |
| 18:11 | it's shifted that many units but opposite to the sign | |
| 18:14 | that's kind of written there , right ? And if | |
| 18:16 | the shift is written on the other side , like | |
| 18:18 | I've kind of introduce things that's actually more easy to | |
| 18:20 | understand . It shifted up and down with the same | |
| 18:23 | side . So in your book sometimes I'll write it | |
| 18:25 | like this and they'll say how many units and what | |
| 18:28 | direction is this problem shifted and you have to look | |
| 18:30 | at the plus and say , well that means it's | |
| 18:31 | shifted down mentally . You can move the to to | |
| 18:34 | the other side , make it a minus sign , | |
| 18:35 | whatever is easier for you . But that is why | |
| 18:38 | it's written that way . All right . So then | |
| 18:40 | now that we have that out of the way , | |
| 18:42 | I can write these are number , these are shifts | |
| 18:45 | with certain numbers . I can write in general , | |
| 18:50 | This is what you would typically see in a book | |
| 18:52 | . Right . In general , the following is true | |
| 18:55 | . If I have why -1 is equal to a | |
| 18:59 | X square where K is just some number ? It | |
| 19:01 | could be 234 whatever . If K is greater than | |
| 19:05 | zero , then you shift up K units . Right | |
| 19:14 | ? But if K is less than zero , you | |
| 19:18 | shift yes , down K units . Yes . All | |
| 19:26 | right . So let's think about this in terms of | |
| 19:28 | what we just talked about a second ago . If | |
| 19:31 | you have let's pick K is equal to one . | |
| 19:33 | K is now one . That's a positive number . | |
| 19:35 | So it's why -1 . And I just told you | |
| 19:38 | when it's written over here , you have to if | |
| 19:40 | it's a minus sign , it actually shifted up its | |
| 19:42 | opposite kind of in your mind of what the signs | |
| 19:45 | kind of lead you to believe . So it's shifted | |
| 19:48 | up one unit . But if K happens to be | |
| 19:50 | negative , let's pick K to be negative to negative | |
| 19:53 | two . Right means why minus a negative two ? | |
| 19:57 | Which means why plus two . So , if you | |
| 19:59 | ever see why plus anything , it's always a shift | |
| 20:01 | uh in the down direction , which means is exactly | |
| 20:04 | what I've written it here . So this kind of | |
| 20:06 | thing is the math gobbledygook that you'll see in most | |
| 20:09 | textbooks , they'll write it like this and I have | |
| 20:11 | these little cases but those are useful and that is | |
| 20:14 | mathematically what it is all true . But it's much | |
| 20:17 | easier to think of it in terms of numbers , | |
| 20:19 | actual numbers . So if you have minus a number | |
| 20:22 | , you shift that problem up that many units . | |
| 20:25 | If you have plus a number next to the y | |
| 20:26 | you shift it down that many units . That's if | |
| 20:29 | the numbers are written on the left hand side of | |
| 20:31 | the equal sign like this after on the other side | |
| 20:33 | of the equal sign , then you just follow the | |
| 20:34 | signs as they're written . So now what I'd like | |
| 20:37 | to do is go off to the computer and show | |
| 20:39 | you more interactively how these shifts work and I can | |
| 20:42 | do a lot more examples , a lot more quickly | |
| 20:44 | when I have the computer going . So let's walk | |
| 20:45 | over there and do that right now . All right | |
| 20:48 | , welcome back . So here we are at the | |
| 20:50 | basic Parabola . I want you to ignore this right | |
| 20:53 | now . I want you to ignore what's written on | |
| 20:55 | top of the axis here . I want you to | |
| 20:56 | look over here . We're graphing the equation why is | |
| 20:59 | equal to X squared . This is the basic problem | |
| 21:02 | . Have been graphing forever . It goes and touches | |
| 21:04 | the origin . And here's the table of values so | |
| 21:07 | you can see that . It's correct . So when | |
| 21:08 | you square this , you get a one . When | |
| 21:10 | you square this , you get a four , you | |
| 21:11 | square the negative three . You get a nine here | |
| 21:13 | . I'm going up All the way to positive and | |
| 21:15 | negative five , squaring it , getting 25 on both | |
| 21:19 | sides . So you can see it's symmetric here . | |
| 21:21 | Now I want to increase this by one and I | |
| 21:25 | don't want you to look at this yet . I | |
| 21:26 | want you to look over here . I have X | |
| 21:27 | squared plus one . You see what's happened . All | |
| 21:30 | I've done is I've added one value to the output | |
| 21:33 | to the y value of my table of values . | |
| 21:35 | And that has shifted the graph up by one , | |
| 21:37 | right ? If I uh increased by two , all | |
| 21:40 | I've done is I've added two values . So this | |
| 21:42 | was 25 before And so on . Originally it was | |
| 21:47 | 16 for the second slot here , but when I | |
| 21:49 | shifted up by two units I get 18 . So | |
| 21:54 | every output here is now added has a number two | |
| 21:56 | added to it . And that is what causes the | |
| 21:58 | curve to go and shift upward . Now I want | |
| 22:01 | you to turn your attention . This is if the | |
| 22:03 | shift is written on the right hand side , you | |
| 22:04 | can pull the two over and you can see how | |
| 22:07 | it's written in terms of uh I don't want to | |
| 22:10 | get into the terminology right now , but when the | |
| 22:12 | when the shift is written on the left hand side | |
| 22:14 | , you just move that to over . So when | |
| 22:16 | it's a minus sign here , you're actually shifting it | |
| 22:18 | up -3 , shifts it up . Uh and of | |
| 22:21 | course this is off the screen there . Uh and | |
| 22:24 | then when -1 is again shifted up and then when | |
| 22:26 | you go back to zero , that means it's not | |
| 22:28 | even there at all . It's just a basic problem | |
| 22:30 | . When you go in the negative direction , you | |
| 22:32 | can see the same thing . When you have a | |
| 22:33 | plus sign as a shift written on the left hand | |
| 22:35 | side , you're shifting the thing down and you can | |
| 22:38 | see what's happening here . When you go back to | |
| 22:39 | the basic problem uh X squared notice we have a | |
| 22:42 | 25 up here , When I go and shift down | |
| 22:45 | one unit . All I'm doing is is subtracting one | |
| 22:47 | from every output . So now I have a 24 | |
| 22:49 | and everything in this column is Basically has a one | |
| 22:52 | subtracted . And of course the I can see the | |
| 22:55 | shift here is a -1 on the right hand side | |
| 22:57 | , so I'm going down but I can move that | |
| 22:59 | one to the left , making why plus one . | |
| 23:03 | And so you need to get in your mind that | |
| 23:04 | when you see these shifts written next to the variable | |
| 23:07 | plus means down minus means up . And as I | |
| 23:10 | go deeper and deeper into negative territory here , plus | |
| 23:13 | three is shifted down three units . And my table | |
| 23:15 | of values reflects that . Now there's one more thing | |
| 23:17 | I want to show you before I close and that | |
| 23:19 | is the following thing . If I change this equation | |
| 23:22 | so that it's not just why is equal to X | |
| 23:24 | squared . It's it's why is equal to two X | |
| 23:26 | squared . Or maybe I changed this equation . So | |
| 23:28 | that is why is equal to four X squared . | |
| 23:30 | You see the table of values updates to reflect . | |
| 23:32 | But the same thing happens with the shifting right . | |
| 23:35 | If I shift up two units like this , all | |
| 23:38 | I've done is add two units to the output , | |
| 23:40 | I've shifted the same exact shape of the curve . | |
| 23:42 | Notice that the shape of this curve is very narrow | |
| 23:45 | . We can crank it up even further and see | |
| 23:47 | that it's really narrow . But any time I have | |
| 23:49 | a shift here of a minus sign over here and | |
| 23:52 | why minus something ? It shifts up that many of | |
| 23:54 | units . And then when it's why plus something I | |
| 23:56 | shift down that many units . Doesn't matter what the | |
| 23:59 | shape of the curve is . Even if I go | |
| 24:02 | upside down let's make it negative four X squared . | |
| 24:05 | Uh This is why is equal to negative four X | |
| 24:07 | squared . So that's an upside down parabola and it's | |
| 24:09 | very narrow whenever I add numbers to it like this | |
| 24:13 | uh then I shift the thing up and of course | |
| 24:15 | I can see it over here is a y minus | |
| 24:18 | in the shift and then of course I can go | |
| 24:20 | the other direction . Why ? Plus in the shift | |
| 24:22 | ? So the thing I want you to really understand | |
| 24:24 | the most of all of this stuff is any parabola | |
| 24:27 | no matter what it shape right ? If you see | |
| 24:31 | the shift written next to the variable like why minus | |
| 24:34 | like this it's shifted up when you see a value | |
| 24:37 | why Plus like this next to the variable , it | |
| 24:39 | shifted down . If you want to mentally move it | |
| 24:41 | to the other side of the equal sign , then | |
| 24:43 | you can see exactly how it affects the table of | |
| 24:45 | values a little more clearly . So now follow me | |
| 24:47 | back to the board where we're going to wrap up | |
| 24:49 | the lesson . Yeah . All right . I hope | |
| 24:51 | you enjoyed the computer demo . It's very easy and | |
| 24:54 | interactive to see how things change . And I put | |
| 24:56 | that table of values in there to show you there's | |
| 24:58 | nothing magical happening with the shifting of parabolas . A | |
| 25:01 | lot of students will just memorize that how to shift | |
| 25:04 | them up and ship them down , but they don't | |
| 25:05 | realize why it's happening . It's basically happening because the | |
| 25:08 | basic shape of a parabola is governed by this . | |
| 25:11 | Here we chose A is equal to one , but | |
| 25:13 | it could be two X squared or three X squared | |
| 25:15 | . It's going to change the basic shape of this | |
| 25:16 | parabola here , here you have a X squared plus | |
| 25:19 | zero invisible right here and that means there's no shift | |
| 25:22 | at all . But when you add one to it | |
| 25:25 | , you shift the thing up . When you add | |
| 25:27 | three to it , you shift it up three units | |
| 25:29 | . When you subtract numbers on the right hand side | |
| 25:31 | , you shift it down . But for all of | |
| 25:33 | those cases , even though we can write it like | |
| 25:35 | this oftentimes in books , you'll see the shift written | |
| 25:38 | over next to the variable and that is so you | |
| 25:41 | can say , okay , the Y direction has shifted | |
| 25:43 | this many units , the Y direction shifted this many | |
| 25:45 | units , the Y direction is shifted this many units | |
| 25:48 | by putting them together , it's easy to see which | |
| 25:51 | which access is being shifted there . But you have | |
| 25:54 | to remember a minus sign means shift up in a | |
| 25:57 | positive sign , means shift down so make sure you | |
| 26:00 | understand this and then go on to me with me | |
| 26:03 | to the next lesson where we're going to no longer | |
| 26:05 | talk about shifting in the Y . Direction direction , | |
| 26:07 | We're gonna grab that parabola and shift it left and | |
| 26:10 | right in the X . Direction . So follow me | |
| 26:12 | and we'll do that right now . |
Summarizer
DESCRIPTION:
Quality Math And Science Videos that feature step-by-step example problems!
OVERVIEW:
06 - Graphing Parabolas - Shifting Vertically (Quadratic Functions) is a free educational video by Math and Science.
This page not only allows students and teachers view 06 - Graphing Parabolas - Shifting Vertically (Quadratic Functions) videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.
GRADES:
STANDARDS:
