06 - Graphing Parabolas - Shifting Vertically (Quadratic Functions) - Free Educational videos for Students in K-12

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 .

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