08 - Graphing Parabolas in Vertex Form & Shifting Horizontally and Vertically - Free Educational videos for Students in K-12

08 - Graphing Parabolas in Vertex Form & Shifting Horizontally and Vertically - By Math and Science

Transcript


00:00	Hello , Welcome back to algebra . I'm Jason with
00:02	math and science dot com . Today we're going to
00:04	be talking about the concept of shifting a parabola in
00:07	the xy plane vertically and horizontally at the same time
00:11	. A couple of things . I want to stay
00:12	here in the beginning of the lesson , number one
00:14	, we've already covered vertical shift , only vertical shifts
00:18	of parables in the previous lesson , We've also already
00:21	covered horizontal shifts of parabolas in the previous listen to
00:25	this one . So if you have not watched those
00:27	two lessons , you have to watch them before this
00:29	one . Otherwise it won't make a lot of sense
00:31	. So we're going to be combining vertical and horizontal
00:33	shifts together in one equation for one shift of a
00:36	probable pretty much anywhere in the xy plane . Second
00:39	thing is right at the beginning I have a computer
00:41	demo that's going to I think solidify and make things
00:43	a lot , a lot easier to understand or visualize
00:46	how these shifting , how the shifting business works .
00:49	But if at any point you're unsure like why is
00:51	it shifted here ? Why is it shifted there ?
00:53	That just means you haven't watched the previous lesson .
00:55	So please do that . And then if you haven't
00:57	already and then come back to this lesson and catch
00:59	up with me here . So first we have to
01:01	talk about Parabolas in vertex form . I have kind
01:04	of avoided talking about that because we haven't quite got
01:07	there yet . But basically we all know that you
01:09	can write parabolas and what we call standard form ,
01:11	that's that's fully blown out . Or you can call
01:14	it expanded form , fully blown up version of a
01:17	parabola for instance , you can have a parabola ,
01:19	I'll go over here , I'm just gonna make one
01:22	up off the top of my head , X squared
01:24	plus two X plus five . You know that's a
01:27	parabola , How do you know ? Well because it's
01:29	got an X . Squared term , that's all ,
01:30	that's all it needs . So this is going to
01:32	be a parabola shape . Somewhere in the xy plane
01:36	, might be shifted here , might be shifted up
01:38	, might be shifted over there . You can't really
01:40	tell from looking at this equation , but you know
01:42	that it's a parabola just because it has an X
01:44	squared term in it . Right ? So that's what
01:46	we called standard form or you can also call it
01:49	expanded form . So I call it standard for most
01:52	books , call it standard form of a parabola or
01:55	also expanded form . But there's another form that I've
01:58	kind of introduced you to without really realizing it .
02:00	It's called the vertex form of a problem . It
02:03	represents the exact same shape of a curve as something
02:07	in standard form , but it's much easier to figure
02:10	out where the thing is because it's easy to read
02:13	the vertex of the problem off of the exact off
02:16	of the equation . So it's called vertex form .
02:19	I'm gonna write that here . So we call this
02:21	a parabola in vertex form . All right . And
02:31	I'm gonna give I'm gonna give it to you in
02:33	its full glory and then we're gonna talk about it
02:34	and we'll do the computer demo . So in vertex
02:37	form might look something like this . Why minus some
02:40	number K . Remember we used K before is equal
02:44	to A . This is the number that tells you
02:46	how big how open and close the problem is we've
02:48	talked about that before . X minus H quantity square
02:52	. The H . Tells you how many units .
02:54	The thing is shifted in the X . Direction .
02:56	The K . Tells you how many units the things
02:58	shifted in the Y direction . The A . Tells
03:00	you how big or small that bigger A . Is
03:03	the more closed up . The parabola is the smaller
03:05	A . Is the more open it up . It
03:07	is like a flower , so to speak right ?
03:10	Um And we also have another . I'm gonna skip
03:12	down a little bit and and tell you that there's
03:14	a different a slightly different way to write this .
03:17	You could write it like this first . I'm gonna
03:19	put the equal sign in the same place . I'm
03:21	gonna say a quantity X minus H . Squared .
03:25	I can take this K . And I can move
03:27	it to the other side by addition by adding Kay
03:29	so you're gonna have a plus K . This equation
03:32	and this equation are exactly the same equation . All
03:35	that's happened is the K . Is now on this
03:36	side instead of on this side . Some books will
03:39	tell you that this is the vertex form of a
03:42	parabola . You can still read how many units the
03:46	parable is shifted an X . By this , it's
03:48	eight units and you can still read how many units
03:51	and K . The thing is shifted in the vertical
03:53	direction from here right ? By having everything on the
03:56	right hand side of the equal sign . A lot
03:58	of books will will do it that way . So
03:59	this is gonna be K units up and and H
04:02	units to the right . But again a lot of
04:04	other books keep the Y shift with the Y .
04:07	Variable and the X shift with the expert just keep
04:09	in mind that you might see it two different ways
04:12	. So what does this K mean ? So this
04:14	guy means the following thing shift . The basic form
04:18	of a parabola . Uh up K . Units And
04:24	this is assuming K is positive . So for instance
04:27	if it's why -1 then it's gonna be shifted up
04:30	one unit . But if it's why plus one it's
04:32	shifted down one unit . We've talked about that in
04:34	great detail already with vertical shifting . So you you
04:37	should have already watched those those lessons before this one
04:41	. The variable A . Or not . The variable
04:43	the constant A . This tells you the steepness and
04:50	it also tells you if it opens if the probable
04:54	opens up or down recall from previous lessons , if
04:59	this number is positive is positive , it means you
05:01	have a smiley face parabola and if it is negative
05:04	you have a frowny face problem . We've all covered
05:06	all this in previous lessons were just putting all the
05:08	ingredients together to make the full blown , fully baked
05:11	equation of a parabola in vertex form . And then
05:15	we have this term where the H . Represents something
05:18	very important . If this represents a shift up vertically
05:23	. So I'll put vertically here , what do you
05:27	think ? This means this is a horizontal , we'll
05:30	be right this way , this means shift um right
05:37	K units . So this is horizontally of course ,
05:41	because it's right . So this is the brilliant form
05:48	of a parabola in vertex form . Why do we
05:52	have a problem in vertex form ? Didn't I just
05:54	tell you that you can write any equation you want
05:56	x squared minus two X minus seven . You know
05:58	, it's a parable . Why do we need another
05:59	form ? Well , it's because when you look at
06:01	this you don't have actually , you know , it's
06:03	a problem but you don't know anywhere where it is
06:05	. But if we can somehow take these equations and
06:08	transform them into equations that look like this , then
06:11	you also know it's a parabola because you have the
06:14	square term . But if you can put it into
06:16	this form , you can immediately read where the problem
06:19	is . Basically you can sketch a curve of it
06:21	right away because you know that the vertex of that
06:23	parable of the bottom most point of it is eight
06:26	units to the right and K units up . So
06:30	it's over and up like that . Right ? So
06:33	just to give you one concrete example before we go
06:36	off and do our computer demo one comparable . We're
06:39	gonna do some more examples after the computer data .
06:41	But just to put one down before Let's say we
06:43	have why -1 is equal to two X minus three
06:48	quantity squared . What does this mean ? It means
06:51	you shift it to the right one unit and up
06:54	one unit . Right ? So this means the problem
06:57	would look something like this . It would be up
06:59	to the right uh I'm sorry , over to the
07:01	right , three units . 23 This is X .
07:04	This is why . And up one unit . This
07:07	is up one unit . So it's a horizontal shift
07:09	to the right three units . In a vertical shift
07:11	up that means the bottom of the parable is hanging
07:13	out here no longer at the origin , it's hanging
07:16	out over here . So that means that this curve
07:18	looks something like this . Not exactly but basically like
07:22	this , there is no way if I give you
07:25	a parabola in standard form that you can tell me
07:27	where it lives in the xy plane , it's just
07:29	not gonna be possible unless you're a human computer to
07:32	do that . But if I give it to you
07:33	in this form which is just rearranging that thing over
07:36	this way , then I can because we've talked about
07:38	vertical and horizontal shifting , you know three years to
07:40	the right one unit . Uh Okay so I have
07:45	more to talk about the shifting . But in the
07:49	back of your mind as we go and do the
07:50	computer um I want you to keep the following in
07:52	the back of your mind . We discussed all of
07:53	these in the previous lessons on horizontal and vertical shifting
07:56	. The reason the thing is shifted up one unit
08:00	is because if you take and mentally and move the
08:02	one to the right hand side like kind of like
08:03	it is right here and put a plus one .
08:05	Then what it means is I take the basic parabola
08:08	values and I just add one unit to the Y
08:11	. Value . So by moving it over I just
08:13	add one unit and I'm shifting it All the values
08:16	up in the Y . Direction , one unit ,
08:19	that's why it's one unit shifted up . But three
08:21	units shifted to the right , a little more complicated
08:23	to understand . And that we explain in the last
08:25	lesson by the following thing . The only way that
08:28	I can get a zero inside of these parentheses so
08:31	that I can square it so that I can have
08:33	a minimum of this . Parabola is if I put
08:36	X . Value in here three units bigger than usual
08:39	. When I say bigger than usual , the base
08:41	graph , the base graph is just why is equal
08:44	to whatever A times X squared . This is the
08:46	base craft with the vertex at 00 right here the
08:49	base craft would be something like this . Right ?
08:52	So if I put a zero in here , I
08:54	square it , I get a zero . That's the
08:55	base graph . The only way that I can get
08:58	a zero here which can be then squared . Which
09:01	will give me a minimum in the Parabola is if
09:04	I put an X value in three units bigger than
09:06	usual . In other words , I have to put
09:07	a three here , three minus 30 I square it
09:11	, I get a zero out . So that's why
09:12	it shifted to the right three units . It shifted
09:15	up three units . Because if you think about it
09:17	, moved over here , whatever I get out of
09:19	this , I just add one unit to it that
09:21	shifts all the points up . And we've talked about
09:23	that at great length before . But now what I
09:26	want you to do is follow me onto the computer
09:27	where well more extensively talk about uh the shifting horizontally
09:31	vertically and then we'll conclude back on the board .
09:34	Okay , welcome back . So here we have our
09:37	handy parable , it's a regular Parabolas shape . You
09:39	can see that it's y is equal to x squared
09:41	. This is the regular table of values because you
09:44	can see negative five squared is 25 negative four squared
09:47	16 . And so on . This version of it
09:49	over here is the more expanded version of this guy
09:52	. You can see I have zeros everywhere , so
09:54	it doesn't really change anything , it's why equals one
09:56	times X square , which is exactly what we already
09:58	have . So let's first introduced a vertical shift .
10:02	We can control the vertical shift by just controlling the
10:07	number kind of on the right hand side of the
10:08	equal sign . Now you can think of this to
10:10	being pulled over next to the Y . That's why
10:12	it's over here . So when we have Y plus
10:14	two , it shifts the thing uh down on and
10:18	when we have minuses , it shifts it up .
10:20	So we talked about that before , you have to
10:22	think of it in terms of opposite signs , right
10:23	? So when you have -3 or -5 it's going
10:26	shifting up and when it's positive and shifting down so
10:29	they reset everything and we can control the other one
10:32	by the other direction by this guy here . So
10:35	we have another slider which is controlling the value of
10:38	the one right there . And uh as we go
10:41	one minus one minus two minus three U c minus
10:44	signs shift the curve to the right where positive signs
10:48	shift the curve to the left . And you can
10:50	also keep an eye on the zero point down here
10:52	. We reset it to zero here . You see
10:54	how the the center of this parable of the vertex
10:57	is at zero comma zero . But as I Um
11:00	move the thing over one unit , the zero point
11:04	has now shifted over to the right because I have
11:06	to put a one in here to make one minus
11:09	12 square zero , which gives me a minimum of
11:11	the problem . When I go over to units ,
11:14	you can see the zero point , the minimum of
11:16	the problem has shifted again to the right , another
11:18	unit . So a total of two units to the
11:19	right , Because now I have to put a two
11:22	into this equation to get to -2 is zero to
11:24	square it . And the same thing happens on the
11:26	other side . When I go to X plus one
11:29	squared now I have to stick a negative one in
11:31	here to give me a uh minimum value because negative
11:35	one plus 10 and I can square it and so
11:38	on . So this is why it shifts left and
11:40	right , because now I have to stick a unit
11:41	value of X , that's three units less than zero
11:44	just to get a zero in here to square it
11:46	. And that's why the zero point shifts left .
11:48	So as I scoot this thing around you can see
11:51	that zero sliding around in the table of values along
11:55	with the graph . So now we said we can
11:58	vertically Uh shift these guys and we can also horizontally
12:03	shift these guys . And now of course the big
12:05	finale as you can put them together , I can
12:07	do a horizontal and a vertical shift of this thing
12:09	and now you can see how it works when you
12:12	have minus signs here , it's X -3 . That
12:14	means it's three units to the right but it's why
12:17	minus two , which means it's two units up .
12:19	So the vertex of this thing which means and all
12:21	the other points as well are shifted over to the
12:24	right . Three units and also up two units as
12:27	well . So here I have uh this guy now
12:30	if I go this direction and pull it and make
12:32	a mismatch like this will have a Y . Plus
12:35	here , that's three units shifted down to unit shifted
12:38	to the right again . It kind of goes opposite
12:40	of the signs when you have a minus sign and
12:41	shifted to the right . Uh And when you have
12:44	a plus sign is shifted in the negative direction which
12:46	is down . And then of course I can reset
12:49	this guy and put it back together and now I
12:51	can go to the left here and I can go
12:53	up here . So I have a negative positive means
12:55	I shift up and why and to the left which
12:58	is the negative direction in terms of X . And
13:01	I can go to this quadrant over here so I
13:03	can move the thing in any quadrant here is a
13:05	Y shift in the negative 33 units down . And
13:08	also an ex shift in negative three or three units
13:10	to the left . So by changing the values of
13:13	these constants , I can move this parabola anywhere I
13:16	want in the xy plane and that's why the vertex
13:19	form is so useful because all I have to do
13:22	is say well what's the shift next to this variable
13:25	? What's the shift next to this variable ? And
13:26	then I can figure out exactly where the problem is
13:30	. Now final thing I want to talk to you
13:31	about is there's one more parameter . See how there's
13:33	a one here which means it's basically why is equal
13:36	to one X squared . Right ? Well I can
13:38	change that too so I can make it for instance
13:41	four . So now the value is let's change it
13:43	to five just to do something I haven't done before
13:45	five X squared but it doesn't change the shifting the
13:49	number in front only changes the shape of the parabola
13:52	. Whenever I uh move this thing to the right
13:55	and up you can see the shape of the problem
13:58	is governed only by the number five . The value
14:01	of the location of the curve is governed by these
14:03	shifting values . The two and the three again two
14:06	units to the right three units up in this case
14:08	. And I can move this thing all over the
14:10	xy plane but it's gonna be shifting the same thing
14:14	the same curve every time . If I can make
14:16	it even more narrow I can make it really fat
14:18	. I can even flip the thing upside down and
14:20	make it look like this . You can see the
14:22	same thing is happening here . When I go here
14:24	it's a negative three units to the three units to
14:26	the right three units up . And it's an upside
14:29	down problem because the negative three in front and then
14:32	I can slide this thing around as well . So
14:34	the shifting , the values of the shifting are completely
14:37	separate from the values of what the actual shape of
14:41	the thing looks like , which is governed by the
14:44	the A N a X squared and y is equal
14:46	to x square . In this case it's a three
14:47	. The way I have it done here , last
14:50	thing I want to say is that this is again
14:52	the vertex form which I've plotted right here , which
14:55	have written down right here . But if you take
14:57	this x minus two and make and square it meaning
15:00	do F O I L do the binomial squaring and
15:02	then multiply that whole thing by three and then you
15:05	have to add to to move it to the other
15:06	side . What you're gonna get is this equation you
15:09	see three X squared minus 12 X plus 14 is
15:12	exactly the same function as what I've written down here
15:16	on the axis . It's the same thing . It's
15:18	just that it's impossible to look at this and really
15:21	see where the what it looks like and where it's
15:23	at . Whereas by looking at this thing over here
15:27	, I can read the vertex directly off of it
15:29	and the three tells me if it's steep or knots
15:31	deep or shallow and that's why vertex form is so
15:34	useful because I can sketch parable is really , really
15:36	fast . So make sure you understand this and then
15:39	follow me back to the board where we will conclude
15:42	the lesson . Yeah . All right , welcome back
15:44	. I hope you've enjoyed the computer demo . We
15:46	finally take the concept of shifting horizontally and also shifting
15:50	vertically and put them together so that we can shift
15:52	anywhere . We call that the vertex form of a
15:54	parabola . So we're just gonna go through a couple
15:56	of quick examples just to make sure we're all on
16:00	the same page before we do , we're gonna do
16:01	a lot more problems . But this is just kind
16:03	of the introduction to things here . What if we
16:05	have the parabola ? Y -1 is equal to X
16:09	-2 , quantity squared ? Well , there's an implied
16:13	one here . I don't have to write the one
16:14	I can but you know , it's a one because
16:17	, you know , it's a times this , but
16:18	so I can just kind of in another colour ,
16:21	I can just write a one to remind you that
16:23	it's one times this . So it's the basic shape
16:25	of the parabola is what the one really governs .
16:28	And so what would this thing look like ? Right
16:30	, where is the shift going to be ? And
16:31	how is the probably gonna look ? Well , the
16:34	minus in the UAE means it's going to be shifted
16:37	up , goes opposite to the sign , the minus
16:39	. And the X . Means it's shifted to the
16:40	right , so it goes in the positive X direction
16:43	, positive Y direction , completely opposite of what these
16:45	signs are . We talked about that many times and
16:48	it goes to units to the right and one unit
16:50	up . This means this is the new vertex of
16:53	the parabola , right . And then the problem will
16:57	look something like this , not an exact curve or
17:01	exact sketch , It's not beautiful , but this is
17:03	basically what it's gonna look like . The how big
17:06	or small the problem opened or closed is governed by
17:08	the number one in front . And to show you
17:11	that , let's do uh kind of a sister equation
17:14	and take a look at the following . What is
17:16	why minus one , two x minus two quantity squared
17:20	what does this look like ? Again , it shifted
17:23	to units to the right and it's also shifted one
17:25	unit up . However , there's one crucial difference between
17:28	this equation and the one before , so this is
17:31	two units , this is one , so the vertex
17:33	is going to be in the same place . The
17:34	difference is that there is now a two in front
17:37	which we saw in the video demo that governs the
17:39	steepness . So the steepness might look something closer to
17:42	this , see how this one's a little bit steeper
17:45	, but it shifted the vertex and everything else has
17:47	shifted in the same spot as before . Uh There
17:51	cruising right along . What if you had , Why
17:54	minus one is negative two x minus two squared .
17:58	Again , same shift two units to the right ,
18:00	one unit up . But now it's not a positive
18:03	to it is a negative two out in front .
18:05	What does that look like ? I think you all
18:07	know what that looks like yet , but shifting again
18:10	stays the same , this is one and this is
18:13	uh -1 . Uh Let me actually write this on
18:16	the bottom here , something like this . But what's
18:20	it going to basically look like ? Well the vertex
18:22	is two units to the right and also one unit
18:26	up . So the vertex is in the same place
18:27	as it always is . But the Parabola now opens
18:31	down because it's negative and also it's it's negative to
18:34	not negative one . So it's kind of steep so
18:36	it's gonna have the same steepness as this one but
18:38	it's gonna open down . So let's see if I
18:40	can try to draw that looks something like this .
18:42	Not exact , I'm trying to draw this about the
18:44	same as this but you can see it's it's definitely
18:46	more closed off than that one . And let's do
18:49	one more . Just because I didn't do this in
18:51	the computer demo , I want to show you what
18:52	one looks like . What if we have , why
18:54	minus one is one half Times X -2 squared .
18:58	I want you to know that these numbers in front
19:01	of the equations , they don't have to be whole
19:03	numbers . They can be fractions . So the vertex
19:05	is in the same location as it always is .
19:08	It's too used to the right and it's one unit
19:11	up . However , it's a one half here ,
19:13	so the vertex is here . So if you see
19:15	the basic problem looks like this , then if it's
19:18	even one half , which means it's even more shallow
19:21	means the probably probably looks something more like this .
19:24	Now you can compare this one compared to this one
19:27	, it's about twice as much open . So the
19:30	number in front governs if it opens up or down
19:33	and the value of the number bigger , the number
19:36	means it's very closed off . And if it's negative
19:38	, of course it goes down opening as an upside
19:41	down frowny face . And of course the more negative
19:44	it is absolute value wise , it's more closed off
19:47	in that direction . These numbers govern completely . Where
19:50	the vertex of the thing is if you see a
19:52	negative sign and ex you shifted to the right .
19:55	If you were to see a positive here , it
19:56	would be two units to the left . If you
19:58	see a negative here , it's one unit up .
20:01	If you were to see a positive here , it's
20:02	one unit down and we'll close it off by just
20:05	giving you uh solidifying what I just said with two
20:09	more really quick ones Uh which are very similar to
20:13	what we have here if I have . Why ?
20:15	Plus one Is equal to X -2 quantity squared .
20:19	This is the same numbers one and two , but
20:22	this is now a plus instead of a minus .
20:24	So what would this one look like ? It's two
20:26	units to the right because there's a minus sign .
20:28	One too , it's one unit down because there's a
20:30	plus sign , so it goes opposite of the sign
20:33	. So the vertex of this Parabola is going to
20:36	be two units to the right and also one unit
20:39	down . And what's in front of this guy is
20:41	a positive one which means it's just the basic shape
20:44	of a Parabola . Not too big , not too
20:46	small , kinda just right , so to speak ,
20:49	and I'm trying to draw this the same as I
20:51	drew , the basic one over there which is right
20:53	here , but instead of basically it's kind of shifted
20:56	down because instead of a minus one we had a
20:59	plus one . So this is the only thing that's
21:01	changed . So we kind of grabbed that shifted down
21:04	, which is what I'm trying to show you here
21:07	and we'll just conclude with one last one , Y
21:10	plus one , same as this equals X plus two
21:15	quantity squared . So what does this one look like
21:18	? We verbally talked about it when you have a
21:20	plus sign , you go to the instead of to
21:22	the right of X , you go to the left
21:24	. So it's two units to the left like this
21:27	and you have a plus sign here which means you
21:30	have one unit down . So let me get rid
21:31	of this , put it here and then we'll go
21:34	negative one like this . So the new vertex is
21:37	two units to the left , one unit down .
21:40	And then you look in the front , you see
21:42	it's just a one , so it's not upside down
21:44	is right side up and it's not too narrow ,
21:46	not too steep or anything , it's just a regular
21:48	shape , Parabolas , same size and shape as this
21:50	one , but now it's shifted . So you've shifted
21:54	this one kind of over to the left here .
21:56	So when you see plus signs , you go in
21:58	the negative direction either down or to the left .
22:01	When you see negative signs , you go to the
22:04	right and so I'm trying to say that over and
22:06	over again , so you'll understand because it is a
22:08	little confusing the first time , but the bottom line
22:11	is positive , wise up , positive excess to the
22:14	right . But when we're doing the shifting , if
22:17	you c minus signs for either X or Y ,
22:19	you're going to be shifting in the positive direction ,
22:21	either positive X . Or positive Y . Up .
22:24	If you c plus signs like this , you're going
22:26	to do the exact opposite way , you're going to
22:28	be going down , or you're gonna be going to
22:30	the left and the negative directions for X or Y
22:33	. I've tried to show you that through problems ,
22:34	I've tried to show you that in the computer demo
22:36	, it takes a little getting used to but make
22:38	sure you understand it and then falling onto the next
22:40	lesson , you're gonna get lots of practice with figuring
22:42	out how to sketch these parabolas using the vertex form
00:0-1	.

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