12 - Writing Quadratic Functions in Vertex Form - Part 1 (Graphing Parabolas) - Free Educational videos for Students in K-12

12 - Writing Quadratic Functions in Vertex Form - Part 1 (Graphing Parabolas) - By Math and Science

Transcript


00:01	Hello . Welcome back to algebra . I'm Jason with
00:02	math science dot com . And today we're going to
00:05	tackle the topic of writing these quadratic functions in vertex
00:09	form . And this is part one . We have
00:10	a couple of lessons here to get this topic of
00:12	cross . So if you've been with me and these
00:14	lessons up until this point we have talked about parameters
00:17	for quite a long time . And most recently we've
00:20	talked about parabolas in this vertex form , which is
00:22	just a different way of writing uh these parabola equations
00:26	down these quadratic equations that make it very easy to
00:29	graph it because the vertex is basically right there in
00:32	front of you in the equation . And then we've
00:34	shown in the last few lessons how to plot the
00:36	vertex and how to plot the rest of the problem
00:39	. So now what we wanna do is tie everything
00:41	together . All right , we want to talk about
00:43	uh the idea that the quadratic equations can be written
00:46	really in two ways . And specifically I want to
00:49	be able to give you any quadratic equation that I
00:51	might write down and I want you to be able
00:53	to then convert it into vertex form because right now
00:56	I've been telling you , hey , here is quadratic
00:58	equations and regular form . Here is quadratic equations in
01:01	vertex form and we've been kind of dealing with them
01:03	separately . Now I'm going to actually teach you how
01:05	to take any quadratic that we have and how to
01:08	convert it into right in terms of vertex form so
01:11	that you can then plotted or graphic or sketch it
01:13	very easily . So we have in general two ways
01:16	to write these quadratic equations . The first one is
01:19	the kind of the first one we learn in algebra
01:20	. I'm gonna call it . Most books , will
01:22	call it the general general role for . I'm gonna
01:25	spell general general form also , this is called in
01:33	some books Standard Form and this is the form of
01:39	a parable that you're basically taught when you very first
01:42	very first time that you're introduced to the concept of
01:44	what a parable is . So the general form or
01:47	the standard form , she'll look really familiar to F
01:49	of X . So some function right , is gonna
01:51	look like this , A X squared plus Bx plus
01:56	C . Now A and B and C are just
01:58	numbers , but specifically A cannot be equal to zero
02:02	because if A is actually equal to zero , then
02:05	it kills this first term completely . And then it's
02:07	not a quadratic equation anymore . Because if A zero
02:10	then all you have is a line Mx plus B
02:12	or B , X plus C . This is just
02:13	a line , right ? But when you add this
02:15	term to it , it changes it into that very
02:17	famous familiar parabolas shape . Now , you know that
02:21	A and B and C can basically be anything that
02:23	could be negative numbers , that can be positive numbers
02:26	. They can be fractions , they can be decimals
02:28	and the different coefficients A B and C when you
02:31	dial them in are going to basically produce a parabola
02:33	which can live anywhere in the xy plane . Might
02:36	be up here , might be over here , might
02:38	be upside down , might be right side up but
02:40	it's very hard to look at these coefficients on paper
02:43	and really be able to look at it and say
02:45	well yeah that's over here because it's just all wrapped
02:47	up in an expanded form basically which is more typically
02:52	called general former standard form . But there's another way
02:55	to write these uh parabolas so I can take any
03:00	parabola in this form and by the way this is
03:02	the general form . So A X . Squared plus
03:04	bx plus C . But of course you know be
03:05	could be zero . So a very a perfectly valid
03:09	. Um Let me just go off off the rails
03:12	here and give you a couple of examples . Okay
03:13	, A parable . It can be in standard form
03:15	, it probably could be a . Would be maybe
03:17	two X squared plus three eggs plus . See that's
03:21	valid . That's a parabola . You can also have
03:23	a parabola which would be maybe you know four X
03:26	. Squared and be can be zero . So there's
03:28	no exit all plus two . This is a parabola
03:30	. So it doesn't have to have all three terms
03:32	. It just has to have the X squared term
03:34	. In fact equally good parabola as you all know
03:37	will be just X squared . So in the case
03:40	of X squared , this the term is zero .
03:42	And also this term is zero . So really A
03:44	, B and C . Or any number you want
03:46	A is the only thing that cannot be zero .
03:48	The other terms can all three of these are parabolas
03:50	in different parts of the xy plane . But by
03:53	looking at these equations , you cannot really tell where
03:56	the parabola is unless you're just a human computer .
03:58	Right ? So we have another way of writing these
04:01	things which we have already learned and it's called vertex
04:05	form , vertex form of a parabola . Another term
04:10	that you might see in books or teachers might tell
04:12	you it's also called the completed square form . Now
04:21	you might say , oh my gosh completing square ,
04:23	what does that mean ? What we've already learned in
04:25	previous lessons how to complete the square . It's a
04:28	topic that we've covered in the past , I've covered
04:30	it extensively . So if you have no idea what
04:32	completing the square is and you probably should go back
04:35	to my previous lessons because I've covered it extensively .
04:37	But this topic , this type of parable , it's
04:40	called vertex form in almost every book . But some
04:43	books call it the completed square form . The reason
04:45	will become obvious in a minute , but it's because
04:47	we have to use completing the square in order to
04:50	write the vertex form . So that's why it's called
04:52	that . Um And just to refresh your memory ,
04:56	the vertex form of a parabola looks something like this
04:59	F . Of X is equal to some constant A
05:03	X minus H quantity squared plus K . Some books
05:08	will write it like this right ? I actually or
05:11	you might instead of F of X , you might
05:12	see it as Y . Right ? So you might
05:15	see it like I can leave F of X so
05:16	that's fine . But I could also write it as
05:17	Y . Okay . But I actually prefer to do
05:20	it like some other books that I use that right
05:23	like this , I just moved the K to the
05:24	other side . So what you have is why minus
05:26	K . Is a X minus H quantity squared .
05:31	So basically these are the same equations . This one
05:33	just has the shift in the Y direction , written
05:36	on the right hand side of the equal sign .
05:37	And this one has the shift in the Y direction
05:40	here . The advantage to writing it like this is
05:43	because the shift in Y is written right next to
05:45	the Y variable and the shift in X has written
05:47	next to the X variable . So it's very easy
05:51	to recognize that this guy is basically easy to graph
05:55	, this is easy to graph and we've done it
06:00	many times before . We had the entire lessons on
06:02	this . The reason it's easy is because the vertex
06:06	is written right into the equation is at the coordinate
06:09	h comma K . In other words , this tells
06:12	you that from the origin this parable is shifted eight
06:15	units to the right and it shifted up K unit
06:18	. So this would be the point H comma K
06:20	. So the vertex is written right there . So
06:22	basically when you have a problem in vertex form ,
06:25	you can write you can put a dot on the
06:27	X Y plane showing its vertex which is its maximum
06:30	or minimum value just right away . You don't have
06:32	to calculate anything , you just read it . So
06:35	then the question is , if we're given an equation
06:38	of a parabola or quadratic any quadratic we want ,
06:42	how do I take it from this form and somehow
06:45	come up with this form ? Because so far I've
06:47	given you some problems in the past with parable is
06:50	written like this , we've done things . And then
06:52	I've also given you lots of problems recently where the
06:55	problem looks like this and we've of course graph them
06:57	. But how can I give you this guy ?
06:59	And then you calculate this ? That is the topic
07:01	of this lesson . So what we're gonna be doing
07:03	is I'm gonna give you a quadratic like this ,
07:05	we're gonna do some mathematical manipulation and we're gonna get
07:08	this and here's a big hint we're gonna end up
07:10	having to complete the square , which is a skill
07:13	that we've learned in the past . So let's go
07:14	over here and kind of inch are way forward into
07:17	that . Let's take a specific example . What if
07:20	I have uh the parabola or the quadratic F .
07:24	Of X is equal to X squared plus two X
07:28	uh minus three . What can I ascertain from this
07:32	? Uh as far as what the graph looks like
07:35	without doing anything else ? Well , pretty much the
07:37	only thing that I can tell is number one that
07:39	it is a quadratic because I have a square term
07:41	. But also I can look at the coefficient in
07:43	front of the X squared . And I see that
07:45	that is just the number one . So the first
07:48	thing you can tell is that it's a positive value
07:50	. So this parabola , even though you don't know
07:52	exactly where it is , you know that it opens
07:54	up because positive values in front of the X ,
07:57	we're always opened up . So you can tell from
08:00	this right away that it opens up instead of opening
08:04	upside down , right ? You can also tell that
08:08	since it's just a one X square , that it's
08:10	gonna have that standard parabolas shape . The basic graph
08:13	is what I've been trying to tell you if it
08:14	were two X squared or three X squared for the
08:16	first time , you know that it would be much
08:18	, much more steep and closed up because that's what
08:20	happens when you have larger and larger coefficients in front
08:24	. But honestly , other than this opening up ,
08:26	you really can't tell anything at all about this problem
08:28	. I don't have any idea if it's way down
08:30	in the bottom of the plane or if it's up
08:32	here or up here . I don't have any idea
08:34	where the crossing points are . I can't tell anything
08:36	from really looking at this . So it's hard to
08:38	graph to graph . So ultimately , I want to
08:46	get this thing into the form of this . I
08:50	want to get into form that looks like this .
08:57	Why minus K A x minus H quantity squared and
09:03	some books , again , I tell you , has
09:05	the K value over there . But this is how
09:06	I generally right , this this guy down now this
09:09	why is this thing called completed square form ? Or
09:13	sometimes it's called the perfect square form . It's because
09:16	this guy right here differs from this quite , quite
09:19	obviously because of this term is a perfect square ,
09:28	it's a perfect square . In other words , I'm
09:30	able to to manipulate this equation into a form where
09:33	I have a binomial squared like that , right ?
09:36	You can always do that with every one of these
09:39	quadratic that can give you any quadratic and I'm gonna
09:41	show you how to manipulate it so that on one
09:43	side of the equal sign , I'm going to have
09:44	a perfect square like that . When I say perfect
09:46	square . What I mean is In terms of binomial
09:49	, it's just a binomial squared , but the analog
09:51	back to numbers would be a perfect square , would
09:53	be like three square or a perfect square be like
09:56	four square . Well this can be written in such
09:58	a way where I can have this perfect binomial square
10:00	. In other words , I don't have a bunch
10:01	of terms running around , expand it out . I
10:04	can compact if I it into something that's basically just
10:07	nice and square and that's why it's called uh sometimes
10:11	you see this uh called vertex form obviously , but
10:13	you might see it's written as perfect square form or
10:16	completed square form . So now we're ready to actually
10:20	jump into it . Let me go . That was
10:22	kind of uh an introduction more than anything . We're
10:25	gonna work with the same exact equation . But I
10:26	want a fresh board and I want to teach you
10:29	how to go from point A to point B .
10:31	So if I give you the function F . Of
10:33	X X squared plus two , X minus three .
10:37	And I want to write it in that other form
10:40	. The first thing you want to do , if
10:41	you see it written in function notation F . Of
10:43	X is I want you to replace F . Of
10:46	X with Y . Because it just ends up becoming
10:48	a lot easier to have wise and excess . You
10:51	could carry the ffx down . It would be fine
10:53	, but then it would just look a little cumbersome
10:54	after a while . All right . So just first
10:57	thing you do replace it with , why ? That's
10:58	all I've done , The next thing I wanna do
11:01	is I want to pretend the why isn't here ?
11:03	I want to complete the square over here . Remember
11:05	we did that a long time ago . And remember
11:07	the very first step of completing the square was take
11:10	the number the constant and move it to the other
11:12	side of the equal side . Right ? So here
11:14	I have a minus three , so I'm gonna add
11:15	three to both sides . So I'll have y plus
11:17	three on the left , an x squared plus two
11:20	X on the right . So all I've done is
11:23	add three to this side and add three to this
11:25	side . Right ? So what I'm trying to do
11:27	is complete the square over here . And if you
11:28	remember the steps , first thing you have to do
11:30	is move the constant term over to the other side
11:33	. Then what you have to do is check .
11:35	It's very important for the next step to check ,
11:37	make sure the coefficient is a one in front of
11:41	the x square term . The coefficient must b one
11:48	in this case . This coefficient already is one .
11:50	So we don't have anything else to do in the
11:52	next problem . I'm gonna show you what happens when
11:54	we don't have that coefficient there . But if you
11:56	remember back to completing the square , we've handled that
11:59	many times before , so for now it's already done
12:02	. So the next step we have to do ,
12:03	remember if you're just completing the square here is we
12:06	have to add one half of this term , The
12:09	coefficient in front of the X . We have to
12:10	add one half of this square , it and we're
12:12	gonna add it to this side and then we have
12:14	to add it to this side . Why do we
12:15	add it to both sides ? Because this is an
12:17	equation , so we can add anything we want to
12:20	both sides of the equation that keeps it balanced .
12:23	So we have to do whatever we're gonna do to
12:24	both sides . So let's just work on the right
12:26	hand side uh for the time being let's go and
12:30	use let's continue with the purple . So we're gonna
12:33	have X . Squared plus two X . But what
12:36	I'm going to add now is another term and that
12:39	term is going to be written like this , you
12:41	open it up and you take this coefficient your divided
12:43	by two and you square this is just a number
12:46	, I know that you know that this is just
12:47	one squared . So it means you're just adding one
12:50	. But I don't want to just put a one
12:52	here because if I just put a one there then
12:53	you don't really know how I got the one .
12:55	So I'm writing it down exactly as I would on
12:57	my paper , I'm saying take this coefficient divided by
13:00	two and square it . And if I'm going to
13:02	add it to this side then of course I have
13:04	to add it to the other side . So it's
13:05	going to be y plus three and you have to
13:08	add exactly the same thing . Take that coefficient divided
13:12	by two quantity squared . Yeah . So now that
13:15	we're in a position where we can see that we're
13:17	just adding some number two both sides . What's gonna
13:20	end up happening is adding this very special number on
13:23	the right hand side . Makes this thing a perfect
13:25	square . Try no meal that I can factor very
13:28	nicely . So on the left I have y plus
13:30	three plus , this is one square which means one
13:33	and then I have X squared plus two X Plus
13:37	. again this comes out to be one on the
13:40	left hand side , I have Y plus four and
13:43	on the right hand side I'm going to try to
13:44	factor this . If I'm unable to factor it ,
13:47	I've already messed up because completing the square makes it
13:50	so that I can always factor the thing , that's
13:52	the whole point of it . So if I cannot
13:54	factor this , I've already messed up . So I'm
13:56	going to open up to parents season and put an
13:57	X right here in an X right here and I
13:59	have a one . The only way that can work
14:01	is if I have one times one and trying to
14:03	make a positive too . So the only way it
14:05	works is with positive sides , check yourself X times
14:08	X is X squared inside terms gives you X .
14:11	Outside terms give you X . Those add to give
14:13	you two X . And then these multiply to give
14:15	you the one . Now you see what's happened .
14:17	I was able to factor it , but I have
14:18	carbon copy exact duplicates of the terms here . So
14:22	I have Y plus four is X plus one quantity
14:25	squared because they're just multiplied together . I should always
14:29	be able when I factor the right hand side after
14:31	I complete the square . Remember we did this a
14:33	lot when we completed the square in those lessons ,
14:36	when you factor it , it should always be duplicate
14:39	terms because you should always be able to write it
14:41	as a square . That's the whole point . You'll
14:44	always be able to do it that way and we
14:45	prove that mathematically back then . But if you notice
14:49	, look what we ended up with were trying to
14:51	get into a form that looks like this , we
14:55	want y minus some number is some constant , A
14:59	X minus some number squared . And we got y
15:03	minus some number equals a constant , which is one
15:05	X plus or minus some number squared . So of
15:09	course we don't have minus signs here . But that
15:11	just means that this parable is shifted in a different
15:14	, you know , opposite lee than than the normal
15:16	uh guy . And so you can kind of see
15:18	that whenever you try to plot this thing . So
15:20	this is the answer . This is what we're trying
15:22	to get to , we're trying to get to vertex
15:24	form , you can read the vertex directly off the
15:27	X shift and the Y shift and the vertex in
15:30	this case Is going to be what shifted from the
15:33	origin . If it was X -1 , it would
15:35	be shifted to the right , but it's X plus
15:38	one . So that means it shifted to the left
15:39	one unit . And if it was why minus the
15:42	number it will be shifted up for . But it's
15:44	why plus this . So it shifted down for which
15:46	means the vertex is at negative one comma negative four
15:50	. Also , we've already done in the past lessons
15:52	when we graph these guys , we look at the
15:54	coefficient in front again , it's positive . So this
15:57	means it opens up and because it's just a positive
16:01	one , we know that it opens up with the
16:03	same shape as y is equal to X squared because
16:05	A is one . If it were three as a
16:08	coefficient out there , it would be much more closed
16:10	off . If it were negative out in front here
16:12	, like negative two or something , it would be
16:14	upside down opening frowny face and if it were negative
16:17	two would be closed in a little bit more because
16:20	of what we've talked about when we graph these things
16:22	. So now we're in a position to attempt to
16:25	sketch this thing . So we're gonna go over here
16:28	and we're gonna try to sketch it . So the
16:30	first thing you want to do when you sketch these
16:32	things obviously is you want to put the vertex down
16:34	. Negative one comma negative four . So here's negative
16:36	one comma one negative two negative three negative four .
16:39	Of course , this is X . And this is
16:41	why so negative one comma negative four means the problem
16:44	is the vertex of it , is there , you
16:46	know , it opens up so this is the bottom
16:49	most point of the problem . So I could just
16:51	sketch it here and say you're done . But really
16:53	when you sketch parable is you want to figure out
16:55	where the crossing points are so that you can plot
16:57	them , You know , effectively there . So let
17:00	me go ahead and try to say to this is
17:03	three , this is four 1 , 2 , 3
17:05	, 4 , put a couple tick marks all over
17:07	the place , you can kind of see it .
17:09	So when we draft these things in the past ,
17:11	what we always needed to do is um find these
17:15	intercepts . So here is the equation . We want
17:18	to find the intercept . We want to figure out
17:19	where are these points that are crossing points ? They're
17:22	probably somewhere around here . But I want to figure
17:25	that out . So I have to set Y equal
17:27	to zero to do that . We've done that skill
17:28	before . So just use this equation and set Y
17:31	equal to zero , Y plus four X plus one
17:35	quantity squared I have to solve for X . So
17:38	what do I have ? I have four equals I
17:40	can expand this out by foil . And this one
17:42	is so easy , it's just gonna be X squared
17:45	plus two times X times one . So two X
17:48	plus one squared like this , whoops forgot plus down
17:52	here . So it's plus one . And then I
17:55	need to subtract the four . So it's going to
17:57	be zero equals X squared plus two X . And
18:00	then one minus four is negative three . So let
18:03	me just double check myself . I have X squared
18:05	plus two X minus three . Right ? So now
18:07	I need to try to factor this and I'm gonna
18:11	factor this guy with an X . Here in an
18:12	X . Here and I have , the only thing
18:15	I can do to make three is one times three
18:16	is three and I have a negative . So it
18:18	needs to be opposite signs and a plus sign here
18:20	. The only way it's gonna work is a plus
18:21	here in the mine is here double check yourself X
18:24	times X is X squared . This gives you negative
18:27	X . This gives you positive three X . I
18:29	add those together . I get the two X .
18:31	And then this gives me negative three . So then
18:33	I have these two guys here . If I set
18:35	this equal to zero , I'll find that X .
18:37	Is equal to one . Move the one over .
18:39	If I set this equal to zero , X .
18:41	Is equal to negative three . So what I've done
18:43	is I've taken the equation of the problem , I
18:45	found I forced Y to be equal to zero and
18:48	I figured out the X values that correspond to that
18:51	and it's at one and three , so uh or
18:55	one and negative three . So here's the first one
18:56	at one and the negative 123 is the other one
18:59	right here . And you got to remember this is
19:00	a free hand parabola . So my tick marks aren't
19:02	exact , but you can see basically what happens is
19:05	this is the problem . Well , that's a very
19:07	bad problem , I'm sorry about that , but you
19:09	get the idea goes down like this , something like
19:14	this , and of course it's a little too crunch
19:16	at the bottom , whatever , but that's the general
19:17	idea . These intercepts should always be equal distant on
19:21	both sides of the vertex . You can see this
19:23	is negative one . So we have to tick marks
19:25	for that one and two tick marks over for that
19:27	one . And the axis of symmetry . If you
19:29	ever asked that The axis of symmetry goes right down
19:32	through this guy , the axis of symmetry has got
19:34	to be access equal to negative one because that's the
19:37	vertical line negative one , that slices the whole thing
19:39	in half . So this is the axis of cemetery
19:42	. I'm not gonna write the cemetery park down .
19:44	So every one of these problems is basically going to
19:47	proceed the same way you're going to be . Given
19:49	some kind of quadratic equation , you're gonna replace this
19:53	with . Why ? Then you need to complete the
19:55	square on the right hand side . So you're gonna
19:56	take the constant term , move it over . You're
19:59	going to check that the coefficient in front of X
20:00	square is one . If it's not , we'll fix
20:03	it , I'll show you how later . But for
20:04	now you proceed , you take the coefficient of the
20:08	X term divided by two and square it . You
20:10	add it to both sides . Then you simplify everything
20:13	. Factor . You should always get a perfect square
20:15	on the right . And then basically you're done ,
20:17	this is the vertex form , it should be y
20:19	plus or minus a number X plus or minus a
20:22	number quantity squared . Then after that you just graph
20:25	it and we already practiced graphing these things many many
20:28	times before . Now in the next problem , the
20:32	process is the same . But completing the square part
20:34	becomes a little bit um uh trickier because there's an
20:40	extra step with the coefficient , I'm gonna have to
20:42	show you . But let's go and actually start this
20:45	problem . Let's start this problem over here so I
20:48	can make sure you have enough room . So the
20:49	problem that we're gonna solve is the following ffx is
20:53	equal to two X squared minus four X plus one
20:58	. Now , if I asked you , what does
21:00	this problem look like ? The only thing you can
21:02	tell me is that it opens up because it's got
21:04	a positive number in front of the X squared because
21:06	it's a positive to , you know , that it's
21:08	closed in on itself a little more than the regular
21:10	X squared parabola . And that's really all , you
21:13	know , you don't have any idea where the thing
21:14	is located or anything . So what we want to
21:17	do is a take this guy and transform it into
21:20	a vertex form . So the thing we have to
21:22	do is the very first thing . Take the F
21:24	of X and replace it with . Why ? So
21:26	two X squared minus four X plus one . Next
21:31	step , you take the constant term on the right
21:33	and you move it over by subtraction or addition or
21:35	whatever you have to do . So in this case
21:36	I have to subtract two , X squared minus four
21:39	X . So that's exactly what we did in the
21:42	previous problem . We just moved the constant term over
21:44	now . In the last problem remember I told you
21:47	but after you do that you have to check that
21:50	the coefficient there is a one in this case it
21:52	was a one . So we were we could proceed
21:54	. But in this case the coefficient in front of
21:57	the x squared term is actually a two . So
21:59	you cannot do that . You can't proceed without doing
22:02	the following steps . So you have y minus one
22:04	. If the coefficient in front is not a one
22:06	, then all you have to do is factor to
22:08	make it a one . So you have x squared
22:09	minus two eggs . So I've just factored out of
22:12	two there and notice that if I cover everything up
22:14	, the coefficient in front of this one is a
22:16	one , that's what you're really trying to do now
22:18	. Yes , we have all this other stuff ,
22:20	we pulled it out , but if you remember the
22:22	completing the square process , what we did to get
22:24	rid of it as we just factor it out and
22:25	we focus on what's on the inside . So the
22:28	coefficient of what's inside here is now one . Okay
22:32	, so let's go on and add one half of
22:36	this term of this coefficient square . So we're gonna
22:38	have why minus one ? I'm gonna give myself a
22:41	lot of space , two X squared minus two X
22:47	. And now I have to add a term .
22:49	What do I add ? I'm going to add to
22:51	it , 1/2 of the coefficient of the x .
22:54	term . So the way you write it is -2
22:57	. That's the coefficient there , divide by two ,
22:59	that's one half of it . And I'm gonna square
23:02	and don't forget that , this whole thing , I
23:04	think I'm adding it inside of these purple parentheses .
23:07	So this is what's inside of here , and I'm
23:09	kind of completing the square of what's on the inside
23:11	. So I do one half of this squared ,
23:13	and that's all living inside of here . So the
23:16	biggest mistake students are going to make is that they're
23:18	going to uh add this term to the right hand
23:21	side . So let me go ahead and do it
23:23	exactly as you probably would think to do it negative
23:25	to over two quantity square . So you would think
23:28	that you're just gonna like uh let's do it like
23:30	this , I'm going to add this guy like this
23:32	and you would be done . But the problem is
23:35	the following . Remember when you're adding things to both
23:37	sides of an equation , you have to add the
23:39	same value to both sides . So it looks like
23:42	I'm adding this over here and I'm adding this over
23:44	here and you're good . But actually think about it
23:46	when I evaluate this is gonna be negative one ,
23:49	negative 2/2 is negative one squared . So I'm gonna
23:51	get a positive one . So I'm adding one over
23:53	here . So you would think that you just add
23:55	one over here also , which is kind of what
23:57	I have . But the problem is I'm not really
23:59	adding one over here , this whole thing is inside
24:02	and the two is multiplying . So really I'm adding
24:05	to , but I've just added it on the inside
24:08	so it looks like I'm just adding this term .
24:10	But really I'm adding this term times too , so
24:12	to make the equation balanced to add it to this
24:15	side , I have to also multi , have to
24:18	multiply the thing by two and this is the term
24:21	that I'm really adding to both sides . So it
24:23	looks like I'm adding something unbalanced cause it looks like
24:26	I'm adding two times something and it's not there .
24:28	But really it is because if I were to blow
24:30	this thing out of multiplying by two anyway , so
24:33	I'm adding the same thing to both sides . Basically
24:35	I'm adding to to both sides two times one ,
24:37	two times one and that's what I'm adding to both
24:40	sides and you can see a little bit more clearly
24:43	in the next lesson um what I have here ,
24:49	so I may have misspoken minute ago , I'm not
24:51	sure I'm adding two to the left and I'm adding
24:53	two to the right because two times one is two
24:55	and two times one is two . I I may
24:57	have misspoke there , I apologize if I did .
24:59	So what you're going to have here is why minus
25:01	one and then on the inside you're gonna have what
25:04	you're going to have one . I'm sorry negative one
25:06	square that gives you one And then one times 2
25:09	is two . So I'm adding just a two right
25:11	there then I'm gonna have to X squared minus two
25:15	X . Plus this is negative one squared , that's
25:17	just adding one . But again two times one is
25:20	I'm really adding to I'm really adding to so the
25:22	equation is completely balanced . So then I have negative
25:27	one plus two is one right and on the right
25:30	hand side to have it too . And now I'm
25:32	an attempt to factor the right hand side . If
25:34	I'm unable to factor it , I've made a mistake
25:36	. So have action X . The only way I
25:38	can get a one is one and a one .
25:40	And the only way I can make this negative is
25:41	to have both of them negative double check yourself X
25:44	times X is X squared . This gives me negative
25:47	X . This gives me negative X . This gives
25:49	me positive one . The negative X plus the negative
25:51	X gives me negative two . X . And of
25:54	course that's what I expect . Just like in the
25:56	previous problem when I factored it , I told you
25:58	when you factor this thing , you'll always get carbon
26:00	copies and be able to write it as a square
26:03	and the exact same thing happened . So what I'm
26:05	able to do is that I have Y plus one
26:08	equals two x minus one . Now it's squared because
26:12	they're multiplied together and look at what I have .
26:14	This is exactly the form that I wanted in .
26:17	This is why plus or minus a number , the
26:20	constant A . Out in front X plus or minus
26:22	some number . It matches exactly the form of the
26:24	equation that I'm trying to get . That's why sometimes
26:27	it's called when I opened the lesson up , I
26:30	said this is called the vertex form of a parabola
26:33	but it's also called the completed square form because you
26:36	have to complete the square to get . So now
26:39	what I want to do to finish the problem out
26:41	, let me re copy this on the right hand
26:43	side . This is actually the answer to the problem
26:45	. So the answer is we checked myself , Y
26:47	plus one is equal to two times x minus one
26:49	quantity squared . If you're asked to take this quadratic
26:52	and change it to a vertex form , you would
26:55	leave it like this and of course if your teacher
26:58	prefers it or your book you could also write it
27:00	with the constant term over here . So I'd have
27:02	a -1 over here and it would be -1 .
27:05	Move to the right hand side like that . But
27:07	I like to leave it in this form . So
27:09	now let's go ahead since we're getting practice with all
27:12	the stuff , we're gonna sketch this problem . So
27:14	let me rewrite it . It's Y plus one is
27:18	two X minus one quantity squared . We want to
27:21	graph this guy . So the first thing you do
27:23	is you say well what's the vertex , what's the
27:27	vertex The X . Value is going to be shifted
27:30	one unit to the right from this , so it's
27:32	gonna be one unit over and it's gonna be one
27:35	unit and why ? But if it was why minus
27:37	it would be up and it's why plus , so
27:39	it's down so it's negative light . So it's one
27:41	common minus one . That's the vertex . It also
27:44	opens up why ? Because it's a positive to if
27:49	this were a negative to , it would open down
27:51	but it's a positive to . It opens up and
27:53	also again it's a little steeper , it's a little
27:55	closed in because it's too as the coefficient instead of
27:58	regular X squared . Parable has coefficient of one .
28:02	So let's take a little sketch of this guy and
28:07	see what it kind of looks like . So the
28:09	vertex is at 1:00 -1 . Which is from doing
28:13	the math right basically down here and then actually I
28:17	want to change the color of it . Just do
28:19	green . So I put the vertex down here one
28:22	comma negative one . And we know it opens up
28:25	so we know this problem is going to go and
28:27	look something like this so we know it's going to
28:28	cut through the X axis . And so in order
28:31	to really graph it properly we probably should find the
28:33	X intercepts . So the find the X intercepts .
28:37	Um What we have to do is take this equation
28:40	that we have and set y equal to zero .
28:43	That's gonna find the points on the X axis when
28:46	y is zero . So we're gonna take this equation
28:48	, we're gonna set y equal to zero . So
28:50	we're gonna set is gonna be zero plus one and
28:53	then it's gonna be too X -1 quantity squared .
28:56	Now we have to find the values of X where
28:58	the crossing points happen . So we have 12 and
29:03	then here it's x minus one . I need to
29:05	expand this out , I need to blow the whole
29:07	thing out again , moves the one over and then
29:09	solve . Now we've already done this stuff , we
29:11	know what this is equal to but let's just do
29:13	it again . Uh It's going to be X squared
29:16	minus X . For the interior term minus X .
29:19	For the exterior term plus number one . So we're
29:23	gonna have one too X squared minus two X plus
29:28	one . Now we're gonna multiply the two in two
29:32	X squared minus , multiply in their four X two
29:37	times , one is two . And now we have
29:39	to move the one over so we'll have a 02
29:42	X square uh minus four X . And then one
29:47	minus two is just a one . Let me double
29:49	check myself . I have two X squared minus four
29:51	X plus one . And that's equal to zero .
29:53	So the first thing that you're always gonna try is
29:56	going to try to factor this thing , see if
29:58	you can factor it and set it equal to zero
29:59	. Right ? So I'm gonna have a two X
30:01	. Here have a two X . And I'm gonna
30:03	have an X . Here . So I multiply those
30:05	two . Give me X squared . The only way
30:06	I can do this is a one in a one
30:09	. So then I try to fiddle with it .
30:10	But you're gonna figure out that it's not gonna work
30:12	because the outside terms are going to be two X
30:15	. When you multiply , this is going to give
30:17	you X . So I can't get four X from
30:20	that . Even if I add them together . Two
30:22	plus one is three . There's no way I'm gonna
30:24	get a four in the middle . So this is
30:25	not factory able . Now when a uh quadratic is
30:29	in fact a rubble , it doesn't mean there's no
30:31	solution . There's definitely a crossing point here . There's
30:34	definitely a solution . But what it means is the
30:36	numbers in here , the factored form of that are
30:39	not whole numbers . The way that it would factor
30:41	out is not gonna be a whole number , Right
30:43	? So I can't just pick whole numbers and find
30:46	the crossing points , basically . So if you come
30:48	up empty and you can't factor it , like in
30:49	the previous problem , we were able to when we
30:52	set it equal to zero and get down to this
30:54	point , we could factor and solve for X .
30:56	But here we are unable to do it . So
30:57	you don't give up what you do is you just
31:00	go and use the quadratic formula , right ? So
31:05	we have to use the quadratic formula on the sky
31:07	. So here we say that A . Is to
31:12	B is negative for and C is one . And
31:17	we have to do the quadratic formula which is negative
31:20	B plus or minus B squared minus four A .
31:23	C . And this is under a radical . Oops
31:28	this whole kind of bought . Shut up . Sorry
31:30	about that . So this whole thing is there to
31:32	A So we have to substitute in for a B
31:36	and C and crank through it . And I know
31:37	you know how to do this . We've done it
31:38	. But let's do it again together . So we
31:40	have negative B . But B is itself negative .
31:42	So we do negative for in parentheses plus or minus
31:46	the be a squared . So it's negative four quantity
31:48	squared minus four times A . Which is to times
31:53	C , which is one . And that whole thing
31:55	is under a radical and on top of two times
31:58	a which is to All right , So then what
32:04	do I have ? I have x equals I have
32:06	four because this is negative times negative plus or minus
32:10	on the inside . I have 16 -8 from inside
32:15	here . There's a radical here and then on the
32:17	bottom two times two is four . Which means that
32:20	X is going to be equal to four Plus or
32:22	minus the square root of eight because the 16 1988
32:27	and then I'm gonna have a four on the bottom
32:30	. Now if you do a factor tree on eight
32:32	, you know , we could go off to the
32:33	side here . I've done it so many times .
32:35	It's kind of crazy . But if you do a
32:37	factor tree on eight it's two times four and two
32:40	times two . So really the square root of eight
32:43	is the two comes out The square root of two
32:46	left over stays on the inside . So eight is
32:50	two times a squared of two . So what you
32:51	have is four plus or minus two times the square
32:55	root of two divided by four . Now I can
32:57	simplify this further by simplifying these coefficients and all that
33:01	, but really I'm trying to sketch it . So
33:03	what I'm gonna do is let you dump that in
33:05	a calculator and say four plus two square root of
33:08	two divided by four when you dump that and you're
33:10	gonna get about 1.7 . This is a approximate ,
33:14	it's not an exact , there's decimals that go on
33:16	after this thing forever . And then the other answer
33:19	is going to be approximately equal two for minus two
33:22	squared of 2/2 , which is 0.3 . These are
33:26	both approximate answers , but they are the places where
33:29	the graph has a y value of zero . So
33:33	if you go back to your graph here and say
33:35	this is one , this is two , this is
33:37	three negative one , negative two negative three . Then
33:40	the crossing points are gonna be at 0.3 Right ?
33:44	Which is right here , here's here's one , here's
33:47	half , so 0.3 is a little bit left of
33:49	half , right , and then 1.7 here's one ,
33:53	here's two , so 1.7 is pretty close to two
33:55	and you can see these dots are on either side
33:58	of the vertex . So the problem is going to
34:00	go something like this is gonna go down cross up
34:03	through those points and go up . Of course it's
34:05	a sketch , it's not exact , but to get
34:07	the idea , it's at 0.3 and 1.7 the vertex
34:10	is down here . And if you ever asked what
34:13	the axis of symmetry is , that's the vertical line
34:15	that goes through the whole thing and that's X equals
34:18	one . Is the axis of symmetry , right ?
34:21	Because it's the value here that cuts in their one
34:24	common negative ones . The vertex of the axis has
34:26	to be one . So that was a long lesson
34:29	. But the process , it depends on so many
34:32	skills that we've learned in the past . First of
34:35	all , you have to understand that they're basically two
34:40	ways to write parabolas are quadratic in general . The
34:42	first way is that the whole thing expanded and blown
34:44	up like this . Right ? The other way is
34:47	a completed square form or alternately called the vertex form
34:50	, which looks like this . Now this looks very
34:53	different than this , but they're the same equation .
34:55	If you take any of these answers that we got
34:57	like this one , right ? And if you multiply
35:00	out with foil multiply by two and move the one
35:03	over , you're going to get what we started with
35:05	, you're gonna get this . So all of these
35:07	work that way . It's basically you have this form
35:10	and it looks different . If you blow this out
35:12	with foil , take the four and move it over
35:14	, you're going to get what we started with .
35:16	They're just two different ways of writing the exact same
35:19	equation , right ? And so we learned that we
35:22	have to complete the square in order to do that
35:24	. And so we've learned the topic of completing the
35:26	square in the past . Here is an application of
35:28	why we care why do we care about completing the
35:30	square ? That kind of rhymes ? Right ? So
35:33	make sure you understand this ? I have one more
35:34	lesson on this topic , It's the same sort of
35:37	thing where the problem complexity to be slightly more involved
35:40	. But ultimately it's all the same steps . So
35:42	follow me on to the next lesson , we're going
35:43	to continue getting practice with writing these quadratic in vertex
35:47	form .

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