09 - Practice Graphing Parabolas - Part 1 (Vertex Form & Standard Form) - Free Educational videos for Students in K-12

09 - Practice Graphing Parabolas - Part 1 (Vertex Form & Standard Form) - By Math and Science

Transcript


00:0-1	Hello . Welcome back to algebra . The title of
00:02	this lesson is practice graphing . Parable is this is
00:05	part one ? We have a few parts to this
00:07	lesson where the problem complexity will gradually increase . So
00:11	we've talked about the general form of the vertex form
00:14	of a parabola . Right ? We all know we
00:15	can graph anything by just making a table of values
00:18	and putting plugging in the table of values . But
00:21	we have this vertex form of a parabola which makes
00:24	it very easy to sketch the thing without making a
00:26	table of values because we can read the vertex directly
00:29	out of the equation . So we're gonna get practice
00:31	without here . We're going to be doing several things
00:34	. Were going to be graphing all of these problems
00:36	. And what I mean by graphing is more like
00:37	sketching it . We don't need any details . We
00:39	just need to be able to sketch the general shape
00:41	. Want to label the vertex , we want to
00:43	label the axis of symmetry of every one of these
00:45	parabolas . and we want to show the intercepts .
00:48	In other words where does the parable across the X
00:50	axis ? This opens up opens down it's going across
00:53	the X axis somewhere . All right . So let's
00:55	go ahead and refresh our memory . So I'm gonna
00:58	say recall that the vertex form of parabola . It
01:04	looks something like this . Why minus K is equal
01:08	to a times X minus H quantity squared . If
01:13	this equation looks foreign or crazy to you it's just
01:16	because you haven't watched my lessons on shifting parameters around
01:19	the xy plane , you should go back and do
01:20	that . But basically what you have here is you
01:23	have the basic parabola A X square . That's the
01:27	basic shape of a parabola . Right ? And you're
01:29	shifting it H units to the right , That's what
01:32	the minus sign means . We talked about that in
01:34	great detail and you're shifting at K units up in
01:37	the Y . Direction . So minus sign means H
01:39	units . This way minus sign here means K units
01:42	up . And of course the A . Here in
01:44	the front determines if it opens up or opens down
01:46	and also the steepness of the problem . So we're
01:49	gonna get a lot of practice with all this stuff
01:51	. What we want to do with every one of
01:52	these problems here is we want to graph it ,
01:55	sketch it I guess is a better word . We
01:57	want to show the vertex of every one of these
02:00	problems . We want to show the access or write
02:02	down the axis of symmetry , which we've defined in
02:06	the past . And we want to show the intercepts
02:12	because if you know all of this stuff uh then
02:14	you can sketch the graph . So that's what you're
02:16	gonna be doing on your exams and you'll be given
02:18	an equation and you'll be asked to sketch it and
02:21	you'll be asked to sketch it without putting a table
02:24	of values , without making a giant table values and
02:27	drawing lots of points . You're not gonna be doing
02:28	that , We're sketching it just by looking at the
02:31	equation . So the first equation we have is something
02:34	like this . Why is equal to negative three times
02:37	X squared ? So we know it's a parable why
02:40	? Because it has a square term . The highest
02:42	power of course is very simple . There are no
02:44	other terms here , but the highest power is a
02:46	square . So we know it's a parable up and
02:49	we also know because we've talked about it before that
02:52	the negative sign in the front is negative three here
02:54	, the negative sign means the parabola opens down so
02:58	whatever is in front of the X . Square term
03:00	, no matter what the number is , if it's
03:01	negative , the Parabola opens down like a like a
03:04	frowny face . If it's a positive then it opens
03:06	up like a smiley face . So we know this
03:08	problem is going to go upside down like this ,
03:11	but we want to obviously talk about more than just
03:14	that we want to sketch it . So let's go
03:16	over here and draw a little access . We don't
03:20	need to be exact about it , this is X
03:22	. And this is why we're going to sketch this
03:24	thing right here . So the first thing we need
03:26	to do is figure out what is the vertex of
03:28	this problem . So what you need to do is
03:30	figure out a way to make this equation look like
03:32	this one because when you have it in this form
03:34	you can read the vertex directly out . So what
03:37	I want you to do is think Think of this
03:40	equation like this instead of why you need to think
03:42	about it as why zero . You don't need to
03:47	have the princes there if you don't want , in
03:48	fact let me take them away . Just so it
03:49	doesn't confuse anybody . Uh and that's going to equal
03:52	to negative three . Now this X square , you
03:54	can think about writing it as X minus zero quantity
03:57	squared . Well , why am I doing that ?
03:59	Because when I subtract zero from why it doesn't change
04:02	anything , it's still why Right ? When I subtract
04:04	zero from exit doesn't change X at all . So
04:07	this is the same as this and this is the
04:09	same as this . So even though it looks different
04:11	, it's actually the same exact equation . But by
04:13	writing in here now it matches this form right here
04:16	. So because of that , you read the vertex
04:19	directly off of this guy , you read the vertex
04:23	so you can see the vertex is going to be
04:27	at . In other words , the parabola here ,
04:29	the basic shape of the problem is shifted zero units
04:32	and X and zero units . And why ? So
04:33	that means it's just located at 00 like a regular
04:36	parabola is so I'm going to put a dot here
04:39	and I can say that the vertex is located at
04:44	zero comma zero . In other words there is no
04:46	shift of this parabola from the basic position of any
04:50	parabola , there's no shift at all . So it's
04:52	going to go here and we also know it's an
04:54	upside down parabola because there is a minus sign in
04:57	the front . So I can say it opens Yeah
05:00	, down and we talked about this in the past
05:03	as well , but because there's a three here ,
05:05	the value of the number in front tells you how
05:08	steep it is . In other words , if it's
05:10	a very large number of positive or negative , then
05:13	the problem is very closed off , very narrow .
05:16	If it's a very uh small number , like a
05:18	one , like a typical one X square , just
05:21	a one , then it's nice and open . And
05:23	of course if you go to one half it opens
05:24	up even more . So it's a fairly narrow steep
05:27	problem uh here and it opens down . So the
05:32	next thing we need to do , we know it's
05:33	gonna go like this uh here . And so probably
05:37	the easiest thing to do since we can see the
05:38	vertex is right on the axis here is go ahead
05:40	and sketch the thing . So we know it's going
05:42	to open a little bit steeper than a traditional problem
05:46	. A regular problem might come and open up a
05:48	little bit more open like this . This one's a
05:50	little bit steeper . The vertex is right here .
05:53	It opens upside down . That's pretty much all the
05:55	information that we have here . But we have one
05:57	more thing we want to talk about . We're gonna
05:59	graphic we're gonna put the vertex , We're going to
06:01	talk about the axis of symmetry . Axis of symmetry
06:03	is what is the line that cuts this parable in
06:06	half ? So the axis of symmetry is going to
06:11	be an equation of a line . A vertical line
06:13	, always a vertical line . So it's going to
06:15	be X is equal to zero . A vertical line
06:18	. Remember , vertical lines are always X equals something
06:21	. A vertical line at X equals zero . Is
06:23	this line the y axis a vertical line and X
06:26	equals five . Is a vertical line over here .
06:28	A vertical line of X equals negative three is a
06:31	line over here at negative three and so on .
06:33	So the axis of symmetry is X equals zero .
06:35	That's a vertical line that goes right and splits this
06:37	thing in half because that is the symmetric , you
06:40	can fold this thing over on itself like that .
06:42	And then the last thing we want to talk about
06:44	is the intercepts here . Usually these problems , like
06:46	if it was upside down over here , it would
06:48	intersect the axis in two points and we would have
06:51	to find those points . But here , because the
06:53	vertex is right here , it only touches the axis
06:55	in one location . We know that the only intercepts
06:58	here . Yeah , are at X is equal to
07:04	zero . There's only basically one intersect . It's a
07:06	double route basically right here because it touches in one
07:08	location . So I don't need to put anything else
07:10	there . So we have the intercepts which is only
07:13	touching at one spot , the axis of symmetry .
07:15	It opens down the vertex and we've sketched it .
07:18	Now , of course you can make a table of
07:20	value and tighten this thing up and get the exact
07:22	shape . That's not the point . The point is
07:24	to sketch it . You're gonna do this kind of
07:26	thing in algebra where you're just sketching things , you're
07:28	also going to do it in calculus , you spend
07:30	a whole chapter sketching curves using the concepts that we
07:33	learn later on in calculus . So sketching is something
07:35	that you just have to get used to . All
07:37	right , let's go onto the next problem . It's
07:39	slightly more complicated than this one , But not complicated
07:42	in general . What if the equation is why -4
07:46	is equal to negative x squared ? So we want
07:49	to sketch it right ? Um Now we already want
07:53	to compare it to this , so this is why
07:56	minus some shift is equal to a X minus some
07:59	shift squared . So this looks almost the same .
08:03	But you wanna think And I'm gonna write the word
08:06	, think . You want to think about this equation
08:08	in a different way , you want to think about
08:09	it is why -4 is equal to the negative sign
08:14	, stays there from above . But this X just
08:16	replace it with X minus zero in other words because
08:19	there's nothing here , there is no shift in the
08:20	X . Direction because X minus zero is what already
08:23	already had there . But I do have a shift
08:25	in the Y . Direction and no shift in the
08:27	X . Direction . So if I come over here
08:29	and draw a or sketch I should say an xy
08:33	playing . So here's X . Here's why I'll just
08:38	put some tick marks 12341234 I'm just sketching 123 If
08:43	I don't need to be exact something like this .
08:45	Where is the vertex of this ? Probably going to
08:47	be ? Well there is no shift in the X
08:50	direction . So that means the X coordinate of the
08:52	vertex is zero , right ? But the y coordinate
08:58	of the vertex is shifted up because remember minus signs
09:02	mean shifted up . If that doesn't make sense to
09:05	you , then it's because you need to go back
09:06	and look at my lessons on shifting parabolas . I
09:08	talk extensively on why the shifting happens when you have
09:12	a minus sign here . This is the minus sign
09:14	Also it just means there's no shift though . But
09:16	here we're shifting upward in the Y direction for units
09:18	. That means the vertex has to be positive for
09:21	So we have 1234 . This problem is shifted up
09:25	for units . I'm putting a doctor because that's the
09:27	vertex , right ? So I'm gonna label this the
09:30	vertex Is at 0:00 for that's the vertex of the
09:35	problem . So the next thing you have to look
09:37	at say , well , is it open up or
09:40	is it open down ? In other words , does
09:41	it look like a smiley face where the vertex here
09:43	? Or does it look like a frowny face with
09:45	the vertex there ? Well , I have a negative
09:48	sign which is in front of the X square term
09:50	. So it opens down and also it's only a
09:55	negative one out here . Seeing the previous problem ,
09:58	it was negative three . This was much steeper ,
10:00	much more closed off , much more constricted like this
10:03	. But this is a negative one out here ,
10:04	which means it's not going to be quite so steep
10:06	. It'll probably open something more like this . Uh
10:10	like a traditional parable of shape , It's just flipped
10:12	upside down . Now , you can sketch it with
10:15	this information honestly because you know the vertex is here
10:18	. You know , it opens down and so you
10:19	could just sketch it . But the problem is you
10:21	don't know exactly where this thing crosses the axis here
10:25	. If I knew that it crossed here and it
10:27	crossed here , then I can sketch through this point
10:30	and I would have a pretty good idea for the
10:31	shape . So that's why in these problems you want
10:33	to sketch the graph , you want to talk about
10:35	If it opens up or down , you want to
10:37	know the vertex , you want to know the axis
10:39	symmetry of course , and you want to know the
10:41	intercepts . That means where does the problem cut into
10:44	the axis ? Because if you know that , then
10:46	you can sketch the thing through those points . So
10:49	we have to figure that out . And unfortunately ,
10:51	the only way to really figure it out is to
10:53	calculate it . So we know this thing is going
10:55	to come down , it's gonna cut somewhere here .
10:58	The function is going to go in both directions ,
10:59	that's going across the access , let's say right here
11:01	and right here in both of these locations , the
11:05	why value of the function is zero . If it
11:08	if it did cross here at this point then the
11:10	y value of this point is gonna be negative ,
11:12	something comma zero . The y value of this point
11:15	or the point here would be let's say two comma
11:17	zero . In other words , both of these points
11:19	have a y value of zero . So the way
11:21	we figure out where they are is we just set
11:24	y is equal to zero , that's all we have
11:26	to do . So we just take a zero and
11:27	we stick it in there , it's gonna be a
11:29	minus four . And then on the right hand side
11:32	it's a negative sign inside here . We just make
11:34	it x minus zero squared . And no , all
11:37	we have to do is soft at the X values
11:39	. These are gonna be the X values where the
11:41	y value is zero . So you're setting the Y
11:45	value equal to zero , forcing yourself to calculate the
11:48	values of X where the Y . Values to zero
11:51	. Those are gonna be the crossing points for the
11:54	call it the X intercept . So it's the X
11:56	intercept . In other words the intercept intersection points with
12:02	the X axis . So then we can calculate this
12:04	, we have negative four on the left and then
12:07	this is X zero . So we just write it
12:09	as negative x squared . And then we have to
12:12	do is we have a negative sign and a negative
12:14	sign . We can divide both sides by -1 .
12:17	Right ? So I divide this by negative one .
12:18	This by negative one . I'm gonna get a positive
12:20	four in a positive X squared here . Just divide
12:22	both sides by negative one . And then we flip
12:25	it around and make it easy to see X squared
12:27	is four . And I think you know what to
12:28	do next ? You take the square root , so
12:30	you get X is plus or minus the square root
12:32	of four . We've done radical equations before , so
12:35	you have X is plus or minus two , right
12:38	? So that means that at plus or minus two
12:41	on the X axis , at plus or minus two
12:44	on the X axis . The y value of the
12:46	function is zero . So that means that right here
12:49	there's an intercept at positive two for x and negative
12:53	two for X . Of course the Y value is
12:54	zero for both of those because it's on the axis
12:56	. And now I have enough information , I can
12:59	go here and just draw this guy going straight through
13:02	like this and I'll try to do my best to
13:04	go down like this and I didn't quite hit it
13:06	. But you get the idea goes through like this
13:08	, that's what this problem looks like . In order
13:10	to do a proper sketch , you have to know
13:11	where the vertex is , so you know where to
13:14	start from and you have to know where the intercepts
13:16	are . So you know where to go . All
13:18	you need is a couple of points like that to
13:19	draw a pretty good sketch of a problem . And
13:22	then the last thing we need to do is what
13:24	is the axis of symmetry ? So the axis of
13:28	symmetry is just the line that cuts the thing in
13:32	half and this thing is centered right here . So
13:34	the y axis just like before is the same thing
13:36	here . The axis of symmetry is X is equal
13:38	to zero . Remember vertical lines X is equal to
13:40	something X is equal to zero . Is a vertical
13:43	line right here , which bisects this guy . So
13:45	that's the axis of symmetry . We have the intercepts
13:48	, we talked about here , the vertex we know
13:50	it opens down . That's everything that was asked of
13:52	us in this drawing and this problem and also to
13:55	sketch the thing . So we're done every one of
13:59	the problems after this . And also in the next
14:01	couple of lessons are going to follow the same basic
14:04	idea . We're going to figure out where the vertex
14:06	is . We're going to calculate the intercepts and we're
14:09	going to then try to sketch it and then look
14:10	at the symmetry and all the other things and write
14:12	them all down . That's what you're gonna be asked
14:14	to do in all of your problems . Okay ,
14:17	next problem . What if we have y equals X
14:22	-3 quantity squared . So we obviously want to figure
14:28	out where the vertex is . So the easiest way
14:30	to do is to think about it this way .
14:34	Really ? This why is a Y minus zero on
14:36	the right hand side ? I'm gonna leave it as
14:38	x minus three quantity squared . This is now in
14:40	the form of that vertex form of of a parabola
14:44	. We can see the vertex has no shift in
14:47	the Y direction , but it does have a shift
14:50	in the positive X direction because negative science means you
14:52	go and shift the thing to the right . So
14:54	if I was going to kind of draw X ,
14:59	why I'll do some tick marks 123 put some tick
15:02	marks all over the place here and have to be
15:04	exact then where is this vertex going to live ?
15:07	Well , again , it shifted three units to the
15:10	right , that's over here , and zero units up
15:13	. That means the vertex lives right here . Uh
15:16	and this parable is either going to go upside down
15:19	like this , or it's going to be right side
15:20	up like this , and we're gonna get to that
15:22	in a second . But for now , let me
15:23	go ahead and write down that there's vertex here at
15:30	three comma 03 extra single three Y is equal to
15:34	zero . And that's something you need to solve the
15:38	problem . The next thing you do is you look
15:41	at the coefficient in front of the X square term
15:43	, whatever is in front of here , In this
15:44	case it's a positive one . So we know because
15:46	it's positive that opens up and you also know that
15:50	there's a one here . So the larger the number
15:53	, the more closed up the problem is , so
15:55	we know it's gonna be a regular shape kind of
15:56	open for a because it's just the one and it
15:59	opens up right there . Alright . The next thing
16:02	we need to do is we need to talk about
16:04	intercepts . Now , in this case you can see
16:06	the vertex is on the axis . So , you
16:08	know , the thing really doesn't have any other intercepts
16:10	besides this point . So you could kind of just
16:13	sketch the thing . In fact , let's just do
16:14	that real quick . The problem is going to look
16:16	something like this , that's probably a little too steep
16:19	. It should be maybe a little bit more open
16:20	than that , but basically that's gonna be the um
16:23	the the problem . But just to be absolutely clear
16:29	about it , let's go find those intercepts the X
16:32	intercepts . Where does this parable across the X axis
16:37	? Of course you can see it right here .
16:38	But let's do it . We're gonna set the y
16:40	value equal to zero . We're gonna force the function
16:42	to have a y value of zero . We're going
16:44	to calculate this . So it's gonna be zero for
16:46	y x minus three quantity square . So we have
16:49	to find the X values . Okay , so then
16:52	we have a zero here and here , we're gonna
16:54	take the square root of both sides . Actually ,
16:57	no , we're not gonna take the square to both
16:58	sides . Let's do it this way . We're gonna
16:59	open it up and say x minus three , this
17:02	is x minus three times x minus three . This
17:03	is a much better way to clean a way to
17:05	do it . And you can see that . To
17:06	solve this equation , x minus three has two equals
17:09	zero and x minus three has equal to zero ,
17:11	so X is equal to three and X is equal
17:14	to three . So what you have is a double
17:15	root at X is equal to three , which is
17:17	exactly what you have . This is 123 there's a
17:20	double root here anytime and parabola goes down to the
17:23	axis and just touches it . Then you still have
17:25	to roots but they're right on top of each other
17:28	, infinitely close together , right on top of each
17:30	other . And so we've shown it mathematically that the
17:33	crossing points are in X is equal to three .
17:35	And we already knew it from the fact that the
17:37	vertex was here as well . So then we need
17:40	to talk about the axis of cemetery . All you
17:46	have to do is go up here and say ,
17:47	well what line cuts this thing in half ? It's
17:49	going to be a line that goes right through the
17:51	middle at X equals 123 It's always going to be
17:55	a vertical line and any vertical line and math always
17:59	has the form X equals something because for us we
18:02	have a vertical line . All of the points on
18:04	that line have the same X value . You know
18:07	the point on the line right here has an x
18:09	value of three . Here has an X value of
18:12	three . Here has an X value of three .
18:14	Any point on this line is three comma something because
18:17	the Y values are all different , but the X
18:19	values are all three . So to show that they're
18:22	all equal to three . The line just as X
18:24	equals three . That means all the points where X
18:26	is equal to three . And that means there's a
18:27	line here of points like this . So this is
18:29	the axis of symmetry intercepts are here . It opens
18:32	up and then the vertex we've already written down .
18:35	All right , we have one more problem . This
18:37	one's gonna take a little more time . Not because
18:39	it's complicated , but just because the math gets a
18:42	little bit cumbersome at the end . So let's go
18:45	and work on that . Right now , let's say
18:46	we have Y plus two is equal to princess X
18:51	minus one quantity square . Now , this is probably
18:55	the 1st 11 of the first ones . Anyway ,
18:57	we don't have to make any changes to the equation
18:59	. It's already in vertex form . We can see
19:02	that we have a shift in the X direction of
19:04	the basic problem shape to the right one unit .
19:07	And we can see we have a shift of the
19:09	problem also down two units because the plus sign means
19:11	down two units . So if we were going to
19:13	do that , uh we're gonna go ahead and draw
19:17	a quick little uh access here . Um Let me
19:23	go and draw a couple tick marks . 123123123123 something
19:29	like this . And let's go ahead and say that
19:32	the vertex of this guy shifted one unit to the
19:34	right , That means here and two units down 12
19:37	So the vertex of this problem is actually right here
19:40	, down below . And the problem can either go
19:44	upside down or it could be right side up .
19:46	You have to look at other parts of the uh
19:49	guy to figure that out . But we can write
19:51	down that the vertex , is that one comma negative
19:55	too ? One unit shifted to the right to unit
19:58	shifted down so from the origin so have one common
20:00	negative too . So then you have to ask yourself
20:03	, okay , is it open up or doesn't open
20:05	down ? So all you do is look at the
20:07	number in front of the princes . But it's an
20:09	invisible one positive one . So it's positive , which
20:12	means it opens up . And that means that this
20:16	equation is going to look something like this . Now
20:18	. I don't want to sketch it yet because the
20:20	problem is I don't know how wide it is or
20:22	how narrow it is . So it could be a
20:24	very steep problem that goes like this or could be
20:26	a really shallow problem that goes like this . We
20:29	need to know the X intercepts . Where does the
20:31	thing cross the X axis here ? So let's find
20:34	the X intercept . Let me switch colors . The
20:38	X intercept you set Why equal to zero ? Because
20:43	we're forcing white to be zero . We're looking for
20:45	points in the function that lie on this axis .
20:48	So why must be zero ? We just stick zero
20:50	into the equation X minus one , quantity squared .
20:54	All you do is take a zero in there and
20:56	then you solve for the values of X . Whatever
20:58	we get is going to be the crossing points on
21:01	the X axis because we set Y is equal to
21:03	zero . So notice we have x minus one square
21:06	. So let's do this properly . It's going to
21:08	be x minus one , X minus one . We're
21:10	gonna have to foil this thing out . Right ?
21:13	So on the right hand side is going to be
21:14	X times X . Giving you X squared . This
21:17	is going to give you negative X . The outside
21:19	terms are also going to give you negative X .
21:21	And these are gonna give you positive one . So
21:23	you're gonna get two is equal to X squared minus
21:26	. You add those together . Get two X plus
21:28	one . All right ? But I have a two
21:30	over here . So I need to solve this thing
21:32	. So I need to move the two over .
21:34	So what I'm gonna have a zero X squared minus
21:37	two X . And then one minus two is gonna
21:40	give you negative one . All I did was subtract
21:42	two from both sides . So let me catch up
21:45	and make sure I'm I'm good here . So what
21:48	I want to do is try to factor this thing
21:50	, I'm gonna set it equal to zero . Of
21:51	course it's set equal to zero . And I'm gonna
21:53	try to factor . So I see I have an
21:55	X squared here , here's an X . And here's
21:56	an X . The only way I can have a
21:58	one is a one time to one like this .
22:01	But then I run into problems because I have a
22:05	negative sign there . So these have to be different
22:08	signs . If I make it , for instance ,
22:09	plus and minus , Like that will make a negative
22:12	one , but they don't have a plus X and
22:14	uh minus X . I add those together . I'm
22:16	not going to get the two X . So then
22:18	I say , okay , well I'll flip it around
22:19	again and you can already see it's not gonna work
22:21	. I'll do this , make it , it has
22:23	to be opposite to multiply these two give me negative
22:25	one , but then I have the same problem .
22:26	These don't add to give me this . So basically
22:30	you just kind of cross through that and say you
22:32	can't factor it . I mean most of the time
22:37	, honestly in real life , most of the time
22:39	you get a quadratic like that and you can't factor
22:41	it most of the time . But it doesn't mean
22:44	it doesn't have solutions . It just means it doesn't
22:45	have solutions that are whole numbers . So because you
22:48	can't factor it doesn't mean you give up , it
22:50	just means you think okay how can I solve it
22:52	? Well we try to factor first that didn't work
22:54	right . The next thing you can do is completing
22:56	the square and then you definitely can factor it .
22:59	But we already learned there's an easier way the quadratic
23:01	formula is always easier . It always works and it
23:04	basically comes from completing the square . So what we
23:07	wanna do , we want to figure out the crossing
23:10	points here . And this is why I said this
23:11	problem becomes it's harder but it's not because it's conceptually
23:16	harder , it's just there's more math to do .
23:18	So now I have to get the X . Values
23:20	by the quadratic formula which remember is it's going to
23:24	be negative B plus or minus B squared minus four
23:27	times A times C . All of that goes under
23:29	a radical and then I have over two times a
23:32	day . That's the quadratic formula A . Is one
23:35	. B is negative to see as negative one .
23:37	So for B I have a negative B but B
23:40	itself is negative two . So I'm gonna wrap that
23:43	in a pregnancy . So it's negative B . And
23:45	I have plus or minus that have be square which
23:47	is negative two , quantity square minus four A C
23:52	minus four times A . Is one C . Is
23:55	negative one . For a C . A radical goes
23:59	around all of that and then this whole thing is
24:01	two times A . Which is one . So that's
24:05	negative B plus or minus B squared minus four times
24:08	a times C . Over to aid . So we
24:10	just have to crank through it and it's just work
24:11	. But you'll get there so negative times negative positive
24:14	too Plus or minus on the inside here . This
24:17	is going to give you a positive four and then
24:21	you have negative times negative gives you positive for also
24:25	have positive for inside of there . You have a
24:27	radical here and then you have on the bottom two
24:30	. So what you have is two plus or minus
24:33	the square root of four plus four is eight .
24:36	And then you have a two here . Now if
24:38	you simplify this radical which is an eight , remember
24:41	we learned how to simplify radicals ? It's two times
24:44	four and four is two times two . And we
24:46	were looking for pair so we can pull out a
24:49	two there . So what we have here is for
24:52	this is gonna be two plus or minus . The
24:54	square debate reduces to a two coming out a square
24:58	of two left underneath . And we've learned all of
25:00	that when we learned radicals . So you have to
25:03	plus or minus two square root of two , divided
25:05	by two , a lot of two . We want
25:07	to simplify as much as possible . We notice we
25:09	have a two , A two and a two here
25:11	. So what we're gonna do is we're gonna split
25:13	this thing up to help us cancel it easier .
25:15	We're gonna split this whole fraction up . It's going
25:17	to be written as 2/2 plus or minus two square
25:21	root of 2/2 . All I've done is taken this
25:24	and break it apart uh into two fractions so that
25:27	I can cancel this and this and cancel this too
25:29	with this too . Now remember when you , when
25:32	you have something that's added on the top here or
25:35	subtracted , then you can split it up because what
25:37	you have , you can think of it going backwards
25:39	if you had a common denominator of two and I
25:41	told you to add these , you would say well
25:43	the denominator has to be too and that will be
25:45	two plus or minus this on the numerator , which
25:48	is exactly what I have . So it's going the
25:50	reverse of kind of fraction addition and it just helps
25:54	to visualize the cancellation . So what you have here
25:56	is this is 12 divided by two is one and
26:00	then this two divided by two is one . So
26:03	you have square root of two , so you have
26:05	two answers , one plus the square root of two
26:07	and one minus the square root of two . But
26:08	for the purpose of sketching and these are the actual
26:10	intercept . So I'm gonna circle this . This is
26:13	the intercepts X intercepts X intercepts , but for the
26:20	purpose of graphing it's kind of hard to figure out
26:22	where that really is . So what we do is
26:24	we say one plus the square root of two is
26:27	approximately equal to one plus in the square to is
26:30	about 1.4 , it's got an infinite decimals after it
26:34	. Um so I'm truncating it but it's about that
26:36	one plus 1.4 and that's going to be approximately equal
26:40	to 2.4 . All right , and then if I
26:44	take the other 11 minus the square root of two
26:46	is approximately equal to one minus 1.4 , which is
26:49	approximately equal to negative 0.4 . These decimals are not
26:53	exact because the square root of two is an infinite
26:56	non repeating decimal goes on forever , it's irrational .
26:59	So the 1.4 is not exactly what this is ,
27:02	that's why I have the squiggly equal signs here .
27:04	But basically it crosses somewhere around 2.4 and somewhere around
27:08	negative 0.4 . So 2.4 would be one to let
27:12	me switch to a different colour . 2.4 would be
27:14	12 here is 2.5 , So 2.4 is gonna be
27:17	just to the left of that and then negative 0.4
27:21	is going to be here is negative 0.5 , negative
27:24	0.4 is a little bit shifted to the right of
27:26	that . So now we have our vertex which is
27:29	right here , this is vertex at one common negative
27:34	too , whoops one common negative to like this and
27:38	this Parabola is going to go through this point down
27:41	here and then through this point up there like this
27:44	it's gonna cross at positive 2.4 and negative 0.4 .
27:49	So you can see that those make a nice symmetric
27:51	parabola and then we can look in the middle and
27:55	say where is the axis of symmetry ? So we
27:57	have to find the dotted line that goes right through
27:59	the center and I know I'm not being perfect with
28:01	this , but the axis of symmetry is that X
28:05	is equal to what X is equal to one because
28:08	that's that's where the line is here is X is
28:10	equal to one right here . So the axis of
28:12	symmetry is that X is equal to one , The
28:15	vertex is at one common negative two . We got
28:18	that from the equation , the X intercepts are really
28:21	exactly equal to one plus or minus the square root
28:23	of two but we changed into decimal so that we
28:25	could actually put it on the graph . The thing
28:27	opens up just because of the coefficient is a positive
28:30	number there and that's it . That's all you really
28:32	have to do . The only reason this one was
28:33	difficult or harder . It was because you couldn't factor
28:36	it . We couldn't figure out how to factor it
28:37	. Now . You can see why we couldn't factor
28:39	it because the crossing points were not nice . Whole
28:41	numbers , they're weird . Long , infinite repeating decimal
28:45	. So you're not going to have whole numbers and
28:47	so that's why we couldn't factor it . So this
28:50	is practice graphing parables . We're putting everything together where
28:53	we're looking at the vertex form of a parabola ,
28:56	we're figuring out where the crossing points are , we're
28:59	looking at the vertex , we're looking at the ,
29:01	you know , the opens up versus opened down .
29:03	The only other thing I'll say before I close is
29:05	that in every one of these problems , let me
29:08	go back to the beginning here . Let me start
29:10	with this one . Like in this problem for instance
29:12	, I'm writing it in my problems like this is
29:15	the vertex form , why minus four equals this .
29:17	You can also write it as just moved forward to
29:21	the other side , Why is negative X squared ?
29:24	And then the plus four comes over . Some people
29:26	will call this a vertex form Right ? Uh some
29:29	books will call that a vertex form or on this
29:32	equation . You might write it as Y is equal
29:34	to X -1 quantity squared . Just take the two
29:38	and subtract it over . Right . So you can
29:41	still read the vertex off here , but you have
29:43	to be careful with the signs because this means shifting
29:45	down and this means shifting to the right and so
29:49	I'm kind of teaching it to you this way because
29:51	negative signs means to the right , positive signs means
29:54	down . But if you have given the equation here
29:57	then you got to be careful because if they why
29:59	shift is written on the right hand side as we've
30:01	talked about many times before , this is a shift
30:03	down , this is a shift to the right ,
30:05	so I'm just trying to kind of like keep you
30:08	aware of all the gouaches here . And that's some
30:09	books have it one way and some books have another
30:11	way . You look on on on the internet and
30:13	you'll see the vertex form one way and the vertex
30:16	form another way . And so there's no right way
30:18	, it's just where do you put that y shift
30:20	on the left or on the right ? So make
30:22	sure you can do every one of these problems and
30:23	then follow me on to the next lesson . We're
30:25	gonna get more practice with practice graphing parabolas .

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