13 - Conic Sections: Parabola, Focus, Directrix, Vertex & Graphing - Part 1 - Free Educational videos for Students in K-12

13 - Conic Sections: Parabola, Focus, Directrix, Vertex & Graphing - Part 1 - By Math and Science

Transcript


00:00	Hello , Welcome back to algebra . The title of
00:02	this lesson is called Connick sections , Parabolas Part one
00:06	. So you might be asking why are we talking
00:08	about Parabolas yet again ? And that is because there
00:11	really is a lot more to the concept of parabolas
00:13	than what we have discussed in the past , in
00:15	the past . We've all learned that , why is
00:17	equal to X squared ? Is that basic parabola shaped
00:20	? And we've grafted , we've sketched them , we've
00:22	talked about how important they are , but now we're
00:24	revisiting in the context of what comic sections are .
00:27	We've already talked about the fact that there are four
00:29	comic sections . We've already talked about circles , that's
00:32	one of the comic sections . Now we're gonna talk
00:34	about Parabolas . Next we'll be talking about ellipses and
00:37	then we'll be talking about hyper Poulos and each one
00:39	of those shapes can be obtained by taking a cone
00:42	. Just a regular old cone and slicing the cone
00:45	in different ways . And I've I've actually done a
00:47	demo of that to show you So again you might
00:49	be saying , why do we care so much about
00:51	Parabolas and specific ? Well it's because Parabolas are all
00:54	around you . The shape of any baseball . When
00:56	you throw it , it kind of arcs up through
00:58	the air and comes back down . That is the
01:00	shape of a parabola . Right ? When you actually
01:02	go into physics and you study motion and how things
01:04	move in gravity , it traces out the shape of
01:07	a parabola . Another example from physics . When you
01:10	learn about energy , kinetic energy , it's called one
01:12	half M is the mass of the object . One
01:15	half M . V squared . So anytime you have
01:17	that square term it's a parabola kind of shape .
01:20	In that case it's energy and velocity . Right ?
01:22	But anyway , these parabolas pop up all over the
01:25	place . So we're going to spend a lot of
01:27	time talking about more detail . Why is that shape
01:30	of the parable is so special ? Why do we
01:32	care ? Why is it so important ? So ,
01:34	we're in the beginning we're gonna review what we already
01:36	know about Parabolas . Then I'm gonna give you kind
01:38	of the basic overview of the equations of Parabola in
01:41	terms of other things we're gonna learn called the directories
01:44	and the directorates and the focus of the problem .
01:46	Uh and then we're gonna go through some derivations .
01:49	I'm gonna show you where that shape comes from .
01:51	If I had to boil down this section Into one
01:53	sentence , it would be why is the shape of
01:55	a parable is so special ? And what is that
01:58	shape ? How do we find out what that special
02:00	shape is ? All right , So let's get started
02:03	. We want to talk about the we want to
02:06	recall uh the basic shape of a problem , right
02:12	? We have done this before , but now we
02:14	are going uh one or two levels deeper into the
02:17	concept of a problem . So we're gonna review really
02:19	quickly what we know . We know . The simplest
02:21	problem that we can have is why is equal to
02:24	X squared ? Or you could replace the Y with
02:26	F . Of X . The function F of X
02:28	is equal to X squared , right ? And this
02:30	parabola is the simplest one that we can have .
02:32	I'm gonna draw a very small autograph here , nothing
02:34	too big here . But if we wanted to draw
02:36	the basic shape of a problem , we know this
02:38	problem touches the X axis , it goes down something
02:41	like this and goes up something like this . It's
02:43	kind of a smiley face notice . Now , this
02:45	is not a perfect drawing , it's got a little
02:46	kink here in the bottom . You can imagine a
02:48	smooth drawing there . The shape of a parabola is
02:51	not a semicircle . A semi circle would be something
02:54	that would go more down like this and more straight
02:57	up . This is much more gradual . Opening is
03:00	what a parabola is now . Of course , that's
03:02	the basic problem . But you know , we can
03:04	change the shape of that Parabola , make it close
03:07	up narrow or open up wider , we can flip
03:09	it upside down so it opens up down and we
03:11	can also take this parabola and we can move it
03:13	anywhere we want in the xy plane just by changing
03:16	the equation of the parabola . So we learned in
03:19	the past that the more general form , the general
03:24	form of the problem , it's something like this ,
03:28	It's why minus K . Is equal to a X
03:31	minus H quantity square . Now , if this looks
03:34	unfamiliar to you or totally your lost already and you've
03:38	never seen it before then it just means you need
03:40	to go back to my lessons on Parabolas . We
03:42	talked about that at great length . I did probably
03:43	10 lessons on it . We talked about what every
03:46	little part of that equation means how to sketch parabolas
03:48	and all of that . But we never talked about
03:50	why the shape of it is so special and why
03:52	it's so important and what the shape of a parabola
03:55	really is . We learned how to sketch it but
03:57	we didn't really talk about why geometrically it works why
04:01	it pops up in nature so much and what's so
04:03	special about it ? So this is the general form
04:06	in an example of that . For an example of
04:12	a practical problem in this form would be something like
04:15	this . Just one example off the top of my
04:16	head . Why minus three is equal to two times
04:19	x minus one quantity squared . So notice you do
04:22	have this x minus one , but this X term
04:25	in general is square . That means it's a parabola
04:27	. The three and the one tell you that this
04:30	parabola is shifted uh in the xy plane , and
04:34	the two in front of the princess tells you that
04:36	it's actually closed up on itself a little more than
04:39	this one right here . So if I were going
04:41	to uh and also because this is a positive number
04:44	out in front , the Parabola opens up if it
04:46	were negative to , it would open down like a
04:48	frowny face . Right ? So the general idea of
04:51	the general shape of this parabola , this is not
04:54	going to be a detailed graph , but just this
04:57	is all review . This is stuff we've done before
04:59	. The x coordinate is shifted one tick mark over
05:03	in the positive one uh direction and then in the
05:06	positive three here and for why 123 So that means
05:10	the bottom of this problem is X is equal to
05:12	one , why is equal to three . We talked
05:15	a long time ago about these minus signs and why
05:17	it shifts it in the positive direction . So we
05:19	know that the vertex the bottom of the problem is
05:21	now here and this problem has a positive too .
05:25	So instead of opening up so broadly like this one
05:28	, I'm just gonna guess here , this is probably
05:30	going to open up a little bit more steeply .
05:32	So you see the higher the number in the front
05:35	of the parentheses , Closes the thing up more .
05:38	If you had 10 or 20 up here it would
05:39	be really , really narrow . And of course if
05:41	it were negative , the whole thing would flip upside
05:43	down and go like a frowny face like this .
05:46	So when this number in front is positive , it
05:48	opens up when this number in front is negative ,
05:49	it opens down everything that we have just discussed ,
05:54	including the vertex the vertex notice you can read the
05:59	vertex directly out of the equation is one comma three
06:02	, everything one comma three , everything that I've just
06:05	talked about , we've covered at great length already .
06:08	If you don't have any idea what I'm talking about
06:09	it already , then please go back to those lessons
06:12	on parabolas . Our goal now is to go a
06:14	little deeper , that's what we know so far ,
06:17	but we want to go a little bit deeper .
06:18	And so to do that , I have a bunch
06:20	of things that I have to cover kind of in
06:21	sequence . Uh and the first thing I wanna do
06:24	is I want to give you a taste . We're
06:26	gonna revisit this board . I almost never ever just
06:29	put things on the board and have you read them
06:31	. I don't like doing that , but in this
06:33	case I have to because there's so much information I
06:35	have to get across , so I have to make
06:37	sure it's all done correctly . Right , So here
06:39	we have the general equations of a problem . I
06:41	want you to ignore everything on the bottom side here
06:44	right now and just focus on the top . Basically
06:47	, this is the general form of a parable equation
06:49	. This purple curve is a parabola , right ?
06:52	The vertex is the lowest point right here . That's
06:55	why it has a V . In the V .
06:56	Is that h comma K . Which is exactly ,
06:59	this is the same form of the equation we wrote
07:01	on the other board . So we had y minus
07:03	K and x minus H . Why minus K ,
07:07	X minus H . This is exactly what I've written
07:09	on the previous board . So this is all stuff
07:11	we've learned . Um But I guess I just want
07:14	to put it all in one place and tell you
07:15	this is the general form of the equation . This
07:18	whole relation between A and C . I'm gonna get
07:20	come back to a little bit later because notice when
07:23	you get over here uh on the on the on
07:26	the graph , you're gonna notice there's a couple of
07:28	additional things that we never learned about in the past
07:30	. We do have the parabola , but now we
07:32	have a special point above the Parabola called the focus
07:35	. We're going to talk about the focus of the
07:37	parabola in just a minute . So you can think
07:39	of the beams of light if you want to think
07:41	of reflected from that Parabola are focused at a point
07:44	, that's why it's called the focus . The focus
07:47	is not here or here or here or here or
07:49	here . There's only one focus of every Parabola and
07:51	it's exactly at that perfect spot , right ? Just
07:53	think of a focusing light or something . And then
07:56	on the other side of the vertex we have a
07:58	blue line which we call the directory X . I
08:01	don't expect you to know what that is , but
08:02	you just need to know that every Parabola we never
08:05	discussed it before , but every Parabola has associated with
08:09	it . Something called the Focus and also a line
08:12	called the directorates . And we're gonna talk at great
08:15	length what the directorate says , don't stress out about
08:17	it right now . But here you have a purple
08:19	parabola , It shifted some distance in X and y
08:22	. So we have the equation of the thing .
08:24	This is the same equation we learned before . The
08:26	vertex is given . The focus is given . We're
08:29	gonna talk about all of this later . The axis
08:31	of symmetry is given that's where you can cut the
08:34	thing in half . We're gonna talk about that later
08:36	. And then the directorate's I've given here as well
08:39	. Don't worry about what these things mean , I'm
08:40	gonna get to it later . And then this C
08:43	relates to the focus of the parabola , related to
08:47	the number that's in front of the parentheses there .
08:49	Remember when A is greater than zero ? When this
08:51	number is greater than zero . The parabola opens up
08:54	like in our example just a second ago . But
08:57	if the number in front of the problem is negative
09:00	, that's what this means . Then this problem goes
09:03	upside down , right , which we've done in the
09:05	past before . So in general you can have two
09:08	kinds of problems , you can have parabolas that open
09:11	up or down , which is really the only time
09:13	we've discussed so far in this class , but you
09:16	can also have problems that can open left and open
09:18	right and we haven't talked about that before . So
09:20	that's why I say we're going a level deeper .
09:22	So here we have a sketch of a problem that
09:25	in this case opens to the right right . So
09:27	you can see the equation of this parabola is very
09:30	similar to this one really . The only difference between
09:32	these equations is the UAE has been replaced with ex
09:36	and the X . Here has been replaced with Y
09:38	. So when you flip something on its side like
09:40	that , what you're doing is you're interchange in the
09:42	X and Y . Variables . And because if you
09:45	tilt your head sideways , really kind of just pretend
09:47	that this is the X . Axis and this is
09:49	the Y axis then it looks the same is the
09:51	other one . So flipping the variables around , does
09:54	the job of tilting the thing on its side .
09:55	When you do your problems in algebra with problems you're
09:58	gonna have some Prabal is tilted to the side and
10:00	some problems that are going up or down . But
10:03	anyhow this is the equation of a sideways probable .
10:06	You have the same relation between the number in front
10:09	and the focus . We're gonna talk about that later
10:11	. We talk about the vertex and the focus of
10:12	the problem . I'll discuss it later . The axis
10:15	of symmetry cuts this thing in half . I'll talk
10:17	about that later . And then there's an equation for
10:19	the directory which is the blue line here . Okay
10:23	now when A . Is greater than zero , when
10:25	this number is greater than zero , the Parabola opens
10:27	to the right because these are the positive values of
10:30	X . And when it opens with a negative A
10:33	then it's flipped around the other way and it opens
10:35	to the left toward the negative values . That's the
10:37	same thing up here when A is positive , the
10:39	thing opens upwards towards the positive Y . When A
10:42	is negative , it opens upside down to the negative
10:45	towards the negative Y values . So I'm putting all
10:48	this on the board to put it all in one
10:50	place , you have the equation of the problem .
10:51	You have the vertex coordinates , you have the focus
10:54	coordinates , you have the axis of symmetry equation and
10:57	you have the equation for the directorate's noticed that this
11:00	this probably goes up and you have a directorates that
11:02	goes horizontally . This equation has a probably going to
11:06	the right with the focus right here and you have
11:08	a director X behind it . So really if you
11:11	tilt your head is exactly the same picture as what's
11:14	above ? You have focus vertex directory X focus vertex
11:18	directory . Ex I have to put it all there
11:21	because you're going to have to do problems with both
11:24	kind of situations . But what I want you to
11:26	do now is kind of forget about this for now
11:28	. Just in the back of your mind , remember
11:30	, okay , we're going to have to talk about
11:31	horizontal and vertical parabolas , we're gonna have to talk
11:34	about focus , vertex directory , X , axis of
11:38	symmetry . Those are all the things you're gonna have
11:40	to do in all of your homework problems . So
11:42	keep that in the back of your mind that we're
11:46	going to get back to that we're gonna circle back
11:48	at the end of this lesson and you will understand
11:51	everything on that board in exquisite detail , but I
11:54	have to guide you there , okay If I just
11:56	throw it at you and say , hey good luck
11:58	, you'll never do , you'll never know what you're
11:59	doing . So go on a journey with me .
12:01	It's gonna be a little bit of a long lesson
12:03	, but we will get there . So the next
12:05	thing I want to talk about going down that road
12:07	is why is the shape of a parable is so
12:10	important . It's important for lots of reasons , but
12:13	one of the biggest reasons honestly that parabolas are so
12:17	useful in everyday real world um situations is every problem
12:23	has what we call the focus of the preamble .
12:28	Now , a focus of a parabola is very easy
12:32	to understand . But in the back of your mind
12:33	, I want you to remember that when we get
12:35	down to ellipses later on , ellipses have foa foa
12:39	foa foa C two focuses as well . And also
12:43	hyperbole to have focuses as well . So it's not
12:46	like problems . And the only thing that has a
12:47	focus , a parable has one focus one dot .
12:50	And the lips actually has two of those focuses ,
12:53	we call emphasize the plural of focus . Uh and
12:56	then hyperbole also have to focus . I so the
12:58	concept of a focus , we're gonna it's gonna stick
13:00	with us as we talk about ellipses , and also
13:03	hyperbole is down the road . And actually the circle
13:06	has a the circle that we've been talking about forever
13:09	. Also has a focus as well . It's just
13:11	the center of the circles . We don't call it
13:13	the focus , we just call it the center .
13:15	Right ? So all of these comic sections have something
13:18	called the focus , right ? But the parable is
13:20	focused is super super important . Now , let me
13:23	try to draw this , It's not gonna be perfect
13:26	, but I'm gonna try to draw a good shape
13:28	problem . Is this a perfect problem ? No ,
13:30	it's not . I can almost guarantee you because the
13:32	bottom here is too flat , but it's my best
13:35	shot at a free hand problem . So if this
13:38	were a problem , the focus would probably be somewhere
13:40	right around here . I'm just guessing because I have
13:43	I don't have graph paper and all that . But
13:44	let's just say the focus is right here somewhere .
13:47	Every parabola shape , whether it's really wide open or
13:51	really , really steep , is gonna have one focus
13:53	at one location , right ? The focus is called
13:56	the focus because it takes if you can imagine this
13:59	thing being a radio dish , which is really one
14:01	of the main reasons we use Parabolas in real world
14:04	. We all of your satellite dishes from the gigantic
14:07	radio telescopes , we have all the way down to
14:09	the small satellite dishes for your television or for your
14:13	whatever kind of whatever dishes you see on a tower
14:15	somewhere . They're all Parabolas and the kind of the
14:19	receiver that's in there , or the transmitter is at
14:21	the focus of that Parabola . It's the thing that's
14:24	kind of suspended in the centre , right ? So
14:26	this thing would be the transmitter and the receiver and
14:29	that means that any light waves or radio waves that
14:31	come in are gonna bounce off of this curve surface
14:34	and they're gonna go this direction towards the focus .
14:38	But I also have a light beam , not just
14:40	right here , but I have light beams everywhere .
14:41	I have a light ray coming in here like this
14:44	and it's gonna bounce off that bottom and hit the
14:47	focus as well notice it's curved differently than it is
14:49	right here . If I take another one to the
14:52	other side , it's gonna bounce off and hit this
14:55	right here and I know that it's a little hard
14:57	to see because my parable is not perfect , but
14:59	you can see because it's constantly curving no matter where
15:01	I stick a light beam , it's gonna bounce and
15:03	it's gonna hit this focus point right here , I
15:06	can take one way over here , in fact ,
15:08	and it's still going to come off and bounce and
15:10	hit into this guy . So that means that if
15:13	I create a parabola in a special shape of the
15:15	problem and I put the transmitter or the receiver ,
15:18	like when you look at the big radio telescopes ,
15:20	there's always like the scaffolding with the thing hanging in
15:24	the middle , like above . That's because that's the
15:27	focus of the problem because if I'm going to receive
15:30	light or radio waves from space , it's gonna concentrate
15:34	them at the focus . So I can I can
15:35	hear them better because I'm concentrating on like a magnifying
15:38	glass would concentrate light , right ? Or if I
15:41	want to broadcast something , this whole thing works in
15:43	reverse . If I'm gonna shoot energy out of here
15:46	, no matter which direction I shoot it towards the
15:48	dish is gonna bounce it this direction . So you
15:50	can think of , you know , like the Death
15:52	Star in Star Wars is not a great example ,
15:54	but that's kind of like the focus of the of
15:56	the Parabola . You can see it kind of bouncing
15:57	in and going out or coming in and bouncing up
16:00	to the receiver . That's the focus . And as
16:03	I said , Parabolas have a focus . Ellipses have
16:05	Phuoc tuy focuses . Hyperbole also have to focus .
16:08	I so focus is a central thing for comic sections
16:11	, circles have a single focus also , it's just
16:13	at the center right of the thing . All right
16:16	, so this is a useful feature of a parabola
16:19	, but what is the special shape ? Clearly the
16:22	special shape is not a circle . This does not
16:24	look like a semi circle . If you could think
16:26	about a circular shape , what does a circle look
16:28	like to you ? A circle looks something kind of
16:31	like this ? So this means a semicircle is the
16:33	bottom of this thing . A semicircle would be something
16:36	kind of like this actually , that's not even a
16:38	semi perfect semicircle , It should be calling more up
16:40	and down right there . This shape will not reflect
16:44	those rays in the proper way towards a focus .
16:46	Like a parabola . Does it has to be opened
16:49	up more into the shape of a parabola in order
16:51	to bounce everything into the focus right here , That's
16:54	called f the focus point right there . Um But
16:57	the question remains , what is the special shape ?
16:59	What is so special about it ? How do we
17:01	define with the special shape is and how do we
17:04	know that ? Why is equal to X squared ?
17:07	Is that shape that is the special shape that focuses
17:10	things ? That's what we really want to know ?
17:13	Okay , so here I'm going to write the definition
17:15	of the problem , I'm gonna write it right here
17:18	, and I'm gonna draw one more picture to set
17:20	up how we're going to derive this green curve .
17:22	I'm gonna we're gonna actually derive the green curve and
17:25	show that it's equal to this equation or to this
17:27	form of an equation . So the definition of a
17:30	parabola says parabola is the set of all points and
17:43	equal . So I'm gonna underline that an equal distance
17:50	from a point . Yeah , I'm gonna call that
17:56	point to focus . Focus is central to the concept
17:59	of a problem . Uh and a line and this
18:05	special line is called the directors , we're going to
18:12	read this a couple of times , we're gonna let
18:13	it sink in and then I'm gonna draw one more
18:15	picture to kind of set it up . A parabola
18:17	is just a shape . It's the set of all
18:20	points that define that shape . What does that mean
18:23	if this is a circle , all of the points
18:25	along this black curve define what the circle is .
18:28	If this is a Parabola , all of the points
18:30	inside the red curve defined what the curve is .
18:33	If this is a Parabola , all of the points
18:35	that define the green curve define the set of points
18:39	that we call this thing a Parabola . So we're
18:40	seeing a Parabola is the set of all points .
18:42	That means the green curve that are in equal distance
18:46	from a point called the focus . And a line
18:49	called the directorate's . Now I haven't drawn the directorate's
18:52	here , but you can see the focus is here
18:53	, the directory is always somewhere behind the parable .
18:56	In fact when you look , I know I told
18:58	you to forget about this , but if you see
19:00	every parable you have you always have a focus up
19:03	above . And then on the other side of the
19:05	rear end of the parabola , you have this blue
19:08	line called the directorate . So you can kind of
19:10	think of the directorate's is just kind of like this
19:12	blast shield that's kind of a high . If you
19:14	think about this thing is like a death ray or
19:16	something like a gun trying to shoot energy off like
19:18	a death star or something into space or something ,
19:21	then it's gonna shoot everything this way and back behind
19:23	it . Is this thing called the Directory . If
19:25	you tilt it off two X . Side , you
19:26	have a focus here . You're shooting your energy out
19:29	this way and you have a directorate that's kind of
19:31	behind it . That's the line that defines the other
19:34	. It's it's in the definition of a parabola to
19:38	define what the shape of a parabola is . In
19:40	other words , another way to say it is if
19:43	I take any line I want and any point I
19:46	want , which I'll call the focus . Then given
19:48	any line and any point I can always choose or
19:52	I can find that beautiful parabolic curve that fits between
19:55	them . Like these do . That is the perfect
19:58	shape of a problem , meaning it will always reflect
20:00	incoming rays to the focus , just like in my
20:03	diagram right here . All right , so let's take
20:07	this definition problem is the set of all points ,
20:10	an equal distance from a point called the focus and
20:13	a line called the directory . So let's take those
20:15	words and let's translate them into a picture . And
20:18	then once we translate them into a picture , we
20:21	will have what we need to derive and figure out
20:25	what the shape of this curve actually looks like ,
20:27	which is what I'm trying to get before we get
20:29	to that . Let's draw one more picture of this
20:33	whole situation . That will make it a little more
20:35	clear . So I'm gonna redraw what I have above
20:38	with the green curve . Right ? I'm gonna draw
20:40	a parabola and it's not going to be perfect ,
20:42	forgive me because I'm not good at drawing things .
20:44	I cannot draw Perfect parable is by hand . So
20:47	, this shape I'm gonna call some kind of problem
20:49	. Okay . Some distance above and kind of inside
20:54	the bowl is some point . A special point called
20:57	the focus that's going to accept all of the incoming
21:00	rays is gonna focus it at that point . So
21:02	this point is called the focus , which we discussed
21:06	before . Right now , the distance between the focus
21:11	and the vertex , which I haven't really talked about
21:13	yet , but there's a point . The lowest point
21:14	of the problem here is called the vertex which we
21:19	discussed when we did parabolas before . So the focus
21:23	is always some distance above the the vertex at some
21:30	distance see . So I'm calling it see , because
21:34	depending on the shape of the parable of the focus
21:36	will be in different locations , but it's got to
21:38	be some distance above it . So we just call
21:40	that distance , see . But whatever distance this is
21:43	right here , there's always a special line on the
21:46	back side underneath the rear end which we call the
21:49	directorate's , it's always a line that goes horizontal .
21:53	If it's a horizontal probable a vertical , if you
21:55	understand what I mean . If it opens upwards ,
21:57	it's a horizontal line . If it opens sideways ,
21:59	it's a vertical line . But the interesting thing and
22:02	the special thing is that the distance between the vertex
22:06	is also a distant sea to the directorate . So
22:09	this thing is called the directorate's Mhm . Okay ,
22:16	so it's crucially important for you to understand that the
22:19	vertex is always halfway between the focus and the directorate's
22:24	I'm gonna say that again , the vertex , the
22:27	lowest point of the problem is always halfway between the
22:31	focus and the directorate's I'm gonna say it a third
22:33	time , the vertex is always halfway between the focus
22:37	and the director . So here's the focus . Here
22:39	is the Director X . This is a distant sea
22:41	, This is a distant sea . So the vertex
22:43	has to be in the middle , right . It's
22:45	always halfway between like this . So I can even
22:48	write that down and I can say the vertex Is
22:50	always 1/2 the way the between focus and direct tricks
23:03	. So now we have a pretty complete picture ,
23:04	we have a parabola drawn on the board , we
23:07	have a focus of the Parabola which is going to
23:09	focus all the incoming light beams or radio waves .
23:11	We have a directorates which sits on the back side
23:14	and we're saying that there's a special shape which is
23:17	the blue curve which we call a parabola . And
23:19	we're saying the definition of a parabola is the set
23:21	of all points . That's the blue line , an
23:23	equal distance from the focus and the Director X an
23:28	equal distance from the focus . And the directorate's ,
23:30	I'm saying it again to get it into your head
23:33	. A parabola is the set of all points for
23:35	that blue curve . That is always every point on
23:38	that curve is always an equal distance between the focus
23:42	and that line that we call the directory . So
23:45	let's talk about how we make that happen . What
23:47	this thing is saying is that this blue curve is
23:52	the set of all points . Were calling a problem
23:54	, The distance between this focus to this point on
23:59	the problem is the same as the distance between this
24:02	and this . Now if you my drawing is not
24:05	perfect . So it looks to me like this line
24:07	is a little bit shorter than this one . But
24:09	if I had opened up my problem or maybe move
24:11	my focus a little bit more , exactly this line
24:14	should be exactly the same as this one . Now
24:16	, in geometry , the way you denote two lines
24:19	being congruent is what we call it . A geometry
24:21	. We're gonna put a little line through there .
24:23	In the line through there . That means this distance
24:24	is the same as this one . This one that
24:27	means that the parable is the set of points .
24:29	That's an equal distance from the focus and the directory
24:32	. So this is an equal distance . This point
24:34	here is an equal distance to the focus . And
24:37	also to the directorate's What if I pick a different
24:39	point on this Parable ? What we're saying is from
24:42	here and then straight down here are equal distances .
24:46	This point on the curve we're calling the parable is
24:49	an equal distance to the focus as it is to
24:51	the directorate's . What if we go crazy , we
24:54	pick a point way over here . Well , that's
24:56	okay , This is farther away . Sure , it
24:58	is . But so is this one this one's farther
24:59	away to the black line . So this is an
25:01	equal distance from there . And they're now of course
25:04	it doesn't just happen on the left hand side .
25:06	It happens on the right hand side too . So
25:07	I'll go ahead and draw one going way over here
25:10	and then won going away over here . This distance
25:13	is the same as this one . So you can
25:15	see that what we have done is effectively we've picked
25:19	a focus in space and then we've picked a Directory
25:22	X . And when you pick a focus and you
25:24	pick a directorates , there always has to be a
25:26	special curved path that goes between them where the bottom
25:30	of the thing goes directly between the focus and the
25:32	directory X . Right . But there has to be
25:34	a special shape so that every single point on this
25:37	blue curve , every point on this curve is an
25:40	equal distance to the focus as it is to the
25:42	directorate's equal distance to the focus as it is to
25:45	the directory at this point at this point this point
25:47	this point this point every point is always an equal
25:49	distance from the focus to the directorates . And that
25:53	is why Parabolas focus energy towards the focus . Focus
25:57	, incoming parallel rays to the focus . Because it's
26:00	especially constructed shape that always is the same distance to
26:04	the focus to the directorates . And when you do
26:06	the math and go through how everything is reflected .
26:08	It turns out that that focuses all of the incoming
26:11	energy into a point we call the focus and that's
26:14	why we call it the focus . This kind of
26:16	thing is not what you learn when you first learned
26:19	about what a parabola is . We talk about a
26:20	parabola , why is equal to X square ? You
26:23	just say , oh , it's a problem . Great
26:24	, we graph it , we talk about it ,
26:26	you know , even in calculus , you learn how
26:28	to do things with parabolas , but until you get
26:31	to a lesson like this , you don't understand why
26:33	we care about perhaps why they're so special . A
26:36	circle is another special shape . It's the set of
26:39	all points in equal distance from the center , an
26:42	equal distance from a single point . That's what we
26:44	call the special shape called a circle . A parabola
26:47	is a special shape where every point on that curve
26:50	is an equal distance to the focus as it is
26:52	to the directorate . So every parabola has a director
26:55	X . Every parabola has a focus and that's something
26:58	you have to get in your head . We first
27:00	learned about Parabolas , we don't talk about the focus
27:03	or the directorate . So it kind of seems like
27:05	we're adding it on . But in fact every parable
27:07	you've ever graft always had a directorates , even if
27:10	you didn't graph it and they always had a focus
27:12	, even if you didn't graphic . But here in
27:14	these problems moving forward , we're always going to talk
27:17	about the focus and the Director X as we graph
27:20	and sketch all of these problems . So we have
27:23	reviewed basically what we knew about Parabolas , we have
27:27	introduced the equations of a parabola , but I haven't
27:31	really explained a lot about this because I need to
27:33	talk about a few more things before I go into
27:35	crazy detail here . But now at least you understand
27:37	a little bit more that every problem has a focus
27:40	, It has a direct tricks , the distance C
27:43	. Is here , the distance C . Is here
27:44	. So the vertex is halfway between the focus and
27:47	the director . Same here . The vertex is halfway
27:50	between the director and the focus for every problem that
27:53	we have and so we've learned all of these things
27:56	and then finally we talk about this special uh property
27:59	of a problem which allows to focus incoming parallel rays
28:02	to a point . And then we talked about the
28:04	definition of a problem , which we've discussed a lot
28:08	so that you can get your brain wrapped around it
28:10	now . But we need to do is we need
28:12	to derive the shape , the equation of a parabola
28:15	. See , here's the geometric description , here's a
28:17	point . We call the focus , here is a
28:19	line called the directorate's . There has to be some
28:21	blue curve called a parabola that every point on this
28:24	curve is an equal distance from the point here to
28:27	the focus . And the point here to the directorate's
28:29	there has to be some special curve we call a
28:31	parable . What is the equation of that curve ?
28:34	Now , you already know the answer . The equations
28:36	of a parable . Always look like this . But
28:38	how do we go from the definition of the parabola
28:41	, which is all about geometry to showing that really
28:44	? This thing is and does describe all parabolas that
28:47	you can graph . How do we know that ?
28:49	And so what we're gonna do is we're gonna go
28:51	through a derivation of that . It's not hard ,
28:53	it's actually really easy to understand , but I do
28:56	have to do some drawings here in the beginning .
28:59	Uh so we're just gonna jump right into it .
29:01	So what we have is we have to draw another
29:03	sketch before we can derive the equation of a problem
29:06	. And this sketch , I'm gonna draw right at
29:08	the top of the board . It's not going to
29:09	be very um uh long here , but I do
29:14	need to get it on the board . So here
29:15	we have the xy plane . I'm trying to give
29:17	myself a lot of space down here to do the
29:19	rest of the work . So here we have the
29:21	xy plane . Now I have to pick some actual
29:24	numbers . So what I'm gonna do is I'm gonna
29:27	put the uh the focus of this parabola at 123
29:33	units along the X axis . So this thing is
29:35	called the focus , it's at three comma zero ,
29:38	right ? And then I'm gonna put the vertex At
29:43	3:02 . So this is the vertex at 3:00 to
29:48	now because this is the vertex and because this is
29:50	the focus , you know that the problem has to
29:53	be opening upside down because the focus is always kind
29:57	of inside the Parabola , the focus is always inside
29:59	the bowl . So if the vertex is here and
30:02	the focus is here , the only way the thing
30:03	works is if it goes something like this and I'm
30:05	gonna try to draw upside down the best I can
30:09	. Is this perfect ? No , it's not .
30:10	I can already tell this kind of opened up a
30:12	little bit weird . But anyway , that's the basic
30:14	problem . This is the highest point of the actually
30:16	, I'm looking at it again , it's completely lopsided
30:18	. Sorry about that actual let's try to let's try
30:20	to fix it just a little bit . So it's
30:23	gonna go off something like something like this . Still
30:27	not great . Sorry about that . Anyway , it's
30:30	an upside down parabola that goes something like this .
30:34	All right now , every problem has a focus and
30:38	every problem also has a directory . So let me
30:40	ask you if the focus is here and the vertex
30:43	is here , where is the director of this problem
30:47	? The Director of the problem has to be on
30:49	the other side has been on the backside . And
30:51	also we said that the vertex is always halfway between
30:54	the focus and the directorates . So if the focus
30:57	is here , the vertex is here , the directorate's
30:59	has to be the same distance away on the other
31:01	side . So that means that uh we have 12
31:07	so this is three , this is four . So
31:09	that means it needs to be a horizontal line up
31:11	here . This is going to be the directory .
31:13	So I'm gonna write this Director X . And what
31:17	is the equation of this directory ? X . Well
31:20	, this is 1 , 3 , 4 . So
31:22	this equation of directions , Directorates is y is equal
31:25	to four , it's a horizontal line , four units
31:27	up like this . How do we know the director
31:29	is actually there ? It's because the vertex always has
31:33	to be in the middle , So , if this
31:34	is two units , that this has to be two
31:35	units . And so when we locked down the vertex
31:38	we are , we know where the director says a
31:40	lot of these problems in algebra , always gonna be
31:42	like , tell me where the vertex is , and
31:44	you'll just have to know how things are set up
31:47	their equal distance on either side of the vertex or
31:49	whatever to write . The equation of the directorate's down
31:51	, which is what we did right here . All
31:55	right . So , what we need to do is
31:57	we need to realize that this blue curve is a
32:02	bunch of points , right ? It's a bunch of
32:04	points . Um And so what we're saying is that
32:08	this point , for instance , right . Whatever this
32:11	point is right here , I don't know exactly where
32:12	it is . It's over here and it's up here
32:14	, but there's some point right there in the curve
32:16	of the problem , right ? But I do know
32:18	one thing and that is the distance from here to
32:20	here to the focus is the same as the distance
32:23	up like this , and I'm gonna put little lines
32:25	to show me this . So this line Here ,
32:29	D one is the point on the direct tricks right
32:32	there . And what I'm saying is that the distance
32:36	between this point and the line directory is the same
32:40	as the distance here to here . Now , I
32:41	can tell you that they don't look the same because
32:43	I'm drawing it freehand . But if you can imagine
32:46	that I opened my problem up a little bit more
32:48	and it was an exact shape , then it would
32:50	exactly be correct , Right ? And then if you
32:53	want to pick another point , let's say at this
32:54	point right here , and this one even looks uh
32:57	looks even worse actually . You can see because this
33:00	one here , what I'm saying is the same as
33:02	the distance up there as well . It doesn't look
33:04	the same . And that is just because my parabola
33:07	is really just to it's too crunched and it's not
33:10	a real Parabolas shape . So if I wanted to
33:11	fix it , I could erase this and you know
33:13	, we could do that if you want . Doesn't
33:15	really matter too much . No , but you know
33:17	why not ? Let's just open it up , just
33:19	a little bit more . Something like this , that's
33:21	probably a little bit closer . Is it still exact
33:23	? No , it's not exact , but it's pretty
33:25	close . So let's go and do something . Like
33:28	let's try to erase this a little bit . So
33:29	the distance here to here is the same as distance
33:32	here . To hear the distance here to here is
33:34	the same as distance here and here . That's pretty
33:36	close . And we'll call this point D . Sub
33:39	two because that's where it hits that line . And
33:42	we'll call we'll do one more point this point here
33:44	. The distance between here and the focus is the
33:47	same as the distance up there to the directory .
33:50	So those lines are there and then this one I'm
33:53	gonna label as an actual point . I'm gonna label
33:56	this one right here , I'm gonna call it P
33:59	. X . Comma Y . And then up here
34:02	this deed is going to be the point on the
34:04	directorate's is an X . Comma . For now I
34:07	need to explain what I'm talking about here because pretty
34:09	soon I'm gonna use this in an equation . What
34:12	I'm saying is the focus is a lockdown point at
34:14	3:00 . The point on the Parabola is just some
34:19	X . Y . Location . If I look at
34:21	all these points here , they are all at different
34:23	X . Y locations . I don't know what this
34:25	point P . Is because I don't know the shape
34:27	of the curve yet but it's at some X .
34:28	Y . Location . Okay , this point here is
34:32	the same X . Value is the point because it's
34:34	straight up . That's why X . Is the same
34:36	here . But it's four units and why ? That's
34:39	why I had to put the number four here .
34:40	So it's an X . Comma fourth . In other
34:43	words the point where I intersect here is that whatever
34:46	the point is here X . Comma four units up
34:50	now . Why do I spend all of this time
34:52	writing this stuff down ? And that is because what
34:54	we need to do is figure out the equation of
34:57	this parabola . We want to derive this thing and
35:00	we know that the definition means that the distance from
35:03	every point on this parabola to the focus and also
35:06	to the directory is the same distance . So that
35:09	means if this is a point F , the distance
35:12	between F and P is the same thing as the
35:16	distance between P and D . This distance is the
35:19	same as this distance . This distance is the same
35:21	as this distance . This one is the same as
35:23	this one . This one is the same as this
35:24	one . This one is the same as this one
35:26	. Every point I pick on this parabola then the
35:29	distance from the focus to the point on the parable
35:32	is the same as from that distance from that same
35:34	point to the directory , X F p is equal
35:37	to P . D . Now we've learned fortunately about
35:39	the distance formula . We know how to calculate points
35:42	between distances between any points in space . Right ?
35:45	So what is the distance FP Well we have to
35:48	use the distance formula . We know this is X
35:50	comma Y and this is three comma zero . So
35:52	let's do the distance formula . It's gonna be the
35:55	difference in the X values . So we're gonna go
35:57	X -3 Quantity squared plus the difference in the y
36:02	values , Y zero quantity squared square root of this
36:07	. This is nothing more than the distance formula .
36:09	It's calculating the distance between f the focus and this
36:13	particular point right here . I don't know what the
36:16	coordinates are . I'm trying I'm going to create an
36:18	equation to figure out what these coordinates are . That's
36:20	what I'm trying to do but I don't know what
36:21	they are now . So I just take it as
36:23	X comma Y difference in the X value squared difference
36:26	in the lives value squared square with the whole thing
36:29	. This is the distance here . But I know
36:31	that this distance has to be the same as this
36:33	distance so I have to put an equal sign and
36:36	I'm gonna now calculate the distance between these two points
36:39	. Again . The difference of the X coordinates is
36:41	x minus x squared . Notice the x coordinates of
36:44	the same and then the distance the difference in the
36:46	y coordinates is Why -4 Quantity Squared ? It is
36:53	crucial that you understand this equation on the board because
36:56	what's going to happen is we're just going to spend
36:58	the rest of this time simplifying it and then it's
37:01	going to end up showing us that this equation of
37:02	a parabola is correct . Okay , So , what
37:05	we have here is the distance between F and P
37:08	. Just the distance formula . The the distance between
37:11	P and D . This is just the distance formula
37:14	. All right . All right . So , let's
37:16	crank through this . Now , we know that x
37:18	minus X is gonna give us zero squared . We
37:20	know that y minus zero is easy as well .
37:22	So , we're just gonna rewrite this X minus three
37:24	quantity squared plus y squared . Because the zero doesn't
37:27	matter . We still have a square root , This
37:30	is just zero squared , it disappears . So you're
37:33	just gonna have Y -4 quantity squared ? We're gonna
37:36	have square root of this whole thing . All right
37:39	. So all we did was simplify this . Now
37:41	, how do we go any farther ? We have
37:43	a square right on the left and a square root
37:45	on the right . So we're just gonna take and
37:47	we're gonna square the left of this equation . And
37:51	when we do that , then we also have to
37:53	square the right hand side of the equation . This
37:56	square is going to cancel with the square root .
37:58	This square is going to cancel with the square root
38:01	. And so what I'm gonna have after I do
38:03	that is just what's underneath X minus three quantity squared
38:08	plus y squared is why minus four quantity squared .
38:15	And so now you can see I'm not quite there
38:16	, but I'm getting it closer to what an equation
38:19	of a problem might look like . So when you
38:21	look at the actual answer , you see the y
38:23	values on the left and the X values are on
38:25	the right . So let's take all the Y values
38:28	and moving to the left and all the X values
38:30	and moving to the right . And so what we're
38:32	gonna have is when I do that is I'll have
38:35	y squared minus why minus four , quantity squared is
38:40	equal to negative x minus three quantity square . Make
38:42	sure you understand I'm holding this the same , I'm
38:45	subtracting this to get it to this side and I'm
38:47	subtracting this to get it to the right side .
38:49	So that's why there's negative in each location . But
38:52	I have all of the wise on the left and
38:54	all of the X . Is on the right .
38:57	All right . So um notice the right hand side
39:01	of this equation is x minus three quantity squared .
39:04	That's what a problem should have X minus something quantity
39:07	squared and there should be a number out in front
39:09	. In this case the number right now is negative
39:11	one . So actually the right hand side looks good
39:13	, the left hand side doesn't look good , it's
39:15	got too many squares and other things going on .
39:18	So what we have to do is expand this .
39:20	So the way you're going to do that is why
39:22	squared minus . And then what you're gonna have is
39:25	why minus four times why minus four . So we're
39:28	gonna use the binomial squaring stuff that we've done in
39:31	the past . We're gonna take the first thing squared
39:33	, Y squared minus two times Why times four plus
39:38	four times four is 16 . Yes . And so
39:43	we just squared this And then on the right hand
39:45	side we have a negative X -3 quantity squared .
39:49	Okay let me just catch up , make sure I'm
39:51	correct . So then we're going to simplify further ,
39:55	we're gonna say well what we're going to have is
39:57	y squared this negative is gonna multiply in making this
40:00	negative Y squared . This negative multiplies in making it
40:04	positive but this is two times y times four .
40:06	So it's gonna be eight , Y negative times negative
40:09	positive . Then we have negative times this gives me
40:11	negative 16 and then this is x minus three quantity
40:16	squared like this now notice we have y squared minus
40:21	y squared , this goes to zero that disappears .
40:24	And then on the left we have uh eight y
40:28	minus 16 . What we wanna do is factor out
40:30	the eight because we have an eight and 16 here
40:32	. So let's factor out the eight . And you
40:34	have y minus two on the left because eight times
40:36	two is the 16 on the right hand side .
40:39	You'll have negative x minus three quantity squared . And
40:43	then let's divide by the eight . So we're gonna
40:46	be left with y minus two is equal to negative
40:49	1/8 . Because I'm gonna divide both sides by eight
40:53	x minus three quantity squared . Now look at what
40:56	we have done , all we did was draw a
40:58	picture and we said the distance from any point on
41:01	this path , which we're calling p to the focus
41:03	is the same thing as this point . The distance
41:06	from the point to the directorate's , we did the
41:08	distance formulas and then all the rest of it was
41:10	just simplifying . And we end up with this ,
41:12	this looks like the equation of a parabola . Why
41:15	minus two is negative 1/8 x minus three quantity squared
41:20	. The example we gave here is in the same
41:21	form why minus something is a constant times X minus
41:24	something quantity squared . In this case we have the
41:27	constant in front that's negative . And so that's why
41:31	it opens upside down . We have a shift that's
41:34	uh 23 units to the right . The vertex will
41:37	be three units to the right . And um and
41:40	uh two units up , so three units to the
41:43	right to units up , three units to the right
41:45	to units up . That's where the vertex is .
41:47	Of this thing . It opens upside down , this
41:49	is the equation of this parabola . So from this
41:52	you can generalize and you can say you can say
41:57	that in general the form is of why minus K
42:00	is a X minus H quantity squared . So what
42:05	we have done is we figured out from the geometric
42:08	definition of what a parabola is , the set of
42:10	all points and equal distance to the focus as it
42:12	is to the directory X . But we've used the
42:16	distance formula to basically make an equation to find out
42:19	that all parabolas have this form . The vertex will
42:21	be shifted according to these X and Y numbers here
42:25	and then the A in the front is going to
42:27	represent how open or close it is . And also
42:30	if it opens up or down . So let's just
42:32	take a second to take a look at what we
42:34	have here and compare it to what we have over
42:39	here . We said the equation of a problem that
42:41	opens up like this looks like this . That's exactly
42:44	what we've written down based on that example . The
42:46	vertex is that h comma K . That's the shift
42:49	in X . The shift and why that's why the
42:51	vertex is H comma K . The focus has to
42:55	be the same distance over in in terms of where
42:58	the vertex is . So the first coordinate has to
43:01	be a church because the vertex is here . So
43:03	for the focus it has to be a church .
43:05	But the y coordinate of the focus has to be
43:07	whatever K is plus this number C . Right ?
43:12	So it's not so helpful to to see it written
43:15	like this , but that's why it's written like this
43:17	. The vertex is hk the focus is H K
43:19	plus C because it's this point plus C units up
43:23	. And why ? That's all it is . The
43:25	axis of symmetry is that X is equal to h
43:28	. It's a vertical line that goes through the focusing
43:30	through the vertex . That's why the axis of symmetry
43:33	is the vertical line X is equal to h this
43:36	coordinate right here . And then the director X is
43:39	the line over here , but it's horizontal line .
43:41	So it has Y equals something . What is going
43:44	to be equal to ? Well , it's gonna equal
43:46	to wherever the vertex is . But because of the
43:49	horizontal line , why is equal to K would go
43:51	right through the vertex . But it's not that line
43:54	, it's k minus C . It's shifted down .
43:56	So the directory is wherever the vertex is a horizontal
44:00	line of wherever the vertex has shifted down , the
44:03	focus is the point , wherever the vertex is shifted
44:05	up , the axis of cemetery goes vertical through both
44:10	of these points there . And then if A is
44:12	greater than zero , it opens up and if A
44:13	is less than zero it opens down . It's a
44:16	monster right to to derive it all and cram it
44:19	all into one lesson . But now you know where
44:21	everything comes from with a practical example . Now I
44:25	would love to be able to stop here and just
44:28	say go do all of your problems . However ,
44:31	every class is going to give you problems that has
44:35	a sideways parabola as well and that's honestly it's just
44:39	not fun to have to do it all in one
44:41	lesson but I need to get that out to you
44:43	as well . So what we're gonna do is we're
44:46	gonna do the same thing that we did before .
44:48	We're gonna derive this equation , which is going to
44:51	be very easy to do . Now , you know
44:52	how we did the first one ? And we're gonna
44:54	find out that the equation of a sideways parabola is
44:57	this one , and we're gonna talk about how the
44:59	vertex and the focus and all that stuff makes sense
45:01	there as well . So in order to do that
45:05	, I have to draw like I did here ,
45:07	I had to draw a picture of a parabola and
45:09	I had to do the calculations . Now we have
45:10	to draw another picture of a parabola . Uh and
45:13	do the calculations on that one as well . So
45:17	what we're going to have to see if I can
45:18	draw this thing right ? So what we're gonna have
45:20	is an X . Y coordinate . It's not going
45:23	to be perfect , I apologize for that in advance
45:27	. So what I wanna do is I want to
45:29	find and this could be some kind of a test
45:31	question . Find the equation of Parabola with focus at
45:44	zero comma negative two and X is equal to three
45:49	as the director X . All right . So what
45:55	we have figured out what the problem tells us is
45:57	find the equation of the problem with the Focus at
45:59	zero comma negative too . So we can write that
46:01	down right away . Zero comma negative two is going
46:04	to be down here . That's going to be the
46:06	focus of this thing . And I could put an
46:08	F there to tell you that that's the focus .
46:10	Uh Zero negative two . X is equal to three
46:13	is the directory . So here's one , here's to
46:15	here's three . So I'll put a it's X is
46:17	equal to three means it's a vertical line . So
46:22	this is X . Is equal to three . This
46:23	is the Director X . Okay . So because this
46:29	thing has a focus here and the directorate's here ,
46:32	you know , the parabola has to open to the
46:34	left because the focus has to be inside the bowl
46:37	, so to speak . And it also has to
46:39	be where the backstop , so to speak . The
46:42	directorate's is in line on the other side of the
46:44	rear end of the problem . So the problem has
46:47	to go something like this . But also we've learned
46:49	many times over . Mhm . That the vertex is
46:52	always halfway between the focus and the directory X .
46:56	So in this problem we actually know what the focus
46:59	is and we know what the directorate's is . And
47:01	so we know there's three points here . So this
47:04	down here , this point right there . In between
47:07	these tick marks , down there has to be the
47:09	vertex . So I'm gonna put here this is the
47:12	vertex . And what is the coordinates of that vertex
47:15	? It has to be 123 units , but I
47:17	have to cut it in half . So the vertex
47:19	is at three halves comma negative to , the vertex
47:23	is at three halves comma negative too because it has
47:25	to be in between the focus and the directorate's right
47:28	now , what does this problem look like ? I
47:29	always mess these things up so I'm gonna just ask
47:33	you to accept my apologies you know already , but
47:36	it has to go something like this and it has
47:38	to go through the vertex so it's going to do
47:40	something kind of like this is gonna go down ,
47:42	it's gonna flat now , it's gonna come out like
47:44	this is the perfect No , I think I drew
47:45	it more where the the vertex is more like here
47:48	, you have to use your imagination and pretend that
47:51	I'm a good artist and I'm not a good artist
47:53	. So sorry about that . So it's gonna be
47:54	something kind of like this that's two kinked . But
47:57	anyway , you see the vertex is right there at
47:59	the lowest point . The thing opens up like that
48:02	. Now the definition says that every point on this
48:05	purple curve is an equal distance to the focus as
48:10	it is to the director . So for instance ,
48:11	down here is a point on the curve . We
48:13	can call this point P . X , comma Y
48:17	. Right . And what this is basically saying is
48:20	that like I messed up my little curve here .
48:22	Sorry about that . What this is basically saying is
48:25	that the distance from this point on the parable to
48:27	the focus is the same as this distance from the
48:30	point to the directory . So I'm gonna put a
48:32	little a little lie a little tick mark line .
48:35	This means that's the same distance as this guy and
48:38	that holds for every other point on here . I
48:40	can draw additional points if you want . I can
48:42	pick a point up here and say , well ,
48:43	this point right here is an equal distance to the
48:46	focus as this point is to the directorate's those are
48:49	equal distances . And this purple curve traces out all
48:52	possible values of this thing we call the Parabola ,
48:56	then every point on here is an equal distance to
48:59	the focus as it is to the director . It's
49:00	that's what it means to be a parabola . All
49:05	right . And so we're gonna do like we did
49:06	in the last problem or we basically said , well
49:09	the distances have to be equal . So we're gonna
49:11	set the exact same thing up and we're going to
49:14	say that if this is a p p m ,
49:17	p X . Y . And this is F .
49:18	And this is the point of the directorates . In
49:21	this case if it's down here it's going to be
49:24	, this point's gonna be d . It's gonna be
49:26	at three comma . Why ? Why three comma ?
49:29	Why ? Because the directory is at 123 ? That's
49:32	the X coordinate at this point right here . The
49:34	Y value . I don't know because it depends on
49:36	basically wherever I'm tracing out , whatever point I pick
49:39	is gonna dictate the Y value . But the X
49:41	value is always gonna be the same because the directory
49:44	is always a vertical line at X . Is equal
49:46	to three . So what we're saying is that the
49:49	focus to the point P . Is the same distance
49:53	as the point P . To the directorate , same
49:55	equation that we use before . So we have to
49:57	use the distance formula . I'm gonna have to scooch
50:00	down a little bit to make sure I have room
50:03	. What is the distance between the point F and
50:06	the point P . Well , the point F .
50:09	Was given in the problem statement . The focus was
50:11	at zero comma negative too . So what we're going
50:15	to find is the distance between FP . So let's
50:18	go and take the difference in the X values here
50:20	. So it's going to be x minus zero quantity
50:23	squared , X minus zero quantity squared plus the difference
50:27	in the Y values . Why minus a negative too
50:30	, quantity squared ? We have to take the square
50:32	root to find the distance between those points . But
50:35	we're saying that that distance is the same as this
50:37	distance . So we're going to take the difference in
50:39	the X values here , X minus three quantity squared
50:43	plus the difference in the y values which is just
50:45	y minus y quantity squared . And we're gonna have
50:49	a radical on top of both of those things .
50:52	Now let's clean it up a little bit . What
50:54	we're going to have , X zero is just gonna
50:56	be x squared . And then this is gonna be
50:58	y plus two squared . We're still going to have
51:03	a radical on the right hand side . Let me
51:06	switch colors . Is giving a little bit hard to
51:08	see X -3 Quantity squared , this becomes zero ,
51:12	there's nothing else there . So we still have the
51:15	square root outside of this guy . Now we have
51:19	this radical on the left and radical on the right
51:21	. So how do we get rid of the radical
51:22	? Same as before ? We just square the entire
51:25	left hand side of the equation and then we're gonna
51:27	have to square the entire right hand side of the
51:29	equation so there's still an equal sign between here like
51:32	this . And so the square is gonna cancel with
51:35	the radical and this square is gonna cancel what the
51:37	radical . So really all you have is what's left
51:39	underneath , the X squared plus y plus two squared
51:43	is equal to x minus three . It's weird .
51:47	Yeah . All right . So now what we want
51:50	to do is we want to rearrange terms . Now
51:52	in the previous time we did , it was a
51:54	vertically oriented , probably we wanted to put all of
51:57	the Y values on the left and all the X
51:59	values on the right , and now we're gonna do
52:01	it exactly in the opposite way . We want to
52:04	take and move this term over here . In this
52:06	term over here . You'll see why in just a
52:08	second . So what we're going to have is the
52:10	x squared minus what's on the right , X minus
52:14	three quantity squared , We'll take this and we'll move
52:17	it to the right so it's going to be negative
52:19	Y plus two quantity squared like this . So all
52:22	we did was we move this term to the left
52:24	, this term to the right . And so now
52:26	we want to simplify this so so we want to
52:29	expand this term out as we did before . We'll
52:31	have a X squared minus two times X times three
52:37	plus three times three is nine . And then we're
52:40	gonna have negative Y plus two quantity squared . All
52:43	we did was square the spine . Amiel We've done
52:45	that so many times . You should be able to
52:47	do that in your sleep by now . Now we
52:50	have to distribute the negative end . We'll have ,
52:52	sorry this is an X square . I forgot to
52:54	write that down . So we'll have an X squared
52:56	minus X squared taking that negative end . This will
53:00	be a positive term two times three is six X
53:03	. And then the negative times the nine is negative
53:05	nine . And on the right will have negative .
53:09	Why ? Plus two Quantity Square ? We don't want
53:12	to expand this right hand side because it's already in
53:14	the form that we want it to be in for
53:17	a parabola . So this gives me zero like this
53:21	. And then I look at this the six x
53:23	minus nine , the six in front . I want
53:25	to factor out of six and you'll see why in
53:28	just a second , let's factor out of six .
53:29	When I do that , it's going to be x
53:31	minus 9/6 uh is equal to negative Y plus two
53:37	quantity squared . If you don't see this , just
53:40	make sure and go backwards six times X six X
53:42	six times this fraction the sixes will cancel . So
53:45	it'll just give you a negative nine . I'm just
53:47	factoring out the six and when you don't have it
53:50	doesn't go in evenly . Sometimes they have to write
53:52	that second thing as a fraction like this . All
53:55	right . So now we're getting very close this coefficient
53:58	in the front , we want to divide by it
54:00	and this 9/6 is also going to be able to
54:03	be written easily as well as uh This 96 is
54:08	the same as 3/2 . We're gonna divide by six
54:12	so it'll be negative 1/6 . Why ? Plus two
54:16	Quantity Squared ? So we have x minus three half
54:19	is equal to negative 1/6 . Y plus two quantity
54:22	square . So is this correct ? Well this is
54:28	similar to the general form of the equation of a
54:33	sideways parabola which is X um minus K . Is
54:39	equal to a y minus h quantity square . So
54:43	you can see what's going on here , right ?
54:46	The parabola that is oriented vertically up and down ,
54:50	so to speak , has the Y value on the
54:52	left . But the X term , the x term
54:54	is on the right and that is what is squared
54:56	for the sideways problem . Everything's flipped around . The
54:59	Y term is what's on the right , that's what
55:01	squared and the X term is on the left .
55:03	So it's totally written backwards to a typical equation .
55:06	And that's because you flip the thing sideways . So
55:09	one way to think of a sideways parabola is just
55:11	the same thing as a vertical parabola . With the
55:13	X and Y variables flipped , you can flip them
55:16	around . So if this were y and this were
55:17	X flip it around . That's going to make a
55:19	sideways version of that parabola but it still has a
55:22	coefficient in front . Uh in this case it's negative
55:25	, so because it's negative , it a positive value
55:29	. It open to the right . A negative value
55:31	opens to the left , which is what our drawing
55:32	had . The vertex of the problem . Is that
55:35	why is equal to negative two and X is equal
55:38	to positive three have , so X is equal to
55:40	positive three halves . Why is equal to negative two
55:43	? That's exactly what we have here . So you
55:44	see the shift and why goes with this , the
55:46	shift and X goes with this . And now that
55:49	we have done that , we can now look at
55:52	the sideways version of the parabola . The equation of
55:55	the problem . It looks exactly like the equation of
55:57	the other vertically oriented parabola . It's just that we
56:00	take the Y . Value and were replaced with X
56:02	. We take the X value and were replaced with
56:04	why ? Okay . The vertex is still at h
56:08	comma K . The X shift goes with the X
56:11	coordinate of the vertex . The Y shift goes with
56:13	the Y coordinate the vertex . So this is the
56:15	vertex right here . The focus has to be this
56:17	point , but plus a little more in the X
56:20	direction . So we have to add , see in
56:22	the X direction , K stays the same . The
56:25	directorate is si units in the other direction . So
56:29	it has to be a vertical line that is uh
56:33	H instead of plus C . It's going to be
56:35	minus E because it's going to be a line that's
56:37	going to be over here . In other words ,
56:38	a vertical line that goes right through this point would
56:40	be X is equal to h right through here .
56:43	But we don't want that line . We want it
56:44	to be si units this way . So we call
56:46	it h minus C . Okay . The axis of
56:50	symmetry for horizontal kind of problems like this is a
56:54	horizontal line , right ? Because it can't be a
56:57	vertical line that goes with vertically oriented parabolas , horizontal
57:00	parabolas have to have a horizontal line and it goes
57:03	through the point K . Why is equal to K
57:05	. Because it goes right through the vertex . And
57:07	then of course when A . Is bigger than zero
57:09	, it opens toward the positive X . Values when
57:12	a . Is less than zero opens towards the other
57:14	direction opposite of that . We have done a tremendous
57:17	amount of this lesson . And to be honest with
57:19	you , I don't like filming lessons this long but
57:22	I had to in this case because if I just
57:24	give you the equations , you won't understand what to
57:26	do with them . And if I just derive the
57:29	shape of the proble you won't know how to solve
57:31	any problems . And if I just review stuff like
57:33	we did in the beginning , then you'll just review
57:36	what we've learned and you won't go any farther .
57:38	So in this lesson we have gone from where we
57:43	have started . We know that problems have this equation
57:45	, we knew that , but then we talked about
57:47	the focus of a problem . We talked about the
57:49	definition being that these points on the parabola are an
57:51	equal distance to the focus as they are to this
57:54	line called the directorate's , which by the way ,
57:56	the bottom part of the problem is always in the
57:58	middle of the focus and the director is always that's
58:02	very useful for you to know that's the definition of
58:04	actually it's halfway right there . Then we apply that
58:07	definition and we say , well , if this distance
58:09	has to be the same as this distance , will
58:11	calculate the distances will set them equal . The rest
58:14	is just algebra . And you get it down to
58:15	the equation of the problem that we have used .
58:17	Why minus some shift on the left , X minus
58:20	some shift on the right squared . This determines if
58:23	it opens up or down . Then we do the
58:25	exact same thing horizontally . We say let's draw horizontal
58:29	problem here is the vertex , right in the middle
58:31	, between the focus and the directorate's , this distance
58:33	has to be the same as this distance will set
58:35	this distance equal to the other distance . We'll do
58:38	all the algebra to get it all down . But
58:39	what we find is that the roles of X and
58:42	Y are flipped . It was why ? On the
58:44	left and X squared on the right now it's X
58:46	on the left with y squared on the right .
58:49	So this would be the general form of the horizontal
58:51	parabola . The shift and Y . For the vertex
58:54	goes here , the shift and X for the vertex
58:56	goes here . Those rules still apply the number in
58:59	front determines if it opens to the right or to
59:02	the left , just as you would expect towards positive
59:04	extra towards negative X . And then finally we have
59:07	this which we're gonna use for all of our problems
59:11	. Every problem has a vertex focus and directorates .
59:15	The equation of the problem is here , the vertex
59:17	focus access the cemetery and directorates . We've all talked
59:20	about them and then we have this other relation A
59:23	. Is equal to the distance . The focus is
59:26	1/4 times that distance A is 1/4 C . That
59:29	is not something I derive for you in this lesson
59:32	. It's something that's usually just given to you .
59:34	I could derive it but it's it's not really worth
59:36	our time . You just need to use this equation
59:38	quite a bit because it relates the parameter A to
59:41	where the focus is located . C units away .
59:44	You have the same relation down here , a similar
59:46	relations for the vertex focus and axis of symmetry and
59:49	so on . I know it looks complicated now ,
59:52	but when we sit down and do our problems ,
59:54	these problems will not be difficult . But you have
59:56	to know that every parabola has a vertex , has
59:59	a focus , has a line called the directorate's what
60:02	the definition of a parabola is and so on .
60:04	And now that , you know all of those things
60:06	, you can conquer any problem I throw at you
60:08	in terms of parabolas . So follow me on to
60:10	the next section . We're going to do that right
60:12	now .

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