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

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

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 .