Hyperbolas - Conic Sections - Free Educational videos for Students in K-12

Hyperbolas - Conic Sections - By The Organic Chemistry Tutor

Transcript


00:0-1	in this video , we're going to focus on hyperbole
00:02	us . They may want to get a sheet of
00:04	paper and a pen to write down some notes .
00:07	So we have a horizontal hyperbole on the left and
00:10	a vertical hyperbole on the right . The formula that
00:14	corresponds to the horizontal hyperbole is x squared over a
00:19	square minus Y squared over B squared . What do
00:23	you think ? What you want ? This is the
00:25	case when the hyperbole is centered at the origin ,
00:30	A . Is the distance between the center of the
00:34	hyperbole and the vertex . So to get the two
00:40	vortices , the coordinates are Plus or -10 for this
00:45	type of hyperbole and the same is true for the
00:50	other one . A . is the distance between the
00:53	vertex and the center of the hyperbole . Now ,
00:58	the equation that corresponds to the vertical hyperba is as
01:02	follows , it's y squared over a squared minus X
01:07	squared over B squared , Which he goes one .
01:12	So when a square is under X squared , it's
01:16	going to open lesson , right ? Or when it's
01:19	a positive sign in front of export , when there
01:24	is a positive sign in front of y squared ,
01:26	it opens up and down and a square is always
01:30	under the positive term . The squad is under the
01:35	negative term . Now the coordinates for the vertex is
01:41	for the vertical formula R zero comma plus or minus
01:45	eight . The coordinates for the first I are plus
01:51	or minus C comma zero on the left . And
01:54	for that purple on the right , it's zero comma
01:56	plus minus C . The hyperbole to I mean the
02:02	full side is always located towards where the hyperbole opens
02:07	up . Mhm and it's C units away from the
02:16	center . Now the equation that correlates a B and
02:24	C . For the hyperbole is c squared is equal
02:28	to a squared plus b squared . In contrast for
02:35	ellipses , the equation is different . C squared is
02:39	equal to a squared minus b squared for an ellipse
02:44	, A is always larger than B . But fiberglas
02:48	A maybe larger than be and sometimes be maybe larger
02:51	than A . So the relative sizes of those two
02:53	are not important . They are important for ellipses but
02:56	not for high prevalence . So make sure you're aware
02:58	of that key difference and we'll go over plenty of
03:02	examples so you'll see this in action . Now ,
03:07	the next thing you need to be familiar with is
03:08	the transverse axis . The transverse axis is a line
03:13	segment That connects the two verses together . So the
03:20	left of the transverse axis is to A . Here
03:29	we have a vertical transverse axis and here on the
03:32	left is the horizontal transfers axis . The distance between
03:39	the folks I or the two focal points is always
03:42	going to be to see . But the length of
03:46	the transverse axis will be to a Now let's talk
03:50	about the equation of the ascent oats . For the
03:56	horizontal hyperbole , it's Y is equal to plus a
03:59	minus B Over eight times x . For the vertical
04:06	hyperba it's Y is equal to plus or minus A
04:10	over B times X . And this is true when
04:14	the hyperbole to is centered at the origin , the
04:18	equation is different if the center is not at the
04:21	origin and we'll talk about that later in the video
04:24	number one graph the hyperbole , identify the coordinates of
04:28	the center vortices and fosse and then write the equations
04:32	of the ascent oats . So what's the value of
04:36	A squared and B squared in this problem , A
04:40	squared is going to be under the variable that has
04:44	a positive coefficient , but one of the negative sign
04:46	is going to be associated with B squared . So
04:52	in this problem a squared is the number under X
04:55	squared . A square is four . A . It's
04:58	going to be the square root of four , which
04:59	is to B squared is equal to nine , And
05:04	B is going to be the square root of nine
05:06	, which is street . So with this information we
05:10	can go ahead and draw a graph by the way
05:13	, the center is going to be the origin for
05:17	this particular problem . So A . Is two and
05:28	it's under X squared . So we're gonna do is
05:31	we're gonna travel to units to the right From the
05:35	center and two units to the left . Let's put
05:37	some marks on his graph 1st . So two units
05:54	are right and then to units that left now be
05:58	a street and it's under Y . So we're going
06:01	to go up three units along the y axis from
06:04	the center and then down three units . Then we're
06:08	gonna do is we're gonna create a rectangle around these
06:12	four points . The purpose of the rectangle is to
06:17	help us to draw the asthma . Totes . The
06:21	assam . Totes will be diagnose that will go right
06:27	through the son of the box . So those are
06:42	the two that to assume totes and what we have
06:47	here represents the vertex is of the hyperbole . So
06:54	whenever you have a positive X squared term , the
06:57	hyperbole is gonna open towards the left and towards the
07:00	right . So it's going to look something like this
07:08	and then it's going to follow the asem toe .
07:19	It's curved . Bye . The vortices . But once
07:23	it follows the ascent , oh , it becomes almost
07:25	linear . So that's how we can graph the hyperbole
07:35	to to the best of our ability . So it's
07:39	initially curve and then it becomes linear as it approaches
07:41	the ascent oats . Now let's write the coordinates of
07:47	the vortices . So the vertex is is going to
07:50	be Plus or -10 for this type of graph And
07:55	a is too . So it's plus or -2 common
07:58	zero . So this one here is that two comma
08:02	zero and this one here Is -20 . So those
08:10	are the coordinates of the vortices . Now , let's
08:13	focus on the coordinates of the full side . This
08:16	is going to be plus or -10 . So we
08:20	need to calculate the value of C and we could
08:23	use this formula to achieve it . C squared is
08:26	equal to a squared plus B squared , A squared
08:30	is four , B squared is nine . So c
08:33	squared is 13 which means see Is equal to the
08:37	square root of 13 . So the coordinates of the
08:43	folks i it's going to be plus or minus route
08:47	13 comma zero . The square root of 13 as
08:56	a decimal value Is approximately 3.6 . So we have
09:02	a focus there and the focus here . So those
09:05	are the two focal points of the hyperbole . As
09:10	you can see , the hyperbole always opens towards the
09:13	focal points . Now let's write the equations of the
09:18	asem totes . Now when you have an asem toe
09:22	centered at the origin and if it opens to the
09:25	right into the left , the equation is going to
09:28	be why is equal to plus or minus be over
09:32	A times X . This value in front of X
09:37	is the slope . Its rise over run going from
09:43	the origin to this point we have a rise of
09:47	three . That's our B . Value and a run
09:49	of A . So the slope is rise over run
09:52	B over A . So that can help you determine
09:56	if it's been over A or ever be if you
09:59	forget so be history . A . Is too so
10:09	why is equal to plus or minus 3/2 X .
10:14	Now there's two equations for the two aspects tools that
10:16	we have here , the one on the right .
10:19	This is going to be why is equal to positive
10:23	3/2 x . As for the one on the left
10:27	. This is why is equal to negative 3/2 X
10:34	. And it makes sense because this line here ,
10:37	the slope is positive because it's increasing as you move
10:40	to the right . As for the other line ,
10:43	the slope is negative because it's decreasing as you move
10:46	to the right . So those are the two equations
10:48	that correspond to the to ask tools that we have
10:51	in this graph and the ascent filter linear . Which
10:55	means that the hyperbole to becomes linear as X increases
11:00	near the vertex is it's kind of curved but as
11:02	X gets larger and it approaches the assent to it
11:05	becomes it behaves like a linear function . Now let's
11:09	work on a similar problem . So we're going to
11:12	do the same thing . We're going to graph the
11:13	hyperbole to identify the coordinates of the center of vortices
11:16	and the folks I and then we're going to write
11:18	the equations of the assam totes . Right now ,
11:23	this hyper Bella is not in standard form . We
11:26	need this number to be a one . So what
11:29	we're gonna do is we're gonna divide everything By 144
11:37	. 144 divided by nine is 16 . So we're
11:42	going to get y squared over 16 . 1:44 divided
11:47	by 16 is nine . So we're going to get
11:49	X squared overnight And 144 divided by itself is one
11:57	. So I'm just going to rewrite this here .
12:04	Mhm . So now that our equation is in standard
12:09	form , we can go ahead and find the details
12:12	that we need . So what's a squared and what's
12:15	B squared in this equation , A square is always
12:18	going to be the one that has the positive coefficient
12:22	. So a squared is going to be 16 .
12:27	That means that A Is the square root of 16
12:30	, which is four . B squared is nine .
12:34	B is going to be the square root of nine
12:36	, which is three . So now let's calculate C
12:40	C squared is equal to a squared plus B squared
12:44	, A squared is 16 , b squared is nine
12:48	, 16 plus nine is 25 And the square root
12:52	of 25 is five . Now let's go ahead and
12:57	draw a rough sketch of this graph . Now ,
13:03	since y squared is listed first , this is going
13:07	to be a vertical hyperbole . So we're gonna travel
13:11	a units along the Y axis since a squared is
13:16	under Y squared and then a units down from the
13:20	center . So just like before the center it's gonna
13:23	be the origin , It's 0,000 , B squared is
13:33	under X squared . And since vehstree we're going to
13:36	travel three units to the left and to the right
13:40	along the X axis . Now let's create our rectangle
13:57	and then let's draw the assam totes which is going
14:01	to go from one side , a direct tangle through
14:05	the center to the other side . Now the coordinates
14:20	of the vertex is it's gonna be zero plus or
14:23	minus A And a . is four , so it's
14:26	zero plus or -4 . Here's the first vertex at
14:30	0:04 And here's the second vertex at 0 -4 .
14:35	And so the graph , it's gonna look like this
14:46	. So that's how we can graph this particular hyperbole
14:51	. So now that we have the coordinates of the
14:52	vertex is and the center , we need to find
14:56	the coordinates of the full side . So it's going
14:59	to be zero comma plus or minus C And C
15:03	is five . So at 05 . Yeah . And
15:09	that 0 -5 , we have two focal points .
15:19	So that's the coordinates for the full side . Now
15:23	, the last thing we need to do is write
15:24	the equations of the ascent oats . So it's gonna
15:27	be y is equal to plus or minus the slope
15:31	times X . The slope is going to be rise
15:34	over run . So the rise is a units the
15:40	run is being its survival run . It's going to
15:43	be a over B A . Is four B.S Stream
15:55	. And so it's going to be Y . Is
15:57	equal to plus or minus 4/3 X . So that's
16:02	how we can graft the hyperbole to identify the coordinates
16:05	of the center advertises in full sight . And that's
16:07	how we can write the equations of the ascent oats
16:11	for this particular equation . Now , let's write down
16:15	some more notes . So let's focus on The Horizontal
16:20	Hyperbole 1st . When this hyperbole is centered at the
16:30	origin , the equation that correlates to it is X
16:34	squared over a squared minus Y squared over B squared
16:38	. But what happens if the center is not at
16:41	the origin ? Let's say it's at some point H
16:44	com McCain the equation changes to this , it becomes
16:48	x minus h squared over a squared minus y minus
16:55	k squared over B squared is equal to one .
16:59	So you may want to write some notes down .
17:01	Even if you don't know how to apply it ,
17:02	you may just want to drop these down and then
17:04	use it later . For reference . Now the next
17:08	thing that we mentioned is that the vertex is for
17:12	horizontal . Hyperbole is centered at the origin Is Plus
17:16	or -10 . When it's not centered at the origin
17:21	is going to change . All we need to do
17:23	is at the center to this to those coordinates .
17:30	So we're gonna add H two plus or minus K
17:33	. I mean plus minus a rather . So it's
17:35	H . Plus or minus A . And then we're
17:37	gonna add zero with K . So it just becomes
17:41	K . So that's how you can right ? The
17:43	new Verdecia simply at the center to the vertex is
17:47	and then you get the the new vortices . When
17:51	the center is not at the origin , the coordinates
17:55	of the folks i is plus or minus C comma
17:58	zero . So when it's shifted and the center is
18:02	not not at the origin simply add H and K
18:05	to this . So it becomes a church plus or
18:10	minus C . Calm . Okay , now the equation
18:17	of the ascent , oats is plus or minus be
18:24	over a times X . When the graph is shifted
18:28	this becomes y minus K is equal to plus or
18:31	minus B over a times x minus age . So
18:37	that's how the equations will shift when the center moves
18:40	from the origin to some point . H comma Kane
18:46	. Now , what if we have the vertical ascent
18:47	on how well the equations change . So first let's
18:53	write the equations when it's centered at the origin .
18:58	So for this one it's going to be y squared
19:00	over a squared minus x squared over b squared .
19:05	And when the center shifts from the origin to h
19:07	comma k , the equation will be adjusted . So
19:11	it's gonna be why minus K squared over a squared
19:15	minus x minus h squared over B squared which equals
19:20	one . The very disease will be zero comma plus
19:26	or minus A . The new vertex is all we're
19:28	gonna do is at the center to this . So
19:31	it becomes h comma K plus or minus A .
19:39	The full side will be zero plus or minus C
19:42	. The new coordinates of the folks , I will
19:44	be H comma K plus or minus C . The
19:51	equation for the ascent oats will be plus or minus
19:55	a over B . X . But the new equation
19:58	becomes y minus K . Is equal to plus or
20:01	minus a over B x minus h . So feel
20:06	free to write those equations . This equation remains the
20:09	same . C squared is equal to a squared over
20:12	b squared . In all cases , number three graph
20:18	the hyperbole . So we're going to follow the same
20:20	instructions that we did for the first two problems .
20:23	But this time we can see that it's not centered
20:25	at the origin . So let's begin by finding the
20:28	center . Here we have ex monastery . We're going
20:32	to change -3 . It's a positive three . And
20:35	here we have y plus two . We're going to
20:37	change positive to to negative through . So the coordinates
20:41	of the center , Our 3 -2 . A squared
20:47	is typically the first number that we see here .
20:50	A squared is four . So a is going to
20:52	be too B squared is nine . So be it's
20:56	gonna be three . Next we need to calculate C
21:03	. A squared is four . B squared is nine
21:06	, four Plus 9 is 13 . So see it's
21:09	going to be the square root of 13 . Now
21:14	let's go ahead and graph the hyperba . So let's
21:41	begin by plotting the center which is that three negative
21:44	two . Now A is to A is associated with
21:49	the X . Variable . So we're going to travel
21:51	to units to the right parallel to the X axis
21:55	and two units to left be history B is B
22:00	squared , was under Y squared . So it's associated
22:03	with why ? So we're gonna go three units up
22:06	parallel to the y axis and three units down from
22:09	the center parallels the y axis . Now let's draw
22:14	a rectangle and then let's draw the ascent oats .
22:44	Now this particular hyperbole to will it open to the
22:48	left and right ? Or will it open up and
22:50	down ? Do we have a horizontal hyperbole ? A
22:53	Or would you say we have a vertical hyperbole ?
22:57	Now we have a positive sign in front of X
22:59	squared . So this is going to open left and
23:01	right . If y squared was positive it would open
23:05	up and down . So the graph is going to
23:09	look something like this . So these two points represent
23:20	the vertex is of the hyperbole . And if you
23:23	recall divert is is for a horizontal hyperbole to that
23:29	is not centered at the origin . It's gonna be
23:32	a church plus or minus a comma Kane H history
23:43	A is to and Kay is negative two . So
23:51	you can always look at the center to find H
23:52	and K . So first we have three plus two
23:57	which is five common negative too . So that's the
24:03	vertex on the right side for this one it's gonna
24:07	be three minus two comma negative 23 minus two is
24:10	one . And so we get the point one negative
24:14	two for this very text on the left . So
24:18	those are the coordinates of the vertex is . Now
24:22	let's find the coordinates of the full side . So
24:26	it's gonna be a church plus or minus C .
24:29	Can McCain hs three . See if the square root
24:35	of 13 K is -2 . So this represents the
24:42	coordinates of the folks I now to get the X
24:47	value Of the folks I we have three plus the
24:50	square to 13 . The square to 13 is 3.6
24:53	. So three plus 3.6 is 6.6 . So somewhere
24:58	in this region we have to focus and then three
25:07	minus Discord 13 , which is negative 130.6 . So
25:11	somewhere in this region is the other focal point .
25:17	So that's the location of the full site on the
25:20	graph . The last thing we need to do is
25:22	write the equations of the ascent oats . So it's
25:26	going to be y minus K is equal to plus
25:31	or minus . Now the rise is the units .
25:38	The run is A . So the slope is going
25:40	to be B over A and then it's x minus
25:45	h . So Kay is -2 . Why minus negative
25:52	two becomes y plus two . So what we have
25:57	here is similar to the original equation . This is
26:00	going to be equal to plus or minus B over
26:02	a . B . History is too and then x
26:05	minus h . So we have x minus tree .
26:12	So this represents the equations of the to assume totes
26:17	. Now let's work on another problem . So we're
26:19	going to do what we did in the last problem
26:21	. We're going to grab the hyperbole to identify the
26:24	coordinates of the center , vortices and foresight . We're
26:28	gonna write the equations of the asthma totes but this
26:30	time we're also going to determine the domain and range
26:33	of the graph . So what's a squared and what's
26:38	B squared ? A squad is usually the number in
26:40	front . So a square is going to be nine
26:45	, which means A . Is a square to nine
26:48	, which is street B squared is going to be
26:50	the other 1 . 16 Be the square to 16
26:54	is four . Now let's calculate C C squared is
26:59	a squared plus B squared , A squared is nine
27:04	, B squared is 16 , nine plus 16 is
27:08	25 And the square root of 25 is five .
27:15	Now , let's determine his center of this particular hyperbole
27:22	. Here we see negative too . So the X
27:24	coordinate of the center will be positive too . Here
27:27	we see -1 . The y coordinate the center will
27:30	be positive one . Now let's go ahead and plot
27:34	it . So let's begin by plotting the center Which
28:07	is at 2:01 . And then we see that a
28:11	history A is under Y squared . So we're gonna
28:15	go up three And then we're gonna go down three
28:26	. Now bs four B . Is under the X
28:28	. Variable to think of left and right when dealing
28:30	with X . So we're gonna go right for from
28:33	the center And then left four units . Now let's
28:38	draw a rectangle . Next let's write out or draw
28:55	the ascent oats . Let's do the same for the
29:03	other one . I think I messed it up .
29:18	Let's do this one more time . I know my
29:21	graph is not drawn to scale but we'll make the
29:30	best of that . So now the graph in part
29:35	, so we know it's going to be curved at
29:39	the the vertex . And then we need to follow
29:43	the aceto . So let's try to draw an accurate
29:50	graph . So that's as best as we can do
30:01	for it . Now , as long as you draw
30:03	up sketch , most teachers will give you , you
30:06	know , the correct answer . But ideally you want
30:09	to follow the aceto as you move away from the
30:13	vertex is so that's the hyperbole that we can draw
30:17	for this particular problem . Now we have the coordinates
30:22	of the center . So now let's focus on devices
30:26	. So for this particular type of graph where we
30:29	have a vertical hyperbole to diversity is is going to
30:33	be according to the verses will be h comma K
30:37	plus or minus A . So H is two ,
30:42	K is one , so it's too common one plus
30:45	or minus A . Which is street . So this
30:49	point here , this vertex that's going to have an
30:52	X . Value of two and then a Y .
30:54	Value of one plus three . Which is for now
30:58	the vertex below that will have the same X value
31:01	but the Y value will be one . Monastery .
31:04	So the point will be -2 2 -2 . So
31:09	those are the coordinates of the vortices to come up
31:12	for And to -2 . Now let's focus on finding
31:17	the coordinates of the false I . So the coordinates
31:21	will be h comma K Plus or -1 , H
31:31	is two , K . is one And C is
31:34	five . So the first one is going to be
31:39	At 2:06 , which will be somewhere here . The
31:44	next one will be too And then it's gonna be
31:47	1 -5 which is -4 . So I'm just gonna
31:51	write that here . So this one is 2:06 And
31:54	this will be 2 -4 . So those are the
31:58	coordinates are the focus on . Now let's focus on
32:02	the equations of the ascent , oats . The form
32:06	that we need is why minus K . Is going
32:09	to equal plus or minus the slope . And so
32:12	we have a rise of a , a run of
32:16	B . So it's gonna be a over B and
32:24	then x minus h . So why minus K ?
32:27	It's gonna be what we see here . That's why
32:31	I -1 is equal to plus or minus AS .
32:35	three , B is four and then x minus h
32:42	is what we see here . So X -2 .
32:47	So those are the equations of the assam totes .
32:52	Now , the last thing we need to do is
32:53	find the domain and range of the graph for a
32:58	vertical hyperbole to the domain is always going to be
33:01	our poll numbers . When you need to find a
33:03	domain . Look at the X valleys , analyze it
33:06	from left to right , the lowest x value will
33:09	be negative infinity and the highest x value is positive
33:13	, singing . So X can be anything along this
33:17	curve . This curve is continuous . There are no
33:19	jumps as you go from left to right . So
33:24	the domain is our phone numbers . Now , the
33:26	range is a different story . We need to look
33:28	at the Y values , so view it from going
33:31	from the bottom to the top , this can go
33:34	all the way down to negative infinity . So the
33:37	lowest Y value is negative infinity . As we go
33:41	up This stops at AY Value of -2 . Now
33:48	between a y value of -2 and four , there
33:52	is no curve , there is no hyperbole . So
33:55	we're going to pick it up back at four and
33:57	then this goes up to infinity . So the range
34:03	from bottom to top is negative infinity to -2 .
34:08	Negative two is included because that's a vertex . So
34:11	we're going to use a bracket union , We're going
34:13	to start back up at four and continue to infinity
34:18	. So that's how we can write the range us
34:22	a hyperbole . And this is the domain for a
34:25	vertical hyperbole .

Summarizer

DESCRIPTION:

This conic sections video tutorial provides a basic introduction into hyperbolas. It explains how to graph hyperbolas and how to find the coordinates of the center, vertices, and foci. In addition, it explains how to write the equations of the asymptotes.

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