How To Graph Polar Equations - Free Educational videos for Students in K-12

How To Graph Polar Equations - By The Organic Chemistry Tutor

Transcript


00:00	in this video , we're going to talk about how
00:02	to graph polar equations . These include circles , lima
00:07	songs , roads , curves and lemon skates . So
00:11	let's start with a circle . The first equation may
00:14	see is always equal to a co sign fada .
00:20	Now , if a is positive , this is going
00:22	to be a circle directed towards the right now ,
00:25	granted my circle is not perfect . So they're with
00:28	me A is basically the diameter of the circle and
00:32	this is gonna be the center of the circle .
00:34	So if you go up to find this point here
00:37	, it's half of a . So if A .
00:40	Is greater than zero , if a is positive ,
00:45	it's gonna open towards the right and if a .
00:48	Is negative , you're gonna get a circle . Let
00:51	me draw good looking circle . This time you can
00:54	get a circle that is directed on the left .
00:57	So keep in mind this is gonna be A and
00:59	this is half of a . So let's try some
01:03	examples and let's say if we have the graph R
01:06	. Is equal to four co signed data . If
01:09	you want to feel free to pause the video and
01:11	try yourself . So we're gonna have a circle on
01:16	the right side . Now A . Is 4.5 of
01:20	a . four divided by two is 2 . So
01:22	that's half a . So what we're gonna do is
01:24	travel four units to the right And then up two
01:28	units and down two units . So the circle is
01:33	going to start at the origin And it ends at
01:36	four on the X . Axis from the center .
01:38	Which is that too . We need to go up
01:41	two units and down two units and then simply just
01:46	connected . So that's how you can graph R .
01:50	Equals for coastline data . Let's try another one .
01:55	Try this one . Let's say that R . Is
01:57	equal to negative six signed data . I mean that's
02:00	signed by coastline data . We can get to sign
02:03	leader now because A . Is negative the circle is
02:09	going to be on the left side on the X
02:12	. Axis . So let's travel six units to the
02:15	right . Since A . Is negative six . one
02:22	half of a is negative three . Now don't worry
02:25	about the negative sign . Too much negative scientist tells
02:27	you if the circle opens to the left now The
02:32	center is gonna be a negative three Which is here
02:36	. So we need to go up three units And
02:40	down three units . So the graph is going to
02:44	be at the origin a negative six negative 33 And
02:49	that -3 -3 . And so that's how you can
02:53	plot the circle . So keep minus is equal to
02:58	a . That distance . And this is also equal
03:02	to a . As well . Which means this part
03:05	is one half of eight so half of a .
03:08	Is basically the radius of the circle . So if
03:15	for some reason you need to find the area of
03:16	the circle . You can use this equation pi r
03:19	squared the radius history . So it's pi times three
03:22	squared Which is nine pi . So now the next
03:28	form we need to know is R equals a sign
03:31	data . Co sign is associated with the X values
03:36	. So as you can see the circle was associated
03:38	with the X axis . Sign is assertion of the
03:41	Y values . And so the circle , it's going
03:44	to be centered on the y axis . So let's
03:48	say if A . Is positive that we're gonna have
03:52	a circle that goes above the X axis , centered
03:57	on the Y axis . So once again the diameter
04:01	will still be equal to A . And this portion
04:05	the radius is half of a . So that's when
04:10	A . Is positive or when A Is greater than
04:13	zero . Now in the other case if A .
04:15	Is negative Or if a . is less than zero
04:20	the circle it's still going to be centered around the
04:23	y . Axis . But it's going to open in
04:26	a negative Y . Direction . So it's gonna be
04:29	below the X . Axis . And so the radius
04:35	as you mentioned before it's just 1/2 of a .
04:42	And the diameter is equal to a . So let's
04:48	try some examples let's say if our is to sign
04:54	feta , go ahead and graph that . So first
05:00	we need to travel up to units A . S
05:03	. Two Half of a . Which is the radius
05:05	is one . So we're gonna have these two points
05:11	. Now let's travel one unit to the right and
05:13	one unit 2 left . So the green dot is
05:17	the center of the circle . So we're gonna travel
05:20	one unit to the right and one unit two left
05:21	from it . And so this is gonna be the
05:24	graph . Try this one . Let's say our is
05:31	negative eight sign fada go ahead and work on that
05:36	example . Now the majority of graph will be below
05:39	the X axis , some are focused on that .
05:42	So let's travel eight units down and then half of
05:48	A . Or four units to the right And four
05:51	years to left . So the point is gonna be
05:56	at the origin and eight units down the center Is
06:01	four units down . So if a . is eight
06:05	The radius is 1/2 of a . Which is for
06:11	. So we got to travel four units to the
06:12	right from the center and forward to the left .
06:16	And so this is going to be the graph .
06:21	So now you know how to graph circles when you're
06:24	given a polar equation . Now the next type of
06:28	graph that we need to go over is the lima
06:31	song and the equation is R . Is equal to
06:39	Yeah A plus or minus be signed data . Now
06:45	if you have positive sign it opens towards the positive
06:48	Y . Axis , that is any upward direction .
06:50	Negative sign opens in a downward direction in a negative
06:54	way direction . You could also have a plus or
07:00	minus be costing data . So if coastline is positive
07:06	it's going to open towards the right , in the
07:09	positive X . Axis direction . And if co sign
07:11	is negative it's gonna open towards the left . So
07:16	let's draw the general shape if it opens towards the
07:18	right , so this is the lima song with the
07:24	inner loop and you get this particular shape if a
07:30	divided by B Is less than one . Now both
07:36	A and B represent positive numbers A and B are
07:42	both greater than zero . So if you get the
07:45	graph three minus for signed data , B . It's
07:51	not negative four . B is positive for an A
07:53	. Is positive three . So let's say if it
07:56	was three plus four coastline data , both A and
07:59	B will still be three and four positive dream positive
08:03	for . So A . And B are not negative
08:10	. Yeah . Now the next shape that we have
08:15	if A divided by B is equal to one is
08:19	the heart . Shapley Masson , also known as the
08:22	cardi order . And here's the generic shape for it
08:28	. So it has like this just simple . So
08:31	it looks something like that . Maybe I could draw
08:32	that better . So that's the cardi origin . Now
08:49	the next one is the dimpled lima sol with no
08:52	inner loop . So that occurs if A . Over
08:57	B . Is between one and two . So let's
09:06	start with the X . Axis . It's a small
09:08	dimple . Sometimes it's hard to notice . So that's
09:13	the dimpled limassol with no inner loop . The next
09:20	one needs to know is if a divided baby Is
09:24	equal to or greater than two . So this limo
09:29	song looks almost like a circle . But it's not
09:32	, there's no dimple and there's no inner loop .
09:37	So I'm going to start from the left . I'm
09:38	going to draw it straight up and then looks like
09:40	this but it's not exactly a circle because as you
09:44	can see the right side it's like bigger than the
09:46	left , but it almost looks like a circle .
09:51	So that's the the lima song without a dimple or
09:55	in in a loop . So those are the four
09:58	shapes need to be familiar with . Yeah . Mhm
10:02	. Let's graph this equation . Let's say our Is
10:06	equal to 3-plus 5 co sign . What do you
10:10	think we need to do here ? We know this
10:14	is a type of lima song . It's in the
10:16	form A . Plus or minus B . Cosign data
10:21	. So first we need to identify A . And
10:23	B . A . is equal to three And b
10:27	. is equal to five . Now we need to
10:29	see if A over B if it's less than one
10:32	if it's between one and two if it's equal to
10:37	one or greater than or equal to two . So
10:40	a over b . That's 3/5 and 3/5 . As
10:44	a decimal is 0.6 which is less than one .
10:47	Now because it's less than one we know we have
10:50	the limousine with the inner loop . Now there's four
10:53	types . The first type is if it's positive co
10:59	sign , this graph will open towards the right .
11:07	The next type is if we have negative coastline and
11:14	in that case this graph what opens towards the left
11:22	, if it's positive sign then it's going to open
11:29	in a positive Y . Direction . And if we
11:35	have negative sign it's going to open towards the negative
11:42	Y . Direction . So it's gonna look something like
11:46	that . So just keep that in mind . That's
11:48	the first english you look forward . So we have
11:51	positive coastline which means it should open towards the right
11:55	side . Now when graph in this type of lima
11:58	song , you want to make sure you get four
12:02	points two x . intercepts and two y intercepts .
12:07	So let's uh draw a sketch of this graph at
12:16	this point is actually positive A . It's a .
12:19	Units relative to the center and this other Y intercept
12:24	is negative A units from the center . The first
12:27	accident step which is associated with the inner loop .
12:30	It's the difference between a modesty . So it's the
12:34	absolute value difference of a modesty . Well , you
12:38	could say is B minus A . Because B is
12:41	gonna be bigger now , the second intercept is the
12:48	sum of A . And B . And that's all
12:51	you need to get a good decent graph . If
12:53	you can plot those four intercepts then you should be
12:58	fine . So let's go ahead and do that .
13:02	So in this case we can see that A is
13:05	equal to three . So we need to go up
13:09	three units And down three units . So those are
13:16	the wider steps . Now , B minus A .
13:20	That's going to give us the first intercept . That's
13:22	5 - stream . That's true . So here's the
13:28	first intercept and then A plus B . That will
13:32	give us a second intercept that street plus five which
13:36	is eight . So that's how you can find the
13:39	two x intercepts . Now let's go ahead and graphic
13:49	. So first let's start with the inner loop and
13:53	then let's go towards the first miner step and then
13:56	the second X intercept , and then towards the other
14:00	uh minus F . So that's a rough sketch of
14:04	this graph . So the points that you need is
14:06	a . Three and negative three on the Y axis
14:10	and two and eight on the X axis . Let's
14:13	try another example . So let's say R . Is
14:16	equal to two -5 signed data . So try this
14:25	one . The first thing I would keep in mind
14:29	is what type of , what direction will it open
14:33	? We know that a over being which is 2/5
14:36	. It's less than one . So this is a
14:38	lima song with an inner loop , but notice that
14:41	we have a negative sign . So therefore has to
14:44	open in a negative Y direction . So we can
14:49	see that A . Is two . That's going to
14:50	give us the Y . Intercepts & b . s
14:54	. five . So let's go ahead and graph it
15:04	well in this case because it opens downward , A
15:07	. Is actually going to be associated with the X
15:08	. Intercept this time instead of the wider steps .
15:11	So it's gonna switch roles . So we need to
15:14	travel to units to the right and to to less
15:21	. If we're dealing with co sign then A .
15:24	Would be associated with the whiteness . S . But
15:25	because we're dealing with sign , the roles are reversed
15:29	. Now A plus B . That's going to be
15:32	two plus five , that's seven And B -A 5
15:37	-2 extremes . So we're gonna travel three units down
15:42	and also seven units . So we're gonna have to
15:46	y intercepts . So let's start with the inner loop
15:53	and then let's when would you do that again ?
15:59	And then let's get the X . Intercept and that's
16:06	it . So make sure you get These two x
16:10	intercepts negative two and 2 And the wind accepts -3
16:14	-7 . And as you mentioned before because we have
16:17	negative sign , it has to open in a negative
16:19	light direction . Now let's try this problem . Let's
16:23	say that are is 3 -7 fill sign data .
16:29	So go ahead and pause the video and work on
16:31	that example . So let's find the value of A
16:35	over B . A . pastry . Be a seven
16:39	and three of the seven is less than one because
16:41	7/7 is one . So what we have is the
16:46	inner loop lima song . Now we're dealing with negative
16:50	co sign , which means it's going to open towards
16:52	the left . So the majority of the graph is
16:56	going to be on the left side . Now A
17:00	. Is associated with the Y intercepts when dealing with
17:03	Co sign when dealing with sign as we saw in
17:06	the last example A . Is associated with the X
17:09	. Intercepts . So we're going to travel three units
17:12	up and three units down to get the Y intercepts
17:18	. When dealing with Co sign . First Sign you
17:22	need to know that is associate with the X intercepts
17:26	. Mhm . So next we find the first accident
17:29	step which is going to be B -A . or
17:32	seven monastery and that's four and then A plus B
17:37	three plus seven is 10 . So because we're going
17:41	towards the left we need to travel four years to
17:44	left . That's gonna give us the first X intercept
17:47	And then 10 minutes to left relative to the origin
17:51	. So that's the second X intercept . Now let's
17:55	go ahead and graph in . So let's start with
17:59	the origin and let's draw the first in the loop
18:03	and then let's focus on the outer loop . And
18:10	so that's it and that's how you can graphic .
18:12	My graph is not perfect , but at least that's
18:15	the general shape . You get the picture . Now
18:19	Let's try this one . Let's say that are is
18:22	3-plus 3 callsign vega . What do we need to
18:29	do here ? Well first we need to determine what
18:31	type of lima . So we have A and B
18:36	. Are the same When A&B are the same .
18:39	RB is equal to one . And in that situation
18:43	we have the heart shaped like Mr . Also known
18:46	as the cardboard . And because CO sign is positive
18:51	, it's going to open towards the right . I'm
18:57	just going to draw the general shape of the cardio
19:00	order , which it looks like this . So there
19:03	is no inner loop . Now I'm dealing with co
19:05	sign the X and Y intercepts are going to be
19:08	a . Again A . And negative eight . Now
19:12	the first sex intercept is just going to be the
19:14	origin . And so we don't have to do anything
19:17	. It's just going to start from the origin .
19:20	Now the second x minus is A plus B .
19:23	We don't have the inner loop which was a minus
19:26	B . For the inner loop limits on . So
19:29	we have to worry about any minor speak or being
19:30	on the same . So now let's go ahead and
19:33	graphic . So in this example a stream . So
19:38	we're gonna go up three units and down three units
19:42	and A plus B . That's three plus three .
19:45	So that's equal to six . So the X intercept
19:49	is going to be six comma zero And a witness
19:56	sips are 03 and 0 -3 . So now that's
20:02	we're gonna have to graft like this and that's it
20:07	. That's how you can graph the heart shaped limassol

Summarizer

DESCRIPTION:

The full version of this precalculus video tutorial focuses on graphing polar equations. It explains how to graph circles, limacons, cardiods, rose curves, and lemniscates. It provides plenty of examples and practice problems.

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How To Graph Polar Equations is a free educational video by The Organic Chemistry Tutor.

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