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.
OVERVIEW:
How To Graph Polar Equations is a free educational video by The Organic Chemistry Tutor.
This page not only allows students and teachers view How To Graph Polar Equations videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.