Direct Variation - By Anywhere Math
Transcript
| 00:0-1 | Welcome anywhere . Math . I'm Jeff Jacobson . And | |
| 00:01 | today we're gonna talk about direct variation . Let's get | |
| 00:05 | started . Okay before we get to an example , | |
| 00:25 | let's talk about direct variation for a moment . Uh | |
| 00:29 | Two quantities X and Y would show direct variation . | |
| 00:34 | If you can write them in an equation like this | |
| 00:36 | where Y equals K . X . If you can | |
| 00:39 | write an equation like that then you would say X | |
| 00:42 | and Y show direct variation . If not if you've | |
| 00:46 | got something extra like minus one or plus seven then | |
| 00:51 | they're not showing direct variation . Okay now this k | |
| 00:55 | uh this case we call the constant of proportionality proportion | |
| 01:04 | . Now let's see it's a long word constant of | |
| 01:08 | proportionality . It's also basically just the slope . Mhm | |
| 01:15 | . Of the line . Okay if you graph it | |
| 01:17 | , that's what that represents . Is just the slope | |
| 01:19 | of the line . Okay let's get to an example | |
| 01:22 | . Example one tell whether X and Y show direct | |
| 01:25 | variation . So we have a 22 problems . We're | |
| 01:28 | going to do first one , we've got a table | |
| 01:30 | X . And Y . We've got some X . | |
| 01:32 | Values , we've got some Y . Values . Uh | |
| 01:35 | notice there's no equation so we're not gonna do it | |
| 01:38 | that way to see if we can if we can | |
| 01:40 | write the equation with Y equals K . X . | |
| 01:43 | Um Instead what we're gonna do is we are going | |
| 01:46 | to graph them . So I'm just gonna make it | |
| 01:49 | quick a quick little sketch uh on my X . | |
| 01:55 | Axis . I just got 123 on my Y . | |
| 01:59 | I guess . I'm sorry I need to go negative | |
| 02:01 | let me redo that . Yeah So I'll do all | |
| 02:05 | four quadrants 123 23 And then let's go uh to | |
| 02:15 | 40 negative two negative four . Okay ? Uh So | |
| 02:21 | 1st 1st ordered pair , one negative two . So | |
| 02:24 | one negative two would be there , Then 2020 would | |
| 02:28 | be there and then three two would be there . | |
| 02:33 | Okay connect my points and the line looks something like | |
| 02:38 | that . Now one other thing for X and Y | |
| 02:42 | to have direct variation . If you graph it , | |
| 02:46 | the line has to go through the origin . Okay | |
| 02:50 | that's really important . You should write that down . | |
| 02:52 | Um It has to go through the origin . And | |
| 02:55 | the reason if you think about it , if the | |
| 02:59 | equation needs to look like that , if X . | |
| 03:03 | Zero . Yeah . Well what is why I have | |
| 03:08 | to be why has to be zero as well because | |
| 03:11 | zero times any number We'll Give You zero . Right | |
| 03:15 | ? So here yeah Notice the line does not go | |
| 03:22 | through my origin , it does not go through 00 | |
| 03:24 | . So X . And Y . Here do they | |
| 03:26 | show direct variation ? And the answer is no they | |
| 03:30 | do not . Okay let's try B . So the | |
| 03:33 | same thing , let's graph the points . And for | |
| 03:37 | this they are both positive , which is nice . | |
| 03:39 | So I'm just going to be the first quadrant . | |
| 03:41 | Uh And that'll be let's go to for 24 1st | |
| 03:49 | point is at the origin 00 right there . And | |
| 03:53 | I've got to to which is there 44 ? Which | |
| 03:57 | is But there draw my line doesn't have to be | |
| 04:03 | perfect . But again notice this time my line does | |
| 04:07 | go through the origin and I can see right here | |
| 04:10 | there's the origin right there . Anyway , so really | |
| 04:13 | I didn't even need to graph it . I could | |
| 04:14 | have noticed that . Um So yes , in this | |
| 04:17 | situation X and Y do show direct variation . Okay | |
| 04:23 | , let's try another example . Alright , example to | |
| 04:26 | same thing tell whether X and Y show direct variation | |
| 04:29 | . But we're gonna tell uh in a different way | |
| 04:32 | . We're not going to graph each of these equations | |
| 04:35 | . We're not gonna grab the line instead , we're | |
| 04:37 | going to look to see if we can get the | |
| 04:39 | equation to look like why equals K . X . | |
| 04:44 | To do that . We're trying to get why alone | |
| 04:47 | , which means we're solving for why . Okay , | |
| 04:51 | so maybe if you want to put a little uh | |
| 04:53 | star here , solve for why . That's what we're | |
| 05:00 | trying to do . We're trying to get why alone | |
| 05:03 | , right ? We're isolating why . So here we | |
| 05:06 | go . Hopefully you remember some a little bit of | |
| 05:08 | algebra uh notice I'm solving for why I'm trying to | |
| 05:12 | get why alone . It's not alone . We have | |
| 05:15 | this plus one , so I need to get rid | |
| 05:17 | of that plus one . And I do that by | |
| 05:20 | subtracting one . Anything I do to one side , | |
| 05:22 | I have to do the other . So subtract one | |
| 05:24 | year uh that becomes zero , goes away and I | |
| 05:28 | get Y equals two X minus one . So now | |
| 05:34 | uh the question , does it show direct variation , | |
| 05:38 | does it look like Y equals K . X . | |
| 05:41 | Almost . It almost does . If I didn't have | |
| 05:44 | that -1 there it would . But because I have | |
| 05:48 | this two X -1 . no X . And Y | |
| 05:53 | . In this situation do not show direct variation because | |
| 05:56 | of this minus one notice there's no minus anything here | |
| 06:00 | . Okay it's just K . X . Um Let's | |
| 06:03 | look at the next one . So one half Y | |
| 06:05 | equals X . Again I'm solving for why I'm trying | |
| 06:09 | to get why alone right now , it's being multiplied | |
| 06:12 | by one half . So to get rid of that | |
| 06:15 | one half , I could divide by a half but | |
| 06:18 | divided by a fraction is the same thing as multiplying | |
| 06:22 | by , it's reciprocal . The reciprocal of one half | |
| 06:25 | is to over one which is two . So to | |
| 06:28 | solve I'm just gonna do I've got one half Y | |
| 06:32 | . I'm gonna multiply by 2/1 . Anything I do | |
| 06:37 | to one side I have to do the other . | |
| 06:39 | So that's gonna be come to X two times X | |
| 06:43 | . And this works because right the twos will cancel | |
| 06:46 | each other out and I basically get one Y . | |
| 06:49 | Which is just why Which is great all by itself | |
| 06:53 | . Which is what I want equals two X . | |
| 06:57 | In this situation , does that look like what I | |
| 07:01 | wanted ? Why it was K . X . Absolutely | |
| 07:05 | . My concept for proportionality is 2K . is to | |
| 07:09 | hear um I have Y equals on the one side | |
| 07:13 | . There's the X . So in this situation yes | |
| 07:18 | X . And Y . Do show correct variation . | |
| 07:21 | Okay here's something to try on your own . Thank | |
| 07:30 | you so much for watching . And if you like | |
| 07:32 | this video please subscribe . |
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