WEBVTT 00:00:00.000 --> 00:00:03.773 Okay, in this video, I wanna talk more about vertical and 00:00:03.773 --> 00:00:06.476 horizontal stretching and reflecting. 00:00:06.476 --> 00:00:10.124 And I'm gonna definitely talk about case 3 and case 4, 00:00:10.124 --> 00:00:13.854 which deals with horizontal stretches or compressions. 00:00:13.854 --> 00:00:18.422 And then 5 or 6 deals with reflections, maybe we'll get to that one, we'll see. 00:00:18.422 --> 00:00:23.918 Again, the basic idea and, 3 with these horizontal compressions or stretches. 00:00:23.918 --> 00:00:27.684 If you multiply by a number bigger than 1 on the inside, 00:00:27.684 --> 00:00:31.144 it's actually gonna squish your graph together. 00:00:31.144 --> 00:00:34.078 If you multiply by a number between 0 and 1, 00:00:34.078 --> 00:00:36.944 it's gonna actually pull it apart, okay? 00:00:36.944 --> 00:00:42.506 So let's look at at least two examples. 00:00:42.506 --> 00:00:45.441 So again, here's my original graph, here at the top, 00:00:45.441 --> 00:00:49.200 my blue graph, this is the one I'm gonna tweak to come up with a new one. 00:00:49.200 --> 00:00:55.196 So it's this little sawtooth function, -4, 0, -3, -2, -2, 00:00:55.196 --> 00:01:01.306 0, -1, 2, 0, 0, and then it's this little step function, okay? 00:01:01.306 --> 00:01:06.168 So, what I'm gonna graph now is the function f of 2x, and 00:01:06.168 --> 00:01:11.129 what this again does, since I'm multiplying the xs by 2, 00:01:11.129 --> 00:01:14.723 it's actually gonna compress the graph. 00:01:14.723 --> 00:01:19.139 It's gonna basically compress it by a factor of 2. 00:01:19.139 --> 00:01:23.744 So instead of going out from 0 to -4 and 0 to 4, 00:01:23.744 --> 00:01:27.927 it's now gonna go from 0 to -2 and 0 to 2. 00:01:27.927 --> 00:01:32.642 I think this is the one that always confuses people as well, cuz, hey, 00:01:32.642 --> 00:01:36.746 you're multiplying by 2, that should make things bigger. 00:01:36.746 --> 00:01:41.127 But you can't really think about it like that, or I guess if you do so, 00:01:41.127 --> 00:01:42.405 it's not correct. 00:01:42.405 --> 00:01:45.933 Think about it this way, if I plug -1 in, 00:01:45.933 --> 00:01:50.254 if I let x equal -1, what would I get on the inside? 00:01:50.254 --> 00:01:55.479 I would get a value of -2, but according to the original graph, 00:01:55.479 --> 00:01:59.475 it says if you plug -2 in, you should get 0 out. 00:01:59.475 --> 00:02:03.824 So if I plug -1 in, I'm gonna get -2 on the inside, 00:02:03.824 --> 00:02:09.609 which should give me an output of 0, okay, and that's the basic idea. 00:02:09.609 --> 00:02:13.583 You're gonna kind of cut the x coordinates in half. 00:02:13.583 --> 00:02:17.612 So, originally, at -2 there was an x coordinate of 0. 00:02:17.612 --> 00:02:24.081 If I cut that x coordinate in half It now becomes -1, keep the same y coordinate. 00:02:24.081 --> 00:02:28.282 At -4, the original x coordinate is -4, 00:02:28.282 --> 00:02:32.271 if you cut that in half, you'll get -2. 00:02:32.271 --> 00:02:36.192 Okay, originally, at- 1 it was up here at a y value of 2. 00:02:36.192 --> 00:02:42.452 If you cut -1 in half you're at negative one-half, and then I'm up here at 2, 00:02:42.452 --> 00:02:48.650 likewise, at negative three-halves, I'm gonna be down here at -2, okay? 00:02:48.650 --> 00:02:53.838 So it still has the same height, but everything has 00:02:53.838 --> 00:02:59.038 gotten squished together by a factor of 2, okay. 00:02:59.038 --> 00:03:02.874 And then that's gonna be the same thing that happens on the right hand side. 00:03:02.874 --> 00:03:08.483 Instead of extending out a distance of 0 to 2, 00:03:08.483 --> 00:03:12.842 it's only gonna go out from 0 to 1. 00:03:12.842 --> 00:03:19.794 And then instead of going out from 2 to 4, it's only gonna go out from now 1 to 2. 00:03:23.247 --> 00:03:28.558 Okay, again, imagine chopping the interval 0 to two-half, you get 0 to 1. 00:03:28.558 --> 00:03:31.857 Imagine chopping the interval 2 to 4 in half, 00:03:31.857 --> 00:03:34.833 you would get the interval 1 to 2, okay? 00:03:34.833 --> 00:03:38.925 So, again, notice the heights are the same, but 00:03:38.925 --> 00:03:43.905 it should look definitely a little more squished together. 00:03:43.905 --> 00:03:46.420 Okay, so let's do another one of these. 00:03:49.795 --> 00:03:53.514 Now, what I'm gonna do is, again, very much a similar thing. 00:03:53.514 --> 00:03:59.137 I'm gonna multiply now the inside by one-half, and now, 00:03:59.137 --> 00:04:06.111 if you multiply by a one half, instead of compressing it by a factor of 2, 00:04:06.111 --> 00:04:10.065 you actually stretch it by a factor of 2. 00:04:10.065 --> 00:04:16.140 So I can't kind of keep the scale correct on this graph, 00:04:16.140 --> 00:04:18.866 cuz I've got -4 to 4. 00:04:18.866 --> 00:04:22.895 So now the idea is, instead of getting squished together, 00:04:22.895 --> 00:04:25.643 you're gonna pull out by a factor of 2. 00:04:25.643 --> 00:04:29.825 So that means, I'm gonna go, instead of from out to -4, 00:04:29.825 --> 00:04:32.753 I'm gonna go all the way out here to -8, 00:04:32.753 --> 00:04:37.207 and instead of +4, I'm gonna go all the way out here to +8. 00:04:37.207 --> 00:04:41.389 Okay, and now basically, you do the same thing, originally, at -2, 00:04:41.389 --> 00:04:42.814 that's where I got a 0. 00:04:42.814 --> 00:04:47.955 If you multiply that x coordinate by 2, cuz we're stretching 00:04:47.955 --> 00:04:52.618 it by a factor of 2, you're gonna go out to -4, okay? 00:04:52.618 --> 00:04:57.857 And at-1, you originally had a y coordinate of 2, well, 00:04:57.857 --> 00:05:03.947 now if you multiply that x coordinate by 2, you're gonna be at -2. 00:05:03.947 --> 00:05:09.398 And then we'll be up here at a height of 2, likewise, 00:05:09.398 --> 00:05:14.861 at -6, we're gonna be down here at a height of -2. 00:05:14.861 --> 00:05:20.181 Again, if you multiply that original x coordinate by 2, 00:05:20.181 --> 00:05:23.980 you're gonna keep that same y value, but 00:05:23.980 --> 00:05:29.424 the x value turns into -6, and the x value stays the same. 00:05:29.424 --> 00:05:31.617 So again, play connect the dots. 00:05:36.450 --> 00:05:39.868 Obviously, again, I'm not a great artist, so forgive my artistry. 00:05:39.868 --> 00:05:43.343 Again, it doesn't really look stretched out because 00:05:43.343 --> 00:05:45.541 just the proportions of my graph. 00:05:45.541 --> 00:05:50.092 But if you had a bigger piece of paper, I think you would definitely see 00:05:50.092 --> 00:05:54.414 this thing looking more elongated, so try that for yourself, and 00:05:54.414 --> 00:05:59.073 then we'll go from, whoops, we'll go all the way out to 4 this time. 00:05:59.073 --> 00:06:04.305 So instead of being this little horizontal line at -1 from 0 to 2, 00:06:04.305 --> 00:06:10.639 again, now I double that, so it's gonna be looking like that all the way out to 4. 00:06:10.639 --> 00:06:16.839 And then I jump down to -2 at 4, and that extends all the way out to the value of 8. 00:06:16.839 --> 00:06:23.797 Okay, so this would be the graph, again, of f one half of x. 00:06:23.797 --> 00:06:26.990 And I'm definitely gonna put all of this stuff together in some more 00:06:26.990 --> 00:06:28.005 concrete examples. 00:06:28.005 --> 00:06:33.142 Again, just trying to give you a general idea of what's going on, let's see. 00:06:33.142 --> 00:06:35.651 I don't know if I can do the other two real quick. 00:06:37.860 --> 00:06:40.066 We'll save the other two for one other video. 00:06:40.066 --> 00:06:47.011 So this, again, deals with horizontal stretches or compressions. 00:06:47.011 --> 00:06:51.347 Next, I'll basically deal with flips about the x-axis and 00:06:51.347 --> 00:06:55.182 flips about the y-axis, so look for another video. 00:06:55.182 --> 00:06:56.302 And then again, 00:06:56.302 --> 00:07:01.344 stay tuned I'll do some more general ones where I do all the compressions, 00:07:01.344 --> 00:07:06.805 and stretching ,and rotating, and shifting, and all of that stuff combined.