Slope - By Anywhere Math
Transcript
| 00:00 | Welcome to anywhere Math . I'm Jeff Jacobson . And | |
| 00:02 | today we're gonna talk about how to find the slope | |
| 00:05 | of a line using any two points on that line | |
| 00:09 | . Let's get started . So before you do an | |
| 00:28 | example what exactly is slope ? Well , slope is | |
| 00:32 | just a way to measure the steepness of a line | |
| 00:34 | . And if you have ever skied or snowboard , | |
| 00:38 | snowboard did you would know uh ski slopes and different | |
| 00:43 | ones are have different steepness . Is some are really | |
| 00:46 | steep and and very difficult and some are a lot | |
| 00:49 | flatter and a lot easier . So that's that's what | |
| 00:51 | slope is , Its just how steep a line is | |
| 00:54 | . Uh It's a ratio and it's a ratio comparing | |
| 00:58 | the change in the UAE which we call the rise | |
| 01:01 | to the change in the X . Which we call | |
| 01:04 | the run between any two points on a line . | |
| 01:06 | We can find the slope with any two points on | |
| 01:09 | a line . Um So here we go . It's | |
| 01:12 | a ratio . So we're gonna write it as a | |
| 01:13 | fraction . So slope again , is the change in | |
| 01:16 | the Y . Over the change in the X . | |
| 01:19 | Or you can think of it as the rise over | |
| 01:22 | the run . So that's what slope is . Now | |
| 01:24 | . Let's do an example to figure out how to | |
| 01:26 | actually find the slope sample . Number one , find | |
| 01:29 | the slope of each line . So for a uh | |
| 01:33 | with this graph here is our line in green . | |
| 01:35 | We've got two points labeled 00 and 34 And again | |
| 01:40 | , like I said before , if we have two | |
| 01:42 | points on the line , we can find the slope | |
| 01:45 | . All we need is two points . And typically | |
| 01:47 | , uh you're gonna want to pick points that are | |
| 01:50 | easy to work with . So if you look at | |
| 01:53 | the line carefully , I could have picked points anywhere | |
| 01:56 | along here . But some of these , like if | |
| 02:00 | I looked here , Well , that looks like it | |
| 02:02 | would be too and then maybe 2.8 , right , | |
| 02:06 | That vessel is not going to be very nice to | |
| 02:08 | work with . So , there's a reason we chose | |
| 02:10 | 34 and 00 because they're both the X and Y | |
| 02:15 | coordinates are both whole numbers . So that's nice and | |
| 02:17 | easy to work with . So slope change in the | |
| 02:21 | why ? Or the rise over change in X . | |
| 02:24 | Which we call the run . Well , from this | |
| 02:27 | point to this point , what was the change in | |
| 02:30 | the Y values ? Well , we went up four | |
| 02:34 | . Okay , so that's gonna go in our numerator | |
| 02:36 | and from 00 to 34 What was our change in | |
| 02:40 | X ? Well , we went over to the right | |
| 02:42 | three and that's a positive three . When we move | |
| 02:46 | to the right , just like when we go up | |
| 02:48 | , that's positive four . So in this situation , | |
| 02:52 | our slope then is 4/3 4/3 . Okay . Uh | |
| 03:00 | now you might be asking , well , mr Jacobson | |
| 03:03 | , what happens if I go the other way ? | |
| 03:05 | What happens if I want to go from 34 down | |
| 03:09 | to 00 ? Would the slope be the same ? | |
| 03:12 | And yes , it would . And the reason is | |
| 03:15 | because if I go from 34,200 , my change and | |
| 03:20 | why is actually negative for it , because I would | |
| 03:23 | be going down for and then I would have to | |
| 03:27 | go over to the left three . That would be | |
| 03:30 | my change in X . So that would be negative | |
| 03:32 | three for my change in X . So that would | |
| 03:35 | look like I'll get rid of that . You would | |
| 03:40 | have negative four because we're going down four and you | |
| 03:45 | would have negative three For your change in X because | |
| 03:49 | we're going to the left 3 ? Well , if | |
| 03:52 | you look , I got negative divided by a negative | |
| 03:55 | , which would simplify to 4/3 , which was the | |
| 03:59 | same as what we had before . So the nice | |
| 04:02 | thing was slope . It doesn't matter if you go | |
| 04:04 | from this point to this point or the other way | |
| 04:07 | around this point to this point , you should get | |
| 04:10 | the same answer if you're doing it correctly . Let's | |
| 04:13 | look at B . So again we're going we have | |
| 04:16 | our our line here That passes through negative to -2 | |
| 04:22 | and passes through 4 1 . So Slope change in | |
| 04:27 | the wide . The vertical change over the change in | |
| 04:30 | the X . The horizontal change . Well , it's | |
| 04:34 | very simple . We've got the arrow here , the | |
| 04:35 | vertical changes three over the horizontal change which is six | |
| 04:40 | . So that's gonna be 3/6 . Now notice that's | |
| 04:44 | not in simplest , simplest form . So what we | |
| 04:46 | need to do is simplify that first and that would | |
| 04:49 | give us one half . So our slope here would | |
| 04:53 | be one half . I'll get rid of that . | |
| 04:56 | And again our slope over here would be a positive | |
| 05:00 | 4/3 . Here's some to try on your own . | |
| 05:11 | All right , here's example , number two , it | |
| 05:13 | says graph the line that passes through the following points | |
| 05:15 | . Then find the slope of the line . So | |
| 05:18 | our two points are negative 35 and four negative six | |
| 05:22 | . So first let's graph those points . So first | |
| 05:26 | thing I'm gonna do is graph the line . I've | |
| 05:27 | got my snazzy little um line graph maker thing here | |
| 05:32 | . Uh So first my first point is negative 35 | |
| 05:36 | So I'm gonna take this point and I'm gonna put | |
| 05:40 | it at negative three and then up five . So | |
| 05:44 | negative 35 is right there . My next 350.4 , | |
| 05:48 | negative six . So again I go over 4 1st | |
| 05:52 | and then down negative six . So 4 -6 is | |
| 05:58 | gonna be right there . So those are my two | |
| 06:01 | points . Now , I am ready to draw my | |
| 06:03 | line , I'm gonna choose those two points . And | |
| 06:08 | there we go , there is my line . Now | |
| 06:11 | let's try to find the slope now that we have | |
| 06:15 | graft our line . Uh Now it's time to find | |
| 06:18 | the slope . So again , like I said before | |
| 06:22 | , it does not matter if I go from this | |
| 06:24 | point to this point or vice versa at this point | |
| 06:27 | to this point , I just need to be consistent | |
| 06:30 | um with going with my change and why am I | |
| 06:33 | changing X . Now , Before I even do that | |
| 06:35 | , let's use a little bit of logic . Looking | |
| 06:38 | at this line , would I expect my slope to | |
| 06:41 | be positive or negative ? Well , if you look | |
| 06:45 | this line is going down as we go from left | |
| 06:48 | to right um which means it should be a negative | |
| 06:52 | slope . If the line is going this way as | |
| 06:56 | we go from left to right , it's going up | |
| 06:58 | , that would be a positive slope . A horizontal | |
| 07:01 | line , zero slope and a vertical line is a | |
| 07:05 | slope that's undefined . Um But let's get back to | |
| 07:09 | this problem at hand , so we're expecting a negative | |
| 07:11 | slope , so let's keep that in mind . Uh | |
| 07:14 | So , first well , what is my change in | |
| 07:17 | ? Why ? Well , I'm going to go from | |
| 07:19 | this point here uh negative 35 Down to four . | |
| 07:23 | Negative 6 . So my change and why is right | |
| 07:28 | here ? Okay , so let's count that . Well | |
| 07:33 | , it goes from five down to zero here , | |
| 07:37 | so that's negative five . And then we go from | |
| 07:40 | zero down six . More . So negative five and | |
| 07:43 | negative six . That gives me negative 11 . Okay | |
| 07:48 | . And if you want , maybe kind of the | |
| 07:49 | slow way would be just to count all the squares | |
| 07:52 | . You could do that , but but really think | |
| 07:55 | uh you're going from five all the way down to | |
| 07:58 | negative six . Okay . And also you can also | |
| 08:01 | look at that over here with our original points , | |
| 08:03 | I'm going from five in my y coordinate down to | |
| 08:07 | negative six In this white coordinate here , and that | |
| 08:10 | is a change of -11 . Uh So next let's | |
| 08:15 | do my change in in X my horizontal change . | |
| 08:19 | Well , that's going to be uh right there And | |
| 08:24 | again . You could count it , but I can | |
| 08:26 | also think , well , I was at negative three | |
| 08:29 | on my X coordinate here . Now I am all | |
| 08:34 | the way up to four . So from negative 3-0 | |
| 08:38 | , that's a change of positive three . And then | |
| 08:41 | from zero for more , that's going to be positive | |
| 08:45 | seven . So from negative three up to four , | |
| 08:48 | that's a change of seven . And because we're going | |
| 08:53 | towards the right , that's positive seven . So now | |
| 08:57 | we're ready to write the slope slope again , is | |
| 09:01 | the vertical change the rise , the change in Y | |
| 09:06 | values Over the change in the X . So my | |
| 09:09 | slope is now negative . 11/7 . Okay , here's | |
| 09:16 | some more to try on your own . Thank you | |
| 09:24 | so much for watching . And as always , if | |
| 09:25 | you like this video , please subscribe . |
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