1 00:00:00,855 --> 00:00:03,521 Find the slope of the line that goes through the ordered 2 00:00:03,521 --> 00:00:08,940 pairs (4, 2) and (-3, 16). 3 00:00:08,940 --> 00:00:16,821 So just as a reminder, slope is defined as rise over run 4 00:00:16,821 --> 00:00:20,405 or you could view that as rise is just a change in y 5 00:00:20,405 --> 00:00:22,821 and run is a change in x 6 00:00:22,821 --> 00:00:25,255 The triangles here, there's the delta symbol 7 00:00:25,255 --> 00:00:27,506 it literally means "change in" 8 00:00:27,506 --> 00:00:30,238 Or another way, and you might see this formula and 9 00:00:30,238 --> 00:00:32,088 it tends to be really complicated 10 00:00:32,088 --> 00:00:34,289 but just remember it's just these two things over here 11 00:00:34,289 --> 00:00:38,055 Sometimes slope will be specified with the variable m 12 00:00:38,055 --> 00:00:40,888 and they'll say m is the same thing 13 00:00:40,888 --> 00:00:43,189 (and this is really the same thing as change in y) 14 00:00:43,189 --> 00:00:48,805 They'll write y2 minus y1 over x2 minus x1 15 00:00:48,805 --> 00:00:51,472 And this notation tends to be kind of complicated 16 00:00:51,472 --> 00:00:55,421 but all this means is you take the y value of your endpoint 17 00:00:55,421 --> 00:00:57,939 and subtract from it the y value of your starting point 18 00:00:57,939 --> 00:01:00,556 That'll essentially give you your change in y 19 00:01:00,556 --> 00:01:04,022 and it says take the x value of your endpoint 20 00:01:04,022 --> 00:01:07,289 and subtract it from the x value of your starting point 21 00:01:07,289 --> 00:01:09,139 and that'll give you your change in x 22 00:01:09,139 --> 00:01:12,156 So whichever of these work for you, let's actually 23 00:01:12,156 --> 00:01:15,206 figure out the slope of the line that goes through these 24 00:01:15,206 --> 00:01:16,889 two points. 25 00:01:16,889 --> 00:01:17,907 So we're starting at 26 00:01:17,907 --> 00:01:19,597 (actually we can do it both ways) 27 00:01:19,623 --> 00:01:22,609 we can start at this point and go to that point and calculate the slope, 28 00:01:22,645 --> 00:01:25,406 or we can start at this point and go to that point and calculate the slope. 29 00:01:25,406 --> 00:01:27,881 So let's do it both ways. 30 00:01:27,881 --> 00:01:35,198 So let's say our starting point is the point (4,2) 31 00:01:35,198 --> 00:01:42,699 And let's say our endpoint is (-3, 16) 32 00:01:42,699 --> 00:01:45,583 So what is the change in x over here 33 00:01:45,583 --> 00:01:48,132 What is the change in x in this scenario 34 00:01:48,132 --> 00:01:50,982 So we're going from 4 to -3 35 00:01:50,982 --> 00:01:54,532 if something goes from 4 to -3, what was its change? 36 00:01:54,532 --> 00:01:58,134 You have to go down 4 to get to zero 37 00:01:58,134 --> 00:02:01,182 Then you have to go down another 3 to get to -3. 38 00:02:01,182 --> 00:02:06,016 So our change in x here is -7 39 00:02:06,016 --> 00:02:10,265 Actually, let me write it this way 40 00:02:10,265 --> 00:02:16,187 Our change in x is equal to -3 - 4 which is equal to 41 00:02:16,187 --> 00:02:18,595 -7 42 00:02:18,595 --> 00:02:22,828 If I'm going from 4 to -3, I went down by 7 43 00:02:22,828 --> 00:02:25,929 Our change in x is -7 44 00:02:25,929 --> 00:02:28,362 Let's do the same thing for the change in y 45 00:02:28,362 --> 00:02:30,712 And notice: I implicitly use this formula over here 46 00:02:30,712 --> 00:02:34,763 our change in x was this value - our endpoint 47 00:02:34,763 --> 00:02:39,230 our end x value minus our starting x value 48 00:02:39,230 --> 00:02:41,796 let's do the same thing for our change in y 49 00:02:41,796 --> 00:02:44,162 Our change in y 50 00:02:44,162 --> 00:02:47,062 If we're starting at 2 and we go to 16 that means 51 00:02:47,062 --> 00:02:49,662 we moved up 14 52 00:02:49,662 --> 00:02:51,429 Or another way you could say it is: 53 00:02:51,429 --> 00:02:53,995 You could take your ending y value 54 00:02:53,995 --> 00:02:56,262 and subtract from the your starting y value 55 00:02:56,262 --> 00:02:58,045 and you get 14 56 00:02:58,045 --> 00:03:00,079 So what is the slope over here? 57 00:03:00,079 --> 00:03:02,794 Well the slope is just change in y over change in x 58 00:03:02,794 --> 00:03:08,475 So the slope over here is change in y over change in x 59 00:03:08,475 --> 00:03:10,909 which is: our change in y is 14 60 00:03:10,909 --> 00:03:14,191 and our change in x is -7 61 00:03:14,191 --> 00:03:21,774 And if we want to simplify this 14 divided by -7 is -2 62 00:03:21,774 --> 00:03:24,044 Now what I want to show you is we could have done it 63 00:03:24,044 --> 00:03:25,957 the other way around 64 00:03:25,957 --> 00:03:28,558 We could have made this the starting point and this the endpoint. 65 00:03:28,558 --> 00:03:30,924 And what we would have gotten is the negative values 66 00:03:30,924 --> 00:03:32,840 of each of these, but they would have cancelled out 67 00:03:32,840 --> 00:03:35,224 and we would still get -2. 68 00:03:35,224 --> 00:03:38,307 Let's try it out. So let's say that our start point 69 00:03:38,307 --> 00:03:45,208 was (-3,16) and let's say that our endpoint is (4,2) 70 00:03:45,208 --> 00:03:49,690 So in this situation, what is our change in x? 71 00:03:49,690 --> 00:03:51,940 Our change in x. 72 00:03:51,940 --> 00:03:56,276 If I start at -3 and I go to 4, that means I went up 7 73 00:03:56,276 --> 00:04:03,309 Or if you want to just calculate that, you would do 4 - -3 74 00:04:03,355 --> 00:04:05,859 but needless to say, we just went up 7 75 00:04:05,859 --> 00:04:08,326 And what is our change in y 76 00:04:08,326 --> 00:04:10,993 Our change in y over here 77 00:04:10,993 --> 00:04:13,077 Or you could say our "rise" 78 00:04:13,077 --> 00:04:15,709 If we start at 16 and we end at 2, that means we 79 00:04:15,709 --> 00:04:18,410 went down 14. 80 00:04:18,410 --> 00:04:22,542 Or you could just say 2 - 16 is -14. 81 00:04:22,542 --> 00:04:25,193 We went down by 14 - this is our run. 82 00:04:25,193 --> 00:04:29,059 So if we say rise over run, which is the same thing as change 83 00:04:29,059 --> 00:04:30,826 in y over change in x. 84 00:04:30,826 --> 00:04:34,776 Our rise is -14 and our run here is 7. 85 00:04:34,776 --> 00:04:37,410 So notice these are just the negatives of these values from 86 00:04:37,410 --> 00:04:39,660 when we swapped them. 87 00:04:39,660 --> 00:04:42,642 So once again, this is equal to -2. 88 00:04:42,642 --> 00:04:47,226 Let's just visualize this - let me do a quick graph here 89 00:04:47,226 --> 00:04:51,026 just to show you what a downward slope would look like 90 00:04:51,026 --> 00:04:53,842 So let me draw our two points 91 00:04:53,842 --> 00:04:56,109 So this is my x axis 92 00:04:56,109 --> 00:04:58,160 That is my y axis 93 00:04:58,160 --> 00:05:01,442 So this point over here, (4,2), so let me graph it 94 00:05:01,442 --> 00:05:05,176 So, we're going to go all the way up to 16, so let me save some space here 95 00:05:05,176 --> 00:05:08,093 So we have 1, 2, 3, 4 96 00:05:08,093 --> 00:05:11,258 it's 4 comma 1 ... 2 97 00:05:11,258 --> 00:05:14,126 So (4,2) is right over here 98 00:05:14,126 --> 00:05:17,926 Then we have the point (-3, 16) 99 00:05:17,926 --> 00:05:19,676 So let me draw that over here 100 00:05:19,676 --> 00:05:21,994 So we have -1 ... 2 ... 3 101 00:05:21,994 --> 00:05:27,976 and we have to go up 16, so this is 2 ... 3 ... 4 ... 5 ... 6 ... 7 ... 8 ... 9 ... 10 ... 11 ... 12 102 00:05:27,976 --> 00:05:30,509 13 ... 14 ... 15 ... 16 103 00:05:30,509 --> 00:05:33,093 So it goes right over here 104 00:05:33,093 --> 00:05:37,991 So this is (-3,16). 105 00:05:37,991 --> 00:05:42,805 So the line that goes between them is going to look something like this 106 00:05:42,805 --> 00:05:47,539 I'm going to try my best to draw a relatively straight line 107 00:05:47,539 --> 00:05:51,603 That line will keep going. 108 00:05:51,603 --> 00:05:53,988 So that's my best attempt. 109 00:05:53,988 --> 00:05:56,854 And notice it's downward sloping - as you increase in x 110 00:05:56,854 --> 00:05:58,689 value the line goes down. 111 00:05:58,689 --> 00:06:01,139 It's going from the top left to the bottom right 112 00:06:01,139 --> 00:06:03,756 As x gets bigger, y gets smaller 113 00:06:03,756 --> 00:06:06,271 that's what a downward sloping line looks like 114 00:06:06,271 --> 00:06:09,257 And just to visualize our change in x's and our change in y's 115 00:06:09,257 --> 00:06:11,506 that we dealt with here, 116 00:06:11,506 --> 00:06:18,537 when we started with (4,2) and ended at (-3,16) 117 00:06:18,537 --> 00:06:22,188 That was analogous to starting here and ending over there 118 00:06:22,188 --> 00:06:25,739 And we said our change in x was -7 119 00:06:25,739 --> 00:06:28,321 We have to move back our run 120 00:06:28,321 --> 00:06:31,122 We had to move in the left direction by 7 121 00:06:31,122 --> 00:06:33,239 That's why it was -7 122 00:06:33,239 --> 00:06:37,923 And then we had to move in the y direction positive 14 123 00:06:37,923 --> 00:06:40,255 That's why our rise was positive 124 00:06:40,255 --> 00:06:42,855 So it's 14 over -7, or -2. 125 00:06:42,855 --> 00:06:48,373 When we did the other way, we started at this point, and ended at this point 126 00:06:48,373 --> 00:06:53,828 Started at (-3,16) and ended at that point. 127 00:06:53,828 --> 00:06:57,045 So in that situation our run was positive 7 128 00:06:57,045 --> 00:06:59,761 And now we had to go down in the y direction since we 129 00:06:59,761 --> 00:07:01,895 switched the starting and the end point. 130 00:07:01,895 --> 00:07:04,394 And now we have to go down -14. 131 00:07:04,394 --> 00:07:08,280 Our run is now positive 7, and our rise is now -14 132 00:07:08,280 --> 99:59:59,999 Either way, we got the same slope.