1 00:00:00,000 --> 00:00:00,410 2 00:00:00,410 --> 00:00:05,429 In this video I'm going to do a bunch of example slope 3 00:00:05,429 --> 00:00:09,449 problems. Just as a bit of review, slope is just a way of 4 00:00:09,449 --> 00:00:11,919 measuring the inclination of a line. 5 00:00:11,919 --> 00:00:13,869 And the definition-- we're going to hopefully get a good 6 00:00:13,869 --> 00:00:16,800 working knowledge of it in this video-- the definition of 7 00:00:16,800 --> 00:00:21,719 it is a change in y divided by change in x. 8 00:00:21,719 --> 00:00:24,960 This may or may not make some sense to you right now, but as 9 00:00:24,960 --> 00:00:27,830 we do more and more examples, I think it'll make a good 10 00:00:27,829 --> 00:00:28,729 amount of sense. 11 00:00:28,730 --> 00:00:30,899 Let's do this first line right here. 12 00:00:30,899 --> 00:00:31,660 Line a. 13 00:00:31,660 --> 00:00:34,460 Let's figure out its slope. 14 00:00:34,460 --> 00:00:37,490 They've actually drawn two points here that we can use as 15 00:00:37,490 --> 00:00:39,130 the reference points. 16 00:00:39,130 --> 00:00:40,140 So first of all, let's look at the 17 00:00:40,140 --> 00:00:41,770 coordinates of those points. 18 00:00:41,770 --> 00:00:44,140 So you have this point right here. 19 00:00:44,140 --> 00:00:45,039 What's its coordinates? 20 00:00:45,039 --> 00:00:48,479 Its x-coordinate is 3. 21 00:00:48,479 --> 00:00:51,069 Its y-coordinate is 6. 22 00:00:51,070 --> 00:00:54,939 And then down here, this point's x-coordinate is 23 00:00:54,939 --> 00:01:00,780 negative 1 and its y-coordinate is negative 6. 24 00:01:00,780 --> 00:01:02,920 So there's a couple of ways we can think about slope. 25 00:01:02,920 --> 00:01:05,329 One is, we could look at it straight up using the formula. 26 00:01:05,329 --> 00:01:10,030 We could say change in y-- so slope is change in y over 27 00:01:10,030 --> 00:01:11,370 change in x. 28 00:01:11,370 --> 00:01:12,850 We can figure it out numerically. 29 00:01:12,849 --> 00:01:15,199 I'll in a second draw it graphically. 30 00:01:15,200 --> 00:01:17,040 So what's our change in y? 31 00:01:17,040 --> 00:01:20,330 Our change in y is literally how much did our y values 32 00:01:20,329 --> 00:01:23,760 change going from this point to that point? 33 00:01:23,760 --> 00:01:25,400 So how much did our y values change? 34 00:01:25,400 --> 00:01:29,570 Our y went from here, y is at negative 6 and it went all the 35 00:01:29,569 --> 00:01:34,059 way up to positive 6. 36 00:01:34,060 --> 00:01:36,150 So what's this distance right here? 37 00:01:36,150 --> 00:01:38,930 It's going to be your end point y value. 38 00:01:38,930 --> 00:01:44,040 It's going to be 6 minus your starting point y value. 39 00:01:44,040 --> 00:01:50,140 Minus negative 6 or 6 plus 6, which is equal to 12. 40 00:01:50,140 --> 00:01:50,959 You could just count this. 41 00:01:50,959 --> 00:01:56,000 You say one, two, three, four, five, six, seven, eight, nine, 42 00:01:56,000 --> 00:01:57,840 ten, eleven, twelve. 43 00:01:57,840 --> 00:02:02,680 So when we changed our y value by 12, we had to change our x 44 00:02:02,680 --> 00:02:06,460 value by-- what was our change it x over the 45 00:02:06,459 --> 00:02:07,799 same change in y? 46 00:02:07,799 --> 00:02:10,569 Well we went from x is equal to negative 1 to 47 00:02:10,569 --> 00:02:14,189 x is equal to 3. 48 00:02:14,189 --> 00:02:16,930 Right? x went from negative 1 to 3. 49 00:02:16,930 --> 00:02:22,110 So we do the end point, which is 3 minus the starting point, 50 00:02:22,110 --> 00:02:24,540 which is negative 1, which is equal to 4. 51 00:02:24,539 --> 00:02:30,269 So our change in y over change in x is equal to 12/4 or if we 52 00:02:30,270 --> 00:02:32,659 want to write this in simplest form, this is the 53 00:02:32,659 --> 00:02:35,030 same thing as 3. 54 00:02:35,030 --> 00:02:38,689 Now the interpretation of this means that for every 1 we move 55 00:02:38,689 --> 00:02:40,520 over-- we could view this, let me write it this way. 56 00:02:40,520 --> 00:02:44,189 Change in y over change in x is equal to-- we could say 57 00:02:44,189 --> 00:02:46,960 it's 3 or we could say it's 3/1. 58 00:02:46,960 --> 00:02:49,590 Which tells us that for every 1 we move in the positive 59 00:02:49,590 --> 00:02:52,650 x-direction, we're going to move up 3 because this is a 60 00:02:52,650 --> 00:02:54,539 positive 3 in the y-direction. 61 00:02:54,539 --> 00:02:55,479 You can see that. 62 00:02:55,479 --> 00:02:59,319 When we moved 1 in the x, we moved up 3 in the y. 63 00:02:59,319 --> 00:03:02,879 When we moved 1 in the x, we moved up 3 in the y. 64 00:03:02,879 --> 00:03:06,500 If you move 2 in the x-direction, you're going to 65 00:03:06,500 --> 00:03:08,250 move 6 in the y. 66 00:03:08,250 --> 00:03:11,210 6/2 is the same thing as 3. 67 00:03:11,210 --> 00:03:17,280 So this 3 tells us how quickly do we go up as we increase x. 68 00:03:17,280 --> 00:03:21,780 Let's do the same thing for the second line on this graph. 69 00:03:21,780 --> 00:03:23,469 Graph b. 70 00:03:23,469 --> 00:03:24,060 Same idea. 71 00:03:24,060 --> 00:03:26,300 I'm going to use the points that they gave us. 72 00:03:26,300 --> 00:03:28,730 But really you could use any points on that line. 73 00:03:28,729 --> 00:03:31,019 So let's see, we have one point here, which is 74 00:03:31,020 --> 00:03:34,550 the point 0, 1. 75 00:03:34,550 --> 00:03:36,540 You have 0, 1. 76 00:03:36,539 --> 00:03:38,780 And then the starting point-- we could call this the finish 77 00:03:38,780 --> 00:03:41,460 point-- the starting point right here, we could view it 78 00:03:41,460 --> 00:03:47,830 as x is negative 6 and y is negative 2. 79 00:03:47,830 --> 00:03:48,990 So same idea. 80 00:03:48,990 --> 00:03:53,230 What is the change in y given some change in x? 81 00:03:53,229 --> 00:03:56,439 So let's do the change in x first. So what is 82 00:03:56,439 --> 00:03:58,060 our change in x? 83 00:03:58,060 --> 00:04:02,930 So in this situation, what is our change in x? delta x. 84 00:04:02,930 --> 00:04:03,849 We could even count it. 85 00:04:03,849 --> 00:04:06,909 It's one, two, three, four, five, six. 86 00:04:06,909 --> 00:04:08,109 It's going to be 6. 87 00:04:08,110 --> 00:04:10,760 But if you didn't have a graph to count from, you could 88 00:04:10,759 --> 00:04:16,370 literally take your finishing x-position, so it's 0, and 89 00:04:16,370 --> 00:04:19,028 subtract from that your starting x-position. 90 00:04:19,028 --> 00:04:22,389 0 minus negative 6. 91 00:04:22,389 --> 00:04:26,360 So when your change in x is equal to-- so this will be 6-- 92 00:04:26,360 --> 00:04:29,180 what is our change in y? 93 00:04:29,180 --> 00:04:31,889 Remember we're taking this as our finishing position. 94 00:04:31,889 --> 00:04:36,199 95 00:04:36,199 --> 00:04:38,519 This is our starting position. 96 00:04:38,519 --> 00:04:40,669 So we took 0 minus negative 6. 97 00:04:40,670 --> 00:04:43,634 So then on the y, we have to do 1 minus negative 2. 98 00:04:43,634 --> 00:04:47,680 99 00:04:47,680 --> 00:04:49,180 What's 1 minus negative 2? 100 00:04:49,180 --> 00:04:51,259 That's the same thing as 1 plus 2. 101 00:04:51,259 --> 00:04:52,409 That is equal to 3. 102 00:04:52,410 --> 00:04:56,450 So it is 3/6 or 1/2. 103 00:04:56,449 --> 00:05:02,079 So notice, when we moved in the x-direction by 6, we moved 104 00:05:02,079 --> 00:05:05,339 in the y-direction by positive 3. 105 00:05:05,339 --> 00:05:09,169 So our change in y was 3 when our change in x was 6. 106 00:05:09,170 --> 00:05:11,430 Now, one of the things that confuses a lot of people is 107 00:05:11,430 --> 00:05:16,110 how do I know what order to-- how did I know to do the 0 108 00:05:16,110 --> 00:05:18,930 first and the negative 6 second and then the 1 first 109 00:05:18,930 --> 00:05:19,920 and then the negative 2 second. 110 00:05:19,920 --> 00:05:22,210 And the answer is you could've done it in either order as 111 00:05:22,209 --> 00:05:23,539 long as you keep them straight. 112 00:05:23,540 --> 00:05:27,300 So you could have also have done change in y 113 00:05:27,300 --> 00:05:28,829 over change in x. 114 00:05:28,829 --> 00:05:34,279 We could have said, it's equal to negative 2 minus 1. 115 00:05:34,279 --> 00:05:37,689 So we're using this coordinate first. Negative 2 minus 1 for 116 00:05:37,689 --> 00:05:43,490 the y over negative 6 minus 0. 117 00:05:43,490 --> 00:05:45,960 Notice this is a negative of that. 118 00:05:45,959 --> 00:05:47,649 That is the negative of that. 119 00:05:47,649 --> 00:05:49,544 But since we have a negative over negative, they're going 120 00:05:49,545 --> 00:05:50,259 to cancel out. 121 00:05:50,259 --> 00:05:54,810 So this is going to be equal to negative 3 over negative 6. 122 00:05:54,810 --> 00:05:56,019 The negatives cancel out. 123 00:05:56,019 --> 00:05:58,250 This is also equal to 1/2. 124 00:05:58,250 --> 00:06:05,250 So the important thing is if you use this y-coordinate 125 00:06:05,250 --> 00:06:09,279 first, then you have to use this 126 00:06:09,279 --> 00:06:11,349 x-coordinate first as well. 127 00:06:11,350 --> 00:06:15,640 If you use this y-coordinate first, as we did here, then 128 00:06:15,639 --> 00:06:17,159 you have to use this x-coordinate 129 00:06:17,160 --> 00:06:18,130 first, as you did there. 130 00:06:18,129 --> 00:06:19,850 You just have to make sure that your change in x and 131 00:06:19,850 --> 00:06:22,750 change in y are-- you're using the same final 132 00:06:22,750 --> 00:06:23,949 and starting points. 133 00:06:23,949 --> 00:06:30,189 Just to interpret this, this is saying that for every minus 134 00:06:30,189 --> 00:06:31,819 6 we go in x. 135 00:06:31,819 --> 00:06:36,480 So if we go minus 6 in x, so that's going backwards, we're 136 00:06:36,480 --> 00:06:40,210 going to go minus 3 in y. 137 00:06:40,209 --> 00:06:41,959 But they're essentially saying the same thing. 138 00:06:41,959 --> 00:06:44,169 The slope of this line is 1/2. 139 00:06:44,170 --> 00:06:50,150 Which tells us for every 2 we travel in x, we go up 1 in y. 140 00:06:50,149 --> 00:06:53,989 Or if we go back 2 in x, we go down 1 in y. 141 00:06:53,990 --> 00:06:56,069 That's what 1/2 slope tells us. 142 00:06:56,069 --> 00:07:00,560 Notice, the line with the 1/2 slope, it is less steep than 143 00:07:00,560 --> 00:07:03,839 the line with a slope of 3. 144 00:07:03,839 --> 00:07:06,489 Let's do a couple more of these. 145 00:07:06,490 --> 00:07:11,199 Let's do line c right here. 146 00:07:11,199 --> 00:07:12,670 I'll do it in pink. 147 00:07:12,670 --> 00:07:15,250 Let's say that the starting point-- I'm just picking this 148 00:07:15,250 --> 00:07:16,120 arbitrarily. 149 00:07:16,120 --> 00:07:18,840 Well, I'm using these points that they've drawn here. 150 00:07:18,839 --> 00:07:23,009 The starting point is at the coordinate negative 1, 6 and 151 00:07:23,009 --> 00:07:29,050 that my finishing point is at the point 5, negative 6. 152 00:07:29,050 --> 00:07:32,939 153 00:07:32,939 --> 00:07:35,980 Our slope is going to be-- let me write this-- slope is going 154 00:07:35,980 --> 00:07:40,480 to be equal to change in x-- sorry, change in y. 155 00:07:40,480 --> 00:07:41,460 I'll never forget that. 156 00:07:41,459 --> 00:07:43,500 Change in y over change in x. 157 00:07:43,500 --> 00:07:46,170 Sometimes it's said rise over run. 158 00:07:46,170 --> 00:07:48,610 Run is how much you're moving in the horizontal direction. 159 00:07:48,610 --> 00:07:51,360 Rise is how much you're moving in the vertical direction. 160 00:07:51,360 --> 00:07:55,840 Then we could say our change in y is our finishing y-point 161 00:07:55,839 --> 00:07:59,560 minus our starting y-point. 162 00:07:59,560 --> 00:08:01,990 This is our finishing y-point. 163 00:08:01,990 --> 00:08:05,990 That's our starting y-point, over our finishing x-point 164 00:08:05,990 --> 00:08:08,800 minus our starting x-point. 165 00:08:08,800 --> 00:08:11,439 If that confuses you, all I'm saying is, it's going to be 166 00:08:11,439 --> 00:08:16,569 equal to our finishing y-point is negative 6 minus our 167 00:08:16,569 --> 00:08:21,990 starting y-point, which is 6, over our finishing x-point, 168 00:08:21,990 --> 00:08:27,379 which is 5, minus our starting x-point, which is negative 1. 169 00:08:27,379 --> 00:08:31,779 So this is equal to negative 6 minus 6 is negative 12. 170 00:08:31,779 --> 00:08:34,279 5 minus negative 1. 171 00:08:34,279 --> 00:08:36,079 That is 6. 172 00:08:36,080 --> 00:08:37,840 So negative 12/6. 173 00:08:37,840 --> 00:08:41,240 That's the same thing as negative 2. 174 00:08:41,240 --> 00:08:45,230 Notice we have a negative slope here. 175 00:08:45,230 --> 00:08:49,670 That's because every time we increase x by 1, we go down in 176 00:08:49,669 --> 00:08:50,740 the y-direction. 177 00:08:50,740 --> 00:08:52,730 So this is a downward sloping line. 178 00:08:52,730 --> 00:08:55,170 It's going from the top left to the bottom right. 179 00:08:55,169 --> 00:08:58,610 As x increases, the y decreases. 180 00:08:58,610 --> 00:09:00,129 And that's why we got a negative slope. 181 00:09:00,129 --> 00:09:03,009 This line over here should have a positive slope. 182 00:09:03,009 --> 00:09:04,860 Let's verify it. 183 00:09:04,860 --> 00:09:07,360 So I'll use the same points that they 184 00:09:07,360 --> 00:09:09,399 use right over there. 185 00:09:09,399 --> 00:09:11,759 So this is line d. 186 00:09:11,759 --> 00:09:17,759 Slope is equal to rise over run. 187 00:09:17,759 --> 00:09:21,139 How much do we rise when we go from that point to that point? 188 00:09:21,139 --> 00:09:21,569 Let's see. 189 00:09:21,570 --> 00:09:22,270 We could do it this way. 190 00:09:22,269 --> 00:09:25,500 We are rising-- I could just count it out. 191 00:09:25,500 --> 00:09:30,110 We are rising one, two, three, four, five, six. 192 00:09:30,110 --> 00:09:32,460 We are rising 6. 193 00:09:32,460 --> 00:09:33,780 How much are we running? 194 00:09:33,779 --> 00:09:36,149 We are running-- I'll do it in a different color. 195 00:09:36,149 --> 00:09:41,579 We're running one, two, three, four, five, six. 196 00:09:41,580 --> 00:09:43,639 We're running 6. 197 00:09:43,639 --> 00:09:46,299 So our slope is 6/6, which is 1. 198 00:09:46,299 --> 00:09:49,509 Which tells us that every time we move 1 in the x-direction-- 199 00:09:49,509 --> 00:09:52,389 positive 1 in the x-direction-- we go positive 1 200 00:09:52,389 --> 00:09:55,389 in the y-direction. 201 00:09:55,389 --> 00:09:59,340 For every x, if we go negative 2 in the x-direction, we're 202 00:09:59,340 --> 00:10:02,340 going to go negative 2 in the y-direction. 203 00:10:02,340 --> 00:10:04,820 So whatever we do in x, we're going to do the same thing in 204 00:10:04,820 --> 00:10:05,710 y in this slope. 205 00:10:05,710 --> 00:10:06,740 Notice, that was pretty easy. 206 00:10:06,740 --> 00:10:09,200 If we wanted to do it mathematically, we could 207 00:10:09,200 --> 00:10:11,259 figure out this coordinate right there. 208 00:10:11,259 --> 00:10:13,569 That we could view as our starting position. 209 00:10:13,570 --> 00:10:17,629 Our starting position is negative 2, negative 4. 210 00:10:17,629 --> 00:10:20,350 211 00:10:20,350 --> 00:10:26,210 Our finishing position is 4, 2. 212 00:10:26,210 --> 00:10:29,720 213 00:10:29,720 --> 00:10:34,850 So our slope, change in y over change in x. 214 00:10:34,850 --> 00:10:42,000 I'll take this point 2 minus negative 4 over 4 minus 215 00:10:42,000 --> 00:10:43,250 negative 2. 216 00:10:43,250 --> 00:10:45,450 217 00:10:45,450 --> 00:10:48,970 2 minus negative 4 is 6. 218 00:10:48,970 --> 00:10:52,000 Remember that was just this distance right there. 219 00:10:52,000 --> 00:10:55,139 Then 4 minus negative 2, that's also 6. 220 00:10:55,139 --> 00:10:57,299 That's that distance right there. 221 00:10:57,299 --> 00:11:00,599 We get a slope of 1. 222 00:11:00,600 --> 00:11:01,379 Let's do another one. 223 00:11:01,379 --> 00:11:02,210 Let's do another couple. 224 00:11:02,210 --> 00:11:03,940 These are interesting. 225 00:11:03,940 --> 00:11:08,200 Let's do the line e right here. 226 00:11:08,200 --> 00:11:11,800 Change in y over change in x. 227 00:11:11,799 --> 00:11:14,419 So our change in y, when we go from this point to this 228 00:11:14,419 --> 00:11:16,159 point-- I'll just count it out. 229 00:11:16,159 --> 00:11:19,209 It's one, two, three, four, five, six, seven, eight. 230 00:11:19,210 --> 00:11:20,530 It's 8. 231 00:11:20,529 --> 00:11:25,509 Or you could even take this y-coordinate 2 minus negative 232 00:11:25,509 --> 00:11:28,460 6 will give you that distance, 8. 233 00:11:28,460 --> 00:11:30,019 What's the change in y? 234 00:11:30,019 --> 00:11:32,639 Well the y-value here is-- oh sorry what's the change in x? 235 00:11:32,639 --> 00:11:34,259 The x-value here is 4. 236 00:11:34,259 --> 00:11:35,389 The x-value there is 4. 237 00:11:35,389 --> 00:11:36,629 X does not change. 238 00:11:36,629 --> 00:11:39,279 So it's 8/0. 239 00:11:39,279 --> 00:11:40,659 Well, we don't know. 240 00:11:40,659 --> 00:11:42,319 8/0 is undefined. 241 00:11:42,320 --> 00:11:44,629 So in this situation the slope is undefined. 242 00:11:44,629 --> 00:11:46,830 When you have a vertical line, you say 243 00:11:46,830 --> 00:11:49,235 your slope is undefined. 244 00:11:49,235 --> 00:11:52,409 245 00:11:52,409 --> 00:11:54,189 Because you're dividing by 0. 246 00:11:54,190 --> 00:11:56,530 But that tells you that you're dealing probably with a 247 00:11:56,529 --> 00:11:58,319 vertical line. 248 00:11:58,320 --> 00:11:59,720 Now finally let's just do this one. 249 00:11:59,720 --> 00:12:02,670 This seems like a pretty straight up vanilla slope 250 00:12:02,669 --> 00:12:04,319 problem right there. 251 00:12:04,320 --> 00:12:08,050 You have that point right there, which is 252 00:12:08,049 --> 00:12:10,089 the point 3, 1. 253 00:12:10,090 --> 00:12:11,090 So this is line f. 254 00:12:11,090 --> 00:12:14,210 You have the point 3, 1. 255 00:12:14,210 --> 00:12:19,750 Then over here you have the point negative 6, negative 2. 256 00:12:19,750 --> 00:12:24,440 So our slope would be equal to change in y. 257 00:12:24,440 --> 00:12:28,960 I'll take this as our ending point, just so you can go in 258 00:12:28,960 --> 00:12:30,129 different directions. 259 00:12:30,129 --> 00:12:31,769 So our change in y-- now we're going to go 260 00:12:31,769 --> 00:12:33,819 down in that direction. 261 00:12:33,820 --> 00:12:36,540 So it's negative 2 minus 1. 262 00:12:36,539 --> 00:12:38,809 That's what this distance is right here. 263 00:12:38,809 --> 00:12:42,239 Negative 2 minus 1, which is equal to negative 3. 264 00:12:42,240 --> 00:12:44,340 Notice we went down 3. 265 00:12:44,340 --> 00:12:47,340 And then what is going to be our change in x? 266 00:12:47,340 --> 00:12:50,480 Well, we're going to go back that amount. 267 00:12:50,480 --> 00:12:51,879 What is that amount? 268 00:12:51,879 --> 00:12:56,200 Well, that is going to be negative 6, that's our end 269 00:12:56,200 --> 00:12:59,920 point, minus 3. 270 00:12:59,919 --> 00:13:04,699 That gives us that distance, which is negative 9. 271 00:13:04,700 --> 00:13:09,140 For every time we go back 9, we're going to go down 3. 272 00:13:09,139 --> 00:13:12,679 273 00:13:12,679 --> 00:13:16,064 Which is the same thing as if we go forward 9, we're going 274 00:13:16,065 --> 00:13:17,370 to go up 3. 275 00:13:17,370 --> 00:13:18,389 All equivalent. 276 00:13:18,389 --> 00:13:21,830 And we see these cancel out and you get a slope of 1/3. 277 00:13:21,830 --> 00:13:22,810 Positive 1/3. 278 00:13:22,809 --> 00:13:25,969 It's an upward sloping line. 279 00:13:25,970 --> 00:13:33,110 Every time we run 3, we rise 1. 280 00:13:33,110 --> 00:13:37,740 281 00:13:37,740 --> 00:13:41,810 Anyway, hopefully that was a good review of slope for you.