00:00This lesson we're going to
00:00look at gradient of a
00:02line. Now, in the formulable
00:03plate, you're given a formula
00:04for the gradient. And these
00:07are the formula for equation
00:09of a straight line. The
00:10M, so you're probably familiar
00:12with the letter M being
00:14used for gradient. So M
00:16in mathematics is the gradient.
00:20And almost always, the formula
00:23y equals Mx plus C,
00:25the M tells you the
00:26gradient. And this formula
00:28y minus y minus y
00:28minus y minus y minus
00:29y minus y minus y
00:29minus y minus y minus
00:29y minus y minus y
00:29minus y minus y minus
00:29minus y minus y minus
00:30minus y minus y minus
00:30y minus y minus y
00:30minus y minus y minus
00:31y minus y minus y
00:31minus y minus y minus
00:37minus y minus y minus
00:38y minus y minus y
00:38minus y minus y minus
00:38minus y minus y minus
00:38y minus y minus y
00:38minus y minus y minus
00:56minus
00:56The change in y over
00:59the change in x and
01:00that's y2 minus y1. It's
01:02the change in y over
01:03x2 minus x1, the change
01:05in x. So imagine I,
01:06let's just draw, let me
01:09just draw, let me just
01:14draw line here. Now, what
01:16is the gradient of this
01:17function? Well, how much has
01:20it changed going up? And
01:23how much has it changed
01:23going up?
01:24across. That's basically what it's
01:26saying. Well, it's changed two
01:27of these and three of
01:29these. Now, these are just
01:30boxes. Our numbers are, they
01:33could actually be meters or
01:34they could be kilometers that
01:35could be anything. It doesn't
01:36really matter. As long as
01:37it's relative, this is two
01:39and this is three. So
01:41the m, the gradient of
01:42this line, is two over
01:45three. The gradient of, let's
01:50say,
01:52this line, well, how much
01:59of a change in the
02:01Y has there been? Well,
02:02let's count these boxes, one,
02:04two, three, four, five, six,
02:05seven, eight. There's a change
02:07of eight, so M is
02:09equal to eight over two.
02:12However, because it's going down,
02:17it's negative, so it's negative
02:19eight over two, and yes,
02:20So you do want to
02:21simplify that eight to the
02:22value of two is four.
02:23So the gradient of this
02:24is negative four. I'll do
02:27two or three more examples
02:28of this. But let me
02:32just give you an example
02:35of gradient in real life.
02:37I like to use the
02:39cycling analogy, because I like
02:40cycling. If you cycle or
02:43if you watch the Tour
02:44de France, you'll know about,
02:46well, if you do road
02:47cycling,
02:48You'll know about gradients of
02:50hills because if you come
02:52if you're cycling along nice
02:53thing a flat road and
02:55then you come across a
02:5610 % Gradient hill and
03:02then you know that goes
03:03on for maybe I don't
03:04know maybe three or four
03:05or sometimes sometimes up to
03:07up to 20 kilometers I
03:10actually cycled up. There's a
03:12famous mountain in France called
03:13long vontu It's very famous
03:15from the Tour de France
03:16It's like a volcano in
03:21the middle of nowhere and
03:24it has like an average
03:25of 10 % for 20
03:26kilometers. It's really, really, really
03:29difficult to cycle up. I
03:30had to stop twice just
03:34to catch my breath. So
03:35what 10 % means is
03:38when it says 10 %
03:39it actually means for every
03:4210 %
03:44is a tenth, so it's
03:461 over 10. So what
03:48that means is for every,
03:50maybe I should do 11%.
03:54Let's say it's 11%. This
03:56is the same as 11
03:57over 100. That means for
03:59every 100, I go across,
04:02I go up 11. Sorry,
04:04I don't know, it might
04:04be something like that. That
04:06might be a gradient of
04:0711%. So if you have
04:09a gradient of, well, in
04:11cycling, like,
04:1216, 17 % is brutal.
04:15It's like even for professionals
04:16that would be very, very
04:17challenging just to cycle up
04:1820%. I think there's hills
04:21like 20, 20, 25 %
04:24and even the professionals complain
04:25that it shouldn't be in
04:26the race. And you'll rarely
04:27see that in a Tour
04:28de France. Skiing is another
04:30good example. I think if
04:34you have 40 % plus,
04:37so 40 % steepness,
04:40which would be 40 %
04:44is obviously 4 out of
04:4610 or 2 fifths. So
04:49if you have a gradient
04:50of 2 fifths, let me
04:51show you what that looks
04:51like here. That means 5
04:53along, 1, 2, 3, 4,
04:555 and 2 up there.
04:59That is a gradient of
05:0040 % or 2 fifths.
05:02Now if I drew this
05:03on for a number of
05:05kilometers and I turned it
05:08into
05:08a ski slope and ports
05:11no all over it that
05:12would be a black slope
05:13or when I say 40
05:15% plus like anything more
05:16than 40 % is considered
05:17a black slope. I think
05:19I think 25 to 40
05:23% is a blue slope.
05:25So if you've got like
05:26a 25 % is a
05:28quarter 25 % is a
05:30quarter. So if you've agreed
05:31in one over four, which
05:33would look like this for
05:35and one one over four
05:36that
05:36would be considered a blue
05:40slope. So I think it
05:41is interesting to think about
05:43where we see a gradient
05:45in real life and what
05:46they actually mean. But in
05:48mathematics, when you see gradients,
05:50this is what they're talking
05:50about. It's the rise over
05:53the run. If they're going
05:55up like this, it's positive,
05:56and if they're going down,
05:57it's negative. It's simple as
05:59that. Let me just do,
06:00let me do one more
06:01of these. This would be,
06:04This would have a gradient
06:06of 1 fourth. So here
06:08m equals 1 fourth. Interestingly,
06:12what would a gradient, what
06:14would a flat, to say
06:16you're cycling along a flat
06:18road, are just a straight
06:19line horizontal line. What gradient
06:21does it have? Well, it's
06:23rise over run. How much
06:24does it rise? Well, it
06:26doesn't rise at all. So
06:27the rise is zero. So
06:28it doesn't matter what the
06:29run is, the gradient. This
06:30is a gradient of zero.
06:33If you really want to
06:33get complicated, let's look at
06:36a vertical line. A vertical
06:38line, what gradient does this
06:40have? Well, this guy's rise
06:43is 4, but it's run
06:44is 0. So here we're
06:46left with 4 over 0.
06:50What happens when you divide
06:51by 0? Well, you can't
06:54do that. So we actually
06:55say, let me get rid
06:57of that, undefined.
07:01So gradient, a vertical line
07:03has an undefined gradient. Okay,
07:07hopefully that clears up what
07:10gradient is. I just want
07:11to apply this formula to
07:13one example to show you
07:14how it works. So I
07:15have the formula nice in
07:17the formula booklet, y2 minus
07:18y1 over x2 minus x1.
07:22I've written x1, y1, x2,
07:24y2 to make it clear
07:25for me. So y2 is
07:272, 2.
07:29minus be careful, it's minus
07:31negative two, so that is
07:32obviously the place people are
07:34gonna make mistakes. Two minus
07:36negative two, which is gonna
07:37become a plus. X two
07:38is five minus negative three,
07:43very, very careful. That's two
07:44plus two over five plus
07:48three equals four over eight
07:52equals a half simplified. So
07:54if the gradient of this
07:55line is a half, and
07:56let me just
07:57to show you if I
07:58was to draw negative three,
07:59negative two here, to five,
08:03two there. Does this have
08:07a gradient of a half?
08:08Well, yes, it does, because
08:10if I go along two
08:12and up one there are
08:15along two and up one.
08:17Or you can go along
08:18eight and up four. It
08:21doesn't make a difference. It's
08:22still going to give you,
08:23this is four rate, this
08:25is one
08:25over two same thing. Okay,
08:27that's the lesson I'm gradient.
08:30Hopefully everything is clear and
08:33understood.