[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.05,0:00:07.67,Default,,0000,0000,0000,,And polynomial functions I would\Nbe looking at functions of X
Dialogue: 0,0:00:07.67,0:00:11.25,Default,,0000,0000,0000,,that represent polynomials of\Nvarying degrees, including
Dialogue: 0,0:00:11.25,0:00:13.29,Default,,0000,0000,0000,,cubics and quadratics and
Dialogue: 0,0:00:13.29,0:00:18.10,Default,,0000,0000,0000,,Quartics. We will look at some\Nof the properties of these
Dialogue: 0,0:00:18.10,0:00:23.03,Default,,0000,0000,0000,,curves and then we will go on to\Nlook at how to deduce the
Dialogue: 0,0:00:23.03,0:00:27.60,Default,,0000,0000,0000,,function of the curve if given\Nthe roots. So first of all, what
Dialogue: 0,0:00:27.60,0:00:31.48,Default,,0000,0000,0000,,is a polynomial with a\Npolynomial of degree add is a
Dialogue: 0,0:00:31.48,0:00:33.94,Default,,0000,0000,0000,,function of the form F of X.
Dialogue: 0,0:00:33.96,0:00:41.93,Default,,0000,0000,0000,,Equals a an X to the power N\Nplus AN minus One X to the power
Dialogue: 0,0:00:41.93,0:00:48.90,Default,,0000,0000,0000,,and minus one plus, and it keeps\Ngoing all the way until we get
Dialogue: 0,0:00:48.90,0:00:55.65,Default,,0000,0000,0000,,to. A2 X squared\Nplus a 1X plus a
Dialogue: 0,0:00:55.65,0:01:01.71,Default,,0000,0000,0000,,zero and here the AA\Nrepresents real numbers and
Dialogue: 0,0:01:01.71,0:01:04.40,Default,,0000,0000,0000,,often called the coefficients.
Dialogue: 0,0:01:05.24,0:01:11.12,Default,,0000,0000,0000,,Now, this may seem complicated\Nat first sight, but it's not and
Dialogue: 0,0:01:11.12,0:01:16.51,Default,,0000,0000,0000,,hopefully a few examples should\Nconvince you of this. So, for
Dialogue: 0,0:01:16.51,0:01:22.39,Default,,0000,0000,0000,,example, let's suppose we had F\Nof X equals 4X cubed, minus
Dialogue: 0,0:01:22.39,0:01:24.35,Default,,0000,0000,0000,,three X squared +2.
Dialogue: 0,0:01:25.72,0:01:30.02,Default,,0000,0000,0000,,Now this is a function and a\Npolynomial of degree three.
Dialogue: 0,0:01:30.02,0:01:35.50,Default,,0000,0000,0000,,Since the highest power of X is\N3 and is often called a cubic.
Dialogue: 0,0:01:36.65,0:01:43.78,Default,,0000,0000,0000,,Let's suppose we hard F of X\Nequals X to the Power 7 -
Dialogue: 0,0:01:43.78,0:01:47.34,Default,,0000,0000,0000,,4 X to the power 5 +
Dialogue: 0,0:01:47.34,0:01:52.20,Default,,0000,0000,0000,,1. This is a polynomial of\Ndegree Seven. Since the highest
Dialogue: 0,0:01:52.20,0:01:56.97,Default,,0000,0000,0000,,power of X is A7, an important\Nthing to notice here is that you
Dialogue: 0,0:01:56.97,0:02:01.75,Default,,0000,0000,0000,,don't need every single power of\NX all the way up to 7. The
Dialogue: 0,0:02:01.75,0:02:05.84,Default,,0000,0000,0000,,important thing is that the\Nhighest power of X is 7, and
Dialogue: 0,0:02:05.84,0:02:07.54,Default,,0000,0000,0000,,that's why it's a polynomial
Dialogue: 0,0:02:07.54,0:02:13.39,Default,,0000,0000,0000,,degree 7. And the final example\NF of X equals.
Dialogue: 0,0:02:13.96,0:02:19.34,Default,,0000,0000,0000,,Four X squared minus two\NX minus 4.
Dialogue: 0,0:02:19.87,0:02:23.44,Default,,0000,0000,0000,,So polynomial of degree 2\Nbecause the highest power of X
Dialogue: 0,0:02:23.44,0:02:26.86,Default,,0000,0000,0000,,is a 2. Less often called a
Dialogue: 0,0:02:26.86,0:02:30.55,Default,,0000,0000,0000,,quadratic. Now it's important\Nwhen we're thinking about
Dialogue: 0,0:02:30.55,0:02:34.61,Default,,0000,0000,0000,,polynomials that we only have\Npositive powers of X, and we
Dialogue: 0,0:02:34.61,0:02:38.30,Default,,0000,0000,0000,,don't have any other kind of\Nfunctions. For example, square
Dialogue: 0,0:02:38.30,0:02:40.15,Default,,0000,0000,0000,,roots or division by X.
Dialogue: 0,0:02:40.74,0:02:42.64,Default,,0000,0000,0000,,So for example, if we had.
Dialogue: 0,0:02:43.23,0:02:46.55,Default,,0000,0000,0000,,F of X equals
Dialogue: 0,0:02:46.55,0:02:52.59,Default,,0000,0000,0000,,4X cubed. Plus the square\Nroot of X minus one.
Dialogue: 0,0:02:53.12,0:02:58.41,Default,,0000,0000,0000,,This is not a polynomial because\Nwe have the square root of X
Dialogue: 0,0:02:58.41,0:03:04.11,Default,,0000,0000,0000,,here and we need all the powers\Nof X to be positive integers to
Dialogue: 0,0:03:04.11,0:03:07.36,Default,,0000,0000,0000,,have function to be a\Npolynomial. Second example.
Dialogue: 0,0:03:07.39,0:03:14.48,Default,,0000,0000,0000,,If we had F of X equals 5X to\Nthe Power 4 - 2 X squared plus 3
Dialogue: 0,0:03:14.48,0:03:19.60,Default,,0000,0000,0000,,divided by X. Once again this is\Nnot a polynomial and this is
Dialogue: 0,0:03:19.60,0:03:24.73,Default,,0000,0000,0000,,because, as I said before, we\Nneed all the powers of X should
Dialogue: 0,0:03:24.73,0:03:30.24,Default,,0000,0000,0000,,be positive integers and so this\N3 divided by X does not fit in
Dialogue: 0,0:03:30.24,0:03:33.00,Default,,0000,0000,0000,,with that. So this is not a
Dialogue: 0,0:03:33.00,0:03:38.26,Default,,0000,0000,0000,,polynomial. OK, we've already\Nmet some basic polynomials, and
Dialogue: 0,0:03:38.26,0:03:44.24,Default,,0000,0000,0000,,you'll recognize these. For\Nexample, F of X equals 2 is
Dialogue: 0,0:03:44.24,0:03:50.21,Default,,0000,0000,0000,,as a constant function, and this\Nis actually a type of
Dialogue: 0,0:03:50.21,0:03:57.27,Default,,0000,0000,0000,,polynomial, and likewise F of X\Nequals 2X plus one, which is a
Dialogue: 0,0:03:57.27,0:04:03.24,Default,,0000,0000,0000,,linear function is also a type\Nof polynomial, and we could
Dialogue: 0,0:04:03.24,0:04:06.00,Default,,0000,0000,0000,,sketch those. So the F of X.
Dialogue: 0,0:04:06.64,0:04:12.82,Default,,0000,0000,0000,,And X and we know that F of X\Nequals 2 is a horizontal line
Dialogue: 0,0:04:12.82,0:04:19.00,Default,,0000,0000,0000,,which passes through two on the\NF of X axis and F of X equals
Dialogue: 0,0:04:19.00,0:04:20.24,Default,,0000,0000,0000,,2X plus one.
Dialogue: 0,0:04:21.29,0:04:26.25,Default,,0000,0000,0000,,Is straight line with gradient\Ntwo which passes through one.
Dialogue: 0,0:04:28.40,0:04:34.42,Default,,0000,0000,0000,,On the F of X axis, stuff of X\Nequals 2 and this is F of X.
Dialogue: 0,0:04:35.24,0:04:38.32,Default,,0000,0000,0000,,Equals 2X plus
Dialogue: 0,0:04:38.32,0:04:43.24,Default,,0000,0000,0000,,one. Important things\Nremember is that all
Dialogue: 0,0:04:43.24,0:04:46.26,Default,,0000,0000,0000,,constant functions are\Nhorizontal straight lines
Dialogue: 0,0:04:46.26,0:04:50.28,Default,,0000,0000,0000,,and all linear functions\Nare straight lines which
Dialogue: 0,0:04:50.28,0:04:51.79,Default,,0000,0000,0000,,are not horizontal.
Dialogue: 0,0:04:52.88,0:04:56.62,Default,,0000,0000,0000,,So let's have a look now at some
Dialogue: 0,0:04:56.62,0:05:02.70,Default,,0000,0000,0000,,quadratic functions. If we had F\Nof X equals X squared, we know
Dialogue: 0,0:05:02.70,0:05:06.77,Default,,0000,0000,0000,,this is a quadratic function\Nbecause it's a polynomial of
Dialogue: 0,0:05:06.77,0:05:11.65,Default,,0000,0000,0000,,degree two and we can sketch.\NThis is a very familiar curve.
Dialogue: 0,0:05:11.65,0:05:14.91,Default,,0000,0000,0000,,If you have F of X on the
Dialogue: 0,0:05:14.91,0:05:19.77,Default,,0000,0000,0000,,vertical axis. An X on\Nour horizontal axis. We
Dialogue: 0,0:05:19.77,0:05:25.30,Default,,0000,0000,0000,,know that the graph F of\NX equals X squared looks
Dialogue: 0,0:05:25.30,0:05:26.80,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:05:29.09,0:05:32.82,Default,,0000,0000,0000,,This is F of X equals
Dialogue: 0,0:05:32.82,0:05:38.48,Default,,0000,0000,0000,,X squared. So what happens as we\Nvary the coefficient of X
Dialogue: 0,0:05:38.48,0:05:42.53,Default,,0000,0000,0000,,squared? That's the number that\Nmultiplies by X squared, so we
Dialogue: 0,0:05:42.53,0:05:47.31,Default,,0000,0000,0000,,had, for example, F of X equals\N2 X squared. How would that
Dialogue: 0,0:05:47.31,0:05:51.73,Default,,0000,0000,0000,,affect our graph? What actually\Nhappens is you can see that all
Dialogue: 0,0:05:51.73,0:05:56.88,Default,,0000,0000,0000,,of our X squared values are the\NF of X values have now been
Dialogue: 0,0:05:56.88,0:06:00.93,Default,,0000,0000,0000,,multiplied by two and hence\Nstretched in the F of X
Dialogue: 0,0:06:00.93,0:06:02.03,Default,,0000,0000,0000,,direction by two.
Dialogue: 0,0:06:02.64,0:06:04.44,Default,,0000,0000,0000,,So you can see we actually get.
Dialogue: 0,0:06:07.27,0:06:09.12,Default,,0000,0000,0000,,Occurs that looks something like
Dialogue: 0,0:06:09.12,0:06:16.14,Default,,0000,0000,0000,,this. And this is F\Nof X equals 2 X squared.
Dialogue: 0,0:06:17.34,0:06:22.43,Default,,0000,0000,0000,,And likewise, we could do that\Nfor any coefficients. So if we
Dialogue: 0,0:06:22.43,0:06:25.82,Default,,0000,0000,0000,,had F of X equals 5 X squared.
Dialogue: 0,0:06:26.45,0:06:29.78,Default,,0000,0000,0000,,It would be stretched five\Ntimes from the value it was
Dialogue: 0,0:06:29.78,0:06:33.12,Default,,0000,0000,0000,,when it was X squared, and it\Nwould look something like
Dialogue: 0,0:06:33.12,0:06:33.42,Default,,0000,0000,0000,,this.
Dialogue: 0,0:06:36.41,0:06:42.90,Default,,0000,0000,0000,,So this would be F of\NX equals 5 X squared.
Dialogue: 0,0:06:43.61,0:06:47.48,Default,,0000,0000,0000,,So this is all for positive\Ncoefficients of X squared. But
Dialogue: 0,0:06:47.48,0:06:50.30,Default,,0000,0000,0000,,what would happen if the\Ncoefficients were negative?
Dialogue: 0,0:06:50.80,0:06:53.13,Default,,0000,0000,0000,,Well, let's have a look.
Dialogue: 0,0:06:53.16,0:06:57.84,Default,,0000,0000,0000,,It starts off with F of X\Nequals minus X squared. Now,
Dialogue: 0,0:06:57.84,0:07:02.52,Default,,0000,0000,0000,,compared with F of X equals X\Nsquared would taken all of
Dialogue: 0,0:07:02.52,0:07:06.03,Default,,0000,0000,0000,,our positive values and we've\Nmultiplied them all by
Dialogue: 0,0:07:06.03,0:07:09.15,Default,,0000,0000,0000,,negative one. So this\Nactually results in a
Dialogue: 0,0:07:09.15,0:07:13.05,Default,,0000,0000,0000,,reflection in the X axis\Nbecause every single value is
Dialogue: 0,0:07:13.05,0:07:17.73,Default,,0000,0000,0000,,now, which was positive has\Nbecome negative. So F of F of
Dialogue: 0,0:07:17.73,0:07:21.63,Default,,0000,0000,0000,,X equals minus X squared will\Nlook something like this.
Dialogue: 0,0:07:24.21,0:07:30.33,Default,,0000,0000,0000,,So this is F of X\Nequals minus X squared.
Dialogue: 0,0:07:31.12,0:07:36.18,Default,,0000,0000,0000,,Likewise, F of X equals minus\Ntwo X squared will be a
Dialogue: 0,0:07:36.18,0:07:42.09,Default,,0000,0000,0000,,reflection in the X axis of F\Nof X equals 2 X squared, and
Dialogue: 0,0:07:42.09,0:07:45.47,Default,,0000,0000,0000,,so we get something which\Nlooks like this.
Dialogue: 0,0:07:48.81,0:07:55.67,Default,,0000,0000,0000,,F of X equals 2 X squared 3 -\N2 X squared and Leslie are going
Dialogue: 0,0:07:55.67,0:08:02.11,Default,,0000,0000,0000,,to look at F of X equals minus\Nfive X squared and you may well
Dialogue: 0,0:08:02.11,0:08:08.12,Default,,0000,0000,0000,,have guessed by now. That's a\Nreflection of F of X equals 5 X
Dialogue: 0,0:08:08.12,0:08:13.26,Default,,0000,0000,0000,,squared in the X axis, which\Ngives us a graph which would
Dialogue: 0,0:08:13.26,0:08:14.55,Default,,0000,0000,0000,,look something like.
Dialogue: 0,0:08:15.48,0:08:16.18,Default,,0000,0000,0000,,This.
Dialogue: 0,0:08:17.51,0:08:23.74,Default,,0000,0000,0000,,We F of X equals\Nminus five X squared.
Dialogue: 0,0:08:24.55,0:08:29.50,Default,,0000,0000,0000,,And in fact, it's true to say\Nthat for any polynomial, if you
Dialogue: 0,0:08:29.50,0:08:33.69,Default,,0000,0000,0000,,multiply the function by minus\None, you will always get the
Dialogue: 0,0:08:33.69,0:08:35.60,Default,,0000,0000,0000,,reflection in the X axis.
Dialogue: 0,0:08:36.24,0:08:41.56,Default,,0000,0000,0000,,I will have a look at some other\Nquadratics and see what happens
Dialogue: 0,0:08:41.56,0:08:46.87,Default,,0000,0000,0000,,if we vary the coefficients of X\Nas opposed to the coefficient of
Dialogue: 0,0:08:46.87,0:08:51.78,Default,,0000,0000,0000,,X squared. And to do this I'm\Ngoing to construct the table.
Dialogue: 0,0:08:52.61,0:08:54.35,Default,,0000,0000,0000,,So we'll have RX value at the
Dialogue: 0,0:08:54.35,0:08:59.67,Default,,0000,0000,0000,,top. And the three functions I'm\Ngoing to look at our X squared
Dialogue: 0,0:08:59.67,0:09:03.17,Default,,0000,0000,0000,,plus X. X squared
Dialogue: 0,0:09:03.17,0:09:09.11,Default,,0000,0000,0000,,plus 4X. And X\Nsquared plus 6X.
Dialogue: 0,0:09:10.04,0:09:17.40,Default,,0000,0000,0000,,And for my X file use, I'm going\Nto choose values of minus 5 - 4
Dialogue: 0,0:09:17.40,0:09:24.76,Default,,0000,0000,0000,,- 3 - 2 - 1 zero. I'm\Ngoing to go all the way up to
Dialogue: 0,0:09:24.76,0:09:26.60,Default,,0000,0000,0000,,a value of two.
Dialogue: 0,0:09:27.48,0:09:31.18,Default,,0000,0000,0000,,Now for each of these functions,\NI'm not necessarily going to
Dialogue: 0,0:09:31.18,0:09:35.54,Default,,0000,0000,0000,,workout the F of X value for\Nevery single value of X. I'm
Dialogue: 0,0:09:35.54,0:09:39.24,Default,,0000,0000,0000,,just going to focus around the\Nturning point of the quadratic,
Dialogue: 0,0:09:39.24,0:09:40.92,Default,,0000,0000,0000,,which is where the quadratic
Dialogue: 0,0:09:40.92,0:09:46.29,Default,,0000,0000,0000,,dips. So just draw some lines in\Non my table.
Dialogue: 0,0:09:46.29,0:09:52.93,Default,,0000,0000,0000,,On some horizontal\Nlines as well.
Dialogue: 0,0:09:55.35,0:09:59.17,Default,,0000,0000,0000,,And now we can fill this
Dialogue: 0,0:09:59.17,0:10:05.90,Default,,0000,0000,0000,,in. So for RF of X equals\NX squared plus X, I'm just going
Dialogue: 0,0:10:05.90,0:10:13.11,Default,,0000,0000,0000,,to focus on minus 2 + 2. So when\NI put X equals minus two in here
Dialogue: 0,0:10:13.11,0:10:19.05,Default,,0000,0000,0000,,we get minus 2 squared, which is\N4 takeaway two which gives Me 2.
Dialogue: 0,0:10:19.54,0:10:24.43,Default,,0000,0000,0000,,When we put X is minus one in we\Nget minus one squared, which is
Dialogue: 0,0:10:24.43,0:10:27.12,Default,,0000,0000,0000,,one. Take away one which gives
Dialogue: 0,0:10:27.12,0:10:34.72,Default,,0000,0000,0000,,us 0. Report this X is zero in\Nhere we just got 0 + 0 which is
Dialogue: 0,0:10:34.72,0:10:41.01,Default,,0000,0000,0000,,0. I put X is one in here.\NWe've got one plus one which
Dialogue: 0,0:10:41.01,0:10:42.23,Default,,0000,0000,0000,,gives Me 2.
Dialogue: 0,0:10:42.89,0:10:47.02,Default,,0000,0000,0000,,And finally, when I pay taxes 2\Nin here, the two squared, which
Dialogue: 0,0:10:47.02,0:10:52.57,Default,,0000,0000,0000,,is 4. +2, which gives me 6 and\Nin fact just for symmetry we
Dialogue: 0,0:10:52.57,0:10:57.26,Default,,0000,0000,0000,,could put X is minus three in\Nhere as well, which gives us
Dialogue: 0,0:10:57.26,0:10:58.71,Default,,0000,0000,0000,,minus 3 squared 9.
Dialogue: 0,0:10:59.35,0:11:02.99,Default,,0000,0000,0000,,Plus minus three. Sorry, which\Ngives us 6.
Dialogue: 0,0:11:03.63,0:11:09.84,Default,,0000,0000,0000,,OK, for my next function,\NX squared plus 4X.
Dialogue: 0,0:11:10.59,0:11:15.58,Default,,0000,0000,0000,,I'm going to look at values from\Nminus four up to 0.
Dialogue: 0,0:11:16.16,0:11:23.17,Default,,0000,0000,0000,,So if I put minus four in\Nhere, we get 16 takeaway 16
Dialogue: 0,0:11:23.17,0:11:25.32,Default,,0000,0000,0000,,which gives me 0.
Dialogue: 0,0:11:26.06,0:11:27.68,Default,,0000,0000,0000,,Put minus three in here.
Dialogue: 0,0:11:28.43,0:11:35.04,Default,,0000,0000,0000,,We get 9 takeaway 12 which\Ngives us minus three.
Dialogue: 0,0:11:35.69,0:11:42.31,Default,,0000,0000,0000,,For put minus two in here, we\Nget four takeaway 8 which is
Dialogue: 0,0:11:42.31,0:11:49.39,Default,,0000,0000,0000,,minus 4. If I put minus one\Nin here, we get one takeaway 4
Dialogue: 0,0:11:49.39,0:11:56.03,Default,,0000,0000,0000,,which is minus three, and if I\Nput zero in here 0 + 0 just
Dialogue: 0,0:11:56.03,0:11:57.36,Default,,0000,0000,0000,,gives me 0.
Dialogue: 0,0:11:58.03,0:12:04.18,Default,,0000,0000,0000,,And Lastly, I'm going to look at\NF of X equals X squared plus 6X.
Dialogue: 0,0:12:04.71,0:12:08.18,Default,,0000,0000,0000,,And I'm going to take this from\Nminus five all the way up to
Dialogue: 0,0:12:08.18,0:12:13.59,Default,,0000,0000,0000,,minus one. So we put minus five\Nin first of all minus 5 squared
Dialogue: 0,0:12:13.59,0:12:17.20,Default,,0000,0000,0000,,is 25. Take away 30 gives us
Dialogue: 0,0:12:17.20,0:12:25.06,Default,,0000,0000,0000,,minus 5. Minus four in\Nhere 16 takeaway 24 which
Dialogue: 0,0:12:25.06,0:12:28.26,Default,,0000,0000,0000,,gives us minus 8.
Dialogue: 0,0:12:29.15,0:12:35.94,Default,,0000,0000,0000,,For X is minus three in. Here we\Nget 9 takeaway 18, which is
Dialogue: 0,0:12:35.94,0:12:38.50,Default,,0000,0000,0000,,minus 9. For X is minus two in.
Dialogue: 0,0:12:39.00,0:12:45.23,Default,,0000,0000,0000,,We get minus two square root of\N4 takeaway 12, which is minus 8.
Dialogue: 0,0:12:45.78,0:12:52.53,Default,,0000,0000,0000,,And we put minus one in. We get\None takeaway 6 which is minus 5.
Dialogue: 0,0:12:53.05,0:12:56.71,Default,,0000,0000,0000,,And you can actually see the\Nsymmetry in each row here, to
Dialogue: 0,0:12:56.71,0:12:59.76,Default,,0000,0000,0000,,show that we've actually focused\Naround that turning point we
Dialogue: 0,0:12:59.76,0:13:02.81,Default,,0000,0000,0000,,talked about. So let's draw the\Ngraph of these functions.
Dialogue: 0,0:13:03.40,0:13:07.18,Default,,0000,0000,0000,,So in a vertical scale will
Dialogue: 0,0:13:07.18,0:13:13.34,Default,,0000,0000,0000,,need. F of X and that's going to\Ntake us from all those values
Dialogue: 0,0:13:13.34,0:13:15.88,Default,,0000,0000,0000,,minus nine up 2 - 6.
Dialogue: 0,0:13:16.43,0:13:20.08,Default,,0000,0000,0000,,So. If we do minus
Dialogue: 0,0:13:20.08,0:13:26.19,Default,,0000,0000,0000,,9. Minus 8 minus Seven\N6 - 5.
Dialogue: 0,0:13:26.84,0:13:30.58,Default,,0000,0000,0000,,For most 3 - 2 -
Dialogue: 0,0:13:30.58,0:13:37.72,Default,,0000,0000,0000,,1 zero. 12345 and we\Ncan just about squeeze 6 in
Dialogue: 0,0:13:37.72,0:13:39.49,Default,,0000,0000,0000,,at the top.
Dialogue: 0,0:13:40.43,0:13:43.24,Default,,0000,0000,0000,,On a longer horizontal axis.
Dialogue: 0,0:13:44.48,0:13:46.93,Default,,0000,0000,0000,,We've gone from minus five to
Dialogue: 0,0:13:46.93,0:13:50.55,Default,,0000,0000,0000,,two. So we just use one
Dialogue: 0,0:13:50.55,0:13:57.64,Default,,0000,0000,0000,,2. Minus 1 -\N2 - 3 -
Dialogue: 0,0:13:57.64,0:14:01.15,Default,,0000,0000,0000,,4. And minus 5.
Dialogue: 0,0:14:01.66,0:14:06.46,Default,,0000,0000,0000,,So first of all, let's look at\Nour function F of X equals X
Dialogue: 0,0:14:06.46,0:14:09.55,Default,,0000,0000,0000,,squared plus X. We've got minus\Nthree and six.
Dialogue: 0,0:14:10.25,0:14:11.46,Default,,0000,0000,0000,,Which is appear.
Dialogue: 0,0:14:11.96,0:14:15.70,Default,,0000,0000,0000,,We've got minus two and two.
Dialogue: 0,0:14:16.55,0:14:20.03,Default,,0000,0000,0000,,Which is here.
Dialogue: 0,0:14:20.26,0:14:22.75,Default,,0000,0000,0000,,Minus one and 0.
Dialogue: 0,0:14:23.68,0:14:25.88,Default,,0000,0000,0000,,We've got zero and zero.
Dialogue: 0,0:14:26.80,0:14:28.10,Default,,0000,0000,0000,,One and two.
Dialogue: 0,0:14:28.66,0:14:32.24,Default,,0000,0000,0000,,And two 16.
Dialogue: 0,0:14:33.65,0:14:36.75,Default,,0000,0000,0000,,And you can see clearly\Nthat this is a parabola
Dialogue: 0,0:14:36.75,0:14:39.85,Default,,0000,0000,0000,,which, as we expected, we\Ncan draw a smooth curve.
Dialogue: 0,0:14:42.26,0:14:43.64,Default,,0000,0000,0000,,Through those points.
Dialogue: 0,0:14:45.77,0:14:49.70,Default,,0000,0000,0000,,So this is F of X equals X
Dialogue: 0,0:14:49.70,0:14:52.17,Default,,0000,0000,0000,,squared. Plus X.
Dialogue: 0,0:14:53.07,0:14:58.11,Default,,0000,0000,0000,,Next, we're going to look at the\Nfunction F of X equals X squared
Dialogue: 0,0:14:58.11,0:15:02.10,Default,,0000,0000,0000,,plus 4X. And the points we had\Nwith minus four and zero.
Dialogue: 0,0:15:02.80,0:15:06.13,Default,,0000,0000,0000,,Minus three and minus
Dialogue: 0,0:15:06.13,0:15:12.51,Default,,0000,0000,0000,,three. Minus two\Nand minus 4.
Dialogue: 0,0:15:14.13,0:15:18.01,Default,,0000,0000,0000,,Minus one and minus three.
Dialogue: 0,0:15:19.57,0:15:23.73,Default,,0000,0000,0000,,And zero and zero which is\Nalready drawn. And once
Dialogue: 0,0:15:23.73,0:15:28.72,Default,,0000,0000,0000,,again you can see this is a\Nsmooth curve is a parabola
Dialogue: 0,0:15:28.72,0:15:29.97,Default,,0000,0000,0000,,as we expected.
Dialogue: 0,0:15:34.20,0:15:42.10,Default,,0000,0000,0000,,And this is F of\NX equals X squared plus
Dialogue: 0,0:15:42.10,0:15:48.34,Default,,0000,0000,0000,,4X. And the final function we're\Ngoing to look at F of X equals X
Dialogue: 0,0:15:48.34,0:15:52.94,Default,,0000,0000,0000,,squared plus 6X. And so we've\Ngot minus 5 - 5.
Dialogue: 0,0:15:53.80,0:15:57.78,Default,,0000,0000,0000,,So over here minus 4 - 8.
Dialogue: 0,0:15:58.77,0:16:00.11,Default,,0000,0000,0000,,Which is down here.
Dialogue: 0,0:16:00.78,0:16:02.67,Default,,0000,0000,0000,,Minus three and minus 9.
Dialogue: 0,0:16:03.48,0:16:06.98,Default,,0000,0000,0000,,Minus 2 - 8.
Dialogue: 0,0:16:07.53,0:16:14.22,Default,,0000,0000,0000,,And minus one and minus five and\Nonce again you can see smooth
Dialogue: 0,0:16:14.22,0:16:16.80,Default,,0000,0000,0000,,curve in the parabola shape.
Dialogue: 0,0:16:21.01,0:16:28.05,Default,,0000,0000,0000,,And this is F of X\Nequals X squared plus 6X.
Dialogue: 0,0:16:28.86,0:16:34.32,Default,,0000,0000,0000,,So we can see that as the\Ncoefficient of X increases from
Dialogue: 0,0:16:34.32,0:16:40.24,Default,,0000,0000,0000,,here to one to four to six, the\Ncurve the parabola is actually
Dialogue: 0,0:16:40.24,0:16:42.96,Default,,0000,0000,0000,,moving down and to the left.
Dialogue: 0,0:16:44.17,0:16:48.55,Default,,0000,0000,0000,,But what would happen if the\Ncoefficients of X was negative?
Dialogue: 0,0:16:48.55,0:16:54.52,Default,,0000,0000,0000,,Well, let's have a look and will\Ndo this in the same way as we've
Dialogue: 0,0:16:54.52,0:16:59.69,Default,,0000,0000,0000,,just on the previous examples.\NAnd we look at a table, and this
Dialogue: 0,0:16:59.69,0:17:04.47,Default,,0000,0000,0000,,time we'll look at how negative\Nvalues of our coefficients of X
Dialogue: 0,0:17:04.47,0:17:09.64,Default,,0000,0000,0000,,affect the graph. So we look at\NX squared minus XX squared minus
Dialogue: 0,0:17:09.64,0:17:15.69,Default,,0000,0000,0000,,4X. And the graph of F of X\Nequals X squared minus 6X.
Dialogue: 0,0:17:18.03,0:17:19.100,Default,,0000,0000,0000,,Now as before, will.
Dialogue: 0,0:17:20.59,0:17:25.50,Default,,0000,0000,0000,,Put a number of values in for X,\Nbut we won't use. All of them,
Dialogue: 0,0:17:25.50,0:17:30.07,Default,,0000,0000,0000,,will only use the ones which are\Nnear to the turning point of the
Dialogue: 0,0:17:30.07,0:17:33.100,Default,,0000,0000,0000,,quadratic, but the values will\Nput in the range will be from
Dialogue: 0,0:17:33.100,0:17:36.29,Default,,0000,0000,0000,,minus two all the way up to
Dialogue: 0,0:17:36.29,0:17:40.53,Default,,0000,0000,0000,,five. Four and\Nfive.
Dialogue: 0,0:17:44.91,0:17:48.68,Default,,0000,0000,0000,,I just put in the lines for
Dialogue: 0,0:17:48.68,0:17:55.38,Default,,0000,0000,0000,,our table. OK, so for the first\None, F of X equals X squared
Dialogue: 0,0:17:55.38,0:18:01.73,Default,,0000,0000,0000,,minus X. We look at what happens\Nwhen we substitute in X is minus
Dialogue: 0,0:18:01.73,0:18:07.39,Default,,0000,0000,0000,,2. So minus 2 squared is 4\Ntakeaway, minus two is the same
Dialogue: 0,0:18:07.39,0:18:12.91,Default,,0000,0000,0000,,as a plus two, which gives us\Nsix when X is minus 1 - 1
Dialogue: 0,0:18:12.91,0:18:17.69,Default,,0000,0000,0000,,squared, is one takeaway minus\None is the same as a plus one
Dialogue: 0,0:18:17.69,0:18:19.53,Default,,0000,0000,0000,,which just gives us 2.
Dialogue: 0,0:18:20.30,0:18:23.69,Default,,0000,0000,0000,,Zero is just zero takeaway
Dialogue: 0,0:18:23.69,0:18:29.96,Default,,0000,0000,0000,,00. When we put one in, we've\Ngot 1. Take away, one which is
Dialogue: 0,0:18:29.96,0:18:36.01,Default,,0000,0000,0000,,0. And two gives us 2 squared\Nfour takeaway two which gives us
Dialogue: 0,0:18:36.01,0:18:38.62,Default,,0000,0000,0000,,2. And again, for symmetry will
Dialogue: 0,0:18:38.62,0:18:44.64,Default,,0000,0000,0000,,do access 3. So 3 squared is 9\Ntakeaway, three is 6.
Dialogue: 0,0:18:45.71,0:18:51.22,Default,,0000,0000,0000,,So our second function F of X\Nequals X squared minus 4X. We're
Dialogue: 0,0:18:51.22,0:18:55.46,Default,,0000,0000,0000,,going to look at going from zero\Nup to five.
Dialogue: 0,0:18:55.98,0:19:01.23,Default,,0000,0000,0000,,And we put zero and we just get\Nzero takeaway 0 which is 0.
Dialogue: 0,0:19:01.85,0:19:03.41,Default,,0000,0000,0000,,When we put X is one.
Dialogue: 0,0:19:04.22,0:19:07.76,Default,,0000,0000,0000,,We get one takeaway 4 which is
Dialogue: 0,0:19:07.76,0:19:14.94,Default,,0000,0000,0000,,minus 3. When we put X\Nis 2, two squared is 4 takeaway
Dialogue: 0,0:19:14.94,0:19:17.78,Default,,0000,0000,0000,,eight, which gives us minus 4.
Dialogue: 0,0:19:18.76,0:19:23.91,Default,,0000,0000,0000,,When X is 3, that gives us 3\Nsquared, which is 9 takeaway 12
Dialogue: 0,0:19:23.91,0:19:27.96,Default,,0000,0000,0000,,which gives us minus three and\Nyou can see the symmetry
Dialogue: 0,0:19:27.96,0:19:32.74,Default,,0000,0000,0000,,starting to form here. Now a\NNexus 4 gives us 4 squared 16
Dialogue: 0,0:19:32.74,0:19:36.42,Default,,0000,0000,0000,,takeaway 16 which as we expect\Nit is a 0.
Dialogue: 0,0:19:37.14,0:19:41.93,Default,,0000,0000,0000,,And for our final function, FX\Nequals X squared minus six X,
Dialogue: 0,0:19:41.93,0:19:47.91,Default,,0000,0000,0000,,we're going to look at coming\Nfrom X is one all the way up to
Dialogue: 0,0:19:47.91,0:19:53.10,Default,,0000,0000,0000,,X equals 5, X equals 1, gives\NUS1 takeaway six, which gives us
Dialogue: 0,0:19:53.10,0:19:59.19,Default,,0000,0000,0000,,minus 5. X equals 2 gives us\N2 squared. Is 4 takeaway 12
Dialogue: 0,0:19:59.19,0:20:01.19,Default,,0000,0000,0000,,which gives us minus 8.
Dialogue: 0,0:20:02.32,0:20:07.58,Default,,0000,0000,0000,,I mean for taxes, three in three\Nsquared is 9 takeaway 18, which
Dialogue: 0,0:20:07.58,0:20:09.20,Default,,0000,0000,0000,,gives us minus 9.
Dialogue: 0,0:20:09.90,0:20:13.72,Default,,0000,0000,0000,,X is 4 + 4 squared.
Dialogue: 0,0:20:14.36,0:20:21.10,Default,,0000,0000,0000,,Take away 24. Which 16\Ntakeaway 24 which is minus
Dialogue: 0,0:20:21.10,0:20:27.08,Default,,0000,0000,0000,,8. I'm waiting for X is 5 in\Nwe get 25 takeaway 30, which
Dialogue: 0,0:20:27.08,0:20:31.49,Default,,0000,0000,0000,,is minus five and once again\Nsymmetry as expected. So as
Dialogue: 0,0:20:31.49,0:20:35.50,Default,,0000,0000,0000,,with our previous examples\Nwe want to draw this graphs
Dialogue: 0,0:20:35.50,0:20:40.32,Default,,0000,0000,0000,,of these functions so we can\Nsee what's going on as our
Dialogue: 0,0:20:40.32,0:20:44.33,Default,,0000,0000,0000,,value of the coefficients of\NX or Y is changing.
Dialogue: 0,0:20:46.26,0:20:52.01,Default,,0000,0000,0000,,So if we got F of X on our\Nvertical axis, this time we're
Dialogue: 0,0:20:52.01,0:20:56.54,Default,,0000,0000,0000,,going from Arlo's values, minus\Nnine. Our highest value is 6.
Dialogue: 0,0:20:57.07,0:21:00.90,Default,,0000,0000,0000,,So we're going from minus 9 - 8
Dialogue: 0,0:21:00.90,0:21:08.06,Default,,0000,0000,0000,,- 7. 6 - 5\N- 4 - 3 - 2
Dialogue: 0,0:21:08.06,0:21:09.92,Default,,0000,0000,0000,,- 1 zero.
Dialogue: 0,0:21:10.63,0:21:18.46,Default,,0000,0000,0000,,12345 I will just squeeze\Nin sex and are horizontal
Dialogue: 0,0:21:18.46,0:21:21.59,Default,,0000,0000,0000,,axis. We've got one.
Dialogue: 0,0:21:23.18,0:21:29.84,Default,,0000,0000,0000,,2. 3. Four\Nand five letter VRX axis, and we
Dialogue: 0,0:21:29.84,0:21:36.08,Default,,0000,0000,0000,,go breakdown some minus 2 - 1 -\N2. So first of all, let's look
Dialogue: 0,0:21:36.08,0:21:41.90,Default,,0000,0000,0000,,at the graph F of X equals X\Nsquared minus X. So minus two
Dialogue: 0,0:21:41.90,0:21:47.31,Default,,0000,0000,0000,,and six was our first point, so\Nminus two and six, which is
Dialogue: 0,0:21:47.31,0:21:51.76,Default,,0000,0000,0000,,here. And we've got minus one\Nand two which is here.
Dialogue: 0,0:21:52.94,0:21:54.84,Default,,0000,0000,0000,,We've got the origin 00.
Dialogue: 0,0:21:55.63,0:21:57.29,Default,,0000,0000,0000,,And we've got 10.
Dialogue: 0,0:21:58.15,0:22:00.79,Default,,0000,0000,0000,,22
Dialogue: 0,0:22:01.44,0:22:07.32,Default,,0000,0000,0000,,I'm 36 And\Nas we expected, you can see we
Dialogue: 0,0:22:07.32,0:22:10.61,Default,,0000,0000,0000,,can join the points of here.\NLet's make smooth curve which
Dialogue: 0,0:22:10.61,0:22:11.81,Default,,0000,0000,0000,,will be a parabola.
Dialogue: 0,0:22:16.74,0:22:22.66,Default,,0000,0000,0000,,So this is F of X equals\NX squared minus X.
Dialogue: 0,0:22:23.52,0:22:29.02,Default,,0000,0000,0000,,Our second function was F of X\Nequals X squared minus 4X. So
Dialogue: 0,0:22:29.02,0:22:31.56,Default,,0000,0000,0000,,first point was 00, which we've
Dialogue: 0,0:22:31.56,0:22:34.48,Default,,0000,0000,0000,,got. One and negative 3.
Dialogue: 0,0:22:35.43,0:22:38.95,Default,,0000,0000,0000,,Say it. We've got\Ntwo and minus 4.
Dialogue: 0,0:22:40.03,0:22:42.87,Default,,0000,0000,0000,,Just hang up three and negative
Dialogue: 0,0:22:42.87,0:22:45.14,Default,,0000,0000,0000,,3. Which is here.
Dialogue: 0,0:22:46.15,0:22:51.33,Default,,0000,0000,0000,,And four and zero, which is here\Nand straight away. We can see a
Dialogue: 0,0:22:51.33,0:22:53.18,Default,,0000,0000,0000,,smooth curve which is a
Dialogue: 0,0:22:53.18,0:22:56.43,Default,,0000,0000,0000,,parabola. Coming through\Nthose points.
Dialogue: 0,0:22:57.97,0:23:00.58,Default,,0000,0000,0000,,And we can label up this is F of
Dialogue: 0,0:23:00.58,0:23:06.79,Default,,0000,0000,0000,,X. Equals X\Nsquared minus 4X.
Dialogue: 0,0:23:08.34,0:23:13.87,Default,,0000,0000,0000,,Lost function to look at is the\Nfunction F of X equals X squared
Dialogue: 0,0:23:13.87,0:23:21.62,Default,,0000,0000,0000,,minus 6X. So our first point\Nwas one and minus five, which is
Dialogue: 0,0:23:21.62,0:23:24.01,Default,,0000,0000,0000,,here. We are two and negative 8.
Dialogue: 0,0:23:25.09,0:23:28.19,Default,,0000,0000,0000,,Yep. Three and minus 9.
Dialogue: 0,0:23:28.89,0:23:31.92,Default,,0000,0000,0000,,4 - 8.
Dialogue: 0,0:23:32.48,0:23:38.19,Default,,0000,0000,0000,,And five and minus five, which\Nis hit, and once again we can
Dialogue: 0,0:23:38.19,0:23:40.82,Default,,0000,0000,0000,,just draw a smooth curve through
Dialogue: 0,0:23:40.82,0:23:48.66,Default,,0000,0000,0000,,these points. To give us the\Nparabola we wanted, this is F
Dialogue: 0,0:23:48.66,0:23:52.50,Default,,0000,0000,0000,,of X equals X squared minus
Dialogue: 0,0:23:52.50,0:23:57.96,Default,,0000,0000,0000,,6X. So what's happening as the\Ncoefficients of X is getting
Dialogue: 0,0:23:57.96,0:24:02.77,Default,,0000,0000,0000,,bigger in absolute terms. So for\Ninstance, we can from minus one
Dialogue: 0,0:24:02.77,0:24:07.98,Default,,0000,0000,0000,,to minus four to minus six, and\Nwe can see straight away that
Dialogue: 0,0:24:07.98,0:24:12.80,Default,,0000,0000,0000,,the actual graph this parabola\Nis moving down and to the right
Dialogue: 0,0:24:12.80,0:24:16.81,Default,,0000,0000,0000,,as the coefficients of X gets\Nbigger in absolute terms.
Dialogue: 0,0:24:18.22,0:24:20.94,Default,,0000,0000,0000,,And that's where the\Ncoefficients of X is negative.
Dialogue: 0,0:24:21.71,0:24:25.97,Default,,0000,0000,0000,,OK, so we know what happens when\Nwe varied coefficients of X
Dialogue: 0,0:24:25.97,0:24:30.23,Default,,0000,0000,0000,,squared and we know what happens\Nwhen we vary the coefficients of
Dialogue: 0,0:24:30.23,0:24:34.49,Default,,0000,0000,0000,,X and that's both of them. For\Nquadratics, what happens when we
Dialogue: 0,0:24:34.49,0:24:38.75,Default,,0000,0000,0000,,vary the constants at the end of\Na quadratic well? Likewise table
Dialogue: 0,0:24:38.75,0:24:42.66,Default,,0000,0000,0000,,of values is a good way to see\Nwhat's going on.
Dialogue: 0,0:24:42.70,0:24:48.78,Default,,0000,0000,0000,,So this time I'm going to use X\Nagain. I'm going to go from
Dialogue: 0,0:24:48.78,0:24:56.15,Default,,0000,0000,0000,,minus two all the way up to +2,\Nso minus 2 - 1 zero one and two,
Dialogue: 0,0:24:56.15,0:25:01.36,Default,,0000,0000,0000,,and for my functions I'm going\Nto use X squared plus X.
Dialogue: 0,0:25:01.94,0:25:05.28,Default,,0000,0000,0000,,X squared plus X
Dialogue: 0,0:25:05.28,0:25:11.93,Default,,0000,0000,0000,,plus one. X\Nsquared plus X
Dialogue: 0,0:25:11.93,0:25:18.40,Default,,0000,0000,0000,,+5. And X squared\Nplus X minus 4.
Dialogue: 0,0:25:19.19,0:25:22.54,Default,,0000,0000,0000,,Now this table is
Dialogue: 0,0:25:22.54,0:25:28.12,Default,,0000,0000,0000,,particularly. Easy for us to\Nworkout compared with the other
Dialogue: 0,0:25:28.12,0:25:33.51,Default,,0000,0000,0000,,ones because we have a slight\Nadvantage in that we have a Head
Dialogue: 0,0:25:33.51,0:25:38.49,Default,,0000,0000,0000,,Start because we already know\Nthe values for X squared plus X
Dialogue: 0,0:25:38.49,0:25:43.47,Default,,0000,0000,0000,,that go here and we can just\Ntake them directly from Aurora,
Dialogue: 0,0:25:43.47,0:25:48.04,Default,,0000,0000,0000,,the table and you'll remember\Nthat they actually gave us the
Dialogue: 0,0:25:48.04,0:25:50.11,Default,,0000,0000,0000,,values 200, two and six.
Dialogue: 0,0:25:50.73,0:25:55.17,Default,,0000,0000,0000,,And in fact, we can use this\Nline in our table to help us
Dialogue: 0,0:25:55.17,0:25:59.29,Default,,0000,0000,0000,,with all of the other lines. The\Nsecond line here, which is X
Dialogue: 0,0:25:59.29,0:26:00.87,Default,,0000,0000,0000,,squared plus X plus one.
Dialogue: 0,0:26:01.50,0:26:05.10,Default,,0000,0000,0000,,His only one bigger than every\Nvalue in the table before. So
Dialogue: 0,0:26:05.10,0:26:06.60,Default,,0000,0000,0000,,all we need to do.
Dialogue: 0,0:26:07.33,0:26:14.28,Default,,0000,0000,0000,,Is just add 1 answer every value\Nfrom the line before, so this
Dialogue: 0,0:26:14.28,0:26:18.03,Default,,0000,0000,0000,,will be a 311, three and Seven.
Dialogue: 0,0:26:18.97,0:26:24.01,Default,,0000,0000,0000,,Likewise, for this line, when\Nwe've got X squared plus X +5.
Dialogue: 0,0:26:24.57,0:26:28.96,Default,,0000,0000,0000,,All this line is just five\Nbigger than our first line, so
Dialogue: 0,0:26:28.96,0:26:30.43,Default,,0000,0000,0000,,we can just 7.
Dialogue: 0,0:26:30.94,0:26:36.34,Default,,0000,0000,0000,,557\Nand
Dialogue: 0,0:26:36.34,0:26:42.68,Default,,0000,0000,0000,,11. And likewise, our\Nlast line is minus four in the
Dialogue: 0,0:26:42.68,0:26:46.88,Default,,0000,0000,0000,,end with exactly the same thing\Nbefore hand, so it's just take
Dialogue: 0,0:26:46.88,0:26:51.08,Default,,0000,0000,0000,,away for from every value from\Nour first line, which gives us
Dialogue: 0,0:26:51.08,0:26:58.27,Default,,0000,0000,0000,,minus 2. Minus 4 -\N4 - 2 and
Dialogue: 0,0:26:58.27,0:27:03.05,Default,,0000,0000,0000,,2. So let's put this information\Non a graph now so we can
Dialogue: 0,0:27:03.05,0:27:06.38,Default,,0000,0000,0000,,actually see what's happening as\Nwe vary the constants at the end
Dialogue: 0,0:27:06.38,0:27:12.14,Default,,0000,0000,0000,,of this quadratic. So you put F\Nof X and are vertical scale so
Dialogue: 0,0:27:12.14,0:27:17.76,Default,,0000,0000,0000,,that we need to go from minus\Nfour all over to 11 so it starts
Dialogue: 0,0:27:17.76,0:27:19.64,Default,,0000,0000,0000,,at minus 4 - 3.
Dialogue: 0,0:27:20.26,0:27:22.28,Default,,0000,0000,0000,,Minus 2 - 1.
Dialogue: 0,0:27:22.87,0:27:29.93,Default,,0000,0000,0000,,0123456789, ten and\Nwe can just
Dialogue: 0,0:27:29.93,0:27:33.46,Default,,0000,0000,0000,,about squeeze already
Dialogue: 0,0:27:33.46,0:27:39.40,Default,,0000,0000,0000,,left it. I no\Nlonger horizontal axis. We're
Dialogue: 0,0:27:39.40,0:27:46.55,Default,,0000,0000,0000,,going from minus 2 + 2\N- 1 - 2 plus one
Dialogue: 0,0:27:46.55,0:27:52.47,Default,,0000,0000,0000,,and +2. So first graph is the\Ngraph of the function F of X
Dialogue: 0,0:27:52.47,0:27:56.14,Default,,0000,0000,0000,,equals X squared plus X, so it's\Nminus two and two.
Dialogue: 0,0:27:57.23,0:28:02.45,Default,,0000,0000,0000,,Which is a point here, minus\None and 0.
Dialogue: 0,0:28:03.03,0:28:07.71,Default,,0000,0000,0000,,So first graph is the graph of\Nthe function F of X equals X
Dialogue: 0,0:28:07.71,0:28:10.71,Default,,0000,0000,0000,,squared plus X, so it's minus\Ntwo and two.
Dialogue: 0,0:28:11.80,0:28:15.71,Default,,0000,0000,0000,,Which is a .8 - 1
Dialogue: 0,0:28:15.71,0:28:23.07,Default,,0000,0000,0000,,and 0. Because the origin\Nnext 00 at the .1 two
Dialogue: 0,0:28:23.07,0:28:26.12,Default,,0000,0000,0000,,and we got the .26.
Dialogue: 0,0:28:27.62,0:28:30.80,Default,,0000,0000,0000,,For our next graph, actually\Nsorry, now we should.
Dialogue: 0,0:28:31.43,0:28:34.03,Default,,0000,0000,0000,,Draw a smooth curve for this one
Dialogue: 0,0:28:34.03,0:28:39.28,Default,,0000,0000,0000,,first. And Labor Lux F of X\Nequals X squared plus X.
Dialogue: 0,0:28:40.34,0:28:45.53,Default,,0000,0000,0000,,For an X graph which is F of X\Nequals X squared plus X plus
Dialogue: 0,0:28:45.53,0:28:49.34,Default,,0000,0000,0000,,one, we need to put the\Nfollowing points minus two and
Dialogue: 0,0:28:49.34,0:28:54.53,Default,,0000,0000,0000,,three. So minus one and one\Nyou can see straight away
Dialogue: 0,0:28:54.53,0:28:58.89,Default,,0000,0000,0000,,that all of these points\Nare just want above what we
Dialogue: 0,0:28:58.89,0:29:00.87,Default,,0000,0000,0000,,had before zero and one.
Dialogue: 0,0:29:01.97,0:29:03.87,Default,,0000,0000,0000,,One and three.
Dialogue: 0,0:29:04.91,0:29:07.81,Default,,0000,0000,0000,,Two and Seven.
Dialogue: 0,0:29:07.98,0:29:11.63,Default,,0000,0000,0000,,So you can imagine our next\Nparabola that we draw
Dialogue: 0,0:29:11.63,0:29:12.72,Default,,0000,0000,0000,,through these points.
Dialogue: 0,0:29:15.26,0:29:20.10,Default,,0000,0000,0000,,Is exactly the same but one\Nabove previous, that's F of X
Dialogue: 0,0:29:20.10,0:29:24.02,Default,,0000,0000,0000,,equals. X squared plus X plus
Dialogue: 0,0:29:24.02,0:29:30.15,Default,,0000,0000,0000,,one. So what about F of X\Nequals X squared plus X +5?
Dialogue: 0,0:29:30.79,0:29:33.89,Default,,0000,0000,0000,,Well, let's have a look minus\Ntwo and Seven.
Dialogue: 0,0:29:35.48,0:29:37.32,Default,,0000,0000,0000,,Gives us a point up here.
Dialogue: 0,0:29:38.28,0:29:40.12,Default,,0000,0000,0000,,Minus one and five.
Dialogue: 0,0:29:41.26,0:29:43.99,Default,,0000,0000,0000,,Zero and five.
Dialogue: 0,0:29:44.73,0:29:47.65,Default,,0000,0000,0000,,One and Seven.
Dialogue: 0,0:29:48.46,0:29:51.73,Default,,0000,0000,0000,,And two and 11.
Dialogue: 0,0:29:53.53,0:29:57.16,Default,,0000,0000,0000,,So we draw a smooth curve\Nthrough these points.
Dialogue: 0,0:30:02.06,0:30:09.03,Default,,0000,0000,0000,,See, that gives us a parabola\Nand this is F of X equals X
Dialogue: 0,0:30:09.03,0:30:11.02,Default,,0000,0000,0000,,squared plus X +5.
Dialogue: 0,0:30:12.32,0:30:17.83,Default,,0000,0000,0000,,So finally we look at the\Nfunction F of X equals X squared
Dialogue: 0,0:30:17.83,0:30:22.92,Default,,0000,0000,0000,,plus X minus four. So we've got\Nminus two and minus 2.
Dialogue: 0,0:30:23.45,0:30:26.94,Default,,0000,0000,0000,,Minus 1 - 4.
Dialogue: 0,0:30:27.96,0:30:29.60,Default,,0000,0000,0000,,0 - 4.
Dialogue: 0,0:30:30.24,0:30:33.93,Default,,0000,0000,0000,,One 1 - 2.
Dialogue: 0,0:30:33.93,0:30:35.01,Default,,0000,0000,0000,,Two and two.
Dialogue: 0,0:30:36.51,0:30:40.07,Default,,0000,0000,0000,,I once again we can\Ndraw a smooth curve.
Dialogue: 0,0:30:41.30,0:30:48.46,Default,,0000,0000,0000,,Through these points and this is\NF of X equals X squared plus
Dialogue: 0,0:30:48.46,0:30:50.12,Default,,0000,0000,0000,,X minus 4.
Dialogue: 0,0:30:50.90,0:30:54.20,Default,,0000,0000,0000,,So we can see that what's\Nactually happening here is from
Dialogue: 0,0:30:54.20,0:30:58.40,Default,,0000,0000,0000,,our original graph of F of X\Nequals X squared plus X, and if
Dialogue: 0,0:30:58.40,0:31:02.30,Default,,0000,0000,0000,,you like, you could have put a\Nplus zero there. And when we
Dialogue: 0,0:31:02.30,0:31:06.20,Default,,0000,0000,0000,,added one, it's moved one up\Nwhen we added five, it's moved 5
Dialogue: 0,0:31:06.20,0:31:11.55,Default,,0000,0000,0000,,or. And when we took away four,\Nit moved 4 down. So it's quite
Dialogue: 0,0:31:11.55,0:31:15.53,Default,,0000,0000,0000,,clear to see the effect that the\Nconstant has on our parabola.
Dialogue: 0,0:31:16.88,0:31:23.20,Default,,0000,0000,0000,,When looking at the graph of a\Nfunction, a turning point is the
Dialogue: 0,0:31:23.20,0:31:28.06,Default,,0000,0000,0000,,point on the curve where the\Ngradient changes from negative
Dialogue: 0,0:31:28.06,0:31:30.97,Default,,0000,0000,0000,,to positive or from positive to
Dialogue: 0,0:31:30.97,0:31:34.92,Default,,0000,0000,0000,,negative. And we're thinking\Nabout polynomials. Are
Dialogue: 0,0:31:34.92,0:31:40.32,Default,,0000,0000,0000,,polynomial of degree an has at\Nmost an minus one turning
Dialogue: 0,0:31:40.32,0:31:44.74,Default,,0000,0000,0000,,points. So for example, a\Nquadratic of degree 2.
Dialogue: 0,0:31:45.47,0:31:47.22,Default,,0000,0000,0000,,Can only have one turning
Dialogue: 0,0:31:47.22,0:31:51.51,Default,,0000,0000,0000,,points. And if we draw just a\Nsketch of a quadratic, you can
Dialogue: 0,0:31:51.51,0:31:54.36,Default,,0000,0000,0000,,see this point here would be\Nat one turning point.
Dialogue: 0,0:31:55.57,0:32:00.06,Default,,0000,0000,0000,,We think about cubics, obviously\Ncubic as a polynomial of degree
Dialogue: 0,0:32:00.06,0:32:05.77,Default,,0000,0000,0000,,3, so that can have at most two\Nturning points, which is why the
Dialogue: 0,0:32:05.77,0:32:09.44,Default,,0000,0000,0000,,general shape of a cubic looks\Nsomething like this.
Dialogue: 0,0:32:10.70,0:32:14.15,Default,,0000,0000,0000,,We see the two turning points\Nare here and here.
Dialogue: 0,0:32:14.75,0:32:20.08,Default,,0000,0000,0000,,But as I said, it can have at\Nmost 2. There's no reason it has
Dialogue: 0,0:32:20.08,0:32:25.40,Default,,0000,0000,0000,,to have two, and a good example\Nis this of this is if you look
Dialogue: 0,0:32:25.40,0:32:27.88,Default,,0000,0000,0000,,at F of X equals X cubed.
Dialogue: 0,0:32:28.47,0:32:31.46,Default,,0000,0000,0000,,And in fact, that would look if\NI do a sketch over here.
Dialogue: 0,0:32:32.13,0:32:38.44,Default,,0000,0000,0000,,F of X&X that would come\Nfrom the bottom left.
Dialogue: 0,0:32:39.87,0:32:42.20,Default,,0000,0000,0000,,Through zero and come up here
Dialogue: 0,0:32:42.20,0:32:47.98,Default,,0000,0000,0000,,like this. That's F of X equals\NX cubed, and so we can see that
Dialogue: 0,0:32:47.98,0:32:52.14,Default,,0000,0000,0000,,the function F of X equals X\Ncubed does not have a turning
Dialogue: 0,0:32:52.14,0:32:56.85,Default,,0000,0000,0000,,point. Another example, if\Nwe were looking at a quartic
Dialogue: 0,0:32:56.85,0:33:01.09,Default,,0000,0000,0000,,curve I a polynomial of\Ndegree four, we know can
Dialogue: 0,0:33:01.09,0:33:05.75,Default,,0000,0000,0000,,have up to at most three\Nturning points, which is why
Dialogue: 0,0:33:05.75,0:33:09.57,Default,,0000,0000,0000,,the general shape of a\Nquartic tends to be
Dialogue: 0,0:33:09.57,0:33:10.84,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:33:12.91,0:33:18.24,Default,,0000,0000,0000,,With the three, turning points\Nare here, here and here.
Dialogue: 0,0:33:19.01,0:33:25.35,Default,,0000,0000,0000,,Now let's suppose I had the\Nfunction F of X equals.
Dialogue: 0,0:33:26.07,0:33:32.31,Default,,0000,0000,0000,,X minus a multiplied by\NX minus B.
Dialogue: 0,0:33:33.27,0:33:37.01,Default,,0000,0000,0000,,Now to find the roots of this\Nfunction, I want to know the
Dialogue: 0,0:33:37.01,0:33:39.61,Default,,0000,0000,0000,,value of X when F of X equals 0.
Dialogue: 0,0:33:40.27,0:33:46.75,Default,,0000,0000,0000,,So when F of X equals 0, we\Nhave 0 equals X minus a
Dialogue: 0,0:33:46.75,0:33:49.07,Default,,0000,0000,0000,,multiplied by X minus B.
Dialogue: 0,0:33:49.79,0:33:55.46,Default,,0000,0000,0000,,So either X minus\Na equals 0.
Dialogue: 0,0:33:55.70,0:34:03.06,Default,,0000,0000,0000,,Or X minus B equals 0,\Nso the roots must be X
Dialogue: 0,0:34:03.06,0:34:06.78,Default,,0000,0000,0000,,equals A. Or X equals
Dialogue: 0,0:34:06.78,0:34:13.69,Default,,0000,0000,0000,,B. Now we can use the\Nconverse of this and say that if
Dialogue: 0,0:34:13.69,0:34:19.76,Default,,0000,0000,0000,,we know the roots are A&B, then\Nthe function must be F of X
Dialogue: 0,0:34:19.76,0:34:25.84,Default,,0000,0000,0000,,equals X minus a times by X\Nminus B or a multiple of that,
Dialogue: 0,0:34:25.84,0:34:31.48,Default,,0000,0000,0000,,and that multiple could be a\Nconstant. Or it could be in fact
Dialogue: 0,0:34:31.48,0:34:36.69,Default,,0000,0000,0000,,any polynomial we choose. So for\Nexample, if we knew that the
Dialogue: 0,0:34:36.69,0:34:38.86,Default,,0000,0000,0000,,roots were three and negative.
Dialogue: 0,0:34:38.89,0:34:46.21,Default,,0000,0000,0000,,2. We would say that F\Nof X would be X minus 3
Dialogue: 0,0:34:46.21,0:34:50.92,Default,,0000,0000,0000,,multiplied by X +2 or\Nmultiple so that multiple
Dialogue: 0,0:34:50.92,0:34:57.21,Default,,0000,0000,0000,,could be three could be 5, or\Nit could be any polynomial.
Dialogue: 0,0:34:58.60,0:35:05.62,Default,,0000,0000,0000,,OK, another example. If my\Nroots were one 2, three
Dialogue: 0,0:35:05.62,0:35:13.25,Default,,0000,0000,0000,,and four. Then my function would\Nhave to be F of X equals X
Dialogue: 0,0:35:13.25,0:35:15.61,Default,,0000,0000,0000,,minus One X minus 2.
Dialogue: 0,0:35:16.30,0:35:18.50,Default,,0000,0000,0000,,X minus three.
Dialogue: 0,0:35:19.09,0:35:24.64,Default,,0000,0000,0000,,At times by X minus four, or it\Ncould be a multiple of that, and
Dialogue: 0,0:35:24.64,0:35:28.34,Default,,0000,0000,0000,,as I keep saying that multiple\Ncould be any polynomial.
Dialogue: 0,0:35:28.96,0:35:36.20,Default,,0000,0000,0000,,Right so Lastly, I'd like to\Nthink about the function F of X.
Dialogue: 0,0:35:36.24,0:35:39.70,Default,,0000,0000,0000,,Equals X minus two all
Dialogue: 0,0:35:39.70,0:35:44.99,Default,,0000,0000,0000,,squared. Now if we try to find\Nthe roots of this function I
Dialogue: 0,0:35:44.99,0:35:49.68,Default,,0000,0000,0000,,when F of X equals 0, look what\Nhappens. We had zero equals and
Dialogue: 0,0:35:49.68,0:35:54.37,Default,,0000,0000,0000,,I'll just rewrite this side as X\Nminus two times by X minus 2.
Dialogue: 0,0:35:55.04,0:35:59.55,Default,,0000,0000,0000,,Which means either X minus two\Nmust be 0.
Dialogue: 0,0:36:00.19,0:36:08.15,Default,,0000,0000,0000,,Or X minus 2 equals 0\Nthe same thing. So X equals
Dialogue: 0,0:36:08.15,0:36:14.46,Default,,0000,0000,0000,,2. Or X equals 2. So\Nthere are two solutions and two
Dialogue: 0,0:36:14.46,0:36:20.92,Default,,0000,0000,0000,,routes at the value X equals 2\Nas what we call a repeated root.
Dialogue: 0,0:36:21.74,0:36:27.97,Default,,0000,0000,0000,,OK, another example. Let's\Nsuppose we have F of
Dialogue: 0,0:36:27.97,0:36:31.43,Default,,0000,0000,0000,,X equals X minus 2
Dialogue: 0,0:36:31.43,0:36:37.02,Default,,0000,0000,0000,,cubed. Multiply by X +4 to\Nthe power 4.
Dialogue: 0,0:36:37.93,0:36:42.67,Default,,0000,0000,0000,,Now, as before, we want to find\Nout what the value of X has to
Dialogue: 0,0:36:42.67,0:36:45.51,Default,,0000,0000,0000,,be to make F of X equal to 0.
Dialogue: 0,0:36:46.17,0:36:51.50,Default,,0000,0000,0000,,And we can see that if we put X\Nequals 2 in here, we will
Dialogue: 0,0:36:51.50,0:36:55.76,Default,,0000,0000,0000,,actually get zero in this\Nbracket. So X equals 2 is one
Dialogue: 0,0:36:55.76,0:36:59.30,Default,,0000,0000,0000,,route and actually there are\Nthree of those because it's
Dialogue: 0,0:36:59.30,0:37:03.56,Default,,0000,0000,0000,,cubed. So this has a repeated\Nroute, three of them, so three
Dialogue: 0,0:37:03.56,0:37:06.76,Default,,0000,0000,0000,,repeated roots. So three roots\Nfor X equals 2.
Dialogue: 0,0:37:07.40,0:37:12.02,Default,,0000,0000,0000,,But also we can see that if X is\Nminus four, then this bracket
Dialogue: 0,0:37:12.02,0:37:16.97,Default,,0000,0000,0000,,will be equal to 0, so X equals\Nminus four is a second route and
Dialogue: 0,0:37:16.97,0:37:20.93,Default,,0000,0000,0000,,there are four of them there. So\Nthere are four repeated roots
Dialogue: 0,0:37:20.93,0:37:24.83,Default,,0000,0000,0000,,there. Now what we say is that.
Dialogue: 0,0:37:25.34,0:37:26.62,Default,,0000,0000,0000,,If a route.
Dialogue: 0,0:37:27.14,0:37:31.44,Default,,0000,0000,0000,,Has an odd number of repeated\Nroots. For instance, this
Dialogue: 0,0:37:31.44,0:37:35.74,Default,,0000,0000,0000,,one's got 3 routes, then it\Nhas an odd multiplicity.
Dialogue: 0,0:37:36.81,0:37:41.14,Default,,0000,0000,0000,,If a root, for instance X\Nequals minus four, has an
Dialogue: 0,0:37:41.14,0:37:44.69,Default,,0000,0000,0000,,even number of roots, then it\Nhasn't even multiplicity.
Dialogue: 0,0:37:45.84,0:37:49.52,Default,,0000,0000,0000,,Why are we interested in\Nmultiplicity at all? Well, for
Dialogue: 0,0:37:49.52,0:37:53.40,Default,,0000,0000,0000,,this reason. If the\Nmultiplicities odd, then that
Dialogue: 0,0:37:53.40,0:37:57.91,Default,,0000,0000,0000,,means the graph actually crosses\Nthe X axis at the roots.
Dialogue: 0,0:37:58.45,0:38:02.95,Default,,0000,0000,0000,,If the multiplicity is even,\Nthen it means that the graph
Dialogue: 0,0:38:02.95,0:38:07.86,Default,,0000,0000,0000,,just touches the X axis and this\Nis very useful tool when
Dialogue: 0,0:38:07.86,0:38:11.40,Default,,0000,0000,0000,,sketching functions. For\Nexample, if we had.
Dialogue: 0,0:38:12.10,0:38:15.11,Default,,0000,0000,0000,,F of X equals.
Dialogue: 0,0:38:16.12,0:38:19.92,Default,,0000,0000,0000,,X minus 3 squared multiplied
Dialogue: 0,0:38:19.92,0:38:25.16,Default,,0000,0000,0000,,by. X plus one to\Nthe power 5.
Dialogue: 0,0:38:25.44,0:38:31.58,Default,,0000,0000,0000,,By X minus 2 cubed times by X\N+2 to the power 4.
Dialogue: 0,0:38:32.37,0:38:35.70,Default,,0000,0000,0000,,Now, first sight, this might\Nlook very complicated, but in
Dialogue: 0,0:38:35.70,0:38:39.36,Default,,0000,0000,0000,,fact we can identify the four\Nroots straight away. The first
Dialogue: 0,0:38:39.36,0:38:43.69,Default,,0000,0000,0000,,One X equals 3 will make this\Nbrackets equal to 0, so that's
Dialogue: 0,0:38:43.69,0:38:48.02,Default,,0000,0000,0000,,our first roots. X equals 3 and\Nwe can see straight away there
Dialogue: 0,0:38:48.02,0:38:54.22,Default,,0000,0000,0000,,are. Two repeated roots there,\Nwhich means that this has an
Dialogue: 0,0:38:54.22,0:38:57.33,Default,,0000,0000,0000,,even multiplicity. Some
Dialogue: 0,0:38:57.33,0:39:02.58,Default,,0000,0000,0000,,even multiplicity.\NOur second route.
Dialogue: 0,0:39:03.13,0:39:07.23,Default,,0000,0000,0000,,He is going to be X equals minus\None because that will make this
Dialogue: 0,0:39:07.23,0:39:09.28,Default,,0000,0000,0000,,bracket equal to 0. So X equals
Dialogue: 0,0:39:09.28,0:39:14.25,Default,,0000,0000,0000,,minus one. I'm not actually five\Nof those repeated root cause.
Dialogue: 0,0:39:14.25,0:39:19.88,Default,,0000,0000,0000,,The power of the bracket is 5,\Nwhich means that this is an odd
Dialogue: 0,0:39:19.88,0:39:26.00,Default,,0000,0000,0000,,multiplicity. We look at the\Nnext One X is 2 will give us a
Dialogue: 0,0:39:26.00,0:39:31.72,Default,,0000,0000,0000,,zero in this bracket. So X\Nequals 2 is a root there and
Dialogue: 0,0:39:31.72,0:39:36.56,Default,,0000,0000,0000,,there are three of them, which\Nmeans is an odd multiplicity.
Dialogue: 0,0:39:37.49,0:39:44.63,Default,,0000,0000,0000,,And finally, our last bracket X\N+2 here. If we put X
Dialogue: 0,0:39:44.63,0:39:47.60,Default,,0000,0000,0000,,equals minus two in there.
Dialogue: 0,0:39:48.18,0:39:52.60,Default,,0000,0000,0000,,That will give us zero in this\Nbracket and there are four of
Dialogue: 0,0:39:52.60,0:39:55.32,Default,,0000,0000,0000,,those repeated roots, which\Nmeans it isn't even.
Dialogue: 0,0:39:55.96,0:40:02.37,Default,,0000,0000,0000,,Multiplicity. So as I said\Nbefore, we can use this to help
Dialogue: 0,0:40:02.37,0:40:07.05,Default,,0000,0000,0000,,us plot a graph or helper sketch\Nthe graph. So for instance, and
Dialogue: 0,0:40:07.05,0:40:11.01,Default,,0000,0000,0000,,even multiplicity here would\Nmean that at the roots X equals
Dialogue: 0,0:40:11.01,0:40:15.33,Default,,0000,0000,0000,,3. This curve would just touch\Nthe access at the roots, X
Dialogue: 0,0:40:15.33,0:40:20.01,Default,,0000,0000,0000,,equals minus one. It would cross\Nthe Axis and at the River X
Dialogue: 0,0:40:20.01,0:40:24.33,Default,,0000,0000,0000,,equals 2 is an odd multiplicity,\Nso it will cross the axis.
Dialogue: 0,0:40:24.88,0:40:29.13,Default,,0000,0000,0000,,And at the roots for X equals\Nminus two is an even more
Dialogue: 0,0:40:29.13,0:40:33.06,Default,,0000,0000,0000,,simplicity, so it would just\Ntouch the axis. So let's have a
Dialogue: 0,0:40:33.06,0:40:36.65,Default,,0000,0000,0000,,look at a function where we can\Nactually sketch this now.
Dialogue: 0,0:40:36.83,0:40:39.76,Default,,0000,0000,0000,,So if we look at the function F
Dialogue: 0,0:40:39.76,0:40:47.60,Default,,0000,0000,0000,,of X. Equals X minus two\Nall squared multiplied by X plus
Dialogue: 0,0:40:47.60,0:40:51.50,Default,,0000,0000,0000,,one. We can identify the two
Dialogue: 0,0:40:51.50,0:40:57.64,Default,,0000,0000,0000,,routes immediately. X equals 2\Nwill make this bracket zero and
Dialogue: 0,0:40:57.64,0:41:01.22,Default,,0000,0000,0000,,we can see that this hasn't even
Dialogue: 0,0:41:01.22,0:41:05.70,Default,,0000,0000,0000,,multiplicity. So even\Nmultiplicity.
Dialogue: 0,0:41:06.82,0:41:13.50,Default,,0000,0000,0000,,And X equals negative one would\Nmake this make this bracket 0.
Dialogue: 0,0:41:13.71,0:41:17.20,Default,,0000,0000,0000,,So this will be an
Dialogue: 0,0:41:17.20,0:41:21.59,Default,,0000,0000,0000,,odd multiplicity. Because\Nthe power of this bracket
Dialogue: 0,0:41:21.59,0:41:22.80,Default,,0000,0000,0000,,is actually one.
Dialogue: 0,0:41:24.33,0:41:28.21,Default,,0000,0000,0000,,Now we talked before about the\Ngeneral shape of a cubic, which
Dialogue: 0,0:41:28.21,0:41:29.82,Default,,0000,0000,0000,,we knew would look something
Dialogue: 0,0:41:29.82,0:41:34.50,Default,,0000,0000,0000,,like this. So how can we combine\Nthis information and this
Dialogue: 0,0:41:34.50,0:41:38.11,Default,,0000,0000,0000,,information to help us sketch\Nthe graph of this function?
Dialogue: 0,0:41:38.11,0:41:40.28,Default,,0000,0000,0000,,Well, let's first of all draw
Dialogue: 0,0:41:40.28,0:41:42.94,Default,,0000,0000,0000,,axes. Graph of X here.
Dialogue: 0,0:41:43.48,0:41:48.47,Default,,0000,0000,0000,,Convert sleep X going\Nhorizontally. Now we know the
Dialogue: 0,0:41:48.47,0:41:55.11,Default,,0000,0000,0000,,two routes ones X equals 2 to\Nthe X equals negative one.
Dialogue: 0,0:41:55.93,0:41:59.74,Default,,0000,0000,0000,,Now about the multiplicity. This\Ntells us that the even
Dialogue: 0,0:41:59.74,0:42:04.31,Default,,0000,0000,0000,,multiplicity is that X equals 2,\Nso that means it just touches
Dialogue: 0,0:42:04.31,0:42:06.22,Default,,0000,0000,0000,,the curve X equals 2.
Dialogue: 0,0:42:07.20,0:42:10.51,Default,,0000,0000,0000,,But it crosses the curve X\Nequals minus one.
Dialogue: 0,0:42:11.28,0:42:14.93,Default,,0000,0000,0000,,I should add here that because\Nwe've got a positive value for
Dialogue: 0,0:42:14.93,0:42:18.27,Default,,0000,0000,0000,,our coefficients of X cubed, if\Nwe multiply this out, we're
Dialogue: 0,0:42:18.27,0:42:20.10,Default,,0000,0000,0000,,definitely going to get curve of
Dialogue: 0,0:42:20.10,0:42:23.27,Default,,0000,0000,0000,,this shape. So we can see\Nit's going to cross through
Dialogue: 0,0:42:23.27,0:42:23.71,Default,,0000,0000,0000,,minus one.
Dialogue: 0,0:42:26.13,0:42:27.49,Default,,0000,0000,0000,,It's going to come down.
Dialogue: 0,0:42:28.00,0:42:31.30,Default,,0000,0000,0000,,And it's just going to touch the\Naccess at minus 2.
Dialogue: 0,0:42:32.33,0:42:38.77,Default,,0000,0000,0000,,So this is a sketch of the\Ncurve F of X equals X minus
Dialogue: 0,0:42:38.77,0:42:42.45,Default,,0000,0000,0000,,two all squared multiplied\Nby X plus one.
Dialogue: 0,0:42:43.68,0:42:46.85,Default,,0000,0000,0000,,We do not know precisely\Nwhere this point is, but
Dialogue: 0,0:42:46.85,0:42:50.02,Default,,0000,0000,0000,,we do know that it lies\Nsomewhere in this region.