WEBVTT 00:00:02.050 --> 00:00:07.671 And polynomial functions I would be looking at functions of X 00:00:07.671 --> 00:00:11.248 that represent polynomials of varying degrees, including 00:00:11.248 --> 00:00:13.292 cubics and quadratics and 00:00:13.292 --> 00:00:18.100 Quartics. We will look at some of the properties of these 00:00:18.100 --> 00:00:23.028 curves and then we will go on to look at how to deduce the 00:00:23.028 --> 00:00:27.604 function of the curve if given the roots. So first of all, what 00:00:27.604 --> 00:00:31.476 is a polynomial with a polynomial of degree add is a 00:00:31.476 --> 00:00:33.940 function of the form F of X. 00:00:33.960 --> 00:00:41.928 Equals a an X to the power N plus AN minus One X to the power 00:00:41.928 --> 00:00:48.900 and minus one plus, and it keeps going all the way until we get 00:00:48.900 --> 00:00:55.654 to. A2 X squared plus a 1X plus a 00:00:55.654 --> 00:01:01.711 zero and here the AA represents real numbers and 00:01:01.711 --> 00:01:04.403 often called the coefficients. 00:01:05.240 --> 00:01:11.120 Now, this may seem complicated at first sight, but it's not and 00:01:11.120 --> 00:01:16.510 hopefully a few examples should convince you of this. So, for 00:01:16.510 --> 00:01:22.390 example, let's suppose we had F of X equals 4X cubed, minus 00:01:22.390 --> 00:01:24.350 three X squared +2. 00:01:25.720 --> 00:01:30.021 Now this is a function and a polynomial of degree three. 00:01:30.021 --> 00:01:35.495 Since the highest power of X is 3 and is often called a cubic. 00:01:36.650 --> 00:01:43.776 Let's suppose we hard F of X equals X to the Power 7 - 00:01:43.776 --> 00:01:47.339 4 X to the power 5 + 00:01:47.339 --> 00:01:52.200 1. This is a polynomial of degree Seven. Since the highest 00:01:52.200 --> 00:01:56.974 power of X is A7, an important thing to notice here is that you 00:01:56.974 --> 00:02:01.748 don't need every single power of X all the way up to 7. The 00:02:01.748 --> 00:02:05.840 important thing is that the highest power of X is 7, and 00:02:05.840 --> 00:02:07.545 that's why it's a polynomial 00:02:07.545 --> 00:02:13.394 degree 7. And the final example F of X equals. 00:02:13.960 --> 00:02:19.344 Four X squared minus two X minus 4. 00:02:19.870 --> 00:02:23.445 So polynomial of degree 2 because the highest power of X 00:02:23.445 --> 00:02:26.862 is a 2. Less often called a 00:02:26.862 --> 00:02:30.553 quadratic. Now it's important when we're thinking about 00:02:30.553 --> 00:02:34.612 polynomials that we only have positive powers of X, and we 00:02:34.612 --> 00:02:38.302 don't have any other kind of functions. For example, square 00:02:38.302 --> 00:02:40.147 roots or division by X. 00:02:40.740 --> 00:02:42.636 So for example, if we had. 00:02:43.230 --> 00:02:46.550 F of X equals 00:02:46.550 --> 00:02:52.586 4X cubed. Plus the square root of X minus one. 00:02:53.120 --> 00:02:58.411 This is not a polynomial because we have the square root of X 00:02:58.411 --> 00:03:04.109 here and we need all the powers of X to be positive integers to 00:03:04.109 --> 00:03:07.365 have function to be a polynomial. Second example. 00:03:07.390 --> 00:03:14.482 If we had F of X equals 5X to the Power 4 - 2 X squared plus 3 00:03:14.482 --> 00:03:19.604 divided by X. Once again this is not a polynomial and this is 00:03:19.604 --> 00:03:24.726 because, as I said before, we need all the powers of X should 00:03:24.726 --> 00:03:30.242 be positive integers and so this 3 divided by X does not fit in 00:03:30.242 --> 00:03:33.000 with that. So this is not a 00:03:33.000 --> 00:03:38.264 polynomial. OK, we've already met some basic polynomials, and 00:03:38.264 --> 00:03:44.237 you'll recognize these. For example, F of X equals 2 is 00:03:44.237 --> 00:03:50.210 as a constant function, and this is actually a type of 00:03:50.210 --> 00:03:57.269 polynomial, and likewise F of X equals 2X plus one, which is a 00:03:57.269 --> 00:04:03.242 linear function is also a type of polynomial, and we could 00:04:03.242 --> 00:04:06.000 sketch those. So the F of X. 00:04:06.640 --> 00:04:12.820 And X and we know that F of X equals 2 is a horizontal line 00:04:12.820 --> 00:04:19.000 which passes through two on the F of X axis and F of X equals 00:04:19.000 --> 00:04:20.236 2X plus one. 00:04:21.290 --> 00:04:26.250 Is straight line with gradient two which passes through one. 00:04:28.400 --> 00:04:34.418 On the F of X axis, stuff of X equals 2 and this is F of X. 00:04:35.240 --> 00:04:38.315 Equals 2X plus 00:04:38.315 --> 00:04:43.238 one. Important things remember is that all 00:04:43.238 --> 00:04:46.256 constant functions are horizontal straight lines 00:04:46.256 --> 00:04:50.280 and all linear functions are straight lines which 00:04:50.280 --> 00:04:51.789 are not horizontal. 00:04:52.880 --> 00:04:56.624 So let's have a look now at some 00:04:56.624 --> 00:05:02.697 quadratic functions. If we had F of X equals X squared, we know 00:05:02.697 --> 00:05:06.767 this is a quadratic function because it's a polynomial of 00:05:06.767 --> 00:05:11.651 degree two and we can sketch. This is a very familiar curve. 00:05:11.651 --> 00:05:14.907 If you have F of X on the 00:05:14.907 --> 00:05:19.774 vertical axis. An X on our horizontal axis. We 00:05:19.774 --> 00:05:25.296 know that the graph F of X equals X squared looks 00:05:25.296 --> 00:05:26.802 something like this. 00:05:29.090 --> 00:05:32.822 This is F of X equals 00:05:32.822 --> 00:05:38.480 X squared. So what happens as we vary the coefficient of X 00:05:38.480 --> 00:05:42.528 squared? That's the number that multiplies by X squared, so we 00:05:42.528 --> 00:05:47.312 had, for example, F of X equals 2 X squared. How would that 00:05:47.312 --> 00:05:51.728 affect our graph? What actually happens is you can see that all 00:05:51.728 --> 00:05:56.880 of our X squared values are the F of X values have now been 00:05:56.880 --> 00:06:00.928 multiplied by two and hence stretched in the F of X 00:06:00.928 --> 00:06:02.032 direction by two. 00:06:02.640 --> 00:06:04.439 So you can see we actually get. 00:06:07.270 --> 00:06:09.120 Occurs that looks something like 00:06:09.120 --> 00:06:16.140 this. And this is F of X equals 2 X squared. 00:06:17.340 --> 00:06:22.428 And likewise, we could do that for any coefficients. So if we 00:06:22.428 --> 00:06:25.820 had F of X equals 5 X squared. 00:06:26.450 --> 00:06:29.783 It would be stretched five times from the value it was 00:06:29.783 --> 00:06:33.116 when it was X squared, and it would look something like 00:06:33.116 --> 00:06:33.419 this. 00:06:36.410 --> 00:06:42.900 So this would be F of X equals 5 X squared. 00:06:43.610 --> 00:06:47.482 So this is all for positive coefficients of X squared. But 00:06:47.482 --> 00:06:50.298 what would happen if the coefficients were negative? 00:06:50.800 --> 00:06:53.130 Well, let's have a look. 00:06:53.160 --> 00:06:57.840 It starts off with F of X equals minus X squared. Now, 00:06:57.840 --> 00:07:02.520 compared with F of X equals X squared would taken all of 00:07:02.520 --> 00:07:06.030 our positive values and we've multiplied them all by 00:07:06.030 --> 00:07:09.150 negative one. So this actually results in a 00:07:09.150 --> 00:07:13.050 reflection in the X axis because every single value is 00:07:13.050 --> 00:07:17.730 now, which was positive has become negative. So F of F of 00:07:17.730 --> 00:07:21.630 X equals minus X squared will look something like this. 00:07:24.210 --> 00:07:30.330 So this is F of X equals minus X squared. 00:07:31.120 --> 00:07:36.184 Likewise, F of X equals minus two X squared will be a 00:07:36.184 --> 00:07:42.092 reflection in the X axis of F of X equals 2 X squared, and 00:07:42.092 --> 00:07:45.468 so we get something which looks like this. 00:07:48.810 --> 00:07:55.674 F of X equals 2 X squared 3 - 2 X squared and Leslie are going 00:07:55.674 --> 00:08:02.109 to look at F of X equals minus five X squared and you may well 00:08:02.109 --> 00:08:08.115 have guessed by now. That's a reflection of F of X equals 5 X 00:08:08.115 --> 00:08:13.263 squared in the X axis, which gives us a graph which would 00:08:13.263 --> 00:08:14.550 look something like. 00:08:15.480 --> 00:08:16.180 This. 00:08:17.510 --> 00:08:23.738 We F of X equals minus five X squared. 00:08:24.550 --> 00:08:29.503 And in fact, it's true to say that for any polynomial, if you 00:08:29.503 --> 00:08:33.694 multiply the function by minus one, you will always get the 00:08:33.694 --> 00:08:35.599 reflection in the X axis. 00:08:36.240 --> 00:08:41.557 I will have a look at some other quadratics and see what happens 00:08:41.557 --> 00:08:46.874 if we vary the coefficients of X as opposed to the coefficient of 00:08:46.874 --> 00:08:51.782 X squared. And to do this I'm going to construct the table. 00:08:52.610 --> 00:08:54.346 So we'll have RX value at the 00:08:54.346 --> 00:08:59.670 top. And the three functions I'm going to look at our X squared 00:08:59.670 --> 00:09:03.174 plus X. X squared 00:09:03.174 --> 00:09:09.110 plus 4X. And X squared plus 6X. 00:09:10.040 --> 00:09:17.400 And for my X file use, I'm going to choose values of minus 5 - 4 00:09:17.400 --> 00:09:24.760 - 3 - 2 - 1 zero. I'm going to go all the way up to 00:09:24.760 --> 00:09:26.600 a value of two. 00:09:27.480 --> 00:09:31.176 Now for each of these functions, I'm not necessarily going to 00:09:31.176 --> 00:09:35.544 workout the F of X value for every single value of X. I'm 00:09:35.544 --> 00:09:39.240 just going to focus around the turning point of the quadratic, 00:09:39.240 --> 00:09:40.920 which is where the quadratic 00:09:40.920 --> 00:09:46.289 dips. So just draw some lines in on my table. 00:09:46.290 --> 00:09:52.926 On some horizontal lines as well. 00:09:55.350 --> 00:09:59.172 And now we can fill this 00:09:59.172 --> 00:10:05.902 in. So for RF of X equals X squared plus X, I'm just going 00:10:05.902 --> 00:10:13.110 to focus on minus 2 + 2. So when I put X equals minus two in here 00:10:13.110 --> 00:10:19.046 we get minus 2 squared, which is 4 takeaway two which gives Me 2. 00:10:19.540 --> 00:10:24.430 When we put X is minus one in we get minus one squared, which is 00:10:24.430 --> 00:10:27.120 one. Take away one which gives 00:10:27.120 --> 00:10:34.715 us 0. Report this X is zero in here we just got 0 + 0 which is 00:10:34.715 --> 00:10:41.008 0. I put X is one in here. We've got one plus one which 00:10:41.008 --> 00:10:42.226 gives Me 2. 00:10:42.890 --> 00:10:47.024 And finally, when I pay taxes 2 in here, the two squared, which 00:10:47.024 --> 00:10:52.572 is 4. +2, which gives me 6 and in fact just for symmetry we 00:10:52.572 --> 00:10:57.265 could put X is minus three in here as well, which gives us 00:10:57.265 --> 00:10:58.709 minus 3 squared 9. 00:10:59.350 --> 00:11:02.990 Plus minus three. Sorry, which gives us 6. 00:11:03.630 --> 00:11:09.840 OK, for my next function, X squared plus 4X. 00:11:10.590 --> 00:11:15.582 I'm going to look at values from minus four up to 0. 00:11:16.160 --> 00:11:23.167 So if I put minus four in here, we get 16 takeaway 16 00:11:23.167 --> 00:11:25.323 which gives me 0. 00:11:26.060 --> 00:11:27.680 Put minus three in here. 00:11:28.430 --> 00:11:35.040 We get 9 takeaway 12 which gives us minus three. 00:11:35.690 --> 00:11:42.307 For put minus two in here, we get four takeaway 8 which is 00:11:42.307 --> 00:11:49.386 minus 4. If I put minus one in here, we get one takeaway 4 00:11:49.386 --> 00:11:56.031 which is minus three, and if I put zero in here 0 + 0 just 00:11:56.031 --> 00:11:57.360 gives me 0. 00:11:58.030 --> 00:12:04.180 And Lastly, I'm going to look at F of X equals X squared plus 6X. 00:12:04.710 --> 00:12:08.182 And I'm going to take this from minus five all the way up to 00:12:08.182 --> 00:12:13.590 minus one. So we put minus five in first of all minus 5 squared 00:12:13.590 --> 00:12:17.200 is 25. Take away 30 gives us 00:12:17.200 --> 00:12:25.064 minus 5. Minus four in here 16 takeaway 24 which 00:12:25.064 --> 00:12:28.256 gives us minus 8. 00:12:29.150 --> 00:12:35.940 For X is minus three in. Here we get 9 takeaway 18, which is 00:12:35.940 --> 00:12:38.500 minus 9. For X is minus two in. 00:12:39.000 --> 00:12:45.230 We get minus two square root of 4 takeaway 12, which is minus 8. 00:12:45.780 --> 00:12:52.530 And we put minus one in. We get one takeaway 6 which is minus 5. 00:12:53.050 --> 00:12:56.710 And you can actually see the symmetry in each row here, to 00:12:56.710 --> 00:12:59.760 show that we've actually focused around that turning point we 00:12:59.760 --> 00:13:02.810 talked about. So let's draw the graph of these functions. 00:13:03.400 --> 00:13:07.180 So in a vertical scale will 00:13:07.180 --> 00:13:13.335 need. F of X and that's going to take us from all those values 00:13:13.335 --> 00:13:15.885 minus nine up 2 - 6. 00:13:16.430 --> 00:13:20.084 So. If we do minus 00:13:20.084 --> 00:13:26.187 9. Minus 8 minus Seven 6 - 5. 00:13:26.840 --> 00:13:30.578 For most 3 - 2 - 00:13:30.578 --> 00:13:37.721 1 zero. 12345 and we can just about squeeze 6 in 00:13:37.721 --> 00:13:39.488 at the top. 00:13:40.430 --> 00:13:43.240 On a longer horizontal axis. 00:13:44.480 --> 00:13:46.928 We've gone from minus five to 00:13:46.928 --> 00:13:50.550 two. So we just use one 00:13:50.550 --> 00:13:57.641 2. Minus 1 - 2 - 3 - 00:13:57.641 --> 00:14:01.150 4. And minus 5. 00:14:01.660 --> 00:14:06.462 So first of all, let's look at our function F of X equals X 00:14:06.462 --> 00:14:09.549 squared plus X. We've got minus three and six. 00:14:10.250 --> 00:14:11.459 Which is appear. 00:14:11.960 --> 00:14:15.698 We've got minus two and two. 00:14:16.550 --> 00:14:20.030 Which is here. 00:14:20.260 --> 00:14:22.748 Minus one and 0. 00:14:23.680 --> 00:14:25.880 We've got zero and zero. 00:14:26.800 --> 00:14:28.099 One and two. 00:14:28.660 --> 00:14:32.239 And two 16. 00:14:33.650 --> 00:14:36.750 And you can see clearly that this is a parabola 00:14:36.750 --> 00:14:39.850 which, as we expected, we can draw a smooth curve. 00:14:42.260 --> 00:14:43.640 Through those points. 00:14:45.770 --> 00:14:49.698 So this is F of X equals X 00:14:49.698 --> 00:14:52.170 squared. Plus X. 00:14:53.070 --> 00:14:58.110 Next, we're going to look at the function F of X equals X squared 00:14:58.110 --> 00:15:02.100 plus 4X. And the points we had with minus four and zero. 00:15:02.800 --> 00:15:06.128 Minus three and minus 00:15:06.128 --> 00:15:12.510 three. Minus two and minus 4. 00:15:14.130 --> 00:15:18.010 Minus one and minus three. 00:15:19.570 --> 00:15:23.730 And zero and zero which is already drawn. And once 00:15:23.730 --> 00:15:28.722 again you can see this is a smooth curve is a parabola 00:15:28.722 --> 00:15:29.970 as we expected. 00:15:34.200 --> 00:15:42.100 And this is F of X equals X squared plus 00:15:42.100 --> 00:15:48.336 4X. And the final function we're going to look at F of X equals X 00:15:48.336 --> 00:15:52.940 squared plus 6X. And so we've got minus 5 - 5. 00:15:53.800 --> 00:15:57.776 So over here minus 4 - 8. 00:15:58.770 --> 00:16:00.110 Which is down here. 00:16:00.780 --> 00:16:02.670 Minus three and minus 9. 00:16:03.480 --> 00:16:06.980 Minus 2 - 8. 00:16:07.530 --> 00:16:14.225 And minus one and minus five and once again you can see smooth 00:16:14.225 --> 00:16:16.800 curve in the parabola shape. 00:16:21.010 --> 00:16:28.050 And this is F of X equals X squared plus 6X. 00:16:28.860 --> 00:16:34.320 So we can see that as the coefficient of X increases from 00:16:34.320 --> 00:16:40.235 here to one to four to six, the curve the parabola is actually 00:16:40.235 --> 00:16:42.965 moving down and to the left. 00:16:44.170 --> 00:16:48.548 But what would happen if the coefficients of X was negative? 00:16:48.548 --> 00:16:54.518 Well, let's have a look and will do this in the same way as we've 00:16:54.518 --> 00:16:59.692 just on the previous examples. And we look at a table, and this 00:16:59.692 --> 00:17:04.468 time we'll look at how negative values of our coefficients of X 00:17:04.468 --> 00:17:09.642 affect the graph. So we look at X squared minus XX squared minus 00:17:09.642 --> 00:17:15.690 4X. And the graph of F of X equals X squared minus 6X. 00:17:18.030 --> 00:17:19.998 Now as before, will. 00:17:20.590 --> 00:17:25.495 Put a number of values in for X, but we won't use. All of them, 00:17:25.495 --> 00:17:30.073 will only use the ones which are near to the turning point of the 00:17:30.073 --> 00:17:33.997 quadratic, but the values will put in the range will be from 00:17:33.997 --> 00:17:36.286 minus two all the way up to 00:17:36.286 --> 00:17:40.528 five. Four and five. 00:17:44.910 --> 00:17:48.676 I just put in the lines for 00:17:48.676 --> 00:17:55.378 our table. OK, so for the first one, F of X equals X squared 00:17:55.378 --> 00:18:01.734 minus X. We look at what happens when we substitute in X is minus 00:18:01.734 --> 00:18:07.386 2. So minus 2 squared is 4 takeaway, minus two is the same 00:18:07.386 --> 00:18:12.906 as a plus two, which gives us six when X is minus 1 - 1 00:18:12.906 --> 00:18:17.690 squared, is one takeaway minus one is the same as a plus one 00:18:17.690 --> 00:18:19.530 which just gives us 2. 00:18:20.300 --> 00:18:23.690 Zero is just zero takeaway 00:18:23.690 --> 00:18:29.961 00. When we put one in, we've got 1. Take away, one which is 00:18:29.961 --> 00:18:36.010 0. And two gives us 2 squared four takeaway two which gives us 00:18:36.010 --> 00:18:38.625 2. And again, for symmetry will 00:18:38.625 --> 00:18:44.639 do access 3. So 3 squared is 9 takeaway, three is 6. 00:18:45.710 --> 00:18:51.222 So our second function F of X equals X squared minus 4X. We're 00:18:51.222 --> 00:18:55.462 going to look at going from zero up to five. 00:18:55.980 --> 00:19:01.230 And we put zero and we just get zero takeaway 0 which is 0. 00:19:01.850 --> 00:19:03.410 When we put X is one. 00:19:04.220 --> 00:19:07.755 We get one takeaway 4 which is 00:19:07.755 --> 00:19:14.935 minus 3. When we put X is 2, two squared is 4 takeaway 00:19:14.935 --> 00:19:17.785 eight, which gives us minus 4. 00:19:18.760 --> 00:19:23.912 When X is 3, that gives us 3 squared, which is 9 takeaway 12 00:19:23.912 --> 00:19:27.960 which gives us minus three and you can see the symmetry 00:19:27.960 --> 00:19:32.744 starting to form here. Now a Nexus 4 gives us 4 squared 16 00:19:32.744 --> 00:19:36.424 takeaway 16 which as we expect it is a 0. 00:19:37.140 --> 00:19:41.928 And for our final function, FX equals X squared minus six X, 00:19:41.928 --> 00:19:47.913 we're going to look at coming from X is one all the way up to 00:19:47.913 --> 00:19:53.100 X equals 5, X equals 1, gives US1 takeaway six, which gives us 00:19:53.100 --> 00:19:59.190 minus 5. X equals 2 gives us 2 squared. Is 4 takeaway 12 00:19:59.190 --> 00:20:01.190 which gives us minus 8. 00:20:02.320 --> 00:20:07.585 I mean for taxes, three in three squared is 9 takeaway 18, which 00:20:07.585 --> 00:20:09.205 gives us minus 9. 00:20:09.900 --> 00:20:13.716 X is 4 + 4 squared. 00:20:14.360 --> 00:20:21.100 Take away 24. Which 16 takeaway 24 which is minus 00:20:21.100 --> 00:20:27.083 8. I'm waiting for X is 5 in we get 25 takeaway 30, which 00:20:27.083 --> 00:20:31.494 is minus five and once again symmetry as expected. So as 00:20:31.494 --> 00:20:35.504 with our previous examples we want to draw this graphs 00:20:35.504 --> 00:20:40.316 of these functions so we can see what's going on as our 00:20:40.316 --> 00:20:44.326 value of the coefficients of X or Y is changing. 00:20:46.260 --> 00:20:52.014 So if we got F of X on our vertical axis, this time we're 00:20:52.014 --> 00:20:56.535 going from Arlo's values, minus nine. Our highest value is 6. 00:20:57.070 --> 00:21:00.902 So we're going from minus 9 - 8 00:21:00.902 --> 00:21:08.060 - 7. 6 - 5 - 4 - 3 - 2 00:21:08.060 --> 00:21:09.920 - 1 zero. 00:21:10.630 --> 00:21:18.460 12345 I will just squeeze in sex and are horizontal 00:21:18.460 --> 00:21:21.592 axis. We've got one. 00:21:23.180 --> 00:21:29.838 2. 3. Four and five letter VRX axis, and we 00:21:29.838 --> 00:21:36.078 go breakdown some minus 2 - 1 - 2. So first of all, let's look 00:21:36.078 --> 00:21:41.902 at the graph F of X equals X squared minus X. So minus two 00:21:41.902 --> 00:21:47.310 and six was our first point, so minus two and six, which is 00:21:47.310 --> 00:21:51.760 here. And we've got minus one and two which is here. 00:21:52.940 --> 00:21:54.840 We've got the origin 00. 00:21:55.630 --> 00:21:57.290 And we've got 10. 00:21:58.150 --> 00:22:00.790 22 00:22:01.440 --> 00:22:07.322 I'm 36 And as we expected, you can see we 00:22:07.322 --> 00:22:10.611 can join the points of here. Let's make smooth curve which 00:22:10.611 --> 00:22:11.807 will be a parabola. 00:22:16.740 --> 00:22:22.658 So this is F of X equals X squared minus X. 00:22:23.520 --> 00:22:29.019 Our second function was F of X equals X squared minus 4X. So 00:22:29.019 --> 00:22:31.557 first point was 00, which we've 00:22:31.557 --> 00:22:34.480 got. One and negative 3. 00:22:35.430 --> 00:22:38.948 Say it. We've got two and minus 4. 00:22:40.030 --> 00:22:42.874 Just hang up three and negative 00:22:42.874 --> 00:22:45.138 3. Which is here. 00:22:46.150 --> 00:22:51.330 And four and zero, which is here and straight away. We can see a 00:22:51.330 --> 00:22:53.180 smooth curve which is a 00:22:53.180 --> 00:22:56.430 parabola. Coming through those points. 00:22:57.970 --> 00:23:00.580 And we can label up this is F of 00:23:00.580 --> 00:23:06.790 X. Equals X squared minus 4X. 00:23:08.340 --> 00:23:13.870 Lost function to look at is the function F of X equals X squared 00:23:13.870 --> 00:23:21.625 minus 6X. So our first point was one and minus five, which is 00:23:21.625 --> 00:23:24.010 here. We are two and negative 8. 00:23:25.090 --> 00:23:28.190 Yep. Three and minus 9. 00:23:28.890 --> 00:23:31.920 4 - 8. 00:23:32.480 --> 00:23:38.187 And five and minus five, which is hit, and once again we can 00:23:38.187 --> 00:23:40.821 just draw a smooth curve through 00:23:40.821 --> 00:23:48.660 these points. To give us the parabola we wanted, this is F 00:23:48.660 --> 00:23:52.500 of X equals X squared minus 00:23:52.500 --> 00:23:57.960 6X. So what's happening as the coefficients of X is getting 00:23:57.960 --> 00:24:02.772 bigger in absolute terms. So for instance, we can from minus one 00:24:02.772 --> 00:24:07.985 to minus four to minus six, and we can see straight away that 00:24:07.985 --> 00:24:12.797 the actual graph this parabola is moving down and to the right 00:24:12.797 --> 00:24:16.807 as the coefficients of X gets bigger in absolute terms. 00:24:18.220 --> 00:24:20.938 And that's where the coefficients of X is negative. 00:24:21.710 --> 00:24:25.970 OK, so we know what happens when we varied coefficients of X 00:24:25.970 --> 00:24:30.230 squared and we know what happens when we vary the coefficients of 00:24:30.230 --> 00:24:34.490 X and that's both of them. For quadratics, what happens when we 00:24:34.490 --> 00:24:38.750 vary the constants at the end of a quadratic well? Likewise table 00:24:38.750 --> 00:24:42.655 of values is a good way to see what's going on. 00:24:42.700 --> 00:24:48.776 So this time I'm going to use X again. I'm going to go from 00:24:48.776 --> 00:24:56.154 minus two all the way up to +2, so minus 2 - 1 zero one and two, 00:24:56.154 --> 00:25:01.362 and for my functions I'm going to use X squared plus X. 00:25:01.940 --> 00:25:05.284 X squared plus X 00:25:05.284 --> 00:25:11.930 plus one. X squared plus X 00:25:11.930 --> 00:25:18.398 +5. And X squared plus X minus 4. 00:25:19.190 --> 00:25:22.542 Now this table is 00:25:22.542 --> 00:25:28.115 particularly. Easy for us to workout compared with the other 00:25:28.115 --> 00:25:33.510 ones because we have a slight advantage in that we have a Head 00:25:33.510 --> 00:25:38.490 Start because we already know the values for X squared plus X 00:25:38.490 --> 00:25:43.470 that go here and we can just take them directly from Aurora, 00:25:43.470 --> 00:25:48.035 the table and you'll remember that they actually gave us the 00:25:48.035 --> 00:25:50.110 values 200, two and six. 00:25:50.730 --> 00:25:55.168 And in fact, we can use this line in our table to help us 00:25:55.168 --> 00:25:59.289 with all of the other lines. The second line here, which is X 00:25:59.289 --> 00:26:00.874 squared plus X plus one. 00:26:01.500 --> 00:26:05.100 His only one bigger than every value in the table before. So 00:26:05.100 --> 00:26:06.600 all we need to do. 00:26:07.330 --> 00:26:14.285 Is just add 1 answer every value from the line before, so this 00:26:14.285 --> 00:26:18.030 will be a 311, three and Seven. 00:26:18.970 --> 00:26:24.010 Likewise, for this line, when we've got X squared plus X +5. 00:26:24.570 --> 00:26:28.962 All this line is just five bigger than our first line, so 00:26:28.962 --> 00:26:30.426 we can just 7. 00:26:30.940 --> 00:26:36.340 557 and 00:26:36.340 --> 00:26:42.680 11. And likewise, our last line is minus four in the 00:26:42.680 --> 00:26:46.880 end with exactly the same thing before hand, so it's just take 00:26:46.880 --> 00:26:51.080 away for from every value from our first line, which gives us 00:26:51.080 --> 00:26:58.274 minus 2. Minus 4 - 4 - 2 and 00:26:58.274 --> 00:27:03.054 2. So let's put this information on a graph now so we can 00:27:03.054 --> 00:27:06.378 actually see what's happening as we vary the constants at the end 00:27:06.378 --> 00:27:12.135 of this quadratic. So you put F of X and are vertical scale so 00:27:12.135 --> 00:27:17.760 that we need to go from minus four all over to 11 so it starts 00:27:17.760 --> 00:27:19.635 at minus 4 - 3. 00:27:20.260 --> 00:27:22.280 Minus 2 - 1. 00:27:22.870 --> 00:27:29.932 0123456789, ten and we can just 00:27:29.932 --> 00:27:33.463 about squeeze already 00:27:33.463 --> 00:27:39.396 left it. I no longer horizontal axis. We're 00:27:39.396 --> 00:27:46.548 going from minus 2 + 2 - 1 - 2 plus one 00:27:46.548 --> 00:27:52.468 and +2. So first graph is the graph of the function F of X 00:27:52.468 --> 00:27:56.142 equals X squared plus X, so it's minus two and two. 00:27:57.230 --> 00:28:02.450 Which is a point here, minus one and 0. 00:28:03.030 --> 00:28:07.706 So first graph is the graph of the function F of X equals X 00:28:07.706 --> 00:28:10.712 squared plus X, so it's minus two and two. 00:28:11.800 --> 00:28:15.712 Which is a .8 - 1 00:28:15.712 --> 00:28:23.069 and 0. Because the origin next 00 at the .1 two 00:28:23.069 --> 00:28:26.124 and we got the .26. 00:28:27.620 --> 00:28:30.797 For our next graph, actually sorry, now we should. 00:28:31.430 --> 00:28:34.027 Draw a smooth curve for this one 00:28:34.027 --> 00:28:39.278 first. And Labor Lux F of X equals X squared plus X. 00:28:40.340 --> 00:28:45.530 For an X graph which is F of X equals X squared plus X plus 00:28:45.530 --> 00:28:49.336 one, we need to put the following points minus two and 00:28:49.336 --> 00:28:54.530 three. So minus one and one you can see straight away 00:28:54.530 --> 00:28:58.886 that all of these points are just want above what we 00:28:58.886 --> 00:29:00.866 had before zero and one. 00:29:01.970 --> 00:29:03.869 One and three. 00:29:04.910 --> 00:29:07.808 Two and Seven. 00:29:07.980 --> 00:29:11.630 So you can imagine our next parabola that we draw 00:29:11.630 --> 00:29:12.725 through these points. 00:29:15.260 --> 00:29:20.096 Is exactly the same but one above previous, that's F of X 00:29:20.096 --> 00:29:24.025 equals. X squared plus X plus 00:29:24.025 --> 00:29:30.152 one. So what about F of X equals X squared plus X +5? 00:29:30.790 --> 00:29:33.886 Well, let's have a look minus two and Seven. 00:29:35.480 --> 00:29:37.316 Gives us a point up here. 00:29:38.280 --> 00:29:40.120 Minus one and five. 00:29:41.260 --> 00:29:43.990 Zero and five. 00:29:44.730 --> 00:29:47.649 One and Seven. 00:29:48.460 --> 00:29:51.728 And two and 11. 00:29:53.530 --> 00:29:57.157 So we draw a smooth curve through these points. 00:30:02.060 --> 00:30:09.032 See, that gives us a parabola and this is F of X equals X 00:30:09.032 --> 00:30:11.024 squared plus X +5. 00:30:12.320 --> 00:30:17.832 So finally we look at the function F of X equals X squared 00:30:17.832 --> 00:30:22.920 plus X minus four. So we've got minus two and minus 2. 00:30:23.450 --> 00:30:26.938 Minus 1 - 4. 00:30:27.960 --> 00:30:29.598 0 - 4. 00:30:30.240 --> 00:30:33.928 One 1 - 2. 00:30:33.930 --> 00:30:35.010 Two and two. 00:30:36.510 --> 00:30:40.074 I once again we can draw a smooth curve. 00:30:41.300 --> 00:30:48.463 Through these points and this is F of X equals X squared plus 00:30:48.463 --> 00:30:50.116 X minus 4. 00:30:50.900 --> 00:30:54.200 So we can see that what's actually happening here is from 00:30:54.200 --> 00:30:58.400 our original graph of F of X equals X squared plus X, and if 00:30:58.400 --> 00:31:02.300 you like, you could have put a plus zero there. And when we 00:31:02.300 --> 00:31:06.200 added one, it's moved one up when we added five, it's moved 5 00:31:06.200 --> 00:31:11.546 or. And when we took away four, it moved 4 down. So it's quite 00:31:11.546 --> 00:31:15.530 clear to see the effect that the constant has on our parabola. 00:31:16.880 --> 00:31:23.198 When looking at the graph of a function, a turning point is the 00:31:23.198 --> 00:31:28.058 point on the curve where the gradient changes from negative 00:31:28.058 --> 00:31:30.974 to positive or from positive to 00:31:30.974 --> 00:31:34.916 negative. And we're thinking about polynomials. Are 00:31:34.916 --> 00:31:40.317 polynomial of degree an has at most an minus one turning 00:31:40.317 --> 00:31:44.736 points. So for example, a quadratic of degree 2. 00:31:45.470 --> 00:31:47.220 Can only have one turning 00:31:47.220 --> 00:31:51.510 points. And if we draw just a sketch of a quadratic, you can 00:31:51.510 --> 00:31:54.360 see this point here would be at one turning point. 00:31:55.570 --> 00:32:00.058 We think about cubics, obviously cubic as a polynomial of degree 00:32:00.058 --> 00:32:05.770 3, so that can have at most two turning points, which is why the 00:32:05.770 --> 00:32:09.442 general shape of a cubic looks something like this. 00:32:10.700 --> 00:32:14.150 We see the two turning points are here and here. 00:32:14.750 --> 00:32:20.075 But as I said, it can have at most 2. There's no reason it has 00:32:20.075 --> 00:32:25.400 to have two, and a good example is this of this is if you look 00:32:25.400 --> 00:32:27.885 at F of X equals X cubed. 00:32:28.470 --> 00:32:31.460 And in fact, that would look if I do a sketch over here. 00:32:32.130 --> 00:32:38.440 F of X&X that would come from the bottom left. 00:32:39.870 --> 00:32:42.198 Through zero and come up here 00:32:42.198 --> 00:32:47.980 like this. That's F of X equals X cubed, and so we can see that 00:32:47.980 --> 00:32:52.140 the function F of X equals X cubed does not have a turning 00:32:52.140 --> 00:32:56.846 point. Another example, if we were looking at a quartic 00:32:56.846 --> 00:33:01.086 curve I a polynomial of degree four, we know can 00:33:01.086 --> 00:33:05.750 have up to at most three turning points, which is why 00:33:05.750 --> 00:33:09.566 the general shape of a quartic tends to be 00:33:09.566 --> 00:33:10.838 something like this. 00:33:12.910 --> 00:33:18.240 With the three, turning points are here, here and here. 00:33:19.010 --> 00:33:25.346 Now let's suppose I had the function F of X equals. 00:33:26.070 --> 00:33:32.310 X minus a multiplied by X minus B. 00:33:33.270 --> 00:33:37.014 Now to find the roots of this function, I want to know the 00:33:37.014 --> 00:33:39.606 value of X when F of X equals 0. 00:33:40.270 --> 00:33:46.752 So when F of X equals 0, we have 0 equals X minus a 00:33:46.752 --> 00:33:49.067 multiplied by X minus B. 00:33:49.790 --> 00:33:55.460 So either X minus a equals 0. 00:33:55.700 --> 00:34:03.056 Or X minus B equals 0, so the roots must be X 00:34:03.056 --> 00:34:06.780 equals A. Or X equals 00:34:06.780 --> 00:34:13.688 B. Now we can use the converse of this and say that if 00:34:13.688 --> 00:34:19.764 we know the roots are A&B, then the function must be F of X 00:34:19.764 --> 00:34:25.840 equals X minus a times by X minus B or a multiple of that, 00:34:25.840 --> 00:34:31.482 and that multiple could be a constant. Or it could be in fact 00:34:31.482 --> 00:34:36.690 any polynomial we choose. So for example, if we knew that the 00:34:36.690 --> 00:34:38.860 roots were three and negative. 00:34:38.890 --> 00:34:46.208 2. We would say that F of X would be X minus 3 00:34:46.208 --> 00:34:50.924 multiplied by X +2 or multiple so that multiple 00:34:50.924 --> 00:34:57.212 could be three could be 5, or it could be any polynomial. 00:34:58.600 --> 00:35:05.620 OK, another example. If my roots were one 2, three 00:35:05.620 --> 00:35:13.252 and four. Then my function would have to be F of X equals X 00:35:13.252 --> 00:35:15.607 minus One X minus 2. 00:35:16.300 --> 00:35:18.499 X minus three. 00:35:19.090 --> 00:35:24.640 At times by X minus four, or it could be a multiple of that, and 00:35:24.640 --> 00:35:28.340 as I keep saying that multiple could be any polynomial. 00:35:28.960 --> 00:35:36.201 Right so Lastly, I'd like to think about the function F of X. 00:35:36.240 --> 00:35:39.695 Equals X minus two all 00:35:39.695 --> 00:35:44.990 squared. Now if we try to find the roots of this function I 00:35:44.990 --> 00:35:49.680 when F of X equals 0, look what happens. We had zero equals and 00:35:49.680 --> 00:35:54.370 I'll just rewrite this side as X minus two times by X minus 2. 00:35:55.040 --> 00:35:59.549 Which means either X minus two must be 0. 00:36:00.190 --> 00:36:08.146 Or X minus 2 equals 0 the same thing. So X equals 00:36:08.146 --> 00:36:14.461 2. Or X equals 2. So there are two solutions and two 00:36:14.461 --> 00:36:20.915 routes at the value X equals 2 as what we call a repeated root. 00:36:21.740 --> 00:36:27.968 OK, another example. Let's suppose we have F of 00:36:27.968 --> 00:36:31.428 X equals X minus 2 00:36:31.428 --> 00:36:37.018 cubed. Multiply by X +4 to the power 4. 00:36:37.930 --> 00:36:42.670 Now, as before, we want to find out what the value of X has to 00:36:42.670 --> 00:36:45.514 be to make F of X equal to 0. 00:36:46.170 --> 00:36:51.495 And we can see that if we put X equals 2 in here, we will 00:36:51.495 --> 00:36:55.755 actually get zero in this bracket. So X equals 2 is one 00:36:55.755 --> 00:36:59.305 route and actually there are three of those because it's 00:36:59.305 --> 00:37:03.565 cubed. So this has a repeated route, three of them, so three 00:37:03.565 --> 00:37:06.760 repeated roots. So three roots for X equals 2. 00:37:07.400 --> 00:37:12.020 But also we can see that if X is minus four, then this bracket 00:37:12.020 --> 00:37:16.970 will be equal to 0, so X equals minus four is a second route and 00:37:16.970 --> 00:37:20.930 there are four of them there. So there are four repeated roots 00:37:20.930 --> 00:37:24.828 there. Now what we say is that. 00:37:25.340 --> 00:37:26.618 If a route. 00:37:27.140 --> 00:37:31.440 Has an odd number of repeated roots. For instance, this 00:37:31.440 --> 00:37:35.740 one's got 3 routes, then it has an odd multiplicity. 00:37:36.810 --> 00:37:41.144 If a root, for instance X equals minus four, has an 00:37:41.144 --> 00:37:44.690 even number of roots, then it hasn't even multiplicity. 00:37:45.840 --> 00:37:49.520 Why are we interested in multiplicity at all? Well, for 00:37:49.520 --> 00:37:53.400 this reason. If the multiplicities odd, then that 00:37:53.400 --> 00:37:57.910 means the graph actually crosses the X axis at the roots. 00:37:58.450 --> 00:38:02.949 If the multiplicity is even, then it means that the graph 00:38:02.949 --> 00:38:07.857 just touches the X axis and this is very useful tool when 00:38:07.857 --> 00:38:11.400 sketching functions. For example, if we had. 00:38:12.100 --> 00:38:15.108 F of X equals. 00:38:16.120 --> 00:38:19.925 X minus 3 squared multiplied 00:38:19.925 --> 00:38:25.160 by. X plus one to the power 5. 00:38:25.440 --> 00:38:31.576 By X minus 2 cubed times by X +2 to the power 4. 00:38:32.370 --> 00:38:35.700 Now, first sight, this might look very complicated, but in 00:38:35.700 --> 00:38:39.363 fact we can identify the four roots straight away. The first 00:38:39.363 --> 00:38:43.692 One X equals 3 will make this brackets equal to 0, so that's 00:38:43.692 --> 00:38:48.021 our first roots. X equals 3 and we can see straight away there 00:38:48.021 --> 00:38:54.220 are. Two repeated roots there, which means that this has an 00:38:54.220 --> 00:38:57.330 even multiplicity. Some 00:38:57.330 --> 00:39:02.579 even multiplicity. Our second route. 00:39:03.130 --> 00:39:07.232 He is going to be X equals minus one because that will make this 00:39:07.232 --> 00:39:09.283 bracket equal to 0. So X equals 00:39:09.283 --> 00:39:14.248 minus one. I'm not actually five of those repeated root cause. 00:39:14.248 --> 00:39:19.876 The power of the bracket is 5, which means that this is an odd 00:39:19.876 --> 00:39:26.000 multiplicity. We look at the next One X is 2 will give us a 00:39:26.000 --> 00:39:31.720 zero in this bracket. So X equals 2 is a root there and 00:39:31.720 --> 00:39:36.560 there are three of them, which means is an odd multiplicity. 00:39:37.490 --> 00:39:44.630 And finally, our last bracket X +2 here. If we put X 00:39:44.630 --> 00:39:47.605 equals minus two in there. 00:39:48.180 --> 00:39:52.600 That will give us zero in this bracket and there are four of 00:39:52.600 --> 00:39:55.320 those repeated roots, which means it isn't even. 00:39:55.960 --> 00:40:02.370 Multiplicity. So as I said before, we can use this to help 00:40:02.370 --> 00:40:07.050 us plot a graph or helper sketch the graph. So for instance, and 00:40:07.050 --> 00:40:11.010 even multiplicity here would mean that at the roots X equals 00:40:11.010 --> 00:40:15.330 3. This curve would just touch the access at the roots, X 00:40:15.330 --> 00:40:20.010 equals minus one. It would cross the Axis and at the River X 00:40:20.010 --> 00:40:24.330 equals 2 is an odd multiplicity, so it will cross the axis. 00:40:24.880 --> 00:40:29.131 And at the roots for X equals minus two is an even more 00:40:29.131 --> 00:40:33.055 simplicity, so it would just touch the axis. So let's have a 00:40:33.055 --> 00:40:36.652 look at a function where we can actually sketch this now. 00:40:36.830 --> 00:40:39.758 So if we look at the function F 00:40:39.758 --> 00:40:47.600 of X. Equals X minus two all squared multiplied by X plus 00:40:47.600 --> 00:40:51.495 one. We can identify the two 00:40:51.495 --> 00:40:57.639 routes immediately. X equals 2 will make this bracket zero and 00:40:57.639 --> 00:41:01.216 we can see that this hasn't even 00:41:01.216 --> 00:41:05.700 multiplicity. So even multiplicity. 00:41:06.820 --> 00:41:13.504 And X equals negative one would make this make this bracket 0. 00:41:13.710 --> 00:41:17.200 So this will be an 00:41:17.200 --> 00:41:21.586 odd multiplicity. Because the power of this bracket 00:41:21.586 --> 00:41:22.804 is actually one. 00:41:24.330 --> 00:41:28.206 Now we talked before about the general shape of a cubic, which 00:41:28.206 --> 00:41:29.821 we knew would look something 00:41:29.821 --> 00:41:34.499 like this. So how can we combine this information and this 00:41:34.499 --> 00:41:38.109 information to help us sketch the graph of this function? 00:41:38.109 --> 00:41:40.275 Well, let's first of all draw 00:41:40.275 --> 00:41:42.940 axes. Graph of X here. 00:41:43.480 --> 00:41:48.466 Convert sleep X going horizontally. Now we know the 00:41:48.466 --> 00:41:55.114 two routes ones X equals 2 to the X equals negative one. 00:41:55.930 --> 00:41:59.740 Now about the multiplicity. This tells us that the even 00:41:59.740 --> 00:42:04.312 multiplicity is that X equals 2, so that means it just touches 00:42:04.312 --> 00:42:06.217 the curve X equals 2. 00:42:07.200 --> 00:42:10.512 But it crosses the curve X equals minus one. 00:42:11.280 --> 00:42:14.928 I should add here that because we've got a positive value for 00:42:14.928 --> 00:42:18.272 our coefficients of X cubed, if we multiply this out, we're 00:42:18.272 --> 00:42:20.096 definitely going to get curve of 00:42:20.096 --> 00:42:23.269 this shape. So we can see it's going to cross through 00:42:23.269 --> 00:42:23.711 minus one. 00:42:26.130 --> 00:42:27.490 It's going to come down. 00:42:28.000 --> 00:42:31.300 And it's just going to touch the access at minus 2. 00:42:32.330 --> 00:42:38.770 So this is a sketch of the curve F of X equals X minus 00:42:38.770 --> 00:42:42.450 two all squared multiplied by X plus one. 00:42:43.680 --> 00:42:46.850 We do not know precisely where this point is, but 00:42:46.850 --> 00:42:50.020 we do know that it lies somewhere in this region.