1 00:00:02,050 --> 00:00:07,671 And polynomial functions I would be looking at functions of X 2 00:00:07,671 --> 00:00:11,248 that represent polynomials of varying degrees, including 3 00:00:11,248 --> 00:00:13,292 cubics and quadratics and 4 00:00:13,292 --> 00:00:18,100 Quartics. We will look at some of the properties of these 5 00:00:18,100 --> 00:00:23,028 curves and then we will go on to look at how to deduce the 6 00:00:23,028 --> 00:00:27,604 function of the curve if given the roots. So first of all, what 7 00:00:27,604 --> 00:00:31,476 is a polynomial with a polynomial of degree add is a 8 00:00:31,476 --> 00:00:33,940 function of the form F of X. 9 00:00:33,960 --> 00:00:41,928 Equals a an X to the power N plus AN minus One X to the power 10 00:00:41,928 --> 00:00:48,900 and minus one plus, and it keeps going all the way until we get 11 00:00:48,900 --> 00:00:55,654 to. A2 X squared plus a 1X plus a 12 00:00:55,654 --> 00:01:01,711 zero and here the AA represents real numbers and 13 00:01:01,711 --> 00:01:04,403 often called the coefficients. 14 00:01:05,240 --> 00:01:11,120 Now, this may seem complicated at first sight, but it's not and 15 00:01:11,120 --> 00:01:16,510 hopefully a few examples should convince you of this. So, for 16 00:01:16,510 --> 00:01:22,390 example, let's suppose we had F of X equals 4X cubed, minus 17 00:01:22,390 --> 00:01:24,350 three X squared +2. 18 00:01:25,720 --> 00:01:30,021 Now this is a function and a polynomial of degree three. 19 00:01:30,021 --> 00:01:35,495 Since the highest power of X is 3 and is often called a cubic. 20 00:01:36,650 --> 00:01:43,776 Let's suppose we hard F of X equals X to the Power 7 - 21 00:01:43,776 --> 00:01:47,339 4 X to the power 5 + 22 00:01:47,339 --> 00:01:52,200 1. This is a polynomial of degree Seven. Since the highest 23 00:01:52,200 --> 00:01:56,974 power of X is A7, an important thing to notice here is that you 24 00:01:56,974 --> 00:02:01,748 don't need every single power of X all the way up to 7. The 25 00:02:01,748 --> 00:02:05,840 important thing is that the highest power of X is 7, and 26 00:02:05,840 --> 00:02:07,545 that's why it's a polynomial 27 00:02:07,545 --> 00:02:13,394 degree 7. And the final example F of X equals. 28 00:02:13,960 --> 00:02:19,344 Four X squared minus two X minus 4. 29 00:02:19,870 --> 00:02:23,445 So polynomial of degree 2 because the highest power of X 30 00:02:23,445 --> 00:02:26,862 is a 2. Less often called a 31 00:02:26,862 --> 00:02:30,553 quadratic. Now it's important when we're thinking about 32 00:02:30,553 --> 00:02:34,612 polynomials that we only have positive powers of X, and we 33 00:02:34,612 --> 00:02:38,302 don't have any other kind of functions. For example, square 34 00:02:38,302 --> 00:02:40,147 roots or division by X. 35 00:02:40,740 --> 00:02:42,636 So for example, if we had. 36 00:02:43,230 --> 00:02:46,550 F of X equals 37 00:02:46,550 --> 00:02:52,586 4X cubed. Plus the square root of X minus one. 38 00:02:53,120 --> 00:02:58,411 This is not a polynomial because we have the square root of X 39 00:02:58,411 --> 00:03:04,109 here and we need all the powers of X to be positive integers to 40 00:03:04,109 --> 00:03:07,365 have function to be a polynomial. Second example. 41 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 42 00:03:14,482 --> 00:03:19,604 divided by X. Once again this is not a polynomial and this is 43 00:03:19,604 --> 00:03:24,726 because, as I said before, we need all the powers of X should 44 00:03:24,726 --> 00:03:30,242 be positive integers and so this 3 divided by X does not fit in 45 00:03:30,242 --> 00:03:33,000 with that. So this is not a 46 00:03:33,000 --> 00:03:38,264 polynomial. OK, we've already met some basic polynomials, and 47 00:03:38,264 --> 00:03:44,237 you'll recognize these. For example, F of X equals 2 is 48 00:03:44,237 --> 00:03:50,210 as a constant function, and this is actually a type of 49 00:03:50,210 --> 00:03:57,269 polynomial, and likewise F of X equals 2X plus one, which is a 50 00:03:57,269 --> 00:04:03,242 linear function is also a type of polynomial, and we could 51 00:04:03,242 --> 00:04:06,000 sketch those. So the F of X. 52 00:04:06,640 --> 00:04:12,820 And X and we know that F of X equals 2 is a horizontal line 53 00:04:12,820 --> 00:04:19,000 which passes through two on the F of X axis and F of X equals 54 00:04:19,000 --> 00:04:20,236 2X plus one. 55 00:04:21,290 --> 00:04:26,250 Is straight line with gradient two which passes through one. 56 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. 57 00:04:35,240 --> 00:04:38,315 Equals 2X plus 58 00:04:38,315 --> 00:04:43,238 one. Important things remember is that all 59 00:04:43,238 --> 00:04:46,256 constant functions are horizontal straight lines 60 00:04:46,256 --> 00:04:50,280 and all linear functions are straight lines which 61 00:04:50,280 --> 00:04:51,789 are not horizontal. 62 00:04:52,880 --> 00:04:56,624 So let's have a look now at some 63 00:04:56,624 --> 00:05:02,697 quadratic functions. If we had F of X equals X squared, we know 64 00:05:02,697 --> 00:05:06,767 this is a quadratic function because it's a polynomial of 65 00:05:06,767 --> 00:05:11,651 degree two and we can sketch. This is a very familiar curve. 66 00:05:11,651 --> 00:05:14,907 If you have F of X on the 67 00:05:14,907 --> 00:05:19,774 vertical axis. An X on our horizontal axis. We 68 00:05:19,774 --> 00:05:25,296 know that the graph F of X equals X squared looks 69 00:05:25,296 --> 00:05:26,802 something like this. 70 00:05:29,090 --> 00:05:32,822 This is F of X equals 71 00:05:32,822 --> 00:05:38,480 X squared. So what happens as we vary the coefficient of X 72 00:05:38,480 --> 00:05:42,528 squared? That's the number that multiplies by X squared, so we 73 00:05:42,528 --> 00:05:47,312 had, for example, F of X equals 2 X squared. How would that 74 00:05:47,312 --> 00:05:51,728 affect our graph? What actually happens is you can see that all 75 00:05:51,728 --> 00:05:56,880 of our X squared values are the F of X values have now been 76 00:05:56,880 --> 00:06:00,928 multiplied by two and hence stretched in the F of X 77 00:06:00,928 --> 00:06:02,032 direction by two. 78 00:06:02,640 --> 00:06:04,439 So you can see we actually get. 79 00:06:07,270 --> 00:06:09,120 Occurs that looks something like 80 00:06:09,120 --> 00:06:16,140 this. And this is F of X equals 2 X squared. 81 00:06:17,340 --> 00:06:22,428 And likewise, we could do that for any coefficients. So if we 82 00:06:22,428 --> 00:06:25,820 had F of X equals 5 X squared. 83 00:06:26,450 --> 00:06:29,783 It would be stretched five times from the value it was 84 00:06:29,783 --> 00:06:33,116 when it was X squared, and it would look something like 85 00:06:33,116 --> 00:06:33,419 this. 86 00:06:36,410 --> 00:06:42,900 So this would be F of X equals 5 X squared. 87 00:06:43,610 --> 00:06:47,482 So this is all for positive coefficients of X squared. But 88 00:06:47,482 --> 00:06:50,298 what would happen if the coefficients were negative? 89 00:06:50,800 --> 00:06:53,130 Well, let's have a look. 90 00:06:53,160 --> 00:06:57,840 It starts off with F of X equals minus X squared. Now, 91 00:06:57,840 --> 00:07:02,520 compared with F of X equals X squared would taken all of 92 00:07:02,520 --> 00:07:06,030 our positive values and we've multiplied them all by 93 00:07:06,030 --> 00:07:09,150 negative one. So this actually results in a 94 00:07:09,150 --> 00:07:13,050 reflection in the X axis because every single value is 95 00:07:13,050 --> 00:07:17,730 now, which was positive has become negative. So F of F of 96 00:07:17,730 --> 00:07:21,630 X equals minus X squared will look something like this. 97 00:07:24,210 --> 00:07:30,330 So this is F of X equals minus X squared. 98 00:07:31,120 --> 00:07:36,184 Likewise, F of X equals minus two X squared will be a 99 00:07:36,184 --> 00:07:42,092 reflection in the X axis of F of X equals 2 X squared, and 100 00:07:42,092 --> 00:07:45,468 so we get something which looks like this. 101 00:07:48,810 --> 00:07:55,674 F of X equals 2 X squared 3 - 2 X squared and Leslie are going 102 00:07:55,674 --> 00:08:02,109 to look at F of X equals minus five X squared and you may well 103 00:08:02,109 --> 00:08:08,115 have guessed by now. That's a reflection of F of X equals 5 X 104 00:08:08,115 --> 00:08:13,263 squared in the X axis, which gives us a graph which would 105 00:08:13,263 --> 00:08:14,550 look something like. 106 00:08:15,480 --> 00:08:16,180 This. 107 00:08:17,510 --> 00:08:23,738 We F of X equals minus five X squared. 108 00:08:24,550 --> 00:08:29,503 And in fact, it's true to say that for any polynomial, if you 109 00:08:29,503 --> 00:08:33,694 multiply the function by minus one, you will always get the 110 00:08:33,694 --> 00:08:35,599 reflection in the X axis. 111 00:08:36,240 --> 00:08:41,557 I will have a look at some other quadratics and see what happens 112 00:08:41,557 --> 00:08:46,874 if we vary the coefficients of X as opposed to the coefficient of 113 00:08:46,874 --> 00:08:51,782 X squared. And to do this I'm going to construct the table. 114 00:08:52,610 --> 00:08:54,346 So we'll have RX value at the 115 00:08:54,346 --> 00:08:59,670 top. And the three functions I'm going to look at our X squared 116 00:08:59,670 --> 00:09:03,174 plus X. X squared 117 00:09:03,174 --> 00:09:09,110 plus 4X. And X squared plus 6X. 118 00:09:10,040 --> 00:09:17,400 And for my X file use, I'm going to choose values of minus 5 - 4 119 00:09:17,400 --> 00:09:24,760 - 3 - 2 - 1 zero. I'm going to go all the way up to 120 00:09:24,760 --> 00:09:26,600 a value of two. 121 00:09:27,480 --> 00:09:31,176 Now for each of these functions, I'm not necessarily going to 122 00:09:31,176 --> 00:09:35,544 workout the F of X value for every single value of X. I'm 123 00:09:35,544 --> 00:09:39,240 just going to focus around the turning point of the quadratic, 124 00:09:39,240 --> 00:09:40,920 which is where the quadratic 125 00:09:40,920 --> 00:09:46,289 dips. So just draw some lines in on my table. 126 00:09:46,290 --> 00:09:52,926 On some horizontal lines as well. 127 00:09:55,350 --> 00:09:59,172 And now we can fill this 128 00:09:59,172 --> 00:10:05,902 in. So for RF of X equals X squared plus X, I'm just going 129 00:10:05,902 --> 00:10:13,110 to focus on minus 2 + 2. So when I put X equals minus two in here 130 00:10:13,110 --> 00:10:19,046 we get minus 2 squared, which is 4 takeaway two which gives Me 2. 131 00:10:19,540 --> 00:10:24,430 When we put X is minus one in we get minus one squared, which is 132 00:10:24,430 --> 00:10:27,120 one. Take away one which gives 133 00:10:27,120 --> 00:10:34,715 us 0. Report this X is zero in here we just got 0 + 0 which is 134 00:10:34,715 --> 00:10:41,008 0. I put X is one in here. We've got one plus one which 135 00:10:41,008 --> 00:10:42,226 gives Me 2. 136 00:10:42,890 --> 00:10:47,024 And finally, when I pay taxes 2 in here, the two squared, which 137 00:10:47,024 --> 00:10:52,572 is 4. +2, which gives me 6 and in fact just for symmetry we 138 00:10:52,572 --> 00:10:57,265 could put X is minus three in here as well, which gives us 139 00:10:57,265 --> 00:10:58,709 minus 3 squared 9. 140 00:10:59,350 --> 00:11:02,990 Plus minus three. Sorry, which gives us 6. 141 00:11:03,630 --> 00:11:09,840 OK, for my next function, X squared plus 4X. 142 00:11:10,590 --> 00:11:15,582 I'm going to look at values from minus four up to 0. 143 00:11:16,160 --> 00:11:23,167 So if I put minus four in here, we get 16 takeaway 16 144 00:11:23,167 --> 00:11:25,323 which gives me 0. 145 00:11:26,060 --> 00:11:27,680 Put minus three in here. 146 00:11:28,430 --> 00:11:35,040 We get 9 takeaway 12 which gives us minus three. 147 00:11:35,690 --> 00:11:42,307 For put minus two in here, we get four takeaway 8 which is 148 00:11:42,307 --> 00:11:49,386 minus 4. If I put minus one in here, we get one takeaway 4 149 00:11:49,386 --> 00:11:56,031 which is minus three, and if I put zero in here 0 + 0 just 150 00:11:56,031 --> 00:11:57,360 gives me 0. 151 00:11:58,030 --> 00:12:04,180 And Lastly, I'm going to look at F of X equals X squared plus 6X. 152 00:12:04,710 --> 00:12:08,182 And I'm going to take this from minus five all the way up to 153 00:12:08,182 --> 00:12:13,590 minus one. So we put minus five in first of all minus 5 squared 154 00:12:13,590 --> 00:12:17,200 is 25. Take away 30 gives us 155 00:12:17,200 --> 00:12:25,064 minus 5. Minus four in here 16 takeaway 24 which 156 00:12:25,064 --> 00:12:28,256 gives us minus 8. 157 00:12:29,150 --> 00:12:35,940 For X is minus three in. Here we get 9 takeaway 18, which is 158 00:12:35,940 --> 00:12:38,500 minus 9. For X is minus two in. 159 00:12:39,000 --> 00:12:45,230 We get minus two square root of 4 takeaway 12, which is minus 8. 160 00:12:45,780 --> 00:12:52,530 And we put minus one in. We get one takeaway 6 which is minus 5. 161 00:12:53,050 --> 00:12:56,710 And you can actually see the symmetry in each row here, to 162 00:12:56,710 --> 00:12:59,760 show that we've actually focused around that turning point we 163 00:12:59,760 --> 00:13:02,810 talked about. So let's draw the graph of these functions. 164 00:13:03,400 --> 00:13:07,180 So in a vertical scale will 165 00:13:07,180 --> 00:13:13,335 need. F of X and that's going to take us from all those values 166 00:13:13,335 --> 00:13:15,885 minus nine up 2 - 6. 167 00:13:16,430 --> 00:13:20,084 So. If we do minus 168 00:13:20,084 --> 00:13:26,187 9. Minus 8 minus Seven 6 - 5. 169 00:13:26,840 --> 00:13:30,578 For most 3 - 2 - 170 00:13:30,578 --> 00:13:37,721 1 zero. 12345 and we can just about squeeze 6 in 171 00:13:37,721 --> 00:13:39,488 at the top. 172 00:13:40,430 --> 00:13:43,240 On a longer horizontal axis. 173 00:13:44,480 --> 00:13:46,928 We've gone from minus five to 174 00:13:46,928 --> 00:13:50,550 two. So we just use one 175 00:13:50,550 --> 00:13:57,641 2. Minus 1 - 2 - 3 - 176 00:13:57,641 --> 00:14:01,150 4. And minus 5. 177 00:14:01,660 --> 00:14:06,462 So first of all, let's look at our function F of X equals X 178 00:14:06,462 --> 00:14:09,549 squared plus X. We've got minus three and six. 179 00:14:10,250 --> 00:14:11,459 Which is appear. 180 00:14:11,960 --> 00:14:15,698 We've got minus two and two. 181 00:14:16,550 --> 00:14:20,030 Which is here. 182 00:14:20,260 --> 00:14:22,748 Minus one and 0. 183 00:14:23,680 --> 00:14:25,880 We've got zero and zero. 184 00:14:26,800 --> 00:14:28,099 One and two. 185 00:14:28,660 --> 00:14:32,239 And two 16. 186 00:14:33,650 --> 00:14:36,750 And you can see clearly that this is a parabola 187 00:14:36,750 --> 00:14:39,850 which, as we expected, we can draw a smooth curve. 188 00:14:42,260 --> 00:14:43,640 Through those points. 189 00:14:45,770 --> 00:14:49,698 So this is F of X equals X 190 00:14:49,698 --> 00:14:52,170 squared. Plus X. 191 00:14:53,070 --> 00:14:58,110 Next, we're going to look at the function F of X equals X squared 192 00:14:58,110 --> 00:15:02,100 plus 4X. And the points we had with minus four and zero. 193 00:15:02,800 --> 00:15:06,128 Minus three and minus 194 00:15:06,128 --> 00:15:12,510 three. Minus two and minus 4. 195 00:15:14,130 --> 00:15:18,010 Minus one and minus three. 196 00:15:19,570 --> 00:15:23,730 And zero and zero which is already drawn. And once 197 00:15:23,730 --> 00:15:28,722 again you can see this is a smooth curve is a parabola 198 00:15:28,722 --> 00:15:29,970 as we expected. 199 00:15:34,200 --> 00:15:42,100 And this is F of X equals X squared plus 200 00:15:42,100 --> 00:15:48,336 4X. And the final function we're going to look at F of X equals X 201 00:15:48,336 --> 00:15:52,940 squared plus 6X. And so we've got minus 5 - 5. 202 00:15:53,800 --> 00:15:57,776 So over here minus 4 - 8. 203 00:15:58,770 --> 00:16:00,110 Which is down here. 204 00:16:00,780 --> 00:16:02,670 Minus three and minus 9. 205 00:16:03,480 --> 00:16:06,980 Minus 2 - 8. 206 00:16:07,530 --> 00:16:14,225 And minus one and minus five and once again you can see smooth 207 00:16:14,225 --> 00:16:16,800 curve in the parabola shape. 208 00:16:21,010 --> 00:16:28,050 And this is F of X equals X squared plus 6X. 209 00:16:28,860 --> 00:16:34,320 So we can see that as the coefficient of X increases from 210 00:16:34,320 --> 00:16:40,235 here to one to four to six, the curve the parabola is actually 211 00:16:40,235 --> 00:16:42,965 moving down and to the left. 212 00:16:44,170 --> 00:16:48,548 But what would happen if the coefficients of X was negative? 213 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 214 00:16:54,518 --> 00:16:59,692 just on the previous examples. And we look at a table, and this 215 00:16:59,692 --> 00:17:04,468 time we'll look at how negative values of our coefficients of X 216 00:17:04,468 --> 00:17:09,642 affect the graph. So we look at X squared minus XX squared minus 217 00:17:09,642 --> 00:17:15,690 4X. And the graph of F of X equals X squared minus 6X. 218 00:17:18,030 --> 00:17:19,998 Now as before, will. 219 00:17:20,590 --> 00:17:25,495 Put a number of values in for X, but we won't use. All of them, 220 00:17:25,495 --> 00:17:30,073 will only use the ones which are near to the turning point of the 221 00:17:30,073 --> 00:17:33,997 quadratic, but the values will put in the range will be from 222 00:17:33,997 --> 00:17:36,286 minus two all the way up to 223 00:17:36,286 --> 00:17:40,528 five. Four and five. 224 00:17:44,910 --> 00:17:48,676 I just put in the lines for 225 00:17:48,676 --> 00:17:55,378 our table. OK, so for the first one, F of X equals X squared 226 00:17:55,378 --> 00:18:01,734 minus X. We look at what happens when we substitute in X is minus 227 00:18:01,734 --> 00:18:07,386 2. So minus 2 squared is 4 takeaway, minus two is the same 228 00:18:07,386 --> 00:18:12,906 as a plus two, which gives us six when X is minus 1 - 1 229 00:18:12,906 --> 00:18:17,690 squared, is one takeaway minus one is the same as a plus one 230 00:18:17,690 --> 00:18:19,530 which just gives us 2. 231 00:18:20,300 --> 00:18:23,690 Zero is just zero takeaway 232 00:18:23,690 --> 00:18:29,961 00. When we put one in, we've got 1. Take away, one which is 233 00:18:29,961 --> 00:18:36,010 0. And two gives us 2 squared four takeaway two which gives us 234 00:18:36,010 --> 00:18:38,625 2. And again, for symmetry will 235 00:18:38,625 --> 00:18:44,639 do access 3. So 3 squared is 9 takeaway, three is 6. 236 00:18:45,710 --> 00:18:51,222 So our second function F of X equals X squared minus 4X. We're 237 00:18:51,222 --> 00:18:55,462 going to look at going from zero up to five. 238 00:18:55,980 --> 00:19:01,230 And we put zero and we just get zero takeaway 0 which is 0. 239 00:19:01,850 --> 00:19:03,410 When we put X is one. 240 00:19:04,220 --> 00:19:07,755 We get one takeaway 4 which is 241 00:19:07,755 --> 00:19:14,935 minus 3. When we put X is 2, two squared is 4 takeaway 242 00:19:14,935 --> 00:19:17,785 eight, which gives us minus 4. 243 00:19:18,760 --> 00:19:23,912 When X is 3, that gives us 3 squared, which is 9 takeaway 12 244 00:19:23,912 --> 00:19:27,960 which gives us minus three and you can see the symmetry 245 00:19:27,960 --> 00:19:32,744 starting to form here. Now a Nexus 4 gives us 4 squared 16 246 00:19:32,744 --> 00:19:36,424 takeaway 16 which as we expect it is a 0. 247 00:19:37,140 --> 00:19:41,928 And for our final function, FX equals X squared minus six X, 248 00:19:41,928 --> 00:19:47,913 we're going to look at coming from X is one all the way up to 249 00:19:47,913 --> 00:19:53,100 X equals 5, X equals 1, gives US1 takeaway six, which gives us 250 00:19:53,100 --> 00:19:59,190 minus 5. X equals 2 gives us 2 squared. Is 4 takeaway 12 251 00:19:59,190 --> 00:20:01,190 which gives us minus 8. 252 00:20:02,320 --> 00:20:07,585 I mean for taxes, three in three squared is 9 takeaway 18, which 253 00:20:07,585 --> 00:20:09,205 gives us minus 9. 254 00:20:09,900 --> 00:20:13,716 X is 4 + 4 squared. 255 00:20:14,360 --> 00:20:21,100 Take away 24. Which 16 takeaway 24 which is minus 256 00:20:21,100 --> 00:20:27,083 8. I'm waiting for X is 5 in we get 25 takeaway 30, which 257 00:20:27,083 --> 00:20:31,494 is minus five and once again symmetry as expected. So as 258 00:20:31,494 --> 00:20:35,504 with our previous examples we want to draw this graphs 259 00:20:35,504 --> 00:20:40,316 of these functions so we can see what's going on as our 260 00:20:40,316 --> 00:20:44,326 value of the coefficients of X or Y is changing. 261 00:20:46,260 --> 00:20:52,014 So if we got F of X on our vertical axis, this time we're 262 00:20:52,014 --> 00:20:56,535 going from Arlo's values, minus nine. Our highest value is 6. 263 00:20:57,070 --> 00:21:00,902 So we're going from minus 9 - 8 264 00:21:00,902 --> 00:21:08,060 - 7. 6 - 5 - 4 - 3 - 2 265 00:21:08,060 --> 00:21:09,920 - 1 zero. 266 00:21:10,630 --> 00:21:18,460 12345 I will just squeeze in sex and are horizontal 267 00:21:18,460 --> 00:21:21,592 axis. We've got one. 268 00:21:23,180 --> 00:21:29,838 2. 3. Four and five letter VRX axis, and we 269 00:21:29,838 --> 00:21:36,078 go breakdown some minus 2 - 1 - 2. So first of all, let's look 270 00:21:36,078 --> 00:21:41,902 at the graph F of X equals X squared minus X. So minus two 271 00:21:41,902 --> 00:21:47,310 and six was our first point, so minus two and six, which is 272 00:21:47,310 --> 00:21:51,760 here. And we've got minus one and two which is here. 273 00:21:52,940 --> 00:21:54,840 We've got the origin 00. 274 00:21:55,630 --> 00:21:57,290 And we've got 10. 275 00:21:58,150 --> 00:22:00,790 22 276 00:22:01,440 --> 00:22:07,322 I'm 36 And as we expected, you can see we 277 00:22:07,322 --> 00:22:10,611 can join the points of here. Let's make smooth curve which 278 00:22:10,611 --> 00:22:11,807 will be a parabola. 279 00:22:16,740 --> 00:22:22,658 So this is F of X equals X squared minus X. 280 00:22:23,520 --> 00:22:29,019 Our second function was F of X equals X squared minus 4X. So 281 00:22:29,019 --> 00:22:31,557 first point was 00, which we've 282 00:22:31,557 --> 00:22:34,480 got. One and negative 3. 283 00:22:35,430 --> 00:22:38,948 Say it. We've got two and minus 4. 284 00:22:40,030 --> 00:22:42,874 Just hang up three and negative 285 00:22:42,874 --> 00:22:45,138 3. Which is here. 286 00:22:46,150 --> 00:22:51,330 And four and zero, which is here and straight away. We can see a 287 00:22:51,330 --> 00:22:53,180 smooth curve which is a 288 00:22:53,180 --> 00:22:56,430 parabola. Coming through those points. 289 00:22:57,970 --> 00:23:00,580 And we can label up this is F of 290 00:23:00,580 --> 00:23:06,790 X. Equals X squared minus 4X. 291 00:23:08,340 --> 00:23:13,870 Lost function to look at is the function F of X equals X squared 292 00:23:13,870 --> 00:23:21,625 minus 6X. So our first point was one and minus five, which is 293 00:23:21,625 --> 00:23:24,010 here. We are two and negative 8. 294 00:23:25,090 --> 00:23:28,190 Yep. Three and minus 9. 295 00:23:28,890 --> 00:23:31,920 4 - 8. 296 00:23:32,480 --> 00:23:38,187 And five and minus five, which is hit, and once again we can 297 00:23:38,187 --> 00:23:40,821 just draw a smooth curve through 298 00:23:40,821 --> 00:23:48,660 these points. To give us the parabola we wanted, this is F 299 00:23:48,660 --> 00:23:52,500 of X equals X squared minus 300 00:23:52,500 --> 00:23:57,960 6X. So what's happening as the coefficients of X is getting 301 00:23:57,960 --> 00:24:02,772 bigger in absolute terms. So for instance, we can from minus one 302 00:24:02,772 --> 00:24:07,985 to minus four to minus six, and we can see straight away that 303 00:24:07,985 --> 00:24:12,797 the actual graph this parabola is moving down and to the right 304 00:24:12,797 --> 00:24:16,807 as the coefficients of X gets bigger in absolute terms. 305 00:24:18,220 --> 00:24:20,938 And that's where the coefficients of X is negative. 306 00:24:21,710 --> 00:24:25,970 OK, so we know what happens when we varied coefficients of X 307 00:24:25,970 --> 00:24:30,230 squared and we know what happens when we vary the coefficients of 308 00:24:30,230 --> 00:24:34,490 X and that's both of them. For quadratics, what happens when we 309 00:24:34,490 --> 00:24:38,750 vary the constants at the end of a quadratic well? Likewise table 310 00:24:38,750 --> 00:24:42,655 of values is a good way to see what's going on. 311 00:24:42,700 --> 00:24:48,776 So this time I'm going to use X again. I'm going to go from 312 00:24:48,776 --> 00:24:56,154 minus two all the way up to +2, so minus 2 - 1 zero one and two, 313 00:24:56,154 --> 00:25:01,362 and for my functions I'm going to use X squared plus X. 314 00:25:01,940 --> 00:25:05,284 X squared plus X 315 00:25:05,284 --> 00:25:11,930 plus one. X squared plus X 316 00:25:11,930 --> 00:25:18,398 +5. And X squared plus X minus 4. 317 00:25:19,190 --> 00:25:22,542 Now this table is 318 00:25:22,542 --> 00:25:28,115 particularly. Easy for us to workout compared with the other 319 00:25:28,115 --> 00:25:33,510 ones because we have a slight advantage in that we have a Head 320 00:25:33,510 --> 00:25:38,490 Start because we already know the values for X squared plus X 321 00:25:38,490 --> 00:25:43,470 that go here and we can just take them directly from Aurora, 322 00:25:43,470 --> 00:25:48,035 the table and you'll remember that they actually gave us the 323 00:25:48,035 --> 00:25:50,110 values 200, two and six. 324 00:25:50,730 --> 00:25:55,168 And in fact, we can use this line in our table to help us 325 00:25:55,168 --> 00:25:59,289 with all of the other lines. The second line here, which is X 326 00:25:59,289 --> 00:26:00,874 squared plus X plus one. 327 00:26:01,500 --> 00:26:05,100 His only one bigger than every value in the table before. So 328 00:26:05,100 --> 00:26:06,600 all we need to do. 329 00:26:07,330 --> 00:26:14,285 Is just add 1 answer every value from the line before, so this 330 00:26:14,285 --> 00:26:18,030 will be a 311, three and Seven. 331 00:26:18,970 --> 00:26:24,010 Likewise, for this line, when we've got X squared plus X +5. 332 00:26:24,570 --> 00:26:28,962 All this line is just five bigger than our first line, so 333 00:26:28,962 --> 00:26:30,426 we can just 7. 334 00:26:30,940 --> 00:26:36,340 557 and 335 00:26:36,340 --> 00:26:42,680 11. And likewise, our last line is minus four in the 336 00:26:42,680 --> 00:26:46,880 end with exactly the same thing before hand, so it's just take 337 00:26:46,880 --> 00:26:51,080 away for from every value from our first line, which gives us 338 00:26:51,080 --> 00:26:58,274 minus 2. Minus 4 - 4 - 2 and 339 00:26:58,274 --> 00:27:03,054 2. So let's put this information on a graph now so we can 340 00:27:03,054 --> 00:27:06,378 actually see what's happening as we vary the constants at the end 341 00:27:06,378 --> 00:27:12,135 of this quadratic. So you put F of X and are vertical scale so 342 00:27:12,135 --> 00:27:17,760 that we need to go from minus four all over to 11 so it starts 343 00:27:17,760 --> 00:27:19,635 at minus 4 - 3. 344 00:27:20,260 --> 00:27:22,280 Minus 2 - 1. 345 00:27:22,870 --> 00:27:29,932 0123456789, ten and we can just 346 00:27:29,932 --> 00:27:33,463 about squeeze already 347 00:27:33,463 --> 00:27:39,396 left it. I no longer horizontal axis. We're 348 00:27:39,396 --> 00:27:46,548 going from minus 2 + 2 - 1 - 2 plus one 349 00:27:46,548 --> 00:27:52,468 and +2. So first graph is the graph of the function F of X 350 00:27:52,468 --> 00:27:56,142 equals X squared plus X, so it's minus two and two. 351 00:27:57,230 --> 00:28:02,450 Which is a point here, minus one and 0. 352 00:28:03,030 --> 00:28:07,706 So first graph is the graph of the function F of X equals X 353 00:28:07,706 --> 00:28:10,712 squared plus X, so it's minus two and two. 354 00:28:11,800 --> 00:28:15,712 Which is a .8 - 1 355 00:28:15,712 --> 00:28:23,069 and 0. Because the origin next 00 at the .1 two 356 00:28:23,069 --> 00:28:26,124 and we got the .26. 357 00:28:27,620 --> 00:28:30,797 For our next graph, actually sorry, now we should. 358 00:28:31,430 --> 00:28:34,027 Draw a smooth curve for this one 359 00:28:34,027 --> 00:28:39,278 first. And Labor Lux F of X equals X squared plus X. 360 00:28:40,340 --> 00:28:45,530 For an X graph which is F of X equals X squared plus X plus 361 00:28:45,530 --> 00:28:49,336 one, we need to put the following points minus two and 362 00:28:49,336 --> 00:28:54,530 three. So minus one and one you can see straight away 363 00:28:54,530 --> 00:28:58,886 that all of these points are just want above what we 364 00:28:58,886 --> 00:29:00,866 had before zero and one. 365 00:29:01,970 --> 00:29:03,869 One and three. 366 00:29:04,910 --> 00:29:07,808 Two and Seven. 367 00:29:07,980 --> 00:29:11,630 So you can imagine our next parabola that we draw 368 00:29:11,630 --> 00:29:12,725 through these points. 369 00:29:15,260 --> 00:29:20,096 Is exactly the same but one above previous, that's F of X 370 00:29:20,096 --> 00:29:24,025 equals. X squared plus X plus 371 00:29:24,025 --> 00:29:30,152 one. So what about F of X equals X squared plus X +5? 372 00:29:30,790 --> 00:29:33,886 Well, let's have a look minus two and Seven. 373 00:29:35,480 --> 00:29:37,316 Gives us a point up here. 374 00:29:38,280 --> 00:29:40,120 Minus one and five. 375 00:29:41,260 --> 00:29:43,990 Zero and five. 376 00:29:44,730 --> 00:29:47,649 One and Seven. 377 00:29:48,460 --> 00:29:51,728 And two and 11. 378 00:29:53,530 --> 00:29:57,157 So we draw a smooth curve through these points. 379 00:30:02,060 --> 00:30:09,032 See, that gives us a parabola and this is F of X equals X 380 00:30:09,032 --> 00:30:11,024 squared plus X +5. 381 00:30:12,320 --> 00:30:17,832 So finally we look at the function F of X equals X squared 382 00:30:17,832 --> 00:30:22,920 plus X minus four. So we've got minus two and minus 2. 383 00:30:23,450 --> 00:30:26,938 Minus 1 - 4. 384 00:30:27,960 --> 00:30:29,598 0 - 4. 385 00:30:30,240 --> 00:30:33,928 One 1 - 2. 386 00:30:33,930 --> 00:30:35,010 Two and two. 387 00:30:36,510 --> 00:30:40,074 I once again we can draw a smooth curve. 388 00:30:41,300 --> 00:30:48,463 Through these points and this is F of X equals X squared plus 389 00:30:48,463 --> 00:30:50,116 X minus 4. 390 00:30:50,900 --> 00:30:54,200 So we can see that what's actually happening here is from 391 00:30:54,200 --> 00:30:58,400 our original graph of F of X equals X squared plus X, and if 392 00:30:58,400 --> 00:31:02,300 you like, you could have put a plus zero there. And when we 393 00:31:02,300 --> 00:31:06,200 added one, it's moved one up when we added five, it's moved 5 394 00:31:06,200 --> 00:31:11,546 or. And when we took away four, it moved 4 down. So it's quite 395 00:31:11,546 --> 00:31:15,530 clear to see the effect that the constant has on our parabola. 396 00:31:16,880 --> 00:31:23,198 When looking at the graph of a function, a turning point is the 397 00:31:23,198 --> 00:31:28,058 point on the curve where the gradient changes from negative 398 00:31:28,058 --> 00:31:30,974 to positive or from positive to 399 00:31:30,974 --> 00:31:34,916 negative. And we're thinking about polynomials. Are 400 00:31:34,916 --> 00:31:40,317 polynomial of degree an has at most an minus one turning 401 00:31:40,317 --> 00:31:44,736 points. So for example, a quadratic of degree 2. 402 00:31:45,470 --> 00:31:47,220 Can only have one turning 403 00:31:47,220 --> 00:31:51,510 points. And if we draw just a sketch of a quadratic, you can 404 00:31:51,510 --> 00:31:54,360 see this point here would be at one turning point. 405 00:31:55,570 --> 00:32:00,058 We think about cubics, obviously cubic as a polynomial of degree 406 00:32:00,058 --> 00:32:05,770 3, so that can have at most two turning points, which is why the 407 00:32:05,770 --> 00:32:09,442 general shape of a cubic looks something like this. 408 00:32:10,700 --> 00:32:14,150 We see the two turning points are here and here. 409 00:32:14,750 --> 00:32:20,075 But as I said, it can have at most 2. There's no reason it has 410 00:32:20,075 --> 00:32:25,400 to have two, and a good example is this of this is if you look 411 00:32:25,400 --> 00:32:27,885 at F of X equals X cubed. 412 00:32:28,470 --> 00:32:31,460 And in fact, that would look if I do a sketch over here. 413 00:32:32,130 --> 00:32:38,440 F of X&X that would come from the bottom left. 414 00:32:39,870 --> 00:32:42,198 Through zero and come up here 415 00:32:42,198 --> 00:32:47,980 like this. That's F of X equals X cubed, and so we can see that 416 00:32:47,980 --> 00:32:52,140 the function F of X equals X cubed does not have a turning 417 00:32:52,140 --> 00:32:56,846 point. Another example, if we were looking at a quartic 418 00:32:56,846 --> 00:33:01,086 curve I a polynomial of degree four, we know can 419 00:33:01,086 --> 00:33:05,750 have up to at most three turning points, which is why 420 00:33:05,750 --> 00:33:09,566 the general shape of a quartic tends to be 421 00:33:09,566 --> 00:33:10,838 something like this. 422 00:33:12,910 --> 00:33:18,240 With the three, turning points are here, here and here. 423 00:33:19,010 --> 00:33:25,346 Now let's suppose I had the function F of X equals. 424 00:33:26,070 --> 00:33:32,310 X minus a multiplied by X minus B. 425 00:33:33,270 --> 00:33:37,014 Now to find the roots of this function, I want to know the 426 00:33:37,014 --> 00:33:39,606 value of X when F of X equals 0. 427 00:33:40,270 --> 00:33:46,752 So when F of X equals 0, we have 0 equals X minus a 428 00:33:46,752 --> 00:33:49,067 multiplied by X minus B. 429 00:33:49,790 --> 00:33:55,460 So either X minus a equals 0. 430 00:33:55,700 --> 00:34:03,056 Or X minus B equals 0, so the roots must be X 431 00:34:03,056 --> 00:34:06,780 equals A. Or X equals 432 00:34:06,780 --> 00:34:13,688 B. Now we can use the converse of this and say that if 433 00:34:13,688 --> 00:34:19,764 we know the roots are A&B, then the function must be F of X 434 00:34:19,764 --> 00:34:25,840 equals X minus a times by X minus B or a multiple of that, 435 00:34:25,840 --> 00:34:31,482 and that multiple could be a constant. Or it could be in fact 436 00:34:31,482 --> 00:34:36,690 any polynomial we choose. So for example, if we knew that the 437 00:34:36,690 --> 00:34:38,860 roots were three and negative. 438 00:34:38,890 --> 00:34:46,208 2. We would say that F of X would be X minus 3 439 00:34:46,208 --> 00:34:50,924 multiplied by X +2 or multiple so that multiple 440 00:34:50,924 --> 00:34:57,212 could be three could be 5, or it could be any polynomial. 441 00:34:58,600 --> 00:35:05,620 OK, another example. If my roots were one 2, three 442 00:35:05,620 --> 00:35:13,252 and four. Then my function would have to be F of X equals X 443 00:35:13,252 --> 00:35:15,607 minus One X minus 2. 444 00:35:16,300 --> 00:35:18,499 X minus three. 445 00:35:19,090 --> 00:35:24,640 At times by X minus four, or it could be a multiple of that, and 446 00:35:24,640 --> 00:35:28,340 as I keep saying that multiple could be any polynomial. 447 00:35:28,960 --> 00:35:36,201 Right so Lastly, I'd like to think about the function F of X. 448 00:35:36,240 --> 00:35:39,695 Equals X minus two all 449 00:35:39,695 --> 00:35:44,990 squared. Now if we try to find the roots of this function I 450 00:35:44,990 --> 00:35:49,680 when F of X equals 0, look what happens. We had zero equals and 451 00:35:49,680 --> 00:35:54,370 I'll just rewrite this side as X minus two times by X minus 2. 452 00:35:55,040 --> 00:35:59,549 Which means either X minus two must be 0. 453 00:36:00,190 --> 00:36:08,146 Or X minus 2 equals 0 the same thing. So X equals 454 00:36:08,146 --> 00:36:14,461 2. Or X equals 2. So there are two solutions and two 455 00:36:14,461 --> 00:36:20,915 routes at the value X equals 2 as what we call a repeated root. 456 00:36:21,740 --> 00:36:27,968 OK, another example. Let's suppose we have F of 457 00:36:27,968 --> 00:36:31,428 X equals X minus 2 458 00:36:31,428 --> 00:36:37,018 cubed. Multiply by X +4 to the power 4. 459 00:36:37,930 --> 00:36:42,670 Now, as before, we want to find out what the value of X has to 460 00:36:42,670 --> 00:36:45,514 be to make F of X equal to 0. 461 00:36:46,170 --> 00:36:51,495 And we can see that if we put X equals 2 in here, we will 462 00:36:51,495 --> 00:36:55,755 actually get zero in this bracket. So X equals 2 is one 463 00:36:55,755 --> 00:36:59,305 route and actually there are three of those because it's 464 00:36:59,305 --> 00:37:03,565 cubed. So this has a repeated route, three of them, so three 465 00:37:03,565 --> 00:37:06,760 repeated roots. So three roots for X equals 2. 466 00:37:07,400 --> 00:37:12,020 But also we can see that if X is minus four, then this bracket 467 00:37:12,020 --> 00:37:16,970 will be equal to 0, so X equals minus four is a second route and 468 00:37:16,970 --> 00:37:20,930 there are four of them there. So there are four repeated roots 469 00:37:20,930 --> 00:37:24,828 there. Now what we say is that. 470 00:37:25,340 --> 00:37:26,618 If a route. 471 00:37:27,140 --> 00:37:31,440 Has an odd number of repeated roots. For instance, this 472 00:37:31,440 --> 00:37:35,740 one's got 3 routes, then it has an odd multiplicity. 473 00:37:36,810 --> 00:37:41,144 If a root, for instance X equals minus four, has an 474 00:37:41,144 --> 00:37:44,690 even number of roots, then it hasn't even multiplicity. 475 00:37:45,840 --> 00:37:49,520 Why are we interested in multiplicity at all? Well, for 476 00:37:49,520 --> 00:37:53,400 this reason. If the multiplicities odd, then that 477 00:37:53,400 --> 00:37:57,910 means the graph actually crosses the X axis at the roots. 478 00:37:58,450 --> 00:38:02,949 If the multiplicity is even, then it means that the graph 479 00:38:02,949 --> 00:38:07,857 just touches the X axis and this is very useful tool when 480 00:38:07,857 --> 00:38:11,400 sketching functions. For example, if we had. 481 00:38:12,100 --> 00:38:15,108 F of X equals. 482 00:38:16,120 --> 00:38:19,925 X minus 3 squared multiplied 483 00:38:19,925 --> 00:38:25,160 by. X plus one to the power 5. 484 00:38:25,440 --> 00:38:31,576 By X minus 2 cubed times by X +2 to the power 4. 485 00:38:32,370 --> 00:38:35,700 Now, first sight, this might look very complicated, but in 486 00:38:35,700 --> 00:38:39,363 fact we can identify the four roots straight away. The first 487 00:38:39,363 --> 00:38:43,692 One X equals 3 will make this brackets equal to 0, so that's 488 00:38:43,692 --> 00:38:48,021 our first roots. X equals 3 and we can see straight away there 489 00:38:48,021 --> 00:38:54,220 are. Two repeated roots there, which means that this has an 490 00:38:54,220 --> 00:38:57,330 even multiplicity. Some 491 00:38:57,330 --> 00:39:02,579 even multiplicity. Our second route. 492 00:39:03,130 --> 00:39:07,232 He is going to be X equals minus one because that will make this 493 00:39:07,232 --> 00:39:09,283 bracket equal to 0. So X equals 494 00:39:09,283 --> 00:39:14,248 minus one. I'm not actually five of those repeated root cause. 495 00:39:14,248 --> 00:39:19,876 The power of the bracket is 5, which means that this is an odd 496 00:39:19,876 --> 00:39:26,000 multiplicity. We look at the next One X is 2 will give us a 497 00:39:26,000 --> 00:39:31,720 zero in this bracket. So X equals 2 is a root there and 498 00:39:31,720 --> 00:39:36,560 there are three of them, which means is an odd multiplicity. 499 00:39:37,490 --> 00:39:44,630 And finally, our last bracket X +2 here. If we put X 500 00:39:44,630 --> 00:39:47,605 equals minus two in there. 501 00:39:48,180 --> 00:39:52,600 That will give us zero in this bracket and there are four of 502 00:39:52,600 --> 00:39:55,320 those repeated roots, which means it isn't even. 503 00:39:55,960 --> 00:40:02,370 Multiplicity. So as I said before, we can use this to help 504 00:40:02,370 --> 00:40:07,050 us plot a graph or helper sketch the graph. So for instance, and 505 00:40:07,050 --> 00:40:11,010 even multiplicity here would mean that at the roots X equals 506 00:40:11,010 --> 00:40:15,330 3. This curve would just touch the access at the roots, X 507 00:40:15,330 --> 00:40:20,010 equals minus one. It would cross the Axis and at the River X 508 00:40:20,010 --> 00:40:24,330 equals 2 is an odd multiplicity, so it will cross the axis. 509 00:40:24,880 --> 00:40:29,131 And at the roots for X equals minus two is an even more 510 00:40:29,131 --> 00:40:33,055 simplicity, so it would just touch the axis. So let's have a 511 00:40:33,055 --> 00:40:36,652 look at a function where we can actually sketch this now. 512 00:40:36,830 --> 00:40:39,758 So if we look at the function F 513 00:40:39,758 --> 00:40:47,600 of X. Equals X minus two all squared multiplied by X plus 514 00:40:47,600 --> 00:40:51,495 one. We can identify the two 515 00:40:51,495 --> 00:40:57,639 routes immediately. X equals 2 will make this bracket zero and 516 00:40:57,639 --> 00:41:01,216 we can see that this hasn't even 517 00:41:01,216 --> 00:41:05,700 multiplicity. So even multiplicity. 518 00:41:06,820 --> 00:41:13,504 And X equals negative one would make this make this bracket 0. 519 00:41:13,710 --> 00:41:17,200 So this will be an 520 00:41:17,200 --> 00:41:21,586 odd multiplicity. Because the power of this bracket 521 00:41:21,586 --> 00:41:22,804 is actually one. 522 00:41:24,330 --> 00:41:28,206 Now we talked before about the general shape of a cubic, which 523 00:41:28,206 --> 00:41:29,821 we knew would look something 524 00:41:29,821 --> 00:41:34,499 like this. So how can we combine this information and this 525 00:41:34,499 --> 00:41:38,109 information to help us sketch the graph of this function? 526 00:41:38,109 --> 00:41:40,275 Well, let's first of all draw 527 00:41:40,275 --> 00:41:42,940 axes. Graph of X here. 528 00:41:43,480 --> 00:41:48,466 Convert sleep X going horizontally. Now we know the 529 00:41:48,466 --> 00:41:55,114 two routes ones X equals 2 to the X equals negative one. 530 00:41:55,930 --> 00:41:59,740 Now about the multiplicity. This tells us that the even 531 00:41:59,740 --> 00:42:04,312 multiplicity is that X equals 2, so that means it just touches 532 00:42:04,312 --> 00:42:06,217 the curve X equals 2. 533 00:42:07,200 --> 00:42:10,512 But it crosses the curve X equals minus one. 534 00:42:11,280 --> 00:42:14,928 I should add here that because we've got a positive value for 535 00:42:14,928 --> 00:42:18,272 our coefficients of X cubed, if we multiply this out, we're 536 00:42:18,272 --> 00:42:20,096 definitely going to get curve of 537 00:42:20,096 --> 00:42:23,269 this shape. So we can see it's going to cross through 538 00:42:23,269 --> 00:42:23,711 minus one. 539 00:42:26,130 --> 00:42:27,490 It's going to come down. 540 00:42:28,000 --> 00:42:31,300 And it's just going to touch the access at minus 2. 541 00:42:32,330 --> 00:42:38,770 So this is a sketch of the curve F of X equals X minus 542 00:42:38,770 --> 00:42:42,450 two all squared multiplied by X plus one. 543 00:42:43,680 --> 00:42:46,850 We do not know precisely where this point is, but 544 00:42:46,850 --> 00:42:50,020 we do know that it lies somewhere in this region.