WEBVTT 00:00:05.190 --> 00:00:15.350 Okay. If you use the square root principle, you get y is plus or 00:00:15.830 --> 00:00:32.811 minus the square root of negative x squared plus nine, which is this and this. These 00:00:32.811 --> 00:00:36.450 two are functions. 00:00:37.590 --> 00:00:46.330 This is not a function. Is this a quadratic function? 00:00:46.660 --> 00:00:54.920 Is the square root of x a polynomial? That same thing is x to what power? One half 00:00:54.920 --> 00:00:57.740 power. Is that a polynomial? 00:01:01.000 --> 00:01:07.951 The exponents of the variables have to be whole numbers for it to be a polynomial. 00:01:07.951 --> 00:01:12.020 These square-root functions are not polynomials. 00:01:13.620 --> 00:01:19.820 So therefore these are functions, but they are not polynomial functions. 00:01:24.140 --> 00:01:28.846 So this is not a function. We could break it up into two separate functions, but it 00:01:28.846 --> 00:01:29.640 is a relation. 00:01:29.640 --> 00:01:41.334 Okay. And it has lots of solutions. Y squared negative x squared plus nine. Let's 00:01:41.334 --> 00:01:46.820 and we often actually write like this. 00:01:46.820 --> 00:01:57.939 X squared plus y squared equals nine. We'll come back to that. What is this gonna 00:01:57.939 --> 00:02:01.920 look like? Let's take a look. 00:02:02.230 --> 00:02:10.853 It turns out x squared plus y squared equals nine is a circle. And what's the radius 00:02:10.853 --> 00:02:12.290 of the circle? 00:02:17.210 --> 00:02:29.205 Three. So why in the world does this give us a circle? I'll kind of give you an 00:02:29.205 --> 00:02:36.190 idea. If I take the square root of both sides. 00:02:36.450 --> 00:02:40.717 Does this thing look familiar from last year or the year before? 00:02:45.690 --> 00:02:54.265 What if I ask you to find the distance between zero, zero, and x, y? What's the 00:02:54.265 --> 00:02:56.110 distance formula? 00:03:03.990 --> 00:03:11.775 If we have two ordered pairs, this is the change in x. This is the change in y. 00:03:11.775 --> 00:03:18.870 Okay. So the distance between them is basically the Pythagorean Theorem. 00:03:18.870 --> 00:03:26.695 The difference of the x's, the difference of the y's squared plus squared is the 00:03:26.695 --> 00:03:34.520 Pythagorean Theorem, which gives us this. Anyways, long story short, this is the 00:03:34.520 --> 00:03:42.541 distance formula between these two ordered pairs, zero, zero, and the ordered pair 00:03:42.541 --> 00:03:43.541 x, y. 00:03:47.570 --> 00:03:54.701 So everything is a solution to this. If it's the distance from zero, zero, the 00:03:54.701 --> 00:04:02.107 distance is equal to three. In fact, that is what a circle is. The set of ordered 00:04:02.107 --> 00:04:07.410 pairs is the same distance away. In this case, zero, zero. 00:04:07.410 --> 00:04:19.287 Anyways, long story short, this is a circle with radius the square root of nine, or 00:04:19.287 --> 00:04:25.870 in other words, three, centered at zero, zero. 00:04:31.950 --> 00:04:44.009 Let me give you a formula for a circle in general. It is (x minus h) squared plus (y 00:04:44.009 --> 00:04:48.890 minus k) squared equals r squared. 00:04:48.890 --> 00:05:08.850 So this is a circle with radius r centered at h, k. 00:05:09.260 --> 00:05:17.920 So, write the equation. 00:05:23.140 --> 00:05:41.920 So, this is a circle centered at three, negative two with a radius of five. 00:05:54.790 --> 00:06:04.350 Where does the three go? Five squared, which is twenty-five. That's it. 00:06:06.510 --> 00:06:13.472 Okay, really, it's just a, this is the transformation, the x direction, so to the 00:06:13.472 --> 00:06:20.606 right by three, up by two, or up by, sorry, up by negative two, which means down by 00:06:20.606 --> 00:06:21.120 two. 00:06:21.320 --> 00:06:34.120 Let's try another one. 00:06:34.120 --> 00:06:42.820 What is the center and radius? 00:06:43.140 --> 00:06:48.840 So, this is x minus what squared? Zero. 00:06:53.540 --> 00:07:10.288 And so, the center is zero, negative five, and the radius is the square root of 00:07:10.288 --> 00:07:11.560 seven. 00:07:11.910 --> 00:07:19.010 Okay, not bad. It's always the opposite of this number. 00:07:22.690 --> 00:07:32.516 Can I write down this formula? Yeah. It's x minus the x coordinate, y minus the y 00:07:32.516 --> 00:07:33.850 coordinate. 00:07:34.730 --> 00:07:45.070 So, it's always just the opposite of that number. So, the opposite is negative five. 00:07:45.400 --> 00:07:54.800 Okay, find the center and radius. 00:08:10.860 --> 00:08:19.729 So, we need to get it in this form. x plus or minus a number, y plus or minus a 00:08:19.729 --> 00:08:22.760 number, equals some number. 00:08:24.940 --> 00:08:26.807 How are we going to do that? 00:08:30.800 --> 00:08:34.840 First, let's rearrange it. 00:08:35.240 --> 00:08:41.360 So, I just grouped the x's together, the y's together, and brought the constant to 00:08:41.360 --> 00:08:42.480 the other side. 00:08:49.840 --> 00:09:00.450 I need to take this and factor it to a perfect square binomial. What do we call that 00:09:00.450 --> 00:09:01.460 process? 00:09:07.320 --> 00:09:14.963 Completing the square. So, we just got to add the correct number to make the same 00:09:14.963 --> 00:09:19.020 binomial, and so we can write it like that. 00:09:19.680 --> 00:09:22.280 So, how do I complete this square here? 00:09:26.000 --> 00:09:31.320 We take the negative 4 and divide it. Divide by 2 and square. 00:09:36.820 --> 00:09:38.590 And we add that to both sides. 00:09:38.810 --> 00:09:45.870 And then we do the same thing here. Square. 10. Divide by 2. Square. 00:09:57.390 --> 00:09:58.790 That's 25. 00:10:00.690 --> 00:10:11.070 So, the center is the opposite of that number. 00:10:19.070 --> 00:10:25.270 Square root of 26. Just barely larger and higher. 00:10:25.660 --> 00:10:28.300 Okay, let's try another one. 00:10:49.880 --> 00:10:51.280 Okay. 00:10:51.600 --> 00:10:55.560 So, what do we do here? We rearrange everything. 00:10:56.860 --> 00:11:02.240 This is called general form. We want to get into standard form. 00:11:19.200 --> 00:11:23.133 So, we're going to complete the square. So, what do we add? 00:11:25.480 --> 00:11:29.220 6 divided by 2. 9. 00:11:38.500 --> 00:11:40.700 Negative 12 divided by 2. 00:12:02.260 --> 00:12:09.300 So, we're going to do the same thing here. 00:12:10.920 --> 00:12:14.140 Negative 3, 6. 00:12:15.880 --> 00:12:16.880 And the radius. 00:12:20.040 --> 00:12:24.620 Okay, what questions do we have on this? 00:12:26.020 --> 00:12:29.106 We just have to complete the square twice to figure it out. It's not too 00:12:29.106 --> 00:12:29.790 complicated.