WEBVTT 00:00:05.110 --> 00:00:16.770 Is this a quadratic function? Yes. How can we tell? We multiply this out. The 00:00:16.770 --> 00:00:20.492 x times the x becomes the x squared. Definitely quadratic. You multiply any two 00:00:20.492 --> 00:00:21.010 linear, you 00:00:21.010 --> 00:00:30.651 get a quadratic. You multiply three linear, what do you get? So our goal is to graph 00:00:30.651 --> 00:00:31.110 this 00:00:31.110 --> 00:00:39.386 function just like we were graphing the quadratic functions. This is non-standard 00:00:39.386 --> 00:00:40.510 form. Is it 00:00:40.510 --> 00:00:51.790 in vertex form? No. Uh-oh. So how are we going to get it in one of those forms? We 00:00:51.790 --> 00:00:52.890 just got 00:00:52.890 --> 00:01:15.990 Multiply it out: x squared plus three x minus x minus three. 00:01:16.350 --> 00:01:22.350 Now it is in standard form. Standard form tells us two things. What are they? 00:01:22.350 --> 00:01:35.090 The y-intercept. We definitely know the y-intercept. And whether it 00:01:35.090 --> 00:01:45.365 opens up or opens down, which one is this? Open up because what's positive? Leading 00:01:45.365 --> 00:01:46.850 coefficient. 00:01:47.250 --> 00:01:49.170 Okay, so if we want to find 00:01:49.810 --> 00:01:57.960 vertex form, how are we going to do that? You could complete the square. The better 00:01:57.960 --> 00:02:00.710 way is to know the vertex is 00:02:00.710 --> 00:02:12.965 always at negative b over 2a comma f of negative b over 2a. Negative b over 2a is, 00:02:12.965 --> 00:02:17.150 what is b? Is 2. And a is 1. 00:02:19.570 --> 00:02:30.590 So that is negative one. That is the x-coordinate of the vertex. 00:02:32.450 --> 00:02:36.170 And how do we find the y coordinate? Plug it in to the function. 00:02:36.460 --> 00:02:42.903 Oh, no, that's right. I'm sorry. I got confused. Negative 4. We agree? Yes. Okay, 00:02:42.903 --> 00:02:43.380 sorry. 00:02:45.700 --> 00:02:49.900 So that is the y coordinate of the vertex. 00:02:52.840 --> 00:03:04.520 So what is our vertex form? The leading coefficient is one. 00:03:05.740 --> 00:03:16.440 Times x minus this number, which is x plus 1 squared minus 4. 00:03:20.060 --> 00:03:25.969 So the vertex (negative one, negative four) is right there. To graph it, follow the 00:03:25.969 --> 00:03:27.820 pattern: over one, up one; 00:03:28.160 --> 00:03:31.200 over two, up four. 00:03:31.200 --> 00:03:33.380 I'm just doing that because it's on the wrong side of the pattern. 00:03:33.610 --> 00:03:42.510 Over 1 up 1, over 2 up 1, 2, 3, 4. 1, 2, 3, up 9. Up here. 00:03:44.270 --> 00:03:47.030 That's going to be the right hand side. And how do we get the left hand side? 00:03:47.030 --> 00:03:53.830 On the other side. According to the axis of symmetry, which goes right through the 00:03:53.830 --> 00:03:54.410 vertex. 00:03:57.430 --> 00:04:00.450 And then we draw a nice curve. 00:04:17.030 --> 00:04:18.630 What are the x-intercepts? 00:04:20.810 --> 00:04:23.590 Negative three and one. 00:04:24.050 --> 00:04:27.390 Going back to the original quadratic. 00:04:30.870 --> 00:04:33.810 How can we look at that and tell the x-intercepts? 00:04:37.250 --> 00:04:45.050 Uh, we could use a zero product property. 00:04:47.450 --> 00:04:52.886 But it is simply the opposite of that number and the opposite of that number. Why is 00:04:52.886 --> 00:04:53.210 that? 00:04:56.670 --> 00:04:59.910 Yeah, right. We plug in the opposite of negative 1, which is 1. 00:05:01.450 --> 00:05:04.070 And you get 0 times whatever. It doesn't matter. It's 0. 00:05:07.050 --> 00:05:15.124 And the opposite of 3. So we plug in negative 3 plus 3. We get 0 times whatever. 00:05:15.124 --> 00:05:15.830 It's 0. 00:05:16.930 --> 00:05:22.630 Okay. So this is a special form. 00:05:25.330 --> 00:05:31.910 And it tells us something right away. What form is this? Oh, can anyone take this? 00:05:31.910 --> 00:05:36.470 Factored form, because the polynomial is factored. 00:05:38.030 --> 00:05:39.970 Okay. So this is called factored form. 00:05:45.770 --> 00:05:55.910 Okay. And what does it tell you? Those are the x-intercepts. 00:05:59.230 --> 00:06:03.490 Also known as zeros. I'll define that for you in a minute. 00:06:09.990 --> 00:06:15.190 Okay. If we have the x-intercepts 00:06:20.230 --> 00:06:25.908 without doing this negative b over 2a, how can we use the x-intercepts to figure out 00:06:25.908 --> 00:06:26.990 the x-coordinate 00:06:26.990 --> 00:06:28.390 of the vertex? 00:06:28.610 --> 00:06:30.510 We find the mean of these two? 00:06:30.930 --> 00:06:36.761 We find the mean, the average of the two, exactly halfway. Okay. Why is it exactly 00:06:36.761 --> 00:06:37.330 halfway? 00:06:41.390 --> 00:06:46.070 Because they are split about the axis of symmetry. 00:06:46.070 --> 00:06:52.750 Any two points on this parabola on the same horizontal line are symmetric. 00:06:52.750 --> 00:07:00.470 Symmetric ordered pairs. So if you find two such points, 00:07:03.230 --> 00:07:07.770 especially when they, like let's say they landed right here and here or something, 00:07:08.470 --> 00:07:14.592 where we couldn't know the zeros, we'll just go halfway. Exactly as the axis of 00:07:14.592 --> 00:07:15.290 symmetry, 00:07:15.750 --> 00:07:19.430 which is the x-coordinate of the vertex. So... 00:07:22.610 --> 00:07:24.810 So that's an alternative way, 00:07:24.960 --> 00:07:31.180 To find this value. And once we have that value, we just plug it into this function 00:07:31.180 --> 00:07:36.600 or this function to figure it out, the y-coordinate. 00:07:38.980 --> 00:07:42.180 So, right, if we plug in what's halfway? Negative one. 00:07:43.200 --> 00:07:48.760 Negative one minus one times negative one plus three, which is 00:07:53.840 --> 00:07:58.540 negative two times two, which is our negative four, our y-coordinate of the vertex. 00:07:58.540 --> 00:08:06.860 Okay, so that is factored form. So factored form 00:08:17.420 --> 00:08:23.573 Factored form is f of x equals the leading coefficient a, the same a as in vertex 00:08:23.573 --> 00:08:26.080 and standard forms, times x minus 00:08:26.960 --> 00:08:34.360 z one, times x minus z two, where 00:08:36.960 --> 00:08:47.360 z one and z two are the zeros, the x-coordinates of the intercepts. 00:08:51.620 --> 00:08:54.720 Okay, let me define the word zero for it. 00:08:54.910 --> 00:08:55.850 a zero 00:09:00.330 --> 00:09:01.550 of a function 00:09:05.130 --> 00:09:08.230 is an x value 00:09:11.690 --> 00:09:13.790 that makes 00:09:16.230 --> 00:09:23.070 the y-value, or function value, 00:09:31.150 --> 00:09:36.370 equal to zero, the number zero. 00:09:44.630 --> 00:09:49.630 It is the same as the x-coordinate of an x-intercept. Sometimes 00:09:54.350 --> 00:10:01.785 we could say the x-intercepts are negative three and one, or you can actually talk 00:10:01.785 --> 00:10:03.870 about the ordered pair. 00:10:13.790 --> 00:10:20.333 Typically, the x-intercept is actually an ordered pair, but we often just refer to 00:10:20.333 --> 00:10:21.530 it as a number. 00:10:25.430 --> 00:10:30.667 But these would be the zeros of the function. Zeros are numbers, x values that make 00:10:30.667 --> 00:10:31.550 the y value or 00:10:31.750 --> 00:10:32.490 the function value zero. 00:10:32.700 --> 00:10:37.380 What form is this in? Factored form. 00:10:46.700 --> 00:10:49.260 Okay, so what do we instantly know? 00:10:49.360 --> 00:10:51.360 What do we know from this form? 00:10:52.960 --> 00:10:53.520 The zeros. 00:10:53.820 --> 00:10:54.480 Zeros. 00:10:55.980 --> 00:11:00.240 Okay, also known as the x-intercepts. 00:11:03.620 --> 00:11:09.634 The zeros are the opposite of that number and the opposite of that number, two, 00:11:09.634 --> 00:11:10.700 negative four. 00:11:16.520 --> 00:11:18.200 Two, negative four. 00:11:20.360 --> 00:11:21.600 Okay. 00:11:23.160 --> 00:11:28.540 And then the vertex, the x-coordinate of the vertex, 00:11:34.240 --> 00:11:38.160 of the vertex is... 00:11:42.520 --> 00:11:50.403 Right, it's z one plus z two divided by two, which is two plus negative four divided 00:11:50.403 --> 00:11:51.060 by two. 00:11:51.860 --> 00:11:57.940 Here that is negative two over two, which is negative one. 00:12:01.140 --> 00:12:04.260 So we know this is the axis of symmetry. 00:12:11.320 --> 00:12:16.980 And then how would we find the y-coordinate of the vertex? 00:12:17.640 --> 00:12:22.880 Plug it in. We don't even need to multiply this out. 00:12:24.100 --> 00:12:32.307 So f of negative one is two times (negative one minus two) times (negative one plus 00:12:32.307 --> 00:12:32.900 four), 00:12:32.900 --> 00:12:39.460 which is two times negative three times three, which is 00:12:44.040 --> 00:12:49.860 negative eighteen. Uh-oh. So the only problem with this is way off my graph. 00:12:52.280 --> 00:13:00.800 Way down here at negative eighteen. Sorry, negative one comma negative eighteen. 00:13:07.960 --> 00:13:14.200 It's going to be a little harder to graph because I don't have a graph paper right 00:13:14.200 --> 00:13:14.580 here. 00:13:19.180 --> 00:13:25.655 But we would follow our pattern over... We're not going to go over one up one. Where 00:13:25.655 --> 00:13:27.120 are we going to go? 00:13:28.340 --> 00:13:38.760 Over one, up two. The parent offsets are (0,0), (1,1), (2,4), and (3,9). 00:13:42.260 --> 00:13:49.700 Our new pattern, instead of x and x squared, uses x and two x squared. 00:13:51.000 --> 00:13:56.040 Multiply those vertical offsets by two. 00:14:01.140 --> 00:14:04.640 Two, eight, eighteen. 00:14:07.680 --> 00:14:10.040 So I go over one up two. 00:14:13.860 --> 00:14:18.140 Over three up... Sorry, over two up eight. 00:14:20.060 --> 00:14:23.540 It's going to be right here at ten. I didn't draw this very well. 00:14:25.000 --> 00:14:28.880 Negative ten. And over three, up eighteen reaches the zeros right there. 00:14:30.460 --> 00:14:32.200 And then we got some symmetric points. 00:14:37.360 --> 00:14:39.020 We get a nice smooth curve. 00:14:52.860 --> 00:14:53.680 Okay. 00:14:53.680 --> 00:14:57.580 Okay. So what is the... 00:15:01.420 --> 00:15:02.800 What is the vertex form? 00:15:09.640 --> 00:15:11.040 Two... 00:15:11.040 --> 00:15:17.280 What was the vertex? 00:15:20.120 --> 00:15:23.360 Was it negative one... 00:15:25.240 --> 00:15:25.740 Negative eighteen. 00:15:26.460 --> 00:15:35.160 It is two times (x plus one) squared, minus eighteen. 00:15:37.320 --> 00:15:42.720 Okay. And then how would we get standard forms? 00:15:45.140 --> 00:15:49.523 We could either multiply this out, multiply it by two, and then subtract 18, or it's 00:15:49.523 --> 00:15:49.940 probably 00:15:49.940 --> 00:16:01.420 easier right here: two times (x squared plus four x minus two x minus eight). 00:16:01.820 --> 00:16:05.440 Which form should we... Let's go with the factored form first. 00:16:15.720 --> 00:16:22.720 Factored form is the leading coefficient times x minus one zero, times 00:16:22.720 --> 00:16:28.640 x minus the other zero. What are the zeros? 00:16:30.680 --> 00:16:32.280 Negative one, negative three. 00:16:32.280 --> 00:16:39.140 Negative one and negative three. So we have x minus negative one, or x plus one, 00:16:39.560 --> 00:16:43.338 and x minus negative three, or x plus three. How do we determine the leading 00:16:43.338 --> 00:16:44.680 coefficient from the graph? 00:16:49.900 --> 00:16:53.040 You start at the vertex. We go one to the right. 00:16:55.140 --> 00:17:00.740 And since we go down two, we know the leading coefficient is negative two. 00:17:04.860 --> 00:17:08.760 We can go right to the vertex form. 00:17:15.160 --> 00:17:21.080 The leading coefficient is negative two, times 00:17:24.440 --> 00:17:30.080 (x plus two) squared, plus two. Lots of twos. 00:17:33.480 --> 00:17:39.080 And then standard form. We know it's going to be 00:17:40.860 --> 00:17:44.420 a coefficient times x squared, plus or minus a coefficient times x. 00:17:45.160 --> 00:17:47.640 We do know the y-intercept, which is what? Negative six. 00:17:50.320 --> 00:17:53.720 But you need to multiply this out. So we got x squared 00:17:57.440 --> 00:18:06.320 negative two times x squared plus three x plus x plus three. 00:18:09.460 --> 00:18:14.260 Three x plus x is four x. Distributing negative two gives negative eight x and 00:18:14.260 --> 00:18:15.060 negative six. 00:18:19.380 --> 00:18:20.180 And that is standard form. 00:18:21.150 --> 00:18:23.190 What two things do we know? 00:18:29.530 --> 00:18:35.133 It opens upward because the leading coefficient is positive, and its y-intercept is 00:18:35.133 --> 00:18:36.010 negative one. 00:18:40.590 --> 00:18:46.910 How can we get this into factored form? One option is to factor it. 00:18:50.470 --> 00:18:51.450 Does it factor? 00:18:54.150 --> 00:18:59.070 The integer factors are one and negative one; those do not add to two. 00:19:00.550 --> 00:19:02.590 So how are we going to factor it? 00:19:03.790 --> 00:19:06.870 Find the zeros. We need to determine the zeros. 00:19:11.150 --> 00:19:13.030 How do we figure out the zeros 00:19:18.730 --> 00:19:21.710 Okay, we could do that, but I have a feeling 00:19:26.770 --> 00:19:31.390 They will not be obvious integers. This is just a rough sketch. 00:19:33.150 --> 00:19:35.490 Okay to find any zeros of a function 00:19:37.230 --> 00:19:40.210 You set the function equal to zero 00:19:43.030 --> 00:19:48.010 So we go zero equals x squared plus 2x minus 1 00:19:54.410 --> 00:19:57.670 And now we have a quadratic equation in one variable 00:20:01.450 --> 00:20:04.290 It cannot be solved by integer factoring. What is the next option? 00:20:05.430 --> 00:20:06.510 Use the quadratic formula. 00:20:07.210 --> 00:20:10.570 We sing it all together. Yes, we ready. Yes 00:20:13.030 --> 00:20:13.990 Negative b, 00:20:16.030 --> 00:20:20.750 the square root of b squared minus four a c. 00:20:20.750 --> 00:20:31.990 Negative two plus or minus the square root of two squared minus four times one times 00:20:32.910 --> 00:20:34.390 negative 1 00:20:35.430 --> 00:20:36.970 All over two. 00:20:52.890 --> 00:20:53.930 And 00:20:53.930 --> 00:20:58.770 The square root of eight is two square roots of two. 00:21:05.290 --> 00:21:09.770 Either factor out a two, or 00:21:10.770 --> 00:21:16.090 divide negative two by two and divide plus or minus two square roots of two by two. 00:21:27.250 --> 00:21:31.490 Those are the zeros, the x-coordinates of the intercepts. 00:21:41.110 --> 00:21:41.950 Oh 00:21:41.950 --> 00:21:43.730 So how do we put it in factored form? 00:21:44.770 --> 00:21:47.650 Well, do we know the leading coefficient which is 00:21:48.290 --> 00:21:48.830 one 00:21:49.810 --> 00:21:52.310 These are two numbers: negative one plus the square root of two 00:21:53.850 --> 00:21:54.690 and 00:21:54.690 --> 00:21:56.830 and negative one minus the square root of two. 00:21:58.370 --> 00:22:00.330 Factored form is 00:22:01.130 --> 00:22:03.530 One times x minus 00:22:04.450 --> 00:22:07.570 (negative one plus the square root of two), 00:22:08.570 --> 00:22:13.770 times x minus (negative one minus the square root of two). 00:22:17.250 --> 00:22:19.250 Distribute the signs to simplify. 00:22:20.370 --> 00:22:22.590 Get x 00:22:23.570 --> 00:22:26.070 plus one minus the square root of two; 00:22:27.250 --> 00:22:29.370 X plus 1 00:22:30.370 --> 00:22:31.070 Plus 00:22:33.130 --> 00:22:34.230 the square root of two. 00:22:44.370 --> 00:22:44.850 So 00:22:45.330 --> 00:22:48.410 So where the heck are those two numbers on the graph 00:22:51.630 --> 00:22:53.090 We'll get there in a second 00:22:55.230 --> 00:22:57.510 What is the square root of two approximately? 00:23:00.610 --> 00:23:01.630 One point something 00:23:03.070 --> 00:23:04.930 Let us use about 00:23:05.770 --> 00:23:07.310 One point four ish 00:23:07.310 --> 00:23:11.730 so negative one plus one point four and 00:23:12.290 --> 00:23:15.130 negative one minus one point four is 00:23:16.350 --> 00:23:20.530 Point four and negative two point four 00:23:23.490 --> 00:23:29.150 So point four and negative two point four 00:23:37.150 --> 00:23:40.670 Okay, the axis of symmetry is halfway 00:23:43.070 --> 00:23:46.290 Okay, which is the average of these two numbers 00:23:48.950 --> 00:23:55.550 Negative one plus the square root of two, plus negative one minus 00:23:56.330 --> 00:23:59.270 the square root of two, all over two. 00:24:00.910 --> 00:24:04.370 The two radical terms cancel to zero. 00:24:05.410 --> 00:24:07.610 So we get negative 2 over 2 00:24:08.390 --> 00:24:09.410 negative 1 00:24:10.830 --> 00:24:13.810 Okay, this is our axis of symmetry 00:24:16.070 --> 00:24:18.370 To get the y-coordinate of the vertex, 00:24:19.790 --> 00:24:21.410 evaluate f at negative one. 00:24:23.130 --> 00:24:29.390 F of negative one is (negative one) squared plus two times 00:24:29.390 --> 00:24:30.150 negative one, 00:24:31.030 --> 00:24:31.870 minus one, 00:24:33.030 --> 00:24:35.030 which is 00:24:39.110 --> 00:24:39.950 negative two. 00:24:42.580 --> 00:24:44.580 Follow our pattern 00:24:58.530 --> 00:25:00.550 Let's put it in vertex form 00:25:03.370 --> 00:25:07.330 The leading coefficient is one. What was the 00:25:08.930 --> 00:25:15.750 x-coordinate of the vertex? Negative one, so use (x plus one) squared, 00:25:17.270 --> 00:25:19.890 minus two. 00:25:24.810 --> 00:25:25.290 Okay 00:25:26.670 --> 00:25:27.150 So 00:25:29.650 --> 00:25:30.610 Almost every 00:25:33.190 --> 00:25:36.911 quadratic we have discussed can be written in all three forms. Each form tells us 00:25:36.911 --> 00:25:37.830 something different. 00:25:40.410 --> 00:25:42.730 We can convert to all the forms 00:25:44.910 --> 00:25:46.170 The only time 00:25:47.690 --> 00:25:49.810 We're not going to be able to write in factored form 00:26:03.710 --> 00:26:04.210 Okay 00:26:05.850 --> 00:26:07.070 Okay is this 00:26:10.350 --> 00:26:12.570 Why can we not write this in factored form 00:26:14.790 --> 00:26:17.090 If there are no x intercepts 00:26:17.770 --> 00:26:19.830 It cannot be factored into real linear factors. 00:26:21.170 --> 00:26:23.290 We know this equation, right? 00:26:24.710 --> 00:26:26.690 Or this function would be 00:26:27.690 --> 00:26:34.410 (x minus three) squared plus two. 00:26:37.330 --> 00:26:38.870 So how can you tell 00:26:48.390 --> 00:26:50.710 Yeah, if you have the graph it's obvious 00:26:51.530 --> 00:26:52.530 if the 00:26:53.290 --> 00:26:55.850 It opens up and above the x-axis 00:26:55.850 --> 00:27:00.402 It has no x-intercepts. Likewise, a downward-opening parabola with a negative 00:27:00.402 --> 00:27:01.230 maximum has no 00:27:01.890 --> 00:27:03.210 x-intercepts. 00:27:03.210 --> 00:27:08.150 What if we only have standard form? 00:27:09.250 --> 00:27:13.250 This is x squared minus 6x 00:27:14.130 --> 00:27:14.770 plus 00:27:15.150 --> 00:27:15.750 11 00:27:15.750 --> 00:27:18.270 If we substitute into the 00:27:20.290 --> 00:27:21.270 Quadratic formula 00:27:21.270 --> 00:27:27.030 You're gonna get negative negative 6 plus or minus 00:27:29.910 --> 00:27:36.310 36 minus 4 times 1 times 11 00:27:36.310 --> 00:27:42.750 Thirty-six minus forty-four. What is the radicand? 00:27:43.450 --> 00:27:45.470 Sorry, what's the discriminant? 00:27:46.410 --> 00:27:46.670 negative 00:27:46.670 --> 00:27:51.310 Negative eight, which means there are no real 00:27:52.230 --> 00:27:58.050 solutions to f of x equals zero: no real zeros and no x-intercepts. 00:28:00.410 --> 00:28:03.330 It cannot be written as a product of real linear factors.