WEBVTT 00:00:05.110 --> 00:00:11.956 I got two different equations up here. What do we call the one on the left? A linear 00:00:11.956 --> 00:00:12.690 equation. 00:00:13.330 --> 00:00:20.750 A linear equation. Why is it called a linear equation? Because the leading term 00:00:23.810 --> 00:00:33.890 has degree one. We have two different polynomials here, one on each side. 00:00:35.350 --> 00:00:41.325 All equations have an expression on the left and an expression on the right. Both of 00:00:41.325 --> 00:00:42.890 these are polynomials. 00:00:44.750 --> 00:00:54.134 This is what type of polynomial? Linear. It happens to be a binomial. This is a 00:00:54.134 --> 00:01:03.043 what? Constant. The degree is one, the degree is zero. So the larger degree 00:01:03.043 --> 00:01:08.270 determines what we call the entire equation. 00:01:08.270 --> 00:01:18.146 So this is called linear equation. We could call it a linear polynomial equation, 00:01:18.146 --> 00:01:27.413 but we just linear implies it's a polynomial. Why is this called a quadratic 00:01:27.413 --> 00:01:28.510 equation? 00:01:32.090 --> 00:01:38.829 The polynomial on the left is what type of polynomial? Quadratic. The polynomial on 00:01:38.829 --> 00:01:45.486 the right is what type? Linear. The larger of the two is the quadratic. So we call 00:01:45.486 --> 00:01:48.490 the whole thing a quadratic equation. 00:01:55.690 --> 00:02:04.636 Are the solutions to these two equations, are they numbers or variables? Numbers. 00:02:04.636 --> 00:02:10.490 Why are they just numbers? There's only one variable. 00:02:10.490 --> 00:02:17.764 So this is a linear equation in one variable. This is a quadratic equation in one 00:02:17.764 --> 00:02:25.037 variable. The solutions are numbers. And how would we solve this one? Isolate the 00:02:25.037 --> 00:02:31.952 variable. We can't do that here. But how would we solve this? We've got three 00:02:31.952 --> 00:02:32.670 options. 00:02:32.670 --> 00:02:46.186 Quadratic formula, factoring, and completing the square. Now I've got two more 00:02:46.186 --> 00:02:57.970 equations. Are the solutions to these two numbers? Are they numbers? 00:02:58.970 --> 00:03:01.230 No, why not? 00:03:01.650 --> 00:03:07.470 So the solutions are ordered pairs. 00:03:08.470 --> 00:03:15.630 What do we call this type of equation? 00:03:22.890 --> 00:03:33.250 This is what type of polynomial? Linear. What is this type of polynomial? Linear. 00:03:35.190 --> 00:03:45.350 But because y equals some other expression of x, we call this a linear function. 00:03:54.150 --> 00:04:00.106 And this is the same exact equation except using different notation. Y and f of x 00:04:00.106 --> 00:04:06.209 mean the same thing. Why do we like this notation better? Or at least why do I like 00:04:06.209 --> 00:04:08.930 this notation? You might not like it. 00:04:12.890 --> 00:04:15.010 Or is there a reason? 00:04:17.490 --> 00:04:31.185 Let me plug in x equals 3 here. Now let's go with 5. We get y equals 3 times 5 minus 00:04:31.185 --> 00:04:33.630 2. Y equals 13. 00:04:38.930 --> 00:04:43.650 And let me plug in 5 into this equation. 00:04:51.770 --> 00:05:00.670 We get f of 5 equals 13. Why is this better than this? 00:05:02.630 --> 00:05:04.070 Does this show you what polynomial? 00:05:05.490 --> 00:05:08.790 Sure. Does this show you what we plug in to get 5? 00:05:09.630 --> 00:05:17.050 Well, not what variable, but what value of x goes with this y value? 00:05:18.050 --> 00:05:26.078 So we know this equation definitely has the solution 5, 13. It's not obvious what x 00:05:26.078 --> 00:05:33.526 value we plugged in unless you look back here and know that that was where we 00:05:33.526 --> 00:05:34.590 plugged in. 00:05:34.590 --> 00:05:40.536 So the function notation tells you what we plugged in for x to get the y value or 00:05:40.536 --> 00:05:46.555 the f of x value. It basically tells you the ordered pair. That's why we like that 00:05:46.555 --> 00:05:47.730 notation better. 00:05:47.730 --> 00:05:48.730 Does that make sense? 00:05:49.020 --> 00:05:57.876 Typically, once we got y equals, we would call this a quadratic function, which 00:05:57.876 --> 00:06:01.800 tells us the two variable equation. 00:06:04.660 --> 00:06:10.524 And this is the same equation, same function as the same solution. I'm just using 00:06:10.524 --> 00:06:11.900 different notation. 00:06:12.900 --> 00:06:22.440 How many solutions does this guy have? 00:06:25.000 --> 00:06:29.840 Uh-oh. One or two? I heard infinitely many. 00:06:33.200 --> 00:06:38.140 Oh boy. We just found one solution. What was it? 5, 13? 00:06:40.120 --> 00:06:46.500 How could I find another one? Plug in any x value. Let's go with zero. 00:06:50.380 --> 00:06:56.640 We get f of zero is negative 2, which tells us zero, negative 2 is the solution. 00:06:58.000 --> 00:07:00.900 So this linear equation has how many solutions? 00:07:02.260 --> 00:07:07.880 Infinitely many. You can plug in any value for x and get a y value. 00:07:09.860 --> 00:07:18.880 And so we don't list all the solutions. What do we do with all the solutions? 00:07:22.540 --> 00:07:25.880 Or how do I write the solutions? And we graph them. 00:07:31.420 --> 00:07:42.366 And that linear function, that linear equation, that graph are all of its, or is the 00:07:42.366 --> 00:07:44.060 solution set. 00:07:44.720 --> 00:07:49.341 Every ordered pair on that line, red line, is a solution. Every ordered pair not on 00:07:49.341 --> 00:07:50.900 that line is not a solution. 00:07:50.900 --> 00:07:53.080 This make sense? 00:07:53.900 --> 00:08:01.235 Okay. So what are we going to do with this? How are we going to write the solution 00:08:01.235 --> 00:08:02.040 set here? 00:08:02.520 --> 00:08:02.740 Graph it. 00:08:03.120 --> 00:08:05.380 We're going to graph the solution set. 00:08:07.320 --> 00:08:11.700 Uh-oh. How are we going to graph this? 00:08:14.520 --> 00:08:15.160 Y intercept. 00:08:19.040 --> 00:08:21.120 We do know where the y intercept, where's the y intercept? 00:08:21.580 --> 00:08:22.480 Negative 4. 00:08:22.480 --> 00:08:24.520 Negative 4. Plug in zero. 00:08:24.820 --> 00:08:24.980 Zero. 00:08:26.120 --> 00:08:26.480 Now what? 00:08:27.800 --> 00:08:31.860 For all of this. Plug in the number for x. We get like two points. 00:08:34.480 --> 00:08:37.080 We can plug another number in. I like this. What's another number? 00:08:37.500 --> 00:08:37.620 One. 00:08:45.160 --> 00:08:45.840 What is that? 00:08:50.060 --> 00:08:50.420 Negative 7? 00:08:51.740 --> 00:08:53.160 I was negative 8. 00:08:53.920 --> 00:08:54.840 Negative 8? I mean? 00:08:55.080 --> 00:08:55.300 Yeah. 00:08:55.580 --> 00:08:55.780 Yes. 00:08:57.080 --> 00:08:57.440 Okay. 00:08:57.640 --> 00:08:58.580 Try negative 1? 00:08:58.900 --> 00:08:59.100 Yeah. 00:08:59.880 --> 00:09:00.140 Sure. 00:09:07.760 --> 00:09:10.880 What is that? 1 plus 5? 00:09:13.220 --> 00:09:15.040 6 minus 4 is 2. Is that right? 00:09:16.680 --> 00:09:17.800 We need to do this, please. 00:09:18.680 --> 00:09:21.620 Okay, now we're there. Uh-oh. Do they fall on the line? 00:09:22.320 --> 00:09:23.280 No, it is going to curve. 00:09:24.640 --> 00:09:26.220 They definitely don't fall on the line. 00:09:27.780 --> 00:09:30.580 There's a reason we call this one a linear equation. 00:09:31.540 --> 00:09:34.840 Because all the ordered pairs fall on the line. 00:09:34.840 --> 00:09:39.080 It's definitely not a linear equation. 00:09:41.320 --> 00:09:44.600 Uh, it's some curve that's not a line. 00:09:45.080 --> 00:09:48.280 We do have a specific name for that curve. We'll tell you later. 00:09:48.860 --> 00:09:55.840 Before we graph this one, let's graph the mother of all quadratic functions. 00:09:56.430 --> 00:10:00.110 Why did I call it the mother of all quadratic functions? 00:10:01.310 --> 00:10:03.210 It's actually known as the parent function. 00:10:04.910 --> 00:10:04.930 Okay. 00:10:06.070 --> 00:10:12.328 Uh, once we get the graph of this, all quadratics are going to look very, very 00:10:12.328 --> 00:10:12.970 similar. 00:10:14.070 --> 00:10:17.070 They're just shifted and stretched a little bit. 00:10:17.810 --> 00:10:17.970 Okay. 00:10:18.410 --> 00:10:22.550 So this is the, called the parent function of all quadratic functions. 00:10:22.550 --> 00:10:29.330 And let's just, uh, just do a bunch of ordered pairs. 00:10:31.050 --> 00:10:40.830 x and f of x. Or x and y. Or f and x and x squared. 00:10:43.350 --> 00:10:45.450 Or the y coordinate, same thing. 00:10:46.570 --> 00:10:47.950 Uh, let's start with zero. 00:10:49.130 --> 00:10:49.470 Zero. 00:10:49.790 --> 00:10:50.610 What is zero squared? 00:10:51.270 --> 00:10:51.390 Zero. 00:10:53.350 --> 00:10:53.950 Zero. 00:10:53.950 --> 00:11:01.490 Uh, let me do this over here. 00:11:05.090 --> 00:11:09.270 Uh, let me call this f of x. 00:11:10.510 --> 00:11:11.630 Zero squared zero. 00:11:12.350 --> 00:11:13.430 Uh, what is one squared? 00:11:15.490 --> 00:11:16.410 Two squared? 00:11:17.390 --> 00:11:17.990 Four. 00:11:17.990 --> 00:11:18.130 Four. 00:11:18.510 --> 00:11:19.170 Three squared? 00:11:19.170 --> 00:11:19.990 Nine. 00:11:21.250 --> 00:11:21.830 Nine. 00:11:23.130 --> 00:11:24.090 Four squared? 00:11:24.770 --> 00:11:25.110 Sixteen. 00:11:25.230 --> 00:11:25.630 Sixteen. 00:11:26.950 --> 00:11:27.170 Okay. 00:11:28.150 --> 00:11:29.910 And let's plug in some negative values. 00:11:30.610 --> 00:11:31.450 Let's go negative one. 00:11:32.190 --> 00:11:32.410 One. 00:11:33.490 --> 00:11:34.090 One. 00:11:34.610 --> 00:11:35.010 Negative two? 00:11:39.350 --> 00:11:40.650 Negative three? 00:11:43.190 --> 00:11:46.050 I don't know why I'm putting them in parentheses. 00:11:49.290 --> 00:11:50.130 Square. 00:11:54.070 --> 00:11:57.450 So these are just a few of the infinite solutions. 00:11:58.190 --> 00:11:59.470 Let's go ahead and plot them. 00:12:02.450 --> 00:12:04.150 One squared is one. 00:12:04.670 --> 00:12:05.650 Two squared is four. 00:12:07.110 --> 00:12:09.510 Three squared is nine. 00:12:26.690 --> 00:12:27.970 Four squared is sixteen. 00:12:28.230 --> 00:12:28.830 That's off our graph. 00:12:31.470 --> 00:12:35.330 And it's a very smooth curve. 00:12:46.290 --> 00:12:47.990 Ah, I screwed up. 00:13:03.490 --> 00:13:05.810 It's almost like a U looking thing. 00:13:09.250 --> 00:13:14.987 So that red are all the solutions to that quadratic function y equals x squared or f 00:13:14.987 --> 00:13:16.490 of x equals x squared. 00:13:20.790 --> 00:13:24.050 All quadratic are going to look something like this, a U. 00:13:24.050 --> 00:13:32.047 It's just going to be stretched or flipped or translated left or right, up and down 00:13:32.047 --> 00:13:33.010 and so on. 00:13:34.410 --> 00:13:38.010 So this specific shape is called a parabola. 00:13:39.210 --> 00:13:41.050 The shape is called... 00:13:46.910 --> 00:13:47.590 Okay. 00:13:48.150 --> 00:13:49.030 It's called a parabola. 00:13:49.030 --> 00:13:54.451 Only quadratics form parabolas. Just because another function looks like a U does 00:13:54.451 --> 00:13:55.990 not make it a parabola. 00:13:58.750 --> 00:14:00.210 It's not a parabola. 00:14:04.150 --> 00:14:10.030 What other properties, just by looking at this, does this have... 00:14:10.430 --> 00:14:13.370 Does this have a minimum value or a maximum value? 00:14:15.230 --> 00:14:20.630 This minimum, or if it's flipped over, you'll see later it'll have a maximum. 00:14:21.250 --> 00:14:23.770 This point's called the vertex. 00:14:25.670 --> 00:14:26.290 Okay. 00:14:26.830 --> 00:14:31.650 It's the same name as what we call these things in a triangle. 00:14:31.650 --> 00:14:34.690 Vertex, vertices, plural. 00:14:36.790 --> 00:14:39.970 When we graph absolute value functions, what do we call this? 00:14:41.190 --> 00:14:41.510 The vertex. 00:14:42.210 --> 00:14:42.410 Okay. 00:14:43.590 --> 00:14:46.750 Parabolas have a vertex as well. 00:14:48.410 --> 00:14:53.310 This case is 0, 0 is the vertex. 00:14:56.170 --> 00:14:59.670 Is this a symmetric function? 00:15:00.450 --> 00:15:00.610 Yes. 00:15:00.610 --> 00:15:01.770 It is. 00:15:04.550 --> 00:15:09.110 Symmetric functions have what we call an axis of symmetry. 00:15:21.370 --> 00:15:23.490 I already spelled it. 00:15:25.290 --> 00:15:25.850 Okay. 00:15:26.570 --> 00:15:30.070 This one's axis of symmetry, so it's a vertical line. 00:15:31.190 --> 00:15:33.410 How would we describe the vertical line? 00:15:36.210 --> 00:15:37.450 You guys did... 00:15:37.450 --> 00:15:42.410 When you did linear functions later on, this vertical line is not a function. 00:15:42.990 --> 00:15:43.410 It's a line. 00:15:43.810 --> 00:15:45.650 What was the equation for this one right here? 00:15:47.830 --> 00:15:50.930 For this one in red is x equals negative 6. 00:15:51.990 --> 00:15:52.230 Okay. 00:15:52.230 --> 00:15:55.210 All vertical lines is x equals a number. 00:15:56.190 --> 00:16:03.210 It's kind of like 0, y plus x equals negative 6. 00:16:05.870 --> 00:16:08.310 So what is the axis of symmetry for this quadratic? 00:16:10.150 --> 00:16:10.850 Not zero. 00:16:11.710 --> 00:16:12.710 x equals zero. 00:16:13.010 --> 00:16:14.270 It's an equation. 00:16:14.670 --> 00:16:15.630 It's not a number. 00:16:16.810 --> 00:16:16.830 Okay. 00:16:17.370 --> 00:16:20.610 That vertical line is x equals zero. 00:16:23.110 --> 00:16:24.150 The third... 00:16:24.150 --> 00:16:25.130 Well, we'll talk about it. 00:16:25.330 --> 00:16:26.490 Don't let me forget the domain range. 00:16:30.830 --> 00:16:34.390 These have what we call symmetric points. 00:16:46.050 --> 00:16:46.570 Okay. 00:16:47.690 --> 00:16:48.990 And what are symmetric points? 00:16:48.990 --> 00:16:54.430 There are two points that are equal distance from the axis of symmetry. 00:16:56.990 --> 00:16:59.230 So this is of distance two away. 00:16:59.730 --> 00:17:02.750 This is the same distance two away from the axis of symmetry. 00:17:04.670 --> 00:17:08.830 So any functions that are symmetric have symmetrical points, 00:17:09.310 --> 00:17:13.070 and they're symmetric about the axis of symmetry. 00:17:14.010 --> 00:17:15.110 Those are the main properties. 00:17:15.110 --> 00:17:18.970 What is the domain of this function? 00:17:25.370 --> 00:17:28.790 Domain is the set of all x coordinates. 00:17:30.230 --> 00:17:32.330 Are there any x values we can't plug in here? 00:17:33.150 --> 00:17:35.690 No, we can plug in any real number. 00:17:36.590 --> 00:17:36.610 Okay. 00:17:37.470 --> 00:17:43.630 So the domain is all real numbers or the set of real numbers. 00:17:46.630 --> 00:17:51.050 But the range here is not all real numbers. 00:17:52.110 --> 00:17:52.170 Okay. 00:17:53.150 --> 00:17:54.510 This has a minimum. 00:17:55.550 --> 00:17:57.250 What is the minimum value? 00:18:00.030 --> 00:18:02.590 Minimums and maximums are numbers. 00:18:03.330 --> 00:18:04.110 Zero is the minimum. 00:18:05.210 --> 00:18:08.090 So when we do our interval notation of the range, 00:18:08.090 --> 00:18:09.850 it has a minimum. 00:18:11.590 --> 00:18:13.550 And is zero in the range? 00:18:13.550 --> 00:18:14.050 Yes. 00:18:14.210 --> 00:18:14.630 It is. 00:18:15.870 --> 00:18:15.990 Okay. 00:18:16.170 --> 00:18:16.910 Is there a maximum? 00:18:17.830 --> 00:18:18.970 No, it goes up forever. 00:18:19.510 --> 00:18:22.290 So zero to infinity, including zero. 00:18:22.850 --> 00:18:24.050 It's the range. 00:18:32.490 --> 00:18:33.210 Okay. 00:18:34.270 --> 00:18:35.690 That's all the properties. 00:18:37.120 --> 00:18:48.440 me do minus four. 00:18:53.060 --> 00:18:54.440 Is this a quadratic function? 00:18:54.440 --> 00:18:55.320 Yes. 00:18:57.640 --> 00:19:00.760 Y equals or F of X equals some quadratic. 00:19:06.440 --> 00:19:08.120 How are we going to graph this? 00:19:13.840 --> 00:19:16.060 We could start by plugging in a bunch of values. 00:19:17.220 --> 00:19:17.900 Okay. 00:19:22.880 --> 00:19:24.920 Let me go backwards. 00:19:25.540 --> 00:19:26.500 I'll come back to this. 00:19:26.840 --> 00:19:27.340 You don't need reds. 00:19:36.480 --> 00:19:41.860 We graphed lots of absolute-value functions a while ago. 00:19:42.220 --> 00:19:42.600 Remember this? 00:19:42.600 --> 00:19:42.740 Six. 00:19:44.320 --> 00:19:47.120 Did we just plug in x values to find y? 00:19:47.720 --> 00:19:48.000 No. 00:19:48.220 --> 00:19:48.360 No. 00:19:48.480 --> 00:19:49.660 How did we go about graphing it? 00:19:53.560 --> 00:19:55.260 The plus three is like minus. 00:19:56.440 --> 00:19:57.540 You just got to do the thing. 00:19:57.820 --> 00:19:58.700 Do the thing? 00:19:59.340 --> 00:20:00.700 What do we call the thing? 00:20:03.040 --> 00:20:06.380 We looked at the different transformations, right? 00:20:09.380 --> 00:20:14.800 When you replace x with x plus three, where is that shifted? 00:20:15.920 --> 00:20:17.820 To the left by three. 00:20:18.560 --> 00:20:23.060 So it shifts absolute value of X left by three. 00:20:23.880 --> 00:20:29.520 And if you add or subtract a number outside of the function, what does that do? 00:20:30.340 --> 00:20:30.640 Lowers down. 00:20:31.060 --> 00:20:31.540 Lowers down. 00:20:32.360 --> 00:20:35.860 So this goes down by five. 00:20:40.300 --> 00:20:42.680 And was there any stretch to it? 00:20:42.920 --> 00:20:43.100 No. 00:20:43.860 --> 00:20:48.140 No reflection or stretch because the coefficient from the function is a one. 00:20:48.900 --> 00:20:49.000 Okay. 00:20:49.020 --> 00:20:50.200 So we found the vertex. 00:20:51.540 --> 00:20:51.940 Left three. 00:20:53.400 --> 00:20:53.780 Down five. 00:20:55.560 --> 00:20:56.020 The vertex. 00:20:56.260 --> 00:20:57.180 And now what do we do? 00:21:00.480 --> 00:21:00.840 Draw. 00:21:01.180 --> 00:21:01.960 Draw what? 00:21:02.860 --> 00:21:03.220 Two lines. 00:21:03.600 --> 00:21:04.940 Yeah, this was a piecewise function. 00:21:07.980 --> 00:21:09.880 This half has a slope of what? 00:21:11.960 --> 00:21:14.320 This half has a slope of negative one. 00:21:14.320 --> 00:21:14.440 Negative one. 00:21:16.320 --> 00:21:20.320 What if this were two? 00:21:21.500 --> 00:21:21.820 What would we do? 00:21:23.640 --> 00:21:24.280 Well, let's move on. 00:21:24.880 --> 00:21:26.600 Two is positive two. 00:21:26.600 --> 00:21:30.760 Yeah, this stretches this function by a factor of two. 00:21:32.140 --> 00:21:32.180 Okay. 00:21:32.800 --> 00:21:35.760 Because they're linear, we just change the slopes. 00:21:35.760 --> 00:21:37.760 We're going to do something different with quadratics. 00:21:39.040 --> 00:21:40.080 We're going to do the same thing. 00:21:40.180 --> 00:21:40.760 We're going to use transformations. 00:21:42.980 --> 00:21:43.020 Okay. 00:21:44.640 --> 00:21:48.100 The first thing we're going to do is find the vertex. 00:21:51.620 --> 00:21:54.060 Well, actually, we'll talk about that in a second. 00:21:54.700 --> 00:21:55.880 For x squared, right? 00:21:56.500 --> 00:21:57.840 The vertex is a zero zero. 00:21:59.440 --> 00:21:59.860 Okay. 00:22:01.200 --> 00:22:03.260 What is this minus four going to do? 00:22:03.740 --> 00:22:04.140 Four. 00:22:04.580 --> 00:22:05.020 Five. 00:22:07.700 --> 00:22:10.160 All this is going to do is take the graph of x squared. 00:22:11.420 --> 00:22:12.640 It goes down by four. 00:22:14.220 --> 00:22:16.100 So the vertex is going to be (0, negative 4). 00:22:22.120 --> 00:22:22.800 The vertex. 00:22:28.000 --> 00:22:28.440 Okay. 00:22:31.960 --> 00:22:40.440 What I like to do is instead of graphing this. 00:22:48.140 --> 00:22:51.680 We're going to follow a pattern. 00:22:52.960 --> 00:22:54.240 I'll show you in a second. 00:22:56.220 --> 00:22:57.440 And this is my pattern. 00:22:59.480 --> 00:23:00.860 I'm calling the pattern. 00:23:03.440 --> 00:23:04.880 Zero squared is zero. 00:23:05.100 --> 00:23:05.920 One squared is one. 00:23:06.100 --> 00:23:07.440 Two squared is four. 00:23:08.480 --> 00:23:09.760 Three squared is nine. 00:23:15.220 --> 00:23:20.860 We could subtract four from all the y's. 00:23:24.720 --> 00:23:25.320 Zero. 00:23:25.740 --> 00:23:27.340 Zero minus four is negative four. 00:23:29.500 --> 00:23:31.200 One minus four. 00:23:31.900 --> 00:23:32.860 Four minus four. 00:23:34.700 --> 00:23:35.400 Oops. 00:23:35.740 --> 00:23:37.260 Yeah. Nine minus four. 00:23:38.220 --> 00:23:39.000 The negative four. 00:23:39.360 --> 00:23:39.940 Negative three. 00:23:40.880 --> 00:23:41.300 Zero. 00:23:45.900 --> 00:23:46.660 Five. 00:23:48.140 --> 00:23:53.080 But with graph paper, we do not need to bother with all of this. 00:23:54.720 --> 00:23:55.920 I just call this. 00:24:02.080 --> 00:24:03.620 This is my vertex. 00:24:07.580 --> 00:24:07.620 Okay. 00:24:08.980 --> 00:24:09.780 And here's what I do. 00:24:11.560 --> 00:24:13.140 I start at the vertex. 00:24:14.860 --> 00:24:16.140 This is my starting point for the zero-zero offsets. 00:24:17.300 --> 00:24:17.780 And I go. 00:24:19.300 --> 00:24:20.360 One to the right. 00:24:20.900 --> 00:24:21.160 One up. 00:24:22.620 --> 00:24:24.220 So we go one to the right, one up. 00:24:24.400 --> 00:24:25.240 This is going to be my next point. 00:24:26.480 --> 00:24:27.860 I go back to the vertex. 00:24:27.860 --> 00:24:29.820 Go two to the right, four up. 00:24:30.820 --> 00:24:33.360 One, two, one, two, three, four. 00:24:35.100 --> 00:24:37.240 Three to the right from the vertex, nine up. 00:24:38.080 --> 00:24:39.160 One, two, three. 00:24:39.820 --> 00:24:43.520 One, two, three, four, five, six, seven, eight, nine. 00:24:46.480 --> 00:24:47.680 That's what I call my pattern. 00:24:49.100 --> 00:24:51.780 Go to our vertex, which is the zero zero. 00:24:51.980 --> 00:24:55.411 Go one to the right, one up, two to the right, four up, three to the right, nine up, 00:24:55.411 --> 00:24:55.820 and so on. 00:24:56.600 --> 00:25:00.260 And then this is the right half of the parabola. 00:25:01.680 --> 00:25:03.480 What's the axis of symmetry here? 00:25:05.240 --> 00:25:05.780 X equals zero. 00:25:05.900 --> 00:25:10.280 X equals zero because we did not shift it left and right. 00:25:13.080 --> 00:25:19.100 So I usually put a dotted line for my axis of symmetry. 00:25:20.700 --> 00:25:25.800 And then we use the fact that it's a line. 00:25:25.820 --> 00:25:27.320 It has symmetric points. 00:25:28.880 --> 00:25:30.780 So any X value, we go same. 00:25:31.700 --> 00:25:33.360 So I went one here and I go one over. 00:25:35.040 --> 00:25:39.180 Two, two, three, one, two, three. 00:25:40.200 --> 00:25:43.160 And there we got enough points. 00:25:44.260 --> 00:25:47.840 So once we find the vertex, we follow the pattern. 00:25:48.320 --> 00:25:51.320 One to the right, one up, two to the right, four up, three to the right, nine up. 00:25:52.360 --> 00:25:54.500 And then we just graph these symmetric points. 00:25:55.820 --> 00:25:59.820 Once we have to add, draw a nice smooth curve. 00:26:09.680 --> 00:26:12.640 And there is the graph of the parabola. 00:26:12.760 --> 00:26:15.020 That is the solution to it. 00:26:15.040 --> 00:26:15.720 It should be quite good. 00:26:16.740 --> 00:26:17.520 Make sense? 00:26:18.980 --> 00:26:19.320 Okay. 00:26:21.060 --> 00:26:22.320 Let us look at another one. 00:26:22.650 --> 00:26:25.050 Is this a quadratic function? 00:26:25.570 --> 00:26:26.290 Yes. 00:26:27.510 --> 00:26:32.410 This is some linear binomial square gives us a quadratic. 00:26:32.990 --> 00:26:34.190 And then we subtract three. 00:26:34.190 --> 00:26:36.930 This is not written in standard form. 00:26:37.150 --> 00:26:38.050 What's standard form? 00:26:43.230 --> 00:26:45.790 The form that we use for the quadratic formula. 00:26:46.810 --> 00:26:48.930 A x squared plus b x plus c. 00:26:49.510 --> 00:26:49.790 Standard form. 00:26:50.750 --> 00:26:52.370 We'll talk about this form in a minute. 00:26:54.830 --> 00:26:58.330 Where's the shift or function X squared? 00:27:00.770 --> 00:27:03.570 Because I replaced X with X minus two. 00:27:04.190 --> 00:27:05.190 What's the shift? 00:27:06.550 --> 00:27:07.170 Left or right? 00:27:08.110 --> 00:27:08.150 Right. 00:27:08.790 --> 00:27:11.190 Right by two. 00:27:12.750 --> 00:27:18.130 And this shifts it down by three. 00:27:21.190 --> 00:27:25.690 So this vertex goes to the right, two, down three. 00:27:29.430 --> 00:27:30.190 So the vertex is. 00:27:34.730 --> 00:27:36.490 Two, negative three. 00:27:38.090 --> 00:27:38.670 Okay. 00:27:41.950 --> 00:27:45.470 Once I find the vertex, I follow my pattern. 00:27:46.830 --> 00:27:53.350 And the pattern is zero, zero, one, one, two, four, three, 00:27:53.350 --> 00:27:53.950 square is nine. 00:27:56.130 --> 00:27:57.570 So I go to the vertex. 00:27:58.910 --> 00:28:03.770 I go to the right one, up one, right two, 00:28:04.190 --> 00:28:11.170 up four, right three, up nine, one, two, three, four, five, 00:28:11.330 --> 00:28:12.870 six, seven, eight, nine. 00:28:16.530 --> 00:28:17.810 And then what do we do? 00:28:25.890 --> 00:28:31.030 We graph these symmetric points along the axis of symmetry. 00:28:31.170 --> 00:28:32.790 The axis of symmetry always goes through the vertex. 00:28:34.190 --> 00:28:36.250 So I got a symmetric point here. 00:28:38.790 --> 00:28:39.570 One, two, one, two. 00:28:41.390 --> 00:28:43.650 One, two, three, one, two, three. 00:28:45.230 --> 00:28:47.450 And now I just draw a nice, smooth curve. 00:28:52.030 --> 00:28:57.630 And the very bottom at the vertex, it becomes very, very flat. 00:28:57.790 --> 00:29:00.250 It is not a sharp V. It is very smooth. 00:29:01.750 --> 00:29:03.030 So don't draw Vs. 00:29:05.190 --> 00:29:06.190 There it is. 00:29:14.810 --> 00:29:17.690 Real quick, what's its domain? 00:29:22.150 --> 00:29:23.670 Yeah, all real numbers. 00:29:25.410 --> 00:29:29.810 Like all polynomials, polynomial functions, the domain is all real numbers. 00:29:31.370 --> 00:29:33.510 Does this have a minimum or a maximum? 00:29:34.190 --> 00:29:37.350 It says a minimum and the minimum is? 00:29:40.270 --> 00:29:40.390 Negative three. 00:29:42.830 --> 00:29:46.390 It is always the y coordinate of the vertex. 00:29:47.730 --> 00:29:51.350 So therefore the range has a minimum. 00:29:53.450 --> 00:29:55.050 Does not have a maximum. 00:30:04.190 --> 00:30:05.890 This is not a standard form. 00:30:07.170 --> 00:30:09.230 Anybody want to guess what form it's in? 00:30:10.950 --> 00:30:12.670 It is called vertex form. 00:30:14.290 --> 00:30:17.190 This is called vertex form of a quadratic function. 00:30:26.650 --> 00:30:29.310 If it's in this form, it's very easy graphically. 00:30:29.670 --> 00:30:30.690 Instantly know the vertex. 00:30:31.690 --> 00:30:35.550 It is the opposite of this number, comma that number. 00:30:38.190 --> 00:30:45.510 We usually call the vertex h,k. 00:30:46.890 --> 00:30:49.390 I do not know why those letters were chosen. 00:30:49.990 --> 00:30:50.710 There they are. 00:30:51.710 --> 00:31:00.710 So vertex form is y equals a. 00:31:03.070 --> 00:31:04.070 We'll talk about it in a second. 00:31:04.990 --> 00:31:08.370 X minus h, squared, plus k. 00:31:15.510 --> 00:31:20.250 The x coordinate of the vertex is the opposite of this number, which is just h. 00:31:23.950 --> 00:31:28.370 The y coordinate is just that, whatever that number is. 00:31:33.820 --> 00:31:35.480 Is this the same? 00:31:36.400 --> 00:31:38.660 So standard form. 00:31:41.940 --> 00:31:47.320 Is y equals a x squared plus b x plus c. 00:31:50.620 --> 00:31:54.820 Is this a in vertex form is same as that a in standard form? 00:31:55.580 --> 00:31:56.540 It is. 00:31:57.240 --> 00:32:03.230 Once you multiply out this binomial squared, you get x squared plus something plus 00:32:03.230 --> 00:32:03.960 something. 00:32:04.760 --> 00:32:06.200 And then you distribute the a. 00:32:07.700 --> 00:32:11.140 It's the same as in standard form. 00:32:11.260 --> 00:32:12.580 It is the leading coefficient. 00:32:16.000 --> 00:32:17.660 So, let's try another one. 00:32:18.020 --> 00:32:19.840 What form is this in? 00:32:21.120 --> 00:32:21.700 Vertex form. 00:32:23.540 --> 00:32:24.340 Okay. 00:32:25.230 --> 00:32:30.490 It takes our x squared, just to the left by three, up by one. 00:32:31.210 --> 00:32:32.910 So our vertex is a negative three one. 00:32:33.490 --> 00:32:34.210 Once I get my vertex. 00:32:35.630 --> 00:32:37.410 Once I get that, what do I do? 00:32:44.170 --> 00:32:47.230 Follow the pattern. 00:32:47.510 --> 00:32:48.370 What do I call it? 00:32:48.370 --> 00:32:52.930 Zero, zero, one squared is one, two squared is four, three squared is nine. 00:32:53.770 --> 00:32:54.870 So go to the vertex. 00:32:57.770 --> 00:32:59.550 One to the right, one up. 00:33:01.010 --> 00:33:03.970 Two to the right, one, two, three, four up. 00:33:05.030 --> 00:33:06.590 Three to the right, nine up. 00:33:13.570 --> 00:33:16.270 That's going to be the right half of our parabola. 00:33:17.270 --> 00:33:19.290 And how do I find the left half? 00:33:22.950 --> 00:33:24.670 Use the what points? 00:33:25.490 --> 00:33:26.190 Symmetric points. 00:33:32.970 --> 00:33:34.510 And draw a nice smooth curve. 00:33:34.510 --> 00:33:34.850 Very good. 00:33:35.530 --> 00:33:36.130 Yes. 00:33:36.130 --> 00:33:36.610 This is a vertex. 00:33:37.870 --> 00:33:38.470 Yes. 00:33:40.130 --> 00:33:41.110 Therefore the vertex is. 00:33:46.930 --> 00:33:50.570 Opposite that number, comma, that number, negative one, negative five. 00:33:54.810 --> 00:33:57.250 So go ahead on your graph paper. 00:33:58.410 --> 00:33:59.470 Draw a couple axes. 00:34:25.510 --> 00:34:28.090 That's what I call my pattern. 00:34:29.390 --> 00:34:32.570 It's just zero, one, two and three squared. 00:34:37.490 --> 00:34:39.830 This leading coefficient of two. 00:34:40.370 --> 00:34:41.710 What is that going to do to it? 00:34:43.230 --> 00:34:46.150 It's going to stretch it by a factor of two. 00:34:48.770 --> 00:34:49.150 So. 00:34:51.690 --> 00:34:54.670 Here's what we do. 00:34:59.630 --> 00:35:03.170 We simply multiply these Y values by two. 00:35:04.490 --> 00:35:05.970 So times two is. 00:35:07.030 --> 00:35:08.570 Let me just do a whole nother one. 00:35:09.350 --> 00:35:10.390 X and two x squared. 00:35:14.550 --> 00:35:16.930 So zero zero. 00:35:18.250 --> 00:35:27.990 Multiplying by two gives the offsets (0,0), (1,2), (2,8), and (3,18). 00:35:30.710 --> 00:35:32.930 Now this is my stretched pattern. 00:35:35.230 --> 00:35:37.850 This is my vertex right here. 00:35:39.690 --> 00:35:42.650 And now I go one to the right, two up. 00:35:43.690 --> 00:35:45.370 One to the right, two up. 00:35:48.050 --> 00:35:50.810 Two to the right, eight up. 00:35:57.690 --> 00:35:59.550 Three to the right, eighteen. 00:35:59.950 --> 00:36:00.990 I don't know if we're going to get there. 00:36:11.210 --> 00:36:11.750 Oh. 00:36:12.750 --> 00:36:13.150 Oh. 00:36:14.750 --> 00:36:15.590 It's up here somewhere. 00:36:28.390 --> 00:36:29.290 Now what? 00:36:31.810 --> 00:36:34.750 We have our right half of our parabola. 00:36:35.510 --> 00:36:36.510 We have the symmetric points. 00:36:38.070 --> 00:36:40.370 We plot the symmetric points. 00:36:41.750 --> 00:36:42.470 About this line. 00:37:18.590 --> 00:37:19.690 And there it is. 00:37:19.870 --> 00:37:23.030 It's just a skinnier looking parabola. 00:37:24.070 --> 00:37:25.550 It's been stretched. 00:37:30.750 --> 00:37:31.430 Okay. 00:37:31.770 --> 00:37:32.970 Questions on this so far? 00:37:34.590 --> 00:37:34.990 Let's try it on. 00:37:52.210 --> 00:37:53.710 The vertex is. 00:37:56.690 --> 00:37:57.750 Negative one. 00:37:58.230 --> 00:37:59.010 The opposite of that number. 00:37:59.270 --> 00:37:59.670 Comma three. 00:38:03.830 --> 00:38:05.290 This is the vertex. 00:38:23.470 --> 00:38:24.150 Okay. 00:38:25.990 --> 00:38:33.410 So the one half is going to compress it or stretch it by a factor one half. 00:38:33.850 --> 00:38:34.890 What does the negative do? 00:38:40.310 --> 00:38:41.870 Negative inside of a function. 00:38:42.090 --> 00:38:43.430 Or sorry, outside of a function. 00:38:44.350 --> 00:38:45.270 It reflects it. 00:38:47.470 --> 00:38:47.570 But. 00:38:48.990 --> 00:38:51.090 We take care of that very easily. 00:38:53.050 --> 00:38:55.990 X and negative one half X squared. 00:38:57.290 --> 00:38:59.050 You just multiply these all by. 00:39:00.090 --> 00:39:01.270 Negative one half. 00:39:13.710 --> 00:39:18.410 Let me add one more for 16. 00:39:20.110 --> 00:39:22.990 Negative one half. 00:39:26.790 --> 00:39:32.550 Literally we are just multiplying this guy by negative one half to get this. 00:39:32.830 --> 00:39:36.730 We get zero zero one negative one half. 00:39:37.870 --> 00:39:43.990 Two negative two three negative nine halves. 00:39:44.830 --> 00:39:47.470 Which is negative four point five. 00:39:52.450 --> 00:39:57.710 OK, so this is our new pattern. 00:39:59.390 --> 00:40:02.470 We multiply the leading coefficient by all the y values in the pattern. 00:40:04.670 --> 00:40:06.730 So this is our vertex, the starting point for the zero-zero offsets. 00:40:07.670 --> 00:40:09.150 Go one to the right. 00:40:10.370 --> 00:40:12.730 But now we go since it's negative. 00:40:12.730 --> 00:40:15.570 We go down by a half. 00:40:16.850 --> 00:40:18.510 Two to the right down by two. 00:40:19.390 --> 00:40:22.090 Three to the right down by four and a half. 00:40:23.030 --> 00:40:27.070 We go one to the right down a half. 00:40:29.670 --> 00:40:32.150 Two to the right down to. 00:40:35.190 --> 00:40:36.570 Three to the right. 00:40:37.890 --> 00:40:39.210 Down four and a half. 00:40:39.430 --> 00:40:41.390 One, two, three, four. 00:40:42.390 --> 00:40:43.030 And a half. 00:40:44.930 --> 00:40:46.610 And four to the right, down eight. 00:40:46.950 --> 00:40:48.770 One, two, three, four. 00:40:57.190 --> 00:41:00.090 Now I have my right hand side. 00:41:06.370 --> 00:41:08.470 This is the axis of symmetry. 00:41:11.730 --> 00:41:15.190 And a nice curve—do not do that yet. 00:41:15.390 --> 00:41:16.910 I need my symmetric points. 00:41:49.390 --> 00:41:49.950 This. 00:41:51.830 --> 00:41:52.690 Not too bad. 00:41:54.070 --> 00:41:55.670 OK, we find the vertex. 00:41:57.330 --> 00:42:00.770 We take our pattern zero zero one one two four three nine. 00:42:01.190 --> 00:42:02.350 Sometimes add four sixteen. 00:42:03.930 --> 00:42:08.350 Then we multiply these y values by the leading coefficient. 00:42:08.350 --> 00:42:10.230 This gives us our new pattern. 00:42:10.850 --> 00:42:12.650 When it's negative, we just go down. 00:42:12.930 --> 00:42:14.030 We always go to the right. 00:42:15.750 --> 00:42:18.090 And now we just go down by whatever this is. 00:42:18.430 --> 00:42:19.350 If it's negative values. 00:42:32.230 --> 00:42:34.750 Now let's do something fun. 00:42:35.050 --> 00:42:37.630 This is not a vertex form. 00:42:37.950 --> 00:42:41.330 What form is this in? Standard form. 00:42:47.850 --> 00:42:51.830 Actually, let me change this. 00:43:16.130 --> 00:43:17.070 Actually, let's leave it. 00:43:17.190 --> 00:43:17.230 Sorry. 00:43:19.010 --> 00:43:20.150 We factor it out. 00:43:23.870 --> 00:43:25.070 Let's try it. 00:43:25.530 --> 00:43:26.250 What is the factor to. 00:43:33.290 --> 00:43:38.210 OK, it could be factored, but we have to use the quadratic formula. 00:43:39.170 --> 00:43:42.350 OK, we're going to get two irrational. 00:43:44.830 --> 00:43:45.970 Numbers right here. 00:43:47.270 --> 00:43:50.810 Actually, let's let's pretend it actually factors it. 00:43:53.830 --> 00:43:54.870 It doesn't. 00:43:54.970 --> 00:43:55.910 But let's say it did. 00:44:04.030 --> 00:44:08.390 What would the numbers three and negative two be? 00:44:12.710 --> 00:44:16.210 OK, let's. 00:44:16.410 --> 00:44:17.850 If I plug in three into here. 00:44:19.030 --> 00:44:28.017 I get three minus three times three plus two, which is equal to zero times five, 00:44:28.017 --> 00:44:29.590 which is zero. 00:44:32.210 --> 00:44:37.890 OK, so we know three comma zero is a solution. 00:44:38.610 --> 00:44:39.750 If I plug in negative two. 00:44:43.130 --> 00:44:44.990 Be negative two minus three. 00:44:47.170 --> 00:44:50.590 Times negative two plus two. 00:44:52.190 --> 00:44:55.030 Negative five times zero, which is zero. 00:45:02.650 --> 00:45:05.210 This is not the function, but let's say it was. 00:45:13.230 --> 00:45:17.630 So that parabola would now look something like this. 00:45:18.250 --> 00:45:22.610 What do we call the points that intersect the X axis? 00:45:25.050 --> 00:45:27.210 X intercepts gives us the X intercepts. 00:45:27.930 --> 00:45:33.430 So if we're able to factor it, it only gives the X intercepts. 00:45:33.430 --> 00:45:35.430 It does give us two symmetric points. 00:45:40.670 --> 00:45:45.710 So if these were the irrational factors, it gives the X intercepts. 00:45:45.750 --> 00:45:47.770 In fact, that has a specific name. 00:45:47.870 --> 00:45:48.530 We'll get to that later. 00:45:50.410 --> 00:45:50.730 So. 00:45:53.550 --> 00:45:55.230 Just factoring is not going to help. 00:46:03.390 --> 00:46:10.430 What do I know the leading coefficient is? 00:46:11.710 --> 00:46:13.790 What's the leading coefficient? 00:46:14.550 --> 00:46:14.630 One. 00:46:15.530 --> 00:46:15.870 OK. 00:46:24.490 --> 00:46:27.570 If I could get it into this form, what do we call this form? 00:46:28.650 --> 00:46:28.890 Vertex. 00:46:29.190 --> 00:46:30.210 Vertex form. 00:46:31.110 --> 00:46:34.190 We would know where the vertex was and could easily graph it. 00:46:41.270 --> 00:46:43.190 I need to make a perfect square. 00:46:45.210 --> 00:46:45.970 Binomial. 00:46:48.050 --> 00:46:48.990 From this. 00:46:50.570 --> 00:46:53.870 How am I going to make this a perfect-square binomial? 00:46:58.070 --> 00:46:59.270 By completing the square. 00:47:06.090 --> 00:47:07.770 So here's what we're going to do. 00:47:07.970 --> 00:47:08.770 So write this. 00:47:10.050 --> 00:47:12.650 X squared minus 2X. 00:47:12.650 --> 00:47:13.850 Leave some space. 00:47:15.250 --> 00:47:16.030 And put a minus 5. 00:47:16.950 --> 00:47:20.510 Why? I'm not going to bring the 5 over here. 00:47:22.470 --> 00:47:23.670 Why am I not going to do that? 00:47:25.450 --> 00:47:27.310 We don't want 5 plus f of X. 00:47:27.790 --> 00:47:29.650 We just want f of X equals. 00:47:29.990 --> 00:47:31.950 So we just leave it to the right here. 00:47:34.290 --> 00:47:34.450 Oops. 00:47:35.210 --> 00:47:35.610 Minus 5. 00:48:02.530 --> 00:48:05.190 So what number am I going to add here? 00:48:07.410 --> 00:48:09.570 We take this number. 00:48:10.950 --> 00:48:12.050 When we complete the square, 00:48:12.650 --> 00:48:14.350 Divide by 2. 00:48:16.070 --> 00:48:16.650 And then do what with it? 00:48:18.810 --> 00:48:20.070 Square it. 00:48:22.890 --> 00:48:23.490 Okay. 00:48:25.370 --> 00:48:26.750 And I'm going to add 1. 00:48:28.370 --> 00:48:30.430 And we're not going to add 1 to both sides. 00:48:32.130 --> 00:48:32.830 We could. 00:48:33.830 --> 00:48:34.770 You can't just add 1. 00:48:35.250 --> 00:48:36.230 But you can add 1. 00:48:37.270 --> 00:48:38.710 And then subtract 1. 00:48:40.230 --> 00:48:41.370 Just adding 0. 00:48:43.210 --> 00:48:45.310 Now this is a perfect square. 00:48:47.810 --> 00:48:48.810 And it's going to factor to what? 00:48:49.130 --> 00:48:53.250 X minus 1 times X minus 1. 00:48:53.770 --> 00:48:56.650 And now the negative 1 minus 5 is negative 6. 00:49:07.490 --> 00:49:11.790 Okay. Now it's in vertex form. 00:49:13.330 --> 00:49:14.430 Okay. 00:49:19.590 --> 00:49:22.210 Now we know the vertex. 00:49:24.450 --> 00:49:25.610 What is the vertex? 00:49:28.870 --> 00:49:30.650 1 negative 6. 00:49:34.770 --> 00:49:36.790 Once I have the vertex. 00:49:37.930 --> 00:49:38.190 Who do I do? 00:49:41.490 --> 00:49:43.910 We follow the pattern. 00:49:47.190 --> 00:49:52.690 0 0 1 1 2 squared is 4 3 squared is 9. 00:50:14.570 --> 00:50:19.030 I'm not following the pattern. 00:50:20.770 --> 00:50:25.730 9 1 2 3 4 5 6 7 8 1. 00:50:28.710 --> 00:50:32.010 I'll get my symmetrical points. 00:50:36.970 --> 00:50:37.950 Okay. 00:50:50.830 --> 00:50:51.430 Okay. 00:50:58.310 --> 00:51:01.730 If you do not like completing the square, there is another option. 00:51:04.230 --> 00:51:04.710 Okay. 00:51:24.130 --> 00:51:26.110 So where did we get this number from? 00:51:26.290 --> 00:51:27.030 The minus 1. 00:51:34.650 --> 00:51:36.570 We took this number. 00:51:37.790 --> 00:51:38.970 Divided by 2. 00:51:39.970 --> 00:51:42.190 We squared it to figure out what we had, right? 00:51:43.410 --> 00:51:45.630 But simply that number divided by 2. 00:51:47.550 --> 00:51:47.570 Okay. 00:51:55.130 --> 00:51:57.710 And then of course, once we get it into here, 00:51:59.890 --> 00:52:02.430 we know the vertex is the opposite of that number. 00:52:03.810 --> 00:52:04.370 Okay. 00:52:04.370 --> 00:52:10.170 So it turns out the vertex, the x coordinate, 00:52:13.070 --> 00:52:16.070 is negative b over 2. 00:52:19.450 --> 00:52:19.630 Okay. 00:52:20.290 --> 00:52:22.610 Unless there's a leading coefficient, 00:52:23.890 --> 00:52:26.450 then we'd have to divide everything by the leading coefficient. 00:52:27.410 --> 00:52:30.170 So it turns out it's negative b over 2a. 00:52:31.330 --> 00:52:33.150 I could show you another reason for that. 00:52:38.010 --> 00:52:40.710 And then how do I get the y coordinate? 00:52:46.210 --> 00:52:48.930 If we have standard form, 00:52:52.450 --> 00:52:54.510 this is the x coordinate. 00:52:55.010 --> 00:52:56.370 We simply plug it into the function. 00:53:01.150 --> 00:53:01.810 Okay. 00:53:02.210 --> 00:53:03.910 So let me show you how to do that. 00:53:06.650 --> 00:53:07.890 Let's do another example. 00:53:08.380 --> 00:53:10.080 What form is this in? 00:53:11.240 --> 00:53:11.560 Standard form. 00:53:11.780 --> 00:53:13.520 Actually, let me change this y to f of x. 00:53:17.480 --> 00:53:21.040 Standard form directly shows two things. 00:53:22.660 --> 00:53:24.100 It tells us the y intercept. 00:53:25.920 --> 00:53:27.720 The only other thing it tells us, 00:53:27.720 --> 00:53:30.820 does it open up or does it open down? 00:53:31.580 --> 00:53:33.140 If the leading coefficient is positive, 00:53:33.460 --> 00:53:34.280 it's going to open up. 00:53:35.080 --> 00:53:35.840 Leading coefficient is negative. 00:53:36.740 --> 00:53:37.120 Okay. 00:53:38.160 --> 00:53:48.800 So the vertex is always at negative b over 2a, 00:53:48.800 --> 00:53:52.100 f of negative b over 2a. 00:53:55.720 --> 00:53:59.800 So negative b over 2a is negative. 00:54:01.020 --> 00:54:01.820 What's our b value? 00:54:03.320 --> 00:54:03.820 Negative 6. 00:54:04.260 --> 00:54:06.460 The negative of negative six, all over two times a. 00:54:06.700 --> 00:54:07.200 What's our a? 00:54:08.020 --> 00:54:08.420 1. 00:54:09.540 --> 00:54:14.600 Which is 6 over 2, which is 3. 00:54:16.160 --> 00:54:22.200 So I now know my vertex is at 3, f of 3. 00:54:24.600 --> 00:54:26.400 And how do I find my f of 3? 00:54:27.820 --> 00:54:29.840 You just plug it into our function. 00:54:31.420 --> 00:54:40.060 F of 3 is 3 squared minus 6 times 3 plus 5. 00:54:51.580 --> 00:54:53.640 Is that negative 4? 00:54:57.080 --> 00:55:00.620 So my vertex is at 3, negative 4. 00:55:03.920 --> 00:55:13.540 Once we know that, we know our function in vertex form is the leading coefficient. 00:55:13.760 --> 00:55:14.440 What's the leading coefficient? 00:55:15.580 --> 00:55:16.380 1. 00:55:16.380 --> 00:55:17.020 1. 00:55:17.600 --> 00:55:22.140 So it's x minus 3 squared minus 4. 00:55:23.300 --> 00:55:26.360 It's the opposite of that number. 00:55:29.620 --> 00:55:32.160 And now we can graph it quickly. 00:55:37.980 --> 00:55:40.620 We go to 3. 00:55:44.580 --> 00:55:45.620 Negative 4. 00:55:47.760 --> 00:55:52.280 Since the leading coefficient is positive, we follow the pattern. 00:56:19.820 --> 00:56:24.400 I'll write my symmetric points. 00:56:34.760 --> 00:56:40.900 So if it's in standard form, 00:56:42.960 --> 00:56:45.740 just write it in vertex form by finding the vertex. 00:56:46.060 --> 00:56:49.380 The vertex’s x coordinate is always negative b over two a. 00:56:49.920 --> 00:56:54.208 Once you get that number, we plug it back into our standard form to get the y 00:56:54.208 --> 00:56:54.820 coordinate. 00:56:56.660 --> 00:57:01.836 And then we write it in vertex form, or you just go with the vertex, follow the 00:57:01.836 --> 00:57:02.360 pattern. 00:57:02.540 --> 00:57:05.940 If we have to adjust the pattern because a is not 1, then we do that. 00:57:07.440 --> 00:57:08.100 Sound good? 00:57:08.400 --> 00:57:09.040 Yes, it is. 00:57:09.040 --> 00:57:13.660 Okay, last question before we start homework. 00:57:14.140 --> 00:57:16.660 Okay, here, I don't give you an equation. 00:57:18.220 --> 00:57:20.100 I give you a graph of a parabola. 00:57:21.300 --> 00:57:22.580 How are we going to find the equation? 00:57:27.080 --> 00:57:28.440 What is the vertex? 00:57:28.440 --> 00:57:33.520 Negative 2, 3. 00:57:33.520 --> 00:57:35.440 Negative 2, 3. 00:57:36.940 --> 00:57:45.280 Okay, vertex form is a x minus h squared plus k. 00:57:50.700 --> 00:57:56.720 So we know y equals a times x plus two, squared, plus three. 00:57:57.880 --> 00:58:00.500 And then how do I figure out my leading coefficient? 00:58:03.800 --> 00:58:06.280 We know it's negative. 00:58:09.740 --> 00:58:11.740 Okay, you see one point. 00:58:12.140 --> 00:58:15.380 We go to the right one and we go down one. 00:58:17.140 --> 00:58:19.260 That tells you the leading coefficient. 00:58:19.560 --> 00:58:21.700 We went down one, leading coefficient negative one. 00:58:22.660 --> 00:58:25.100 If we had to go down two, it would be negative two. 00:58:27.380 --> 00:58:30.520 Okay, it's all based on going over one. 00:58:31.280 --> 00:58:32.020 Do we go up one? 00:58:32.080 --> 00:58:32.780 Do we go up two? 00:58:32.920 --> 00:58:34.960 Do we go down one, down two, whatever it is. 00:58:34.960 --> 00:58:42.620 So we know it is y equals negative one times x plus two, squared, plus three. 00:58:43.800 --> 00:58:46.740 If we want standard form, we just have to multiply this out.