1 00:00:05,040 --> 00:00:14,440 Okay. We are going over the different properties of quadratics based on the graph, 2 00:00:14,660 --> 00:00:22,800 based on the equation or a certain form of the equation. What is 3 00:00:22,800 --> 00:00:30,360 the domain of all polynomials? All real numbers. Again, you could write it like that 4 00:00:30,360 --> 00:00:30,900 or you 5 00:00:30,900 --> 00:00:41,020 could write from negative infinity to positive infinity. Okay. Does this quadratic 6 00:00:41,020 --> 00:00:41,760 have a 7 00:00:41,760 --> 00:00:48,947 minimum or a maximum? Minimum. Minimum. Okay. So, and what is that minimum value? 8 00:00:48,947 --> 00:00:50,100 Negative one. 9 00:00:50,190 --> 00:01:00,386 Yes, negative one. So go to the min, circle the min, or whatever. Min equals 10 00:01:00,386 --> 00:01:02,130 negative one. 11 00:01:03,110 --> 00:01:11,395 Minimums and maximums are y values or function values. It's just a number. 12 00:01:11,395 --> 00:01:13,410 Therefore, we know 13 00:01:13,410 --> 00:01:20,412 the range. What's the range? Negative one all the way to infinity, including 14 00:01:20,412 --> 00:01:21,610 negative one. 15 00:01:22,250 --> 00:01:27,972 We think this is the y intercept, but we don't know yet for sure. Maybe it's just 16 00:01:27,972 --> 00:01:28,890 very close to 17 00:01:28,890 --> 00:01:38,230 seven. Can we tell what the zeros are yet? Yeah. No. No. We could approximate it, 18 00:01:38,290 --> 00:01:56,462 but we definitely can't tell. It's probably going to be some irrational numbers. Do 19 00:01:56,462 --> 00:01:58,870 we know the 20 00:02:00,890 --> 00:02:09,137 leading coefficient? Go over. So if we have the graph, go to the vertex, you go over 21 00:02:09,137 --> 00:02:09,930 one, 22 00:02:09,930 --> 00:02:14,472 and however many you go up or down, that is the coefficient. So we go over one, up 23 00:02:14,472 --> 00:02:15,472 one, two. 24 00:02:15,910 --> 00:02:21,521 So the leading coefficient, our a value is two. That's not on the list, but let me 25 00:02:21,521 --> 00:02:22,410 write it over 26 00:02:22,410 --> 00:02:30,787 here. Okay. And now we can write the equation for this in what form? The vertex 27 00:02:30,787 --> 00:02:32,590 form. We have the 28 00:02:32,590 --> 00:02:41,875 We know the vertex and the leading coefficient, so use vertex form: 29 00:02:41,875 --> 00:02:44,370 f(x)=2(x−(−2))²−1, 30 00:02:44,370 --> 00:02:51,070 or 2(x+2)²−1. 31 00:02:51,590 --> 00:02:55,386 Axis of symmetry. Actually, we could definitely have done that before. What is the 32 00:02:55,386 --> 00:02:55,950 axis of 33 00:02:55,950 --> 00:03:04,030 symmetry? It's an equation x equals negative two, the x coordinate of the vertex. 34 00:03:04,610 --> 00:03:07,943 We can put it in standard form. How do we do that? 35 00:03:09,890 --> 00:03:12,490 Yeah. Multiply this out. So we got 36 00:03:16,470 --> 00:03:21,430 Two times (x+2)(x+2), minus one. 37 00:03:24,950 --> 00:03:30,295 Inside, x squared plus two x plus two x plus four; then multiply by two and subtract 38 00:03:30,295 --> 00:03:31,295 one. 39 00:03:34,470 --> 00:03:42,690 Two x squared plus eight x plus eight minus one. 40 00:03:50,590 --> 00:03:52,600 There is our standard form: 2x²+8x+7. 41 00:03:52,910 --> 00:04:03,124 There are other ways to figure out these properties. There is more than one method. 42 00:04:03,124 --> 00:04:06,570 Now we know the y-intercept. 43 00:04:07,210 --> 00:04:11,650 seven. This guy right there. Seven. 44 00:04:13,910 --> 00:04:18,570 Okay. Or you could write zero comma seven. Either one's fine. 45 00:04:19,090 --> 00:04:22,130 Factor form. How are we going to do that? 46 00:04:26,630 --> 00:04:34,170 Okay. Okay. Yeah. Let's find the zeros, which are when we set this equal to zero. 47 00:04:36,930 --> 00:04:40,730 Go zeros. We set Y equal to zero. 48 00:04:45,090 --> 00:04:48,412 Solve. You can either solve by factoring, but I don't know if this is going to 49 00:04:48,412 --> 00:04:49,412 factor. 50 00:04:50,030 --> 00:04:54,310 Okay. In fact, if we can't figure out what those are, it's not going to factor 51 00:04:54,310 --> 00:04:57,000 easily. So we got to use quadratic formula. 52 00:05:03,030 --> 00:05:08,050 64 minus 56. 53 00:05:08,870 --> 00:05:10,850 Is that rad eight? Yeah. 54 00:05:13,650 --> 00:05:19,150 And rad eight simplifies to two rad two. 55 00:05:26,730 --> 00:05:33,970 And we could divide them both by two. 56 00:05:36,690 --> 00:05:46,159 Divide: negative eight over four, plus or minus two root two over four. Negative 57 00:05:46,159 --> 00:05:48,290 two, plus or minus 58 00:05:50,170 --> 00:05:53,870 root two over two. Is that right? 59 00:05:54,470 --> 00:05:59,670 You could write it all as one big fraction or two separate numbers like I did. 60 00:06:01,530 --> 00:06:04,410 So those are our zeros. 61 00:06:10,490 --> 00:06:14,410 Negative two plus rad two over two. 62 00:06:16,570 --> 00:06:20,170 And negative two minus rad two over two. 63 00:06:20,830 --> 00:06:26,070 Therefore, we can write it in factor form. What's factor form going to be? 64 00:06:30,050 --> 00:06:34,210 X plus two. 65 00:06:37,410 --> 00:06:41,770 F of X equals the leading coefficient, which is what? 66 00:06:43,510 --> 00:06:55,590 Two times X minus first zero, which so it's going to be X minus this whole thing. 67 00:06:56,890 --> 00:07:08,410 Subtract the whole zero: x minus (−2+√2/2). The first factor is x+2−√2/2. 68 00:07:09,910 --> 00:07:17,010 I will write it once so we do not have to do that again. The factors are x+2−√2/2 69 00:07:17,010 --> 00:07:21,130 and x+2+√2/2, with the leading coefficient two. 70 00:07:21,720 --> 00:07:30,800 Yeah, there's factor form. Last thing is new is called intervals of increase. 71 00:07:31,520 --> 00:07:36,980 So when is the function increasing and when is it decreasing? 72 00:07:44,660 --> 00:07:48,660 This part of the graph, is that increasing or decreasing? 73 00:07:50,860 --> 00:07:52,260 Okay, how do we know? 74 00:07:53,700 --> 00:07:56,840 Okay, what about this half? 75 00:07:58,840 --> 00:08:00,220 What is going up? 76 00:08:02,840 --> 00:08:10,260 Okay, so a function is increasing or decreasing from left to right. 77 00:08:12,680 --> 00:08:20,785 Okay, so here from left to right, the Y values are going down. And from here, left 78 00:08:20,785 --> 00:08:26,320 to right, the Y values are going up. They're increasing. 79 00:08:27,000 --> 00:08:33,140 It has to do with as X is increasing, what is Y doing? 80 00:08:34,980 --> 00:08:45,194 Okay, and the number line, X increases left to right. So as X from here on, as X is 81 00:08:45,194 --> 00:08:48,640 increasing, Y is increasing. 82 00:08:49,580 --> 00:08:58,220 So the interval of increase is going to be from negative two to infinity. 83 00:09:07,380 --> 00:09:11,280 Okay, that function is increasing on that interval. 84 00:09:13,380 --> 00:09:16,513 Okay, we'll talk about Y at negative two later. 85 00:09:17,140 --> 00:09:28,840 And then from left to right, it's decreasing from negative infinity to negative two. 86 00:09:29,380 --> 00:09:34,500 Okay, why do I include negative two in both of them? 87 00:09:40,280 --> 00:09:43,680 Okay, in my saying it's increasing at negative two. 88 00:09:44,880 --> 00:09:49,552 We're not talking about whether a function is increasing, it doesn't increase or 89 00:09:49,552 --> 00:09:50,885 decrease at a point. 90 00:09:52,520 --> 00:09:55,900 Okay, it's only increasing on an interval. 91 00:09:58,200 --> 00:10:03,306 Okay, because everything on that interval, which is an interval of X values, the Y 92 00:10:03,306 --> 00:10:04,906 value is getting larger. 93 00:10:05,900 --> 00:10:11,011 Okay, or the actual definition of increasing is if you have any two numbers in the 94 00:10:11,011 --> 00:10:12,011 set, 95 00:10:15,500 --> 00:10:17,520 okay, let's call it A and B, 96 00:10:19,340 --> 00:10:27,200 a and b, with a less than b, then f(a) is less than f(b). 97 00:10:29,700 --> 00:10:35,340 Okay, so if you take any two numbers in this set, this is going to be true. 98 00:10:37,280 --> 00:10:41,682 Okay, including at negative two, and the only possibility for negative two would be 99 00:10:41,682 --> 00:10:42,682 the A. 100 00:10:43,000 --> 00:10:46,867 Okay, you take any other number, this is going to be true. 101 00:10:47,310 --> 00:10:53,514 Long story short: if that does not make total sense, remember the intervals of 102 00:10:53,514 --> 00:10:54,150 increase 103 00:10:54,150 --> 00:11:05,750 and decrease, we include the endpoints. Okay, okay, let's go to the next problem. 104 00:11:06,350 --> 00:11:12,790 Sorry, what form is this in? Factor form. So where does this tell us immediately? 105 00:11:13,430 --> 00:11:22,790 Zero. Zeros, and they are negative one and three. 106 00:11:23,410 --> 00:11:30,240 Okay, what else do we know automatically? It opens down because the leading 107 00:11:30,240 --> 00:11:32,790 coefficient is negative one. 108 00:11:34,710 --> 00:11:38,777 We need to find some more properties. What should we do next? 109 00:11:39,850 --> 00:11:43,983 The axis of symmetry, we can do that. What's that going to be? 110 00:11:45,090 --> 00:11:49,830 So negative one plus three divided by two, the average of the zeros. 111 00:11:51,990 --> 00:11:53,050 Okay, which is 112 00:11:57,770 --> 00:11:59,570 two over two, which is one. 113 00:12:03,390 --> 00:12:09,876 Okay, so the axis of symmetry is an equation. X equals one. Okay, and how do I find 114 00:12:09,876 --> 00:12:13,470 the vertex now that we have the axis symmetry? 115 00:12:16,230 --> 00:12:17,497 You can plug it in. 116 00:12:21,910 --> 00:12:25,830 Substitute one: one plus one, and one minus three. 117 00:12:25,830 --> 00:12:37,630 Negative two times negative two, which is four. 118 00:12:38,680 --> 00:12:40,260 The vertex is 119 00:12:46,520 --> 00:12:47,340 one four. 120 00:12:47,670 --> 00:12:49,590 We do need the 121 00:12:55,650 --> 00:13:01,502 leading coefficient. Actually, do we know the leading coefficient? Yeah, it's 122 00:13:01,502 --> 00:13:05,530 negative one. So we can write vertex form now, right? 123 00:13:07,690 --> 00:13:15,810 f(x)=−(x−1)²+4. 124 00:13:16,490 --> 00:13:20,390 We know the domain. What's the domain? All real numbers. 125 00:13:22,870 --> 00:13:29,622 And the range, we said it has a maximum. Did we say that already? Yes. The maximum 126 00:13:29,622 --> 00:13:30,622 was the max. 127 00:13:31,190 --> 00:13:41,190 That was four. Therefore, we know the range. Four is the maximum. 128 00:13:41,880 --> 00:13:44,460 We need to get standard form. 129 00:13:47,640 --> 00:13:56,691 Distribute the product, then apply the outside negative sign. Inside the parentheses 130 00:13:56,691 --> 00:13:58,200 are x²−3x+x−3. 131 00:14:08,120 --> 00:14:09,853 That gives positive two x, 132 00:14:12,840 --> 00:14:13,840 plus three. 133 00:14:14,200 --> 00:14:17,240 The y-intercept is three. 134 00:14:27,380 --> 00:14:34,093 And lastly, the interval, where is it increasing? When is it going left to right 135 00:14:34,093 --> 00:14:35,093 uphill? 136 00:14:38,260 --> 00:14:40,060 Negative infinity to one. 137 00:14:49,080 --> 00:14:51,280 And decreasing on the other half.