1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:03,470 We need to find the vertex and the axis of 3 00:00:03,470 --> 00:00:06,390 symmetry of this graph. 4 00:00:06,389 --> 00:00:07,949 The whole point of doing this problem is so that you 5 00:00:07,950 --> 00:00:10,240 understand what the vertex and axis of symmetry is. 6 00:00:10,240 --> 00:00:13,690 And just as a bit of a refresher, if a parabola looks 7 00:00:13,689 --> 00:00:17,280 like this, the vertex is the lowest point here, so this 8 00:00:17,280 --> 00:00:20,260 minimum point here, for an upward opening problem. 9 00:00:20,260 --> 00:00:23,610 If the parabola opens downward like this, the vertex is the 10 00:00:23,609 --> 00:00:25,710 topmost point right like that. 11 00:00:25,710 --> 00:00:26,970 It's the maximum point. 12 00:00:26,969 --> 00:00:29,109 And the axis of symmetry is the line that you could 13 00:00:29,109 --> 00:00:31,899 reflect the parabola around, and it's symmetric. 14 00:00:31,899 --> 00:00:33,609 So that's the axis of symmetry. 15 00:00:33,609 --> 00:00:37,064 That is a reflection of the left-hand side along that axis 16 00:00:37,064 --> 00:00:37,710 of symmetry. 17 00:00:37,710 --> 00:00:40,329 Same thing if it's a downward-opening parabola. 18 00:00:40,329 --> 00:00:42,369 And the general way of telling the difference between an 19 00:00:42,369 --> 00:00:44,959 upward-opening and a downward-opening parabola is 20 00:00:44,960 --> 00:00:50,420 that this will have a positive coefficient on the x squared 21 00:00:50,420 --> 00:00:53,260 term, and this will have a negative coefficient. 22 00:00:53,259 --> 00:00:54,909 And we'll see that in a little bit more detail. 23 00:00:54,909 --> 00:00:56,459 So let's just work on this. 24 00:00:56,460 --> 00:00:59,450 Now, in order to figure out the vertex, there's a quick 25 00:00:59,450 --> 00:01:02,600 and dirty formula, but I'm not going to do the formula here 26 00:01:02,600 --> 00:01:05,000 because the formula really tells you nothing about how 27 00:01:05,000 --> 00:01:05,540 you got it. 28 00:01:05,540 --> 00:01:07,740 But I'll show you how to apply the formula at the end of this 29 00:01:07,739 --> 00:01:10,244 video, if you see this on a math test and just want to do 30 00:01:10,245 --> 00:01:11,109 it really quickly. 31 00:01:11,109 --> 00:01:14,659 But we're going to do it the slow, intuitive way first. 32 00:01:14,659 --> 00:01:17,939 So let's think about how we can find either the maximum or 33 00:01:17,939 --> 00:01:19,709 the minimum point of this parabola. 34 00:01:19,709 --> 00:01:21,799 So the best way I can think of doing it is to 35 00:01:21,799 --> 00:01:22,899 complete the square. 36 00:01:22,900 --> 00:01:25,400 And it might seem like a very foreign concept right now, but 37 00:01:25,400 --> 00:01:27,490 let's just do it one step at a time. 38 00:01:27,489 --> 00:01:31,339 So I can rewrite this as y is equal to-- well, I can factor 39 00:01:31,340 --> 00:01:32,329 out a negative 2. 40 00:01:32,329 --> 00:01:39,319 It's equal to negative 2 times x squared minus 4x minus 4. 41 00:01:39,319 --> 00:01:41,064 And I'm going to put the minus 4 out here. 42 00:01:41,064 --> 00:01:43,989 And this is where I'm going to complete the square. 43 00:01:43,989 --> 00:01:46,809 Now, what I want to do is express the stuff in the 44 00:01:46,810 --> 00:01:50,430 parentheses as a sum of a perfect square and then some 45 00:01:50,430 --> 00:01:51,610 number over here. 46 00:01:51,609 --> 00:01:54,400 And I have x squared minus 4x. 47 00:01:54,400 --> 00:01:57,969 If I wanted this to be a perfect square, it would be a 48 00:01:57,969 --> 00:02:00,549 perfect square if I had a positive 4 over here. 49 00:02:00,549 --> 00:02:02,869 If I had a positive 4 over there, then this would be a 50 00:02:02,870 --> 00:02:03,870 perfect square. 51 00:02:03,870 --> 00:02:07,070 It would be x minus 2 squared. 52 00:02:07,069 --> 00:02:09,780 And I got the 4, because I said, well, I want whatever 53 00:02:09,780 --> 00:02:12,110 half of this number is, so half of negative 54 00:02:12,110 --> 00:02:13,530 4 is negative 2. 55 00:02:13,530 --> 00:02:14,500 Let me square it. 56 00:02:14,500 --> 00:02:16,469 That'll give me a positive 4 right there. 57 00:02:16,469 --> 00:02:18,469 But I can't just add a 4 willy-nilly to 58 00:02:18,469 --> 00:02:19,680 one side of an equation. 59 00:02:19,680 --> 00:02:21,550 I either have to add it to the other side or I would have to 60 00:02:21,550 --> 00:02:22,900 then just subtract it. 61 00:02:22,900 --> 00:02:24,849 So here I haven't changed equation. 62 00:02:24,849 --> 00:02:26,889 I added 4 and then I subtracted 4. 63 00:02:26,889 --> 00:02:30,049 I just added zero to this little expression here, so it 64 00:02:30,050 --> 00:02:30,880 didn't change it. 65 00:02:30,879 --> 00:02:34,269 But what it does allow me to do is express this part right 66 00:02:34,270 --> 00:02:42,170 here as a perfect square. x squared minus 4x plus 4 is x 67 00:02:42,169 --> 00:02:45,469 minus 2 squared. 68 00:02:45,469 --> 00:02:48,379 It is x minus 2 squared. 69 00:02:48,379 --> 00:02:53,340 And then you have this negative 2 out front 70 00:02:53,340 --> 00:02:55,420 multiplying everything, and then you have a negative 4 71 00:02:55,419 --> 00:02:59,089 minus negative 4, minus 8, just like that. 72 00:02:59,090 --> 00:03:02,490 So you have y is equal to negative 2 times this entire 73 00:03:02,490 --> 00:03:05,200 thing, and now we can multiply out the negative 2 again. 74 00:03:05,199 --> 00:03:06,810 So we can distribute it. 75 00:03:06,810 --> 00:03:11,050 Y is equal to negative 2 times x minus 2 squared. 76 00:03:11,050 --> 00:03:17,420 And then negative 2 times negative 8 is plus 16. 77 00:03:17,419 --> 00:03:19,269 Now, all I did is algebraically 78 00:03:19,270 --> 00:03:21,110 arrange this equation. 79 00:03:21,110 --> 00:03:23,970 But what this allows us to do is think about what the 80 00:03:23,969 --> 00:03:26,370 maximum or minimum point of this equation is. 81 00:03:26,370 --> 00:03:29,250 So let's just explore this a little bit. 82 00:03:29,250 --> 00:03:32,300 This quantity right here, x minus 2 squared, if you're 83 00:03:32,300 --> 00:03:34,360 squaring anything, this is always going to 84 00:03:34,360 --> 00:03:36,380 be a positive quantity. 85 00:03:36,379 --> 00:03:41,219 That right there is always positive. 86 00:03:41,219 --> 00:03:43,300 But it's being multiplied by a negative number. 87 00:03:43,300 --> 00:03:46,210 So if you look at the larger context, if you look at the 88 00:03:46,210 --> 00:03:49,159 always positive multiplied by the negative 2, that's going 89 00:03:49,159 --> 00:03:53,449 to be always negative. 90 00:03:53,449 --> 00:03:59,149 And the more positive that this number becomes when you 91 00:03:59,150 --> 00:04:01,740 multiply it by a negative, the more negative this entire 92 00:04:01,740 --> 00:04:02,990 expression becomes. 93 00:04:02,990 --> 00:04:05,340 94 00:04:05,340 --> 00:04:07,560 So if you think about it, this is going to be a 95 00:04:07,560 --> 00:04:10,009 downward-opening parabola. 96 00:04:10,009 --> 00:04:12,039 We have a negative coefficient out here. 97 00:04:12,039 --> 00:04:15,919 And the maximum point on this downward-opening parabola is 98 00:04:15,919 --> 00:04:21,639 when this expression right here is as small as possible. 99 00:04:21,639 --> 00:04:24,289 If this gets any larger, it's just multiplied by a negative 100 00:04:24,290 --> 00:04:26,629 number, and then you subtract it from 16. 101 00:04:26,629 --> 00:04:29,920 So if this expression right here is 0, then we have our 102 00:04:29,920 --> 00:04:32,490 maximum y value, which is 16. 103 00:04:32,490 --> 00:04:35,250 So how do we get x is equal to 0 here? 104 00:04:35,250 --> 00:04:37,860 Well, the way to get x minus 2 equal to 0-- so 105 00:04:37,860 --> 00:04:39,040 let's just do it. 106 00:04:39,040 --> 00:04:41,400 x minus 2 is equal to 0, so that happens when 107 00:04:41,399 --> 00:04:43,219 x is equal to 2. 108 00:04:43,220 --> 00:04:46,020 So when x is equal to 2, this expression is 0. 109 00:04:46,019 --> 00:04:49,240 0 times a negative number, it's all 0, and then y is 110 00:04:49,240 --> 00:04:52,240 equal to 16. 111 00:04:52,240 --> 00:04:55,319 This is our vertex, this is our maximum point. 112 00:04:55,319 --> 00:04:58,170 We just reasoned through it, just looking at the algebra, 113 00:04:58,170 --> 00:05:01,300 that the highest value this can take on is 16. 114 00:05:01,300 --> 00:05:04,389 As x moves away from 2 in the positive or negative 115 00:05:04,389 --> 00:05:07,419 direction, this quantity right here, it might be negative or 116 00:05:07,420 --> 00:05:09,629 positive, but when you square it, it's going to be positive. 117 00:05:09,629 --> 00:05:11,240 And when you multiply it by negative 2, it's going to 118 00:05:11,240 --> 00:05:13,439 become negative and it's going to subtract from 16. 119 00:05:13,439 --> 00:05:18,009 So our vertex right here is x is equal to 2. 120 00:05:18,009 --> 00:05:20,170 Actually, let's say each of these units are 2. 121 00:05:20,170 --> 00:05:28,060 So this is 2, 4, 6, 8, 10, 12, 14, 16. 122 00:05:28,060 --> 00:05:29,069 So my vertex is here. 123 00:05:29,069 --> 00:05:33,149 That is the absolute maximum point for this parabola. 124 00:05:33,149 --> 00:05:37,019 And its axis of symmetry is going to be along the line x 125 00:05:37,019 --> 00:05:40,469 is equal to 2, along the vertical line x is equal to 2. 126 00:05:40,470 --> 00:05:46,300 That is going to be its axis of symmetry. 127 00:05:46,300 --> 00:05:48,550 And now if we're just curious for a couple of other points, 128 00:05:48,550 --> 00:05:51,060 just because we want to plot this thing, we could say, 129 00:05:51,060 --> 00:05:54,300 well, what happens when x is equal to 0? 130 00:05:54,300 --> 00:05:54,990 That's an easy one. 131 00:05:54,990 --> 00:05:58,420 When x is equal to 0, y is equal to 8. 132 00:05:58,420 --> 00:06:01,930 So when x is equal to 0, we have 1, 2, 3, 4-- oh, well, 133 00:06:01,930 --> 00:06:02,530 these are 2. 134 00:06:02,529 --> 00:06:04,049 2, 4, 6, 8. 135 00:06:04,050 --> 00:06:05,090 It's right there. 136 00:06:05,089 --> 00:06:06,250 This is an axis of symmetry. 137 00:06:06,250 --> 00:06:09,759 So when x is equal to 3, y is also going to be equal to 8. 138 00:06:09,759 --> 00:06:13,389 So this parabola is a really steep and narrow one that 139 00:06:13,389 --> 00:06:16,180 looks something like this, where this right here is the 140 00:06:16,180 --> 00:06:17,370 maximum point. 141 00:06:17,370 --> 00:06:20,259 Now I told you this is the slow and intuitive way to do 142 00:06:20,259 --> 00:06:20,949 the problem. 143 00:06:20,949 --> 00:06:23,349 If you wanted a quick and dirty way to figure out a 144 00:06:23,350 --> 00:06:25,300 vertex, there is a formula that you can derive it 145 00:06:25,300 --> 00:06:28,329 actually, doing this exact same process we just did, but 146 00:06:28,329 --> 00:06:33,129 the formula for the vertex, or the x-value of the vertex, or 147 00:06:33,129 --> 00:06:37,500 the axis of symmetry, is x is equal to negative b over 2a. 148 00:06:37,500 --> 00:06:40,139 So if we just apply this-- but, you know, this is just 149 00:06:40,139 --> 00:06:42,659 kind of mindless application of a formula. 150 00:06:42,660 --> 00:06:45,130 I wanted to show you the intuition why this formula 151 00:06:45,129 --> 00:06:46,009 even exists. 152 00:06:46,009 --> 00:06:49,240 But if you just mindlessly apply this, you'll get-- 153 00:06:49,240 --> 00:06:50,490 what's b here? 154 00:06:50,490 --> 00:06:55,340 So x is equal to negative-- b here is 8. 155 00:06:55,339 --> 00:06:59,239 8 over 2 times a. 156 00:06:59,240 --> 00:07:01,689 a right here is a negative 2. 157 00:07:01,689 --> 00:07:03,519 2 times negative 2. 158 00:07:03,519 --> 00:07:05,269 So what is that going to be equal to? 159 00:07:05,269 --> 00:07:10,229 It is negative 8 over negative 4, which is equal to 2, which 160 00:07:10,230 --> 00:07:13,390 is the exact same thing we got by reasoning it out. 161 00:07:13,389 --> 00:07:16,399 And when x is equal to 2, y is equal to 16. 162 00:07:16,399 --> 00:07:18,089 Same exact result there. 163 00:07:18,089 --> 00:07:21,250 That's the point 2 comma 16. 164 00:07:21,250 --> 00:07:22,065