1 00:00:00,000 --> 00:00:00,540 2 00:00:00,540 --> 00:00:03,870 We're asked to graph this function, finding the roots 3 00:00:03,870 --> 00:00:06,660 and the vertex of this parabola. 4 00:00:06,660 --> 00:00:10,150 So to find the roots, when people talk about roots, 5 00:00:10,150 --> 00:00:12,870 they're talking about the points of intersection of this 6 00:00:12,869 --> 00:00:14,469 function and the x-axis. 7 00:00:14,470 --> 00:00:17,740 So this is the x-axis here, and this is the y-axis. 8 00:00:17,739 --> 00:00:20,109 And you're going to intersect the x-axis when 9 00:00:20,109 --> 00:00:21,480 y is equal to 0. 10 00:00:21,480 --> 00:00:27,730 So to find the roots, we said y is equal to 0, so we solve 11 00:00:27,730 --> 00:00:33,230 the equation, x squared plus 4x minus 12 is equal to 0. 12 00:00:33,229 --> 00:00:35,750 All I did is I said y is equal to 0, and I swapped the left- 13 00:00:35,750 --> 00:00:37,270 and right-hand sides. 14 00:00:37,270 --> 00:00:40,080 Now, to factor this, we have a 1 as a leading coefficient, so 15 00:00:40,079 --> 00:00:42,759 we don't have to do the all out grouping. 16 00:00:42,759 --> 00:00:46,609 We can just think, are there two numbers whose products are 17 00:00:46,609 --> 00:00:49,850 negative 12, and if I add them I get 4? 18 00:00:49,850 --> 00:00:50,960 So let's think about it. 19 00:00:50,960 --> 00:00:56,060 It looks like 2, positive 6, and negative 2 would work, if 20 00:00:56,060 --> 00:00:58,600 I go through all of negative 12's factors. 21 00:00:58,600 --> 00:01:04,540 So this can be factored as x plus 6, times x minus 2. 22 00:01:04,540 --> 00:01:06,310 6 minus 2 is 4. 23 00:01:06,310 --> 00:01:09,500 6 times negative 2 is negative 12, so that's going to be 24 00:01:09,500 --> 00:01:10,519 equal to 0. 25 00:01:10,519 --> 00:01:12,839 And if we have two numbers, and if, when you multiply 26 00:01:12,840 --> 00:01:16,290 them, they equal 0, that means that either x plus 6 is equal 27 00:01:16,290 --> 00:01:21,490 to 0, or x minus 2 is equal to 0. 28 00:01:21,489 --> 00:01:24,599 Subtract 6 from both sides of this equation, you get x is 29 00:01:24,599 --> 00:01:26,429 equal to negative 6. 30 00:01:26,430 --> 00:01:29,790 Add 2 to both sides of this equation, and you get x is 31 00:01:29,790 --> 00:01:31,140 equal to 2. 32 00:01:31,140 --> 00:01:35,010 So y is equal to 0 when x is equal to negative 6, or x is 33 00:01:35,010 --> 00:01:35,910 equal to 2. 34 00:01:35,909 --> 00:01:38,679 So let's graph both of those points. 35 00:01:38,680 --> 00:01:41,130 x is equal to negative 6, y will be equal to 0. 36 00:01:41,129 --> 00:01:44,599 So x is 1, 2, 3, 4, 5, 6. 37 00:01:44,599 --> 00:01:47,599 So negative 6 comma 0. 38 00:01:47,599 --> 00:01:50,939 That will be on the graph of this function, or the graph of 39 00:01:50,939 --> 00:01:52,379 this parabola. 40 00:01:52,379 --> 00:01:56,359 And also the point, x is equal to 2. 41 00:01:56,359 --> 00:01:58,679 So 1, 2. 42 00:01:58,680 --> 00:02:01,840 Just like that. x is equal to 2. 43 00:02:01,840 --> 00:02:03,579 So this is the point 2, 0. 44 00:02:03,579 --> 00:02:06,359 Now to find the x value of the vertex, there's actually 45 00:02:06,359 --> 00:02:07,959 several ways of doing it. 46 00:02:07,959 --> 00:02:09,810 There is a formula that gives it to you. 47 00:02:09,810 --> 00:02:12,479 But actually, the easiest way to figure it out is it's 48 00:02:12,479 --> 00:02:16,239 actually going to be the midpoint between the two 0's. 49 00:02:16,240 --> 00:02:19,280 If you already figured out the 0's, the x value where the 50 00:02:19,280 --> 00:02:24,840 vertex was going to lie-- because the two 0's are always 51 00:02:24,840 --> 00:02:28,629 symmetric around the axis of symmetry, which is the same x 52 00:02:28,629 --> 00:02:30,340 value as the vertex. 53 00:02:30,340 --> 00:02:32,759 You can just find the midpoint of these two points, and that 54 00:02:32,759 --> 00:02:34,500 will be the x value of the vertex. 55 00:02:34,500 --> 00:02:35,900 So let's just do that. 56 00:02:35,900 --> 00:02:38,280 So if we take the midpoint, we can literally just average 57 00:02:38,280 --> 00:02:39,099 these two numbers. 58 00:02:39,099 --> 00:02:44,264 So negative 6 plus 2, over 2 is negative 4 over 2, which is 59 00:02:44,264 --> 00:02:45,719 equal to negative 2. 60 00:02:45,719 --> 00:02:48,680 So the vertex is x is equal to negative 2. 61 00:02:48,680 --> 00:02:51,780 And when x is equal to negative 2, what is y? 62 00:02:51,780 --> 00:02:56,020 So x is the vertex, and then y will be equal to negative 2 63 00:02:56,020 --> 00:02:58,560 squared, is positive 4. 64 00:02:58,560 --> 00:03:01,569 4 times negative 2 is negative 8. 65 00:03:01,569 --> 00:03:04,840 And then we have a minus 12 there. 66 00:03:04,840 --> 00:03:09,280 So 4 minus 8 gets us to negative 4, minus another 12 67 00:03:09,280 --> 00:03:11,939 is negative 16. 68 00:03:11,939 --> 00:03:17,759 So the point negative 2, negative 16 is 69 00:03:17,759 --> 00:03:20,500 going to be our vertex. 70 00:03:20,500 --> 00:03:21,330 So let me graph that. 71 00:03:21,330 --> 00:03:23,060 I think it's going to fall off of this graph a little bit. 72 00:03:23,060 --> 00:03:27,469 So negative 2, and then we're going to go 1, 2, 3, 4, 5, 6, 73 00:03:27,469 --> 00:03:34,750 7, 8, 9, 10 11, 12, 13, 14, 15, 16. 74 00:03:34,750 --> 00:03:36,210 So it's going to be right around here. 75 00:03:36,210 --> 00:03:37,830 We went off the graph paper. 76 00:03:37,830 --> 00:03:47,020 But negative 2 all the way down to negative 16. 77 00:03:47,020 --> 00:03:48,630 So that right there is the vertex. 78 00:03:48,629 --> 00:03:51,579 Negative 2, negative 16. 79 00:03:51,580 --> 00:03:52,550 All the way down here. 80 00:03:52,550 --> 00:03:54,540 We can even lower the y-axis. 81 00:03:54,539 --> 00:03:56,250 And so if I were going to graph the parabola, it would 82 00:03:56,250 --> 00:03:57,689 look something like this. 83 00:03:57,689 --> 00:04:01,900 It would look something like that. 84 00:04:01,900 --> 00:04:04,469 Just like that and obviously it would keep going. 85 00:04:04,469 --> 00:04:06,979 Now, I figured out the vertex in this problem just by 86 00:04:06,979 --> 00:04:09,530 averaging the x values of the two routes. 87 00:04:09,530 --> 00:04:12,699 That gave me the x value of the axis of symmetry, which is 88 00:04:12,699 --> 00:04:16,459 this right here. x is equal to negative 2. 89 00:04:16,459 --> 00:04:19,399 That's the axis around which the parabola is symmetric. 90 00:04:19,399 --> 00:04:21,860 The other way to figure out the x value of the vertex, is 91 00:04:21,860 --> 00:04:25,555 to use the formula, negative b over 2a. 92 00:04:25,555 --> 00:04:29,800 So the x value of the vertex is equal to negative b-- 93 00:04:29,800 --> 00:04:33,230 negative 4-- over 2 times a. 94 00:04:33,230 --> 00:04:36,860 a is 1, so over 2, which is also going to give you x is 95 00:04:36,860 --> 00:04:39,150 equal to negative 2. 96 00:04:39,149 --> 00:04:43,379 Now, the last way to solve for the vertex is to complete the 97 00:04:43,379 --> 00:04:48,790 square on this parabola right here. 98 00:04:48,790 --> 00:04:50,710 So what we can do is we can rewrite it. 99 00:04:50,709 --> 00:04:55,909 We can say this is y is equal to x squared plus 4x. 100 00:04:55,910 --> 00:04:57,955 I'm going to put the minus 12 out here. 101 00:04:57,954 --> 00:05:00,680 And what we want to do is add and subtract the same number, 102 00:05:00,680 --> 00:05:02,889 so I can express this as a perfect square. 103 00:05:02,889 --> 00:05:07,229 So if I take 1/2 of 4x, or if I take 1/2 of 4, I get 2 if I 104 00:05:07,230 --> 00:05:08,439 square that. 105 00:05:08,439 --> 00:05:11,810 I could add a 4 here, and then subtract a 4. 106 00:05:11,810 --> 00:05:18,110 And then what that does is it turns this into x plus 2 107 00:05:18,110 --> 00:05:20,939 squared, right? x plus 2 squared is x squared 108 00:05:20,939 --> 00:05:21,569 plus 4x, plus 4. 109 00:05:21,569 --> 00:05:23,159 We've seen that multiple times. 110 00:05:23,160 --> 00:05:24,870 So you have x plus 2 squared. 111 00:05:24,870 --> 00:05:28,740 And then you have the minus 4 minus 12, minus 16. 112 00:05:28,740 --> 00:05:30,540 So this is just another way of writing 113 00:05:30,540 --> 00:05:31,910 this exact same function. 114 00:05:31,910 --> 00:05:37,790 But when you write it this way you see that the lowest point, 115 00:05:37,790 --> 00:05:43,300 this part of the function right here is always positive. 116 00:05:43,300 --> 00:05:44,650 It's a quantity squared. 117 00:05:44,649 --> 00:05:47,099 The lowest value that it can be is 0. 118 00:05:47,100 --> 00:05:49,040 But then it can only go up from 0. 119 00:05:49,040 --> 00:05:52,840 So the lowest point, the lowest y value we can take on, 120 00:05:52,839 --> 00:05:58,649 is when this expression right here is 0. 121 00:05:58,649 --> 00:06:01,989 And when that happens, the y value is negative 16. 122 00:06:01,990 --> 00:06:05,069 Now, what x makes this value 0? 123 00:06:05,069 --> 00:06:07,579 Well, x equals negative 2 will make this 0. 124 00:06:07,579 --> 00:06:12,310 If you wanted to say, x plus 2 is equal to 0, this expression 125 00:06:12,310 --> 00:06:15,170 right here is going to be 0 when x plus 2 is equal to 0. 126 00:06:15,170 --> 00:06:18,460 Subtract 2 from both sides, x is equal to negative 2. 127 00:06:18,459 --> 00:06:21,729 So that's another way to find the vertex, all of them 128 00:06:21,730 --> 00:06:23,379 equally valid.