1 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 2 00:00:11,956 --> 00:00:12,690 equation. 3 00:00:13,330 --> 00:00:20,750 A linear equation. Why is it called a linear equation? Because the leading term 4 00:00:23,810 --> 00:00:33,890 has degree one. We have two different polynomials here, one on each side. 5 00:00:35,350 --> 00:00:41,325 All equations have an expression on the left and an expression on the right. Both of 6 00:00:41,325 --> 00:00:42,890 these are polynomials. 7 00:00:44,750 --> 00:00:54,134 This is what type of polynomial? Linear. It happens to be a binomial. This is a 8 00:00:54,134 --> 00:01:03,043 what? Constant. The degree is one, the degree is zero. So the larger degree 9 00:01:03,043 --> 00:01:08,270 determines what we call the entire equation. 10 00:01:08,270 --> 00:01:18,146 So this is called linear equation. We could call it a linear polynomial equation, 11 00:01:18,146 --> 00:01:27,413 but we just linear implies it's a polynomial. Why is this called a quadratic 12 00:01:27,413 --> 00:01:28,510 equation? 13 00:01:32,090 --> 00:01:38,829 The polynomial on the left is what type of polynomial? Quadratic. The polynomial on 14 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 15 00:01:45,486 --> 00:01:48,490 the whole thing a quadratic equation. 16 00:01:55,690 --> 00:02:04,636 Are the solutions to these two equations, are they numbers or variables? Numbers. 17 00:02:04,636 --> 00:02:10,490 Why are they just numbers? There's only one variable. 18 00:02:10,490 --> 00:02:17,764 So this is a linear equation in one variable. This is a quadratic equation in one 19 00:02:17,764 --> 00:02:25,037 variable. The solutions are numbers. And how would we solve this one? Isolate the 20 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 21 00:02:31,952 --> 00:02:32,670 options. 22 00:02:32,670 --> 00:02:46,186 Quadratic formula, factoring, and completing the square. Now I've got two more 23 00:02:46,186 --> 00:02:57,970 equations. Are the solutions to these two numbers? Are they numbers? 24 00:02:58,970 --> 00:03:01,230 No, why not? 25 00:03:01,650 --> 00:03:07,470 So the solutions are ordered pairs. 26 00:03:08,470 --> 00:03:15,630 What do we call this type of equation? 27 00:03:22,890 --> 00:03:33,250 This is what type of polynomial? Linear. What is this type of polynomial? Linear. 28 00:03:35,190 --> 00:03:45,350 But because y equals some other expression of x, we call this a linear function. 29 00:03:54,150 --> 00:04:00,106 And this is the same exact equation except using different notation. Y and f of x 30 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 31 00:04:06,209 --> 00:04:08,930 this notation? You might not like it. 32 00:04:12,890 --> 00:04:15,010 Or is there a reason? 33 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 34 00:04:31,185 --> 00:04:33,630 2. Y equals 13. 35 00:04:38,930 --> 00:04:43,650 And let me plug in 5 into this equation. 36 00:04:51,770 --> 00:05:00,670 We get f of 5 equals 13. Why is this better than this? 37 00:05:02,630 --> 00:05:04,070 Does this show you what polynomial? 38 00:05:05,490 --> 00:05:08,790 Sure. Does this show you what we plug in to get 5? 39 00:05:09,630 --> 00:05:17,050 Well, not what variable, but what value of x goes with this y value? 40 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 41 00:05:26,078 --> 00:05:33,526 value we plugged in unless you look back here and know that that was where we 42 00:05:33,526 --> 00:05:34,590 plugged in. 43 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 44 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 45 00:05:46,555 --> 00:05:47,730 notation better. 46 00:05:47,730 --> 00:05:48,730 Does that make sense? 47 00:05:49,020 --> 00:05:57,876 Typically, once we got y equals, we would call this a quadratic function, which 48 00:05:57,876 --> 00:06:01,800 tells us the two variable equation. 49 00:06:04,660 --> 00:06:10,524 And this is the same equation, same function as the same solution. I'm just using 50 00:06:10,524 --> 00:06:11,900 different notation. 51 00:06:12,900 --> 00:06:22,440 How many solutions does this guy have? 52 00:06:25,000 --> 00:06:29,840 Uh-oh. One or two? I heard infinitely many. 53 00:06:33,200 --> 00:06:38,140 Oh boy. We just found one solution. What was it? 5, 13? 54 00:06:40,120 --> 00:06:46,500 How could I find another one? Plug in any x value. Let's go with zero. 55 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. 56 00:06:58,000 --> 00:07:00,900 So this linear equation has how many solutions? 57 00:07:02,260 --> 00:07:07,880 Infinitely many. You can plug in any value for x and get a y value. 58 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? 59 00:07:22,540 --> 00:07:25,880 Or how do I write the solutions? And we graph them. 60 00:07:31,420 --> 00:07:42,366 And that linear function, that linear equation, that graph are all of its, or is the 61 00:07:42,366 --> 00:07:44,060 solution set. 62 00:07:44,720 --> 00:07:49,341 Every ordered pair on that line, red line, is a solution. Every ordered pair not on 63 00:07:49,341 --> 00:07:50,900 that line is not a solution. 64 00:07:50,900 --> 00:07:53,080 This make sense? 65 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 66 00:08:01,235 --> 00:08:02,040 set here? 67 00:08:02,520 --> 00:08:02,740 Graph it. 68 00:08:03,120 --> 00:08:05,380 We're going to graph the solution set. 69 00:08:07,320 --> 00:08:11,700 Uh-oh. How are we going to graph this? 70 00:08:14,520 --> 00:08:15,160 Y intercept. 71 00:08:19,040 --> 00:08:21,120 We do know where the y intercept, where's the y intercept? 72 00:08:21,580 --> 00:08:22,480 Negative 4. 73 00:08:22,480 --> 00:08:24,520 Negative 4. Plug in zero. 74 00:08:24,820 --> 00:08:24,980 Zero. 75 00:08:26,120 --> 00:08:26,480 Now what? 76 00:08:27,800 --> 00:08:31,860 For all of this. Plug in the number for x. We get like two points. 77 00:08:34,480 --> 00:08:37,080 We can plug another number in. I like this. What's another number? 78 00:08:37,500 --> 00:08:37,620 One. 79 00:08:45,160 --> 00:08:45,840 What is that? 80 00:08:50,060 --> 00:08:50,420 Negative 7? 81 00:08:51,740 --> 00:08:53,160 I was negative 8. 82 00:08:53,920 --> 00:08:54,840 Negative 8? I mean? 83 00:08:55,080 --> 00:08:55,300 Yeah. 84 00:08:55,580 --> 00:08:55,780 Yes. 85 00:08:57,080 --> 00:08:57,440 Okay. 86 00:08:57,640 --> 00:08:58,580 Try negative 1? 87 00:08:58,900 --> 00:08:59,100 Yeah. 88 00:08:59,880 --> 00:09:00,140 Sure. 89 00:09:07,760 --> 00:09:10,880 What is that? 1 plus 5? 90 00:09:13,220 --> 00:09:15,040 6 minus 4 is 2. Is that right? 91 00:09:16,680 --> 00:09:17,800 We need to do this, please. 92 00:09:18,680 --> 00:09:21,620 Okay, now we're there. Uh-oh. Do they fall on the line? 93 00:09:22,320 --> 00:09:23,280 No, it is going to curve. 94 00:09:24,640 --> 00:09:26,220 They definitely don't fall on the line. 95 00:09:27,780 --> 00:09:30,580 There's a reason we call this one a linear equation. 96 00:09:31,540 --> 00:09:34,840 Because all the ordered pairs fall on the line. 97 00:09:34,840 --> 00:09:39,080 It's definitely not a linear equation. 98 00:09:41,320 --> 00:09:44,600 Uh, it's some curve that's not a line. 99 00:09:45,080 --> 00:09:48,280 We do have a specific name for that curve. We'll tell you later. 100 00:09:48,860 --> 00:09:55,840 Before we graph this one, let's graph the mother of all quadratic functions. 101 00:09:56,430 --> 00:10:00,110 Why did I call it the mother of all quadratic functions? 102 00:10:01,310 --> 00:10:03,210 It's actually known as the parent function. 103 00:10:04,910 --> 00:10:04,930 Okay. 104 00:10:06,070 --> 00:10:12,328 Uh, once we get the graph of this, all quadratics are going to look very, very 105 00:10:12,328 --> 00:10:12,970 similar. 106 00:10:14,070 --> 00:10:17,070 They're just shifted and stretched a little bit. 107 00:10:17,810 --> 00:10:17,970 Okay. 108 00:10:18,410 --> 00:10:22,550 So this is the, called the parent function of all quadratic functions. 109 00:10:22,550 --> 00:10:29,330 And let's just, uh, just do a bunch of ordered pairs. 110 00:10:31,050 --> 00:10:40,830 x and f of x. Or x and y. Or f and x and x squared. 111 00:10:43,350 --> 00:10:45,450 Or the y coordinate, same thing. 112 00:10:46,570 --> 00:10:47,950 Uh, let's start with zero. 113 00:10:49,130 --> 00:10:49,470 Zero. 114 00:10:49,790 --> 00:10:50,610 What is zero squared? 115 00:10:51,270 --> 00:10:51,390 Zero. 116 00:10:53,350 --> 00:10:53,950 Zero. 117 00:10:53,950 --> 00:11:01,490 Uh, let me do this over here. 118 00:11:05,090 --> 00:11:09,270 Uh, let me call this f of x. 119 00:11:10,510 --> 00:11:11,630 Zero squared zero. 120 00:11:12,350 --> 00:11:13,430 Uh, what is one squared? 121 00:11:15,490 --> 00:11:16,410 Two squared? 122 00:11:17,390 --> 00:11:17,990 Four. 123 00:11:17,990 --> 00:11:18,130 Four. 124 00:11:18,510 --> 00:11:19,170 Three squared? 125 00:11:19,170 --> 00:11:19,990 Nine. 126 00:11:21,250 --> 00:11:21,830 Nine. 127 00:11:23,130 --> 00:11:24,090 Four squared? 128 00:11:24,770 --> 00:11:25,110 Sixteen. 129 00:11:25,230 --> 00:11:25,630 Sixteen. 130 00:11:26,950 --> 00:11:27,170 Okay. 131 00:11:28,150 --> 00:11:29,910 And let's plug in some negative values. 132 00:11:30,610 --> 00:11:31,450 Let's go negative one. 133 00:11:32,190 --> 00:11:32,410 One. 134 00:11:33,490 --> 00:11:34,090 One. 135 00:11:34,610 --> 00:11:35,010 Negative two? 136 00:11:39,350 --> 00:11:40,650 Negative three? 137 00:11:43,190 --> 00:11:46,050 I don't know why I'm putting them in parentheses. 138 00:11:49,290 --> 00:11:50,130 Square. 139 00:11:54,070 --> 00:11:57,450 So these are just a few of the infinite solutions. 140 00:11:58,190 --> 00:11:59,470 Let's go ahead and plot them. 141 00:12:02,450 --> 00:12:04,150 One squared is one. 142 00:12:04,670 --> 00:12:05,650 Two squared is four. 143 00:12:07,110 --> 00:12:09,510 Three squared is nine. 144 00:12:26,690 --> 00:12:27,970 Four squared is sixteen. 145 00:12:28,230 --> 00:12:28,830 That's off our graph. 146 00:12:31,470 --> 00:12:35,330 And it's a very smooth curve. 147 00:12:46,290 --> 00:12:47,990 Ah, I screwed up. 148 00:13:03,490 --> 00:13:05,810 It's almost like a U looking thing. 149 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 150 00:13:14,987 --> 00:13:16,490 of x equals x squared. 151 00:13:20,790 --> 00:13:24,050 All quadratic are going to look something like this, a U. 152 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 153 00:13:32,047 --> 00:13:33,010 and so on. 154 00:13:34,410 --> 00:13:38,010 So this specific shape is called a parabola. 155 00:13:39,210 --> 00:13:41,050 The shape is called... 156 00:13:46,910 --> 00:13:47,590 Okay. 157 00:13:48,150 --> 00:13:49,030 It's called a parabola. 158 00:13:49,030 --> 00:13:54,451 Only quadratics form parabolas. Just because another function looks like a U does 159 00:13:54,451 --> 00:13:55,990 not make it a parabola. 160 00:13:58,750 --> 00:14:00,210 It's not a parabola. 161 00:14:04,150 --> 00:14:10,030 What other properties, just by looking at this, does this have... 162 00:14:10,430 --> 00:14:13,370 Does this have a minimum value or a maximum value? 163 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. 164 00:14:21,250 --> 00:14:23,770 This point's called the vertex. 165 00:14:25,670 --> 00:14:26,290 Okay. 166 00:14:26,830 --> 00:14:31,650 It's the same name as what we call these things in a triangle. 167 00:14:31,650 --> 00:14:34,690 Vertex, vertices, plural. 168 00:14:36,790 --> 00:14:39,970 When we graph absolute value functions, what do we call this? 169 00:14:41,190 --> 00:14:41,510 The vertex. 170 00:14:42,210 --> 00:14:42,410 Okay. 171 00:14:43,590 --> 00:14:46,750 Parabolas have a vertex as well. 172 00:14:48,410 --> 00:14:53,310 This case is 0, 0 is the vertex. 173 00:14:56,170 --> 00:14:59,670 Is this a symmetric function? 174 00:15:00,450 --> 00:15:00,610 Yes. 175 00:15:00,610 --> 00:15:01,770 It is. 176 00:15:04,550 --> 00:15:09,110 Symmetric functions have what we call an axis of symmetry. 177 00:15:21,370 --> 00:15:23,490 I already spelled it. 178 00:15:25,290 --> 00:15:25,850 Okay. 179 00:15:26,570 --> 00:15:30,070 This one's axis of symmetry, so it's a vertical line. 180 00:15:31,190 --> 00:15:33,410 How would we describe the vertical line? 181 00:15:36,210 --> 00:15:37,450 You guys did... 182 00:15:37,450 --> 00:15:42,410 When you did linear functions later on, this vertical line is not a function. 183 00:15:42,990 --> 00:15:43,410 It's a line. 184 00:15:43,810 --> 00:15:45,650 What was the equation for this one right here? 185 00:15:47,830 --> 00:15:50,930 For this one in red is x equals negative 6. 186 00:15:51,990 --> 00:15:52,230 Okay. 187 00:15:52,230 --> 00:15:55,210 All vertical lines is x equals a number. 188 00:15:56,190 --> 00:16:03,210 It's kind of like 0, y plus x equals negative 6. 189 00:16:05,870 --> 00:16:08,310 So what is the axis of symmetry for this quadratic? 190 00:16:10,150 --> 00:16:10,850 Not zero. 191 00:16:11,710 --> 00:16:12,710 x equals zero. 192 00:16:13,010 --> 00:16:14,270 It's an equation. 193 00:16:14,670 --> 00:16:15,630 It's not a number. 194 00:16:16,810 --> 00:16:16,830 Okay. 195 00:16:17,370 --> 00:16:20,610 That vertical line is x equals zero. 196 00:16:23,110 --> 00:16:24,150 The third... 197 00:16:24,150 --> 00:16:25,130 Well, we'll talk about it. 198 00:16:25,330 --> 00:16:26,490 Don't let me forget the domain range. 199 00:16:30,830 --> 00:16:34,390 These have what we call symmetric points. 200 00:16:46,050 --> 00:16:46,570 Okay. 201 00:16:47,690 --> 00:16:48,990 And what are symmetric points? 202 00:16:48,990 --> 00:16:54,430 There are two points that are equal distance from the axis of symmetry. 203 00:16:56,990 --> 00:16:59,230 So this is of distance two away. 204 00:16:59,730 --> 00:17:02,750 This is the same distance two away from the axis of symmetry. 205 00:17:04,670 --> 00:17:08,830 So any functions that are symmetric have symmetrical points, 206 00:17:09,310 --> 00:17:13,070 and they're symmetric about the axis of symmetry. 207 00:17:14,010 --> 00:17:15,110 Those are the main properties. 208 00:17:15,110 --> 00:17:18,970 What is the domain of this function? 209 00:17:25,370 --> 00:17:28,790 Domain is the set of all x coordinates. 210 00:17:30,230 --> 00:17:32,330 Are there any x values we can't plug in here? 211 00:17:33,150 --> 00:17:35,690 No, we can plug in any real number. 212 00:17:36,590 --> 00:17:36,610 Okay. 213 00:17:37,470 --> 00:17:43,630 So the domain is all real numbers or the set of real numbers. 214 00:17:46,630 --> 00:17:51,050 But the range here is not all real numbers. 215 00:17:52,110 --> 00:17:52,170 Okay. 216 00:17:53,150 --> 00:17:54,510 This has a minimum. 217 00:17:55,550 --> 00:17:57,250 What is the minimum value? 218 00:18:00,030 --> 00:18:02,590 Minimums and maximums are numbers. 219 00:18:03,330 --> 00:18:04,110 Zero is the minimum. 220 00:18:05,210 --> 00:18:08,090 So when we do our interval notation of the range, 221 00:18:08,090 --> 00:18:09,850 it has a minimum. 222 00:18:11,590 --> 00:18:13,550 And is zero in the range? 223 00:18:13,550 --> 00:18:14,050 Yes. 224 00:18:14,210 --> 00:18:14,630 It is. 225 00:18:15,870 --> 00:18:15,990 Okay. 226 00:18:16,170 --> 00:18:16,910 Is there a maximum? 227 00:18:17,830 --> 00:18:18,970 No, it goes up forever. 228 00:18:19,510 --> 00:18:22,290 So zero to infinity, including zero. 229 00:18:22,850 --> 00:18:24,050 It's the range. 230 00:18:32,490 --> 00:18:33,210 Okay. 231 00:18:34,270 --> 00:18:35,690 That's all the properties. 232 00:18:37,120 --> 00:18:48,440 me do minus four. 233 00:18:53,060 --> 00:18:54,440 Is this a quadratic function? 234 00:18:54,440 --> 00:18:55,320 Yes. 235 00:18:57,640 --> 00:19:00,760 Y equals or F of X equals some quadratic. 236 00:19:06,440 --> 00:19:08,120 How are we going to graph this? 237 00:19:13,840 --> 00:19:16,060 We could start by plugging in a bunch of values. 238 00:19:17,220 --> 00:19:17,900 Okay. 239 00:19:22,880 --> 00:19:24,920 Let me go backwards. 240 00:19:25,540 --> 00:19:26,500 I'll come back to this. 241 00:19:26,840 --> 00:19:27,340 You don't need reds. 242 00:19:36,480 --> 00:19:41,860 We graphed lots of absolute-value functions a while ago. 243 00:19:42,220 --> 00:19:42,600 Remember this? 244 00:19:42,600 --> 00:19:42,740 Six. 245 00:19:44,320 --> 00:19:47,120 Did we just plug in x values to find y? 246 00:19:47,720 --> 00:19:48,000 No. 247 00:19:48,220 --> 00:19:48,360 No. 248 00:19:48,480 --> 00:19:49,660 How did we go about graphing it? 249 00:19:53,560 --> 00:19:55,260 The plus three is like minus. 250 00:19:56,440 --> 00:19:57,540 You just got to do the thing. 251 00:19:57,820 --> 00:19:58,700 Do the thing? 252 00:19:59,340 --> 00:20:00,700 What do we call the thing? 253 00:20:03,040 --> 00:20:06,380 We looked at the different transformations, right? 254 00:20:09,380 --> 00:20:14,800 When you replace x with x plus three, where is that shifted? 255 00:20:15,920 --> 00:20:17,820 To the left by three. 256 00:20:18,560 --> 00:20:23,060 So it shifts absolute value of X left by three. 257 00:20:23,880 --> 00:20:29,520 And if you add or subtract a number outside of the function, what does that do? 258 00:20:30,340 --> 00:20:30,640 Lowers down. 259 00:20:31,060 --> 00:20:31,540 Lowers down. 260 00:20:32,360 --> 00:20:35,860 So this goes down by five. 261 00:20:40,300 --> 00:20:42,680 And was there any stretch to it? 262 00:20:42,920 --> 00:20:43,100 No. 263 00:20:43,860 --> 00:20:48,140 No reflection or stretch because the coefficient from the function is a one. 264 00:20:48,900 --> 00:20:49,000 Okay. 265 00:20:49,020 --> 00:20:50,200 So we found the vertex. 266 00:20:51,540 --> 00:20:51,940 Left three. 267 00:20:53,400 --> 00:20:53,780 Down five. 268 00:20:55,560 --> 00:20:56,020 The vertex. 269 00:20:56,260 --> 00:20:57,180 And now what do we do? 270 00:21:00,480 --> 00:21:00,840 Draw. 271 00:21:01,180 --> 00:21:01,960 Draw what? 272 00:21:02,860 --> 00:21:03,220 Two lines. 273 00:21:03,600 --> 00:21:04,940 Yeah, this was a piecewise function. 274 00:21:07,980 --> 00:21:09,880 This half has a slope of what? 275 00:21:11,960 --> 00:21:14,320 This half has a slope of negative one. 276 00:21:14,320 --> 00:21:14,440 Negative one. 277 00:21:16,320 --> 00:21:20,320 What if this were two? 278 00:21:21,500 --> 00:21:21,820 What would we do? 279 00:21:23,640 --> 00:21:24,280 Well, let's move on. 280 00:21:24,880 --> 00:21:26,600 Two is positive two. 281 00:21:26,600 --> 00:21:30,760 Yeah, this stretches this function by a factor of two. 282 00:21:32,140 --> 00:21:32,180 Okay. 283 00:21:32,800 --> 00:21:35,760 Because they're linear, we just change the slopes. 284 00:21:35,760 --> 00:21:37,760 We're going to do something different with quadratics. 285 00:21:39,040 --> 00:21:40,080 We're going to do the same thing. 286 00:21:40,180 --> 00:21:40,760 We're going to use transformations. 287 00:21:42,980 --> 00:21:43,020 Okay. 288 00:21:44,640 --> 00:21:48,100 The first thing we're going to do is find the vertex. 289 00:21:51,620 --> 00:21:54,060 Well, actually, we'll talk about that in a second. 290 00:21:54,700 --> 00:21:55,880 For x squared, right? 291 00:21:56,500 --> 00:21:57,840 The vertex is a zero zero. 292 00:21:59,440 --> 00:21:59,860 Okay. 293 00:22:01,200 --> 00:22:03,260 What is this minus four going to do? 294 00:22:03,740 --> 00:22:04,140 Four. 295 00:22:04,580 --> 00:22:05,020 Five. 296 00:22:07,700 --> 00:22:10,160 All this is going to do is take the graph of x squared. 297 00:22:11,420 --> 00:22:12,640 It goes down by four. 298 00:22:14,220 --> 00:22:16,100 So the vertex is going to be (0, negative 4). 299 00:22:22,120 --> 00:22:22,800 The vertex. 300 00:22:28,000 --> 00:22:28,440 Okay. 301 00:22:31,960 --> 00:22:40,440 What I like to do is instead of graphing this. 302 00:22:48,140 --> 00:22:51,680 We're going to follow a pattern. 303 00:22:52,960 --> 00:22:54,240 I'll show you in a second. 304 00:22:56,220 --> 00:22:57,440 And this is my pattern. 305 00:22:59,480 --> 00:23:00,860 I'm calling the pattern. 306 00:23:03,440 --> 00:23:04,880 Zero squared is zero. 307 00:23:05,100 --> 00:23:05,920 One squared is one. 308 00:23:06,100 --> 00:23:07,440 Two squared is four. 309 00:23:08,480 --> 00:23:09,760 Three squared is nine. 310 00:23:15,220 --> 00:23:20,860 We could subtract four from all the y's. 311 00:23:24,720 --> 00:23:25,320 Zero. 312 00:23:25,740 --> 00:23:27,340 Zero minus four is negative four. 313 00:23:29,500 --> 00:23:31,200 One minus four. 314 00:23:31,900 --> 00:23:32,860 Four minus four. 315 00:23:34,700 --> 00:23:35,400 Oops. 316 00:23:35,740 --> 00:23:37,260 Yeah. Nine minus four. 317 00:23:38,220 --> 00:23:39,000 The negative four. 318 00:23:39,360 --> 00:23:39,940 Negative three. 319 00:23:40,880 --> 00:23:41,300 Zero. 320 00:23:45,900 --> 00:23:46,660 Five. 321 00:23:48,140 --> 00:23:53,080 But with graph paper, we do not need to bother with all of this. 322 00:23:54,720 --> 00:23:55,920 I just call this. 323 00:24:02,080 --> 00:24:03,620 This is my vertex. 324 00:24:07,580 --> 00:24:07,620 Okay. 325 00:24:08,980 --> 00:24:09,780 And here's what I do. 326 00:24:11,560 --> 00:24:13,140 I start at the vertex. 327 00:24:14,860 --> 00:24:16,140 This is my starting point for the zero-zero offsets. 328 00:24:17,300 --> 00:24:17,780 And I go. 329 00:24:19,300 --> 00:24:20,360 One to the right. 330 00:24:20,900 --> 00:24:21,160 One up. 331 00:24:22,620 --> 00:24:24,220 So we go one to the right, one up. 332 00:24:24,400 --> 00:24:25,240 This is going to be my next point. 333 00:24:26,480 --> 00:24:27,860 I go back to the vertex. 334 00:24:27,860 --> 00:24:29,820 Go two to the right, four up. 335 00:24:30,820 --> 00:24:33,360 One, two, one, two, three, four. 336 00:24:35,100 --> 00:24:37,240 Three to the right from the vertex, nine up. 337 00:24:38,080 --> 00:24:39,160 One, two, three. 338 00:24:39,820 --> 00:24:43,520 One, two, three, four, five, six, seven, eight, nine. 339 00:24:46,480 --> 00:24:47,680 That's what I call my pattern. 340 00:24:49,100 --> 00:24:51,780 Go to our vertex, which is the zero zero. 341 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, 342 00:24:55,411 --> 00:24:55,820 and so on. 343 00:24:56,600 --> 00:25:00,260 And then this is the right half of the parabola. 344 00:25:01,680 --> 00:25:03,480 What's the axis of symmetry here? 345 00:25:05,240 --> 00:25:05,780 X equals zero. 346 00:25:05,900 --> 00:25:10,280 X equals zero because we did not shift it left and right. 347 00:25:13,080 --> 00:25:19,100 So I usually put a dotted line for my axis of symmetry. 348 00:25:20,700 --> 00:25:25,800 And then we use the fact that it's a line. 349 00:25:25,820 --> 00:25:27,320 It has symmetric points. 350 00:25:28,880 --> 00:25:30,780 So any X value, we go same. 351 00:25:31,700 --> 00:25:33,360 So I went one here and I go one over. 352 00:25:35,040 --> 00:25:39,180 Two, two, three, one, two, three. 353 00:25:40,200 --> 00:25:43,160 And there we got enough points. 354 00:25:44,260 --> 00:25:47,840 So once we find the vertex, we follow the pattern. 355 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. 356 00:25:52,360 --> 00:25:54,500 And then we just graph these symmetric points. 357 00:25:55,820 --> 00:25:59,820 Once we have to add, draw a nice smooth curve. 358 00:26:09,680 --> 00:26:12,640 And there is the graph of the parabola. 359 00:26:12,760 --> 00:26:15,020 That is the solution to it. 360 00:26:15,040 --> 00:26:15,720 It should be quite good. 361 00:26:16,740 --> 00:26:17,520 Make sense? 362 00:26:18,980 --> 00:26:19,320 Okay. 363 00:26:21,060 --> 00:26:22,320 Let us look at another one. 364 00:26:22,650 --> 00:26:25,050 Is this a quadratic function? 365 00:26:25,570 --> 00:26:26,290 Yes. 366 00:26:27,510 --> 00:26:32,410 This is some linear binomial square gives us a quadratic. 367 00:26:32,990 --> 00:26:34,190 And then we subtract three. 368 00:26:34,190 --> 00:26:36,930 This is not written in standard form. 369 00:26:37,150 --> 00:26:38,050 What's standard form? 370 00:26:43,230 --> 00:26:45,790 The form that we use for the quadratic formula. 371 00:26:46,810 --> 00:26:48,930 A x squared plus b x plus c. 372 00:26:49,510 --> 00:26:49,790 Standard form. 373 00:26:50,750 --> 00:26:52,370 We'll talk about this form in a minute. 374 00:26:54,830 --> 00:26:58,330 Where's the shift or function X squared? 375 00:27:00,770 --> 00:27:03,570 Because I replaced X with X minus two. 376 00:27:04,190 --> 00:27:05,190 What's the shift? 377 00:27:06,550 --> 00:27:07,170 Left or right? 378 00:27:08,110 --> 00:27:08,150 Right. 379 00:27:08,790 --> 00:27:11,190 Right by two. 380 00:27:12,750 --> 00:27:18,130 And this shifts it down by three. 381 00:27:21,190 --> 00:27:25,690 So this vertex goes to the right, two, down three. 382 00:27:29,430 --> 00:27:30,190 So the vertex is. 383 00:27:34,730 --> 00:27:36,490 Two, negative three. 384 00:27:38,090 --> 00:27:38,670 Okay. 385 00:27:41,950 --> 00:27:45,470 Once I find the vertex, I follow my pattern. 386 00:27:46,830 --> 00:27:53,350 And the pattern is zero, zero, one, one, two, four, three, 387 00:27:53,350 --> 00:27:53,950 square is nine. 388 00:27:56,130 --> 00:27:57,570 So I go to the vertex. 389 00:27:58,910 --> 00:28:03,770 I go to the right one, up one, right two, 390 00:28:04,190 --> 00:28:11,170 up four, right three, up nine, one, two, three, four, five, 391 00:28:11,330 --> 00:28:12,870 six, seven, eight, nine. 392 00:28:16,530 --> 00:28:17,810 And then what do we do? 393 00:28:25,890 --> 00:28:31,030 We graph these symmetric points along the axis of symmetry. 394 00:28:31,170 --> 00:28:32,790 The axis of symmetry always goes through the vertex. 395 00:28:34,190 --> 00:28:36,250 So I got a symmetric point here. 396 00:28:38,790 --> 00:28:39,570 One, two, one, two. 397 00:28:41,390 --> 00:28:43,650 One, two, three, one, two, three. 398 00:28:45,230 --> 00:28:47,450 And now I just draw a nice, smooth curve. 399 00:28:52,030 --> 00:28:57,630 And the very bottom at the vertex, it becomes very, very flat. 400 00:28:57,790 --> 00:29:00,250 It is not a sharp V. It is very smooth. 401 00:29:01,750 --> 00:29:03,030 So don't draw Vs. 402 00:29:05,190 --> 00:29:06,190 There it is. 403 00:29:14,810 --> 00:29:17,690 Real quick, what's its domain? 404 00:29:22,150 --> 00:29:23,670 Yeah, all real numbers. 405 00:29:25,410 --> 00:29:29,810 Like all polynomials, polynomial functions, the domain is all real numbers. 406 00:29:31,370 --> 00:29:33,510 Does this have a minimum or a maximum? 407 00:29:34,190 --> 00:29:37,350 It says a minimum and the minimum is? 408 00:29:40,270 --> 00:29:40,390 Negative three. 409 00:29:42,830 --> 00:29:46,390 It is always the y coordinate of the vertex. 410 00:29:47,730 --> 00:29:51,350 So therefore the range has a minimum. 411 00:29:53,450 --> 00:29:55,050 Does not have a maximum. 412 00:30:04,190 --> 00:30:05,890 This is not a standard form. 413 00:30:07,170 --> 00:30:09,230 Anybody want to guess what form it's in? 414 00:30:10,950 --> 00:30:12,670 It is called vertex form. 415 00:30:14,290 --> 00:30:17,190 This is called vertex form of a quadratic function. 416 00:30:26,650 --> 00:30:29,310 If it's in this form, it's very easy graphically. 417 00:30:29,670 --> 00:30:30,690 Instantly know the vertex. 418 00:30:31,690 --> 00:30:35,550 It is the opposite of this number, comma that number. 419 00:30:38,190 --> 00:30:45,510 We usually call the vertex h,k. 420 00:30:46,890 --> 00:30:49,390 I do not know why those letters were chosen. 421 00:30:49,990 --> 00:30:50,710 There they are. 422 00:30:51,710 --> 00:31:00,710 So vertex form is y equals a. 423 00:31:03,070 --> 00:31:04,070 We'll talk about it in a second. 424 00:31:04,990 --> 00:31:08,370 X minus h, squared, plus k. 425 00:31:15,510 --> 00:31:20,250 The x coordinate of the vertex is the opposite of this number, which is just h. 426 00:31:23,950 --> 00:31:28,370 The y coordinate is just that, whatever that number is. 427 00:31:33,820 --> 00:31:35,480 Is this the same? 428 00:31:36,400 --> 00:31:38,660 So standard form. 429 00:31:41,940 --> 00:31:47,320 Is y equals a x squared plus b x plus c. 430 00:31:50,620 --> 00:31:54,820 Is this a in vertex form is same as that a in standard form? 431 00:31:55,580 --> 00:31:56,540 It is. 432 00:31:57,240 --> 00:32:03,230 Once you multiply out this binomial squared, you get x squared plus something plus 433 00:32:03,230 --> 00:32:03,960 something. 434 00:32:04,760 --> 00:32:06,200 And then you distribute the a. 435 00:32:07,700 --> 00:32:11,140 It's the same as in standard form. 436 00:32:11,260 --> 00:32:12,580 It is the leading coefficient. 437 00:32:16,000 --> 00:32:17,660 So, let's try another one. 438 00:32:18,020 --> 00:32:19,840 What form is this in? 439 00:32:21,120 --> 00:32:21,700 Vertex form. 440 00:32:23,540 --> 00:32:24,340 Okay. 441 00:32:25,230 --> 00:32:30,490 It takes our x squared, just to the left by three, up by one. 442 00:32:31,210 --> 00:32:32,910 So our vertex is a negative three one. 443 00:32:33,490 --> 00:32:34,210 Once I get my vertex. 444 00:32:35,630 --> 00:32:37,410 Once I get that, what do I do? 445 00:32:44,170 --> 00:32:47,230 Follow the pattern. 446 00:32:47,510 --> 00:32:48,370 What do I call it? 447 00:32:48,370 --> 00:32:52,930 Zero, zero, one squared is one, two squared is four, three squared is nine. 448 00:32:53,770 --> 00:32:54,870 So go to the vertex. 449 00:32:57,770 --> 00:32:59,550 One to the right, one up. 450 00:33:01,010 --> 00:33:03,970 Two to the right, one, two, three, four up. 451 00:33:05,030 --> 00:33:06,590 Three to the right, nine up. 452 00:33:13,570 --> 00:33:16,270 That's going to be the right half of our parabola. 453 00:33:17,270 --> 00:33:19,290 And how do I find the left half? 454 00:33:22,950 --> 00:33:24,670 Use the what points? 455 00:33:25,490 --> 00:33:26,190 Symmetric points. 456 00:33:32,970 --> 00:33:34,510 And draw a nice smooth curve. 457 00:33:34,510 --> 00:33:34,850 Very good. 458 00:33:35,530 --> 00:33:36,130 Yes. 459 00:33:36,130 --> 00:33:36,610 This is a vertex. 460 00:33:37,870 --> 00:33:38,470 Yes. 461 00:33:40,130 --> 00:33:41,110 Therefore the vertex is. 462 00:33:46,930 --> 00:33:50,570 Opposite that number, comma, that number, negative one, negative five. 463 00:33:54,810 --> 00:33:57,250 So go ahead on your graph paper. 464 00:33:58,410 --> 00:33:59,470 Draw a couple axes. 465 00:34:25,510 --> 00:34:28,090 That's what I call my pattern. 466 00:34:29,390 --> 00:34:32,570 It's just zero, one, two and three squared. 467 00:34:37,490 --> 00:34:39,830 This leading coefficient of two. 468 00:34:40,370 --> 00:34:41,710 What is that going to do to it? 469 00:34:43,230 --> 00:34:46,150 It's going to stretch it by a factor of two. 470 00:34:48,770 --> 00:34:49,150 So. 471 00:34:51,690 --> 00:34:54,670 Here's what we do. 472 00:34:59,630 --> 00:35:03,170 We simply multiply these Y values by two. 473 00:35:04,490 --> 00:35:05,970 So times two is. 474 00:35:07,030 --> 00:35:08,570 Let me just do a whole nother one. 475 00:35:09,350 --> 00:35:10,390 X and two x squared. 476 00:35:14,550 --> 00:35:16,930 So zero zero. 477 00:35:18,250 --> 00:35:27,990 Multiplying by two gives the offsets (0,0), (1,2), (2,8), and (3,18). 478 00:35:30,710 --> 00:35:32,930 Now this is my stretched pattern. 479 00:35:35,230 --> 00:35:37,850 This is my vertex right here. 480 00:35:39,690 --> 00:35:42,650 And now I go one to the right, two up. 481 00:35:43,690 --> 00:35:45,370 One to the right, two up. 482 00:35:48,050 --> 00:35:50,810 Two to the right, eight up. 483 00:35:57,690 --> 00:35:59,550 Three to the right, eighteen. 484 00:35:59,950 --> 00:36:00,990 I don't know if we're going to get there. 485 00:36:11,210 --> 00:36:11,750 Oh. 486 00:36:12,750 --> 00:36:13,150 Oh. 487 00:36:14,750 --> 00:36:15,590 It's up here somewhere. 488 00:36:28,390 --> 00:36:29,290 Now what? 489 00:36:31,810 --> 00:36:34,750 We have our right half of our parabola. 490 00:36:35,510 --> 00:36:36,510 We have the symmetric points. 491 00:36:38,070 --> 00:36:40,370 We plot the symmetric points. 492 00:36:41,750 --> 00:36:42,470 About this line. 493 00:37:18,590 --> 00:37:19,690 And there it is. 494 00:37:19,870 --> 00:37:23,030 It's just a skinnier looking parabola. 495 00:37:24,070 --> 00:37:25,550 It's been stretched. 496 00:37:30,750 --> 00:37:31,430 Okay. 497 00:37:31,770 --> 00:37:32,970 Questions on this so far? 498 00:37:34,590 --> 00:37:34,990 Let's try it on. 499 00:37:52,210 --> 00:37:53,710 The vertex is. 500 00:37:56,690 --> 00:37:57,750 Negative one. 501 00:37:58,230 --> 00:37:59,010 The opposite of that number. 502 00:37:59,270 --> 00:37:59,670 Comma three. 503 00:38:03,830 --> 00:38:05,290 This is the vertex. 504 00:38:23,470 --> 00:38:24,150 Okay. 505 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. 506 00:38:33,850 --> 00:38:34,890 What does the negative do? 507 00:38:40,310 --> 00:38:41,870 Negative inside of a function. 508 00:38:42,090 --> 00:38:43,430 Or sorry, outside of a function. 509 00:38:44,350 --> 00:38:45,270 It reflects it. 510 00:38:47,470 --> 00:38:47,570 But. 511 00:38:48,990 --> 00:38:51,090 We take care of that very easily. 512 00:38:53,050 --> 00:38:55,990 X and negative one half X squared. 513 00:38:57,290 --> 00:38:59,050 You just multiply these all by. 514 00:39:00,090 --> 00:39:01,270 Negative one half. 515 00:39:13,710 --> 00:39:18,410 Let me add one more for 16. 516 00:39:20,110 --> 00:39:22,990 Negative one half. 517 00:39:26,790 --> 00:39:32,550 Literally we are just multiplying this guy by negative one half to get this. 518 00:39:32,830 --> 00:39:36,730 We get zero zero one negative one half. 519 00:39:37,870 --> 00:39:43,990 Two negative two three negative nine halves. 520 00:39:44,830 --> 00:39:47,470 Which is negative four point five. 521 00:39:52,450 --> 00:39:57,710 OK, so this is our new pattern. 522 00:39:59,390 --> 00:40:02,470 We multiply the leading coefficient by all the y values in the pattern. 523 00:40:04,670 --> 00:40:06,730 So this is our vertex, the starting point for the zero-zero offsets. 524 00:40:07,670 --> 00:40:09,150 Go one to the right. 525 00:40:10,370 --> 00:40:12,730 But now we go since it's negative. 526 00:40:12,730 --> 00:40:15,570 We go down by a half. 527 00:40:16,850 --> 00:40:18,510 Two to the right down by two. 528 00:40:19,390 --> 00:40:22,090 Three to the right down by four and a half. 529 00:40:23,030 --> 00:40:27,070 We go one to the right down a half. 530 00:40:29,670 --> 00:40:32,150 Two to the right down to. 531 00:40:35,190 --> 00:40:36,570 Three to the right. 532 00:40:37,890 --> 00:40:39,210 Down four and a half. 533 00:40:39,430 --> 00:40:41,390 One, two, three, four. 534 00:40:42,390 --> 00:40:43,030 And a half. 535 00:40:44,930 --> 00:40:46,610 And four to the right, down eight. 536 00:40:46,950 --> 00:40:48,770 One, two, three, four. 537 00:40:57,190 --> 00:41:00,090 Now I have my right hand side. 538 00:41:06,370 --> 00:41:08,470 This is the axis of symmetry. 539 00:41:11,730 --> 00:41:15,190 And a nice curve—do not do that yet. 540 00:41:15,390 --> 00:41:16,910 I need my symmetric points. 541 00:41:49,390 --> 00:41:49,950 This. 542 00:41:51,830 --> 00:41:52,690 Not too bad. 543 00:41:54,070 --> 00:41:55,670 OK, we find the vertex. 544 00:41:57,330 --> 00:42:00,770 We take our pattern zero zero one one two four three nine. 545 00:42:01,190 --> 00:42:02,350 Sometimes add four sixteen. 546 00:42:03,930 --> 00:42:08,350 Then we multiply these y values by the leading coefficient. 547 00:42:08,350 --> 00:42:10,230 This gives us our new pattern. 548 00:42:10,850 --> 00:42:12,650 When it's negative, we just go down. 549 00:42:12,930 --> 00:42:14,030 We always go to the right. 550 00:42:15,750 --> 00:42:18,090 And now we just go down by whatever this is. 551 00:42:18,430 --> 00:42:19,350 If it's negative values. 552 00:42:32,230 --> 00:42:34,750 Now let's do something fun. 553 00:42:35,050 --> 00:42:37,630 This is not a vertex form. 554 00:42:37,950 --> 00:42:41,330 What form is this in? Standard form. 555 00:42:47,850 --> 00:42:51,830 Actually, let me change this. 556 00:43:16,130 --> 00:43:17,070 Actually, let's leave it. 557 00:43:17,190 --> 00:43:17,230 Sorry. 558 00:43:19,010 --> 00:43:20,150 We factor it out. 559 00:43:23,870 --> 00:43:25,070 Let's try it. 560 00:43:25,530 --> 00:43:26,250 What is the factor to. 561 00:43:33,290 --> 00:43:38,210 OK, it could be factored, but we have to use the quadratic formula. 562 00:43:39,170 --> 00:43:42,350 OK, we're going to get two irrational. 563 00:43:44,830 --> 00:43:45,970 Numbers right here. 564 00:43:47,270 --> 00:43:50,810 Actually, let's let's pretend it actually factors it. 565 00:43:53,830 --> 00:43:54,870 It doesn't. 566 00:43:54,970 --> 00:43:55,910 But let's say it did. 567 00:44:04,030 --> 00:44:08,390 What would the numbers three and negative two be? 568 00:44:12,710 --> 00:44:16,210 OK, let's. 569 00:44:16,410 --> 00:44:17,850 If I plug in three into here. 570 00:44:19,030 --> 00:44:28,017 I get three minus three times three plus two, which is equal to zero times five, 571 00:44:28,017 --> 00:44:29,590 which is zero. 572 00:44:32,210 --> 00:44:37,890 OK, so we know three comma zero is a solution. 573 00:44:38,610 --> 00:44:39,750 If I plug in negative two. 574 00:44:43,130 --> 00:44:44,990 Be negative two minus three. 575 00:44:47,170 --> 00:44:50,590 Times negative two plus two. 576 00:44:52,190 --> 00:44:55,030 Negative five times zero, which is zero. 577 00:45:02,650 --> 00:45:05,210 This is not the function, but let's say it was. 578 00:45:13,230 --> 00:45:17,630 So that parabola would now look something like this. 579 00:45:18,250 --> 00:45:22,610 What do we call the points that intersect the X axis? 580 00:45:25,050 --> 00:45:27,210 X intercepts gives us the X intercepts. 581 00:45:27,930 --> 00:45:33,430 So if we're able to factor it, it only gives the X intercepts. 582 00:45:33,430 --> 00:45:35,430 It does give us two symmetric points. 583 00:45:40,670 --> 00:45:45,710 So if these were the irrational factors, it gives the X intercepts. 584 00:45:45,750 --> 00:45:47,770 In fact, that has a specific name. 585 00:45:47,870 --> 00:45:48,530 We'll get to that later. 586 00:45:50,410 --> 00:45:50,730 So. 587 00:45:53,550 --> 00:45:55,230 Just factoring is not going to help. 588 00:46:03,390 --> 00:46:10,430 What do I know the leading coefficient is? 589 00:46:11,710 --> 00:46:13,790 What's the leading coefficient? 590 00:46:14,550 --> 00:46:14,630 One. 591 00:46:15,530 --> 00:46:15,870 OK. 592 00:46:24,490 --> 00:46:27,570 If I could get it into this form, what do we call this form? 593 00:46:28,650 --> 00:46:28,890 Vertex. 594 00:46:29,190 --> 00:46:30,210 Vertex form. 595 00:46:31,110 --> 00:46:34,190 We would know where the vertex was and could easily graph it. 596 00:46:41,270 --> 00:46:43,190 I need to make a perfect square. 597 00:46:45,210 --> 00:46:45,970 Binomial. 598 00:46:48,050 --> 00:46:48,990 From this. 599 00:46:50,570 --> 00:46:53,870 How am I going to make this a perfect-square binomial? 600 00:46:58,070 --> 00:46:59,270 By completing the square. 601 00:47:06,090 --> 00:47:07,770 So here's what we're going to do. 602 00:47:07,970 --> 00:47:08,770 So write this. 603 00:47:10,050 --> 00:47:12,650 X squared minus 2X. 604 00:47:12,650 --> 00:47:13,850 Leave some space. 605 00:47:15,250 --> 00:47:16,030 And put a minus 5. 606 00:47:16,950 --> 00:47:20,510 Why? I'm not going to bring the 5 over here. 607 00:47:22,470 --> 00:47:23,670 Why am I not going to do that? 608 00:47:25,450 --> 00:47:27,310 We don't want 5 plus f of X. 609 00:47:27,790 --> 00:47:29,650 We just want f of X equals. 610 00:47:29,990 --> 00:47:31,950 So we just leave it to the right here. 611 00:47:34,290 --> 00:47:34,450 Oops. 612 00:47:35,210 --> 00:47:35,610 Minus 5. 613 00:48:02,530 --> 00:48:05,190 So what number am I going to add here? 614 00:48:07,410 --> 00:48:09,570 We take this number. 615 00:48:10,950 --> 00:48:12,050 When we complete the square, 616 00:48:12,650 --> 00:48:14,350 Divide by 2. 617 00:48:16,070 --> 00:48:16,650 And then do what with it? 618 00:48:18,810 --> 00:48:20,070 Square it. 619 00:48:22,890 --> 00:48:23,490 Okay. 620 00:48:25,370 --> 00:48:26,750 And I'm going to add 1. 621 00:48:28,370 --> 00:48:30,430 And we're not going to add 1 to both sides. 622 00:48:32,130 --> 00:48:32,830 We could. 623 00:48:33,830 --> 00:48:34,770 You can't just add 1. 624 00:48:35,250 --> 00:48:36,230 But you can add 1. 625 00:48:37,270 --> 00:48:38,710 And then subtract 1. 626 00:48:40,230 --> 00:48:41,370 Just adding 0. 627 00:48:43,210 --> 00:48:45,310 Now this is a perfect square. 628 00:48:47,810 --> 00:48:48,810 And it's going to factor to what? 629 00:48:49,130 --> 00:48:53,250 X minus 1 times X minus 1. 630 00:48:53,770 --> 00:48:56,650 And now the negative 1 minus 5 is negative 6. 631 00:49:07,490 --> 00:49:11,790 Okay. Now it's in vertex form. 632 00:49:13,330 --> 00:49:14,430 Okay. 633 00:49:19,590 --> 00:49:22,210 Now we know the vertex. 634 00:49:24,450 --> 00:49:25,610 What is the vertex? 635 00:49:28,870 --> 00:49:30,650 1 negative 6. 636 00:49:34,770 --> 00:49:36,790 Once I have the vertex. 637 00:49:37,930 --> 00:49:38,190 Who do I do? 638 00:49:41,490 --> 00:49:43,910 We follow the pattern. 639 00:49:47,190 --> 00:49:52,690 0 0 1 1 2 squared is 4 3 squared is 9. 640 00:50:14,570 --> 00:50:19,030 I'm not following the pattern. 641 00:50:20,770 --> 00:50:25,730 9 1 2 3 4 5 6 7 8 1. 642 00:50:28,710 --> 00:50:32,010 I'll get my symmetrical points. 643 00:50:36,970 --> 00:50:37,950 Okay. 644 00:50:50,830 --> 00:50:51,430 Okay. 645 00:50:58,310 --> 00:51:01,730 If you do not like completing the square, there is another option. 646 00:51:04,230 --> 00:51:04,710 Okay. 647 00:51:24,130 --> 00:51:26,110 So where did we get this number from? 648 00:51:26,290 --> 00:51:27,030 The minus 1. 649 00:51:34,650 --> 00:51:36,570 We took this number. 650 00:51:37,790 --> 00:51:38,970 Divided by 2. 651 00:51:39,970 --> 00:51:42,190 We squared it to figure out what we had, right? 652 00:51:43,410 --> 00:51:45,630 But simply that number divided by 2. 653 00:51:47,550 --> 00:51:47,570 Okay. 654 00:51:55,130 --> 00:51:57,710 And then of course, once we get it into here, 655 00:51:59,890 --> 00:52:02,430 we know the vertex is the opposite of that number. 656 00:52:03,810 --> 00:52:04,370 Okay. 657 00:52:04,370 --> 00:52:10,170 So it turns out the vertex, the x coordinate, 658 00:52:13,070 --> 00:52:16,070 is negative b over 2. 659 00:52:19,450 --> 00:52:19,630 Okay. 660 00:52:20,290 --> 00:52:22,610 Unless there's a leading coefficient, 661 00:52:23,890 --> 00:52:26,450 then we'd have to divide everything by the leading coefficient. 662 00:52:27,410 --> 00:52:30,170 So it turns out it's negative b over 2a. 663 00:52:31,330 --> 00:52:33,150 I could show you another reason for that. 664 00:52:38,010 --> 00:52:40,710 And then how do I get the y coordinate? 665 00:52:46,210 --> 00:52:48,930 If we have standard form, 666 00:52:52,450 --> 00:52:54,510 this is the x coordinate. 667 00:52:55,010 --> 00:52:56,370 We simply plug it into the function. 668 00:53:01,150 --> 00:53:01,810 Okay. 669 00:53:02,210 --> 00:53:03,910 So let me show you how to do that. 670 00:53:06,650 --> 00:53:07,890 Let's do another example. 671 00:53:08,380 --> 00:53:10,080 What form is this in? 672 00:53:11,240 --> 00:53:11,560 Standard form. 673 00:53:11,780 --> 00:53:13,520 Actually, let me change this y to f of x. 674 00:53:17,480 --> 00:53:21,040 Standard form directly shows two things. 675 00:53:22,660 --> 00:53:24,100 It tells us the y intercept. 676 00:53:25,920 --> 00:53:27,720 The only other thing it tells us, 677 00:53:27,720 --> 00:53:30,820 does it open up or does it open down? 678 00:53:31,580 --> 00:53:33,140 If the leading coefficient is positive, 679 00:53:33,460 --> 00:53:34,280 it's going to open up. 680 00:53:35,080 --> 00:53:35,840 Leading coefficient is negative. 681 00:53:36,740 --> 00:53:37,120 Okay. 682 00:53:38,160 --> 00:53:48,800 So the vertex is always at negative b over 2a, 683 00:53:48,800 --> 00:53:52,100 f of negative b over 2a. 684 00:53:55,720 --> 00:53:59,800 So negative b over 2a is negative. 685 00:54:01,020 --> 00:54:01,820 What's our b value? 686 00:54:03,320 --> 00:54:03,820 Negative 6. 687 00:54:04,260 --> 00:54:06,460 The negative of negative six, all over two times a. 688 00:54:06,700 --> 00:54:07,200 What's our a? 689 00:54:08,020 --> 00:54:08,420 1. 690 00:54:09,540 --> 00:54:14,600 Which is 6 over 2, which is 3. 691 00:54:16,160 --> 00:54:22,200 So I now know my vertex is at 3, f of 3. 692 00:54:24,600 --> 00:54:26,400 And how do I find my f of 3? 693 00:54:27,820 --> 00:54:29,840 You just plug it into our function. 694 00:54:31,420 --> 00:54:40,060 F of 3 is 3 squared minus 6 times 3 plus 5. 695 00:54:51,580 --> 00:54:53,640 Is that negative 4? 696 00:54:57,080 --> 00:55:00,620 So my vertex is at 3, negative 4. 697 00:55:03,920 --> 00:55:13,540 Once we know that, we know our function in vertex form is the leading coefficient. 698 00:55:13,760 --> 00:55:14,440 What's the leading coefficient? 699 00:55:15,580 --> 00:55:16,380 1. 700 00:55:16,380 --> 00:55:17,020 1. 701 00:55:17,600 --> 00:55:22,140 So it's x minus 3 squared minus 4. 702 00:55:23,300 --> 00:55:26,360 It's the opposite of that number. 703 00:55:29,620 --> 00:55:32,160 And now we can graph it quickly. 704 00:55:37,980 --> 00:55:40,620 We go to 3. 705 00:55:44,580 --> 00:55:45,620 Negative 4. 706 00:55:47,760 --> 00:55:52,280 Since the leading coefficient is positive, we follow the pattern. 707 00:56:19,820 --> 00:56:24,400 I'll write my symmetric points. 708 00:56:34,760 --> 00:56:40,900 So if it's in standard form, 709 00:56:42,960 --> 00:56:45,740 just write it in vertex form by finding the vertex. 710 00:56:46,060 --> 00:56:49,380 The vertex’s x coordinate is always negative b over two a. 711 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 712 00:56:54,208 --> 00:56:54,820 coordinate. 713 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 714 00:57:01,836 --> 00:57:02,360 pattern. 715 00:57:02,540 --> 00:57:05,940 If we have to adjust the pattern because a is not 1, then we do that. 716 00:57:07,440 --> 00:57:08,100 Sound good? 717 00:57:08,400 --> 00:57:09,040 Yes, it is. 718 00:57:09,040 --> 00:57:13,660 Okay, last question before we start homework. 719 00:57:14,140 --> 00:57:16,660 Okay, here, I don't give you an equation. 720 00:57:18,220 --> 00:57:20,100 I give you a graph of a parabola. 721 00:57:21,300 --> 00:57:22,580 How are we going to find the equation? 722 00:57:27,080 --> 00:57:28,440 What is the vertex? 723 00:57:28,440 --> 00:57:33,520 Negative 2, 3. 724 00:57:33,520 --> 00:57:35,440 Negative 2, 3. 725 00:57:36,940 --> 00:57:45,280 Okay, vertex form is a x minus h squared plus k. 726 00:57:50,700 --> 00:57:56,720 So we know y equals a times x plus two, squared, plus three. 727 00:57:57,880 --> 00:58:00,500 And then how do I figure out my leading coefficient? 728 00:58:03,800 --> 00:58:06,280 We know it's negative. 729 00:58:09,740 --> 00:58:11,740 Okay, you see one point. 730 00:58:12,140 --> 00:58:15,380 We go to the right one and we go down one. 731 00:58:17,140 --> 00:58:19,260 That tells you the leading coefficient. 732 00:58:19,560 --> 00:58:21,700 We went down one, leading coefficient negative one. 733 00:58:22,660 --> 00:58:25,100 If we had to go down two, it would be negative two. 734 00:58:27,380 --> 00:58:30,520 Okay, it's all based on going over one. 735 00:58:31,280 --> 00:58:32,020 Do we go up one? 736 00:58:32,080 --> 00:58:32,780 Do we go up two? 737 00:58:32,920 --> 00:58:34,960 Do we go down one, down two, whatever it is. 738 00:58:34,960 --> 00:58:42,620 So we know it is y equals negative one times x plus two, squared, plus three. 739 00:58:43,800 --> 00:58:46,740 If we want standard form, we just have to multiply this out.