1 00:00:00,000 --> 00:00:00,750 2 00:00:00,750 --> 00:00:02,200 We're on problem 36. 3 00:00:02,200 --> 00:00:04,129 It says which of the following sentences is true about the 4 00:00:04,129 --> 00:00:08,230 graphs of y is equal to 3 times x minus 5 squared plus 1 5 00:00:08,230 --> 00:00:12,269 and y equals 3 times x plus 5 squared plus 1? 6 00:00:12,269 --> 00:00:15,119 So let's do something very similar to what we did in the 7 00:00:15,119 --> 00:00:18,429 past. If you think about it, both of these equations, y is 8 00:00:18,429 --> 00:00:20,089 going to be 1 or greater. 9 00:00:20,089 --> 00:00:23,109 Let's just analyze this a little bit. 10 00:00:23,109 --> 00:00:28,140 This term right here, since we're squaring, is always 11 00:00:28,140 --> 00:00:30,170 going to be positive. 12 00:00:30,170 --> 00:00:33,469 Even if what's inside the parentheses becomes negative, 13 00:00:33,469 --> 00:00:36,030 if we have x is minus 10, inside the parenthesis becomes 14 00:00:36,030 --> 00:00:38,179 negative, but when you square it, it 15 00:00:38,179 --> 00:00:38,869 always becomes positive. 16 00:00:38,869 --> 00:00:40,809 You're going to multiply 3 times a positive number, so 17 00:00:40,810 --> 00:00:42,250 you're going to get a positive number. 18 00:00:42,250 --> 00:00:44,560 So the lowest value that this could be is 0. 19 00:00:44,560 --> 00:00:48,420 The lowest value that y could be is actually 1. 20 00:00:48,420 --> 00:00:54,469 The same thing here, this number can become very 21 00:00:54,469 --> 00:00:55,789 negative, but when you square it, it's 22 00:00:55,789 --> 00:00:57,070 going to become positive. 23 00:00:57,070 --> 00:01:00,250 So this expression with the squared here is going to be 24 00:01:00,250 --> 00:01:01,320 positive, and you multiply it by 3, and 25 00:01:01,320 --> 00:01:02,429 it's going to be positive. 26 00:01:02,429 --> 00:01:05,126 So the lowest value here is always going to be zero when 27 00:01:05,126 --> 00:01:08,210 you include this whole term. 28 00:01:08,209 --> 00:01:10,159 So similarly, the lowest value y can be is 1. 29 00:01:10,159 --> 00:01:11,759 I just want to think about it a little bit just give you a 30 00:01:11,760 --> 00:01:13,380 little bit of an intuition. 31 00:01:13,379 --> 00:01:15,750 Let's think of this in the context of what we learned 32 00:01:15,750 --> 00:01:17,281 last time with the shifting. 33 00:01:17,281 --> 00:01:21,390 So let me draw it in a color that you can see. 34 00:01:21,390 --> 00:01:26,219 So if that is the y-axis, and I'll just draw mainly the 35 00:01:26,219 --> 00:01:34,239 positive area, so if I were to just draw y is equal to x 36 00:01:34,239 --> 00:01:39,099 squared plus 1, it would look like this where this is 1. 37 00:01:39,099 --> 00:01:40,119 That's y is equal to 1. 38 00:01:40,120 --> 00:01:43,329 The graph would look something along this. 39 00:01:43,329 --> 00:01:45,599 y is equal to x squared plus 1. 40 00:01:45,599 --> 00:01:47,030 That's a horrible drawing. 41 00:01:47,030 --> 00:01:49,790 Normally, I wouldn't redo it, but that was just atrocious. 42 00:01:49,790 --> 00:01:52,070 y is equal to x squared plus 1 looks something like that. 43 00:01:52,069 --> 00:01:53,539 It's symmetric. 44 00:01:53,540 --> 00:01:54,180 You get the idea. 45 00:01:54,180 --> 00:01:55,890 You've seen these parabolas before. 46 00:01:55,890 --> 00:01:57,969 This is y is equal to x squared plus 1. 47 00:01:57,969 --> 00:02:00,719 48 00:02:00,719 --> 00:02:07,719 Now, if we were to do x minus 5 squared plus 1, 49 00:02:07,719 --> 00:02:08,520 what happens to it? 50 00:02:08,520 --> 00:02:09,189 Well, let me think about it. 51 00:02:09,189 --> 00:02:10,909 What is 3x squared plus 1? 52 00:02:10,909 --> 00:02:12,939 Well, then it just increases a little bit faster. 53 00:02:12,939 --> 00:02:15,340 So if I were say y equals 3x squared plus 1, it might look 54 00:02:15,340 --> 00:02:16,860 something like this. 55 00:02:16,860 --> 00:02:19,670 It'll just increase a little bit faster, three times as 56 00:02:19,669 --> 00:02:21,549 fast actually. 57 00:02:21,550 --> 00:02:24,630 So that would be 3x squared plus 1. 58 00:02:24,629 --> 00:02:26,590 The rate of increase in both directions just goes faster 59 00:02:26,590 --> 00:02:29,680 because you have this constant term 3 out there multiplying 60 00:02:29,680 --> 00:02:30,140 the numbers. 61 00:02:30,139 --> 00:02:33,729 OK, now what happens when you shift it? 62 00:02:33,729 --> 00:02:35,979 So let's do x minus 5. 63 00:02:35,979 --> 00:02:40,829 So where x equals 0 was the minimum point before, now if 64 00:02:40,830 --> 00:02:44,490 we substitute a 5 here, that'll be our minimum point. 65 00:02:44,490 --> 00:02:47,469 Because then that whole term becomes zero. 66 00:02:47,469 --> 00:02:50,784 So this vertex will now be shifted to the right. 67 00:02:50,784 --> 00:02:53,370 Let me do it in another color. 68 00:02:53,370 --> 00:02:56,560 So if this is the point 5, now this would be the graph. 69 00:02:56,560 --> 00:02:58,870 If you just took this graph and you shifted it over to the 70 00:02:58,870 --> 00:03:04,240 right by 5-- I won't draw the whole thing-- that graph right 71 00:03:04,240 --> 00:03:10,230 there would be 3 times x minus 5 squared plus 1. 72 00:03:10,229 --> 00:03:12,989 Remember, the y shift is always intuitive. 73 00:03:12,990 --> 00:03:14,400 If you add 1, you're shifting it up. 74 00:03:14,400 --> 00:03:16,250 If you subtract 1, you're shifting it down. 75 00:03:16,250 --> 00:03:17,099 The x shift isn't. 76 00:03:17,099 --> 00:03:19,799 We subtracted 5, x minus 5. 77 00:03:19,800 --> 00:03:23,080 We replaced x with x minus 5, but we shifted to the right. 78 00:03:23,080 --> 00:03:25,900 The intuition is there, because now plus 5 makes this 79 00:03:25,900 --> 00:03:27,980 expression zero. 80 00:03:27,979 --> 00:03:30,590 So that's 3x minus 5 squared. 81 00:03:30,590 --> 00:03:34,430 In the same logic, 3 times x plus 5 squared is going to be 82 00:03:34,430 --> 00:03:37,830 to here, plus 1. 83 00:03:37,830 --> 00:03:40,719 That's going to be shifted to-- let me pick a good 84 00:03:40,719 --> 00:03:43,740 color-- to the left. 85 00:03:43,740 --> 00:03:44,840 This is going to look something like this. 86 00:03:44,840 --> 00:03:46,890 It's going to be this blue graph shifted to the left. 87 00:03:46,889 --> 00:03:48,369 This is minus 5. 88 00:03:48,370 --> 00:03:53,670 So this is the graph right here of 3 times x plus 5 89 00:03:53,669 --> 00:03:55,569 squared plus 1 90 00:03:55,569 --> 00:03:56,810 Now, hopefully, you have an intuition. 91 00:03:56,810 --> 00:03:59,189 So let's read their statements and see which one makes sense. 92 00:03:59,189 --> 00:04:00,579 Which of the following is true? 93 00:04:00,580 --> 00:04:02,500 Their vertices are maximums. No, that's not 94 00:04:02,500 --> 00:04:03,544 true of any of these. 95 00:04:03,544 --> 00:04:06,609 Because the vertices is that point right there. 96 00:04:06,610 --> 00:04:09,120 It's actually the minimum point. 97 00:04:09,120 --> 00:04:11,900 A maximum point would look something like that. 98 00:04:11,900 --> 00:04:13,539 We know that, because you just go positive. 99 00:04:13,539 --> 00:04:15,509 This term can only be positive. 100 00:04:15,509 --> 00:04:18,399 If this was a negative 3, then it would flip it over. 101 00:04:18,399 --> 00:04:20,220 So it's not choice A. 102 00:04:20,220 --> 00:04:22,820 The graphs have the same shape with different vertices. 103 00:04:22,819 --> 00:04:27,329 Yeah, both of these graphs have the shape of 3x squared, 104 00:04:27,329 --> 00:04:30,370 but 1 vertices is 10 to the left of the other one. 105 00:04:30,370 --> 00:04:31,660 So I think B is our choice. 106 00:04:31,660 --> 00:04:32,860 Let's read the other ones. 107 00:04:32,860 --> 00:04:33,960 The graphs have different shapes 108 00:04:33,959 --> 00:04:34,609 with different vertices. 109 00:04:34,610 --> 00:04:35,879 No, they have the same shape. 110 00:04:35,879 --> 00:04:37,100 They definitely have the same shape. 111 00:04:37,100 --> 00:04:40,040 They both have this 3x squared shape. 112 00:04:40,040 --> 00:04:42,580 One graph has a vertex that is a maximum, while the other 113 00:04:42,579 --> 00:04:43,569 has-- no, that's not right. 114 00:04:43,569 --> 00:04:45,409 They both are upward facing, so they both 115 00:04:45,410 --> 00:04:46,270 have minimum points. 116 00:04:46,269 --> 00:04:49,709 So it's choice B. 117 00:04:49,709 --> 00:04:59,419 Next problem, problem 37. 118 00:04:59,420 --> 00:05:02,530 Let me see what it says. 119 00:05:02,529 --> 00:05:03,964 What are the x-intercepts? 120 00:05:03,964 --> 00:05:05,579 Let me copy and paste that. 121 00:05:05,579 --> 00:05:08,620 122 00:05:08,620 --> 00:05:12,530 OK, I'll paste it there. 123 00:05:12,529 --> 00:05:15,079 What are the x-intercepts of the graph of that? 124 00:05:15,079 --> 00:05:17,189 Well, the x-intercepts, whatever this graph looks 125 00:05:17,189 --> 00:05:18,850 like, I don't know exactly what it looks like. 126 00:05:18,850 --> 00:05:22,730 127 00:05:22,730 --> 00:05:26,470 This graph is going to look something like this. 128 00:05:26,470 --> 00:05:27,800 I actually have no idea what it looks like 129 00:05:27,800 --> 00:05:28,540 until I solve it. 130 00:05:28,540 --> 00:05:29,550 It's going to look something like this. 131 00:05:29,550 --> 00:05:31,819 When they say x-intercepts, they're like, where does it 132 00:05:31,819 --> 00:05:33,120 intersect the x-axis? 133 00:05:33,120 --> 00:05:35,079 So that's like there and there. 134 00:05:35,079 --> 00:05:38,180 I don't know if those are the actual points, right? 135 00:05:38,180 --> 00:05:41,680 To do that, we set the function equal to zero, 136 00:05:41,680 --> 00:05:43,629 because this is the point y is equal to 0. 137 00:05:43,629 --> 00:05:47,439 You're essentially saying when does this function equal zero 138 00:05:47,439 --> 00:05:50,230 because that's the x-axis when y is equal to 0. 139 00:05:50,230 --> 00:05:55,080 So you set y is equal to 0, and you get 0 is equal to 12x 140 00:05:55,079 --> 00:06:00,029 squared minus 5x minus 2. 141 00:06:00,029 --> 00:06:03,259 Whenever I have a coefficient larger than 1 in front of the 142 00:06:03,259 --> 00:06:06,539 x squared term, I find that very hard to just eyeball and 143 00:06:06,540 --> 00:06:08,920 factor, so I use the quadratic equation. 144 00:06:08,920 --> 00:06:12,199 So negative B, this is the B. 145 00:06:12,199 --> 00:06:13,649 B is minus 5. 146 00:06:13,649 --> 00:06:16,310 So negative negative 5 is plus 5. 147 00:06:16,310 --> 00:06:20,720 Negative B plus or minus the square root of B squared, 148 00:06:20,720 --> 00:06:27,530 negative 5 squared is 25, minus 4 times A, which is 12, 149 00:06:27,529 --> 00:06:29,774 times C, which is minus 2. 150 00:06:29,774 --> 00:06:32,439 151 00:06:32,439 --> 00:06:34,730 So let's just make that times plus 2 and put 152 00:06:34,730 --> 00:06:35,600 the plus out there. 153 00:06:35,600 --> 00:06:37,370 A minus times a minus is a plus. 154 00:06:37,370 --> 00:06:42,649 All of that over 2A, all of that over 24, 2 times A. 155 00:06:42,649 --> 00:06:48,439 So that is equal to 5 plus or minus the square root-- let's 156 00:06:48,439 --> 00:06:58,129 see, it was 25 plus 4 times 12 times 2. 157 00:06:58,129 --> 00:07:00,079 Because that was a minus 2, but we had a minus there 158 00:07:00,079 --> 00:07:05,919 before, so 8 times 12, so 96, all of that over 24. 159 00:07:05,920 --> 00:07:07,830 What's 25 plus 96? 160 00:07:07,829 --> 00:07:13,490 It's 121, which is 11 squared. 161 00:07:13,490 --> 00:07:21,819 So this becomes 5 plus or minus 11 over 24. 162 00:07:21,819 --> 00:07:26,250 Remember, these are the x-values where that original 163 00:07:26,250 --> 00:07:28,050 function will equal zero. 164 00:07:28,050 --> 00:07:29,050 It's always important to remember 165 00:07:29,050 --> 00:07:31,319 what we're even doing. 166 00:07:31,319 --> 00:07:36,750 So let's see, if x is equal to 5 plus 11 over 24, that is 167 00:07:36,750 --> 00:07:41,889 equal to 16/24, which is equal to 2/3. 168 00:07:41,889 --> 00:07:43,849 That's one potential intercept. 169 00:07:43,850 --> 00:07:46,610 So maybe that's right here. 170 00:07:46,610 --> 00:07:50,500 That's x is equal to 2/3 and y is equal to 0. 171 00:07:50,500 --> 00:07:55,459 The other value is x is equal to 5 minus 11 over 24. 172 00:07:55,459 --> 00:08:00,859 That's minus 6/24, which is equal to minus 1/4, which 173 00:08:00,860 --> 00:08:01,650 could be this point. 174 00:08:01,649 --> 00:08:03,939 I actually drew the graph not that far off of 175 00:08:03,939 --> 00:08:04,469 what it could be. 176 00:08:04,470 --> 00:08:06,480 So this would be x is equal to minus 1/4. 177 00:08:06,480 --> 00:08:09,370 Those are the x-intercepts of that graph. 178 00:08:09,370 --> 00:08:17,240 So 2/3 and minus 1/4 is choice C on the test. 179 00:08:17,240 --> 00:08:18,699 We have time for at least one more. 180 00:08:18,699 --> 00:08:22,289 181 00:08:22,290 --> 00:08:24,900 Oh boy, they drew us all these this graphs. 182 00:08:24,899 --> 00:08:26,250 Let me shrink it. 183 00:08:26,250 --> 00:08:29,790 184 00:08:29,790 --> 00:08:32,029 I want to be able to fit all the graphs. 185 00:08:32,029 --> 00:08:35,918 So let me copy and paste their graphs. 186 00:08:35,918 --> 00:08:40,288 So this is one where the clipboard is definitely going 187 00:08:40,288 --> 00:08:41,538 to come in useful. 188 00:08:41,538 --> 00:08:52,449 189 00:08:52,450 --> 00:08:53,700 OK that's good enough. 190 00:08:53,700 --> 00:09:02,030 191 00:09:02,029 --> 00:09:05,399 I've never done something this graphical. 192 00:09:05,399 --> 00:09:10,740 So the graph they say is y is equal to minus 2 times x minus 193 00:09:10,740 --> 00:09:13,639 1 squared plus 1. 194 00:09:13,639 --> 00:09:15,379 So that's what we have to find the graph of. 195 00:09:15,379 --> 00:09:18,669 196 00:09:18,669 --> 00:09:21,219 So immediately when you look at it, you say, OK, this is 197 00:09:21,220 --> 00:09:26,639 like the same thing as y is equal to minus 2x squared plus 198 00:09:26,639 --> 00:09:28,970 1, but they shifted the x. 199 00:09:28,970 --> 00:09:31,779 They shifted the x to the right by 1. 200 00:09:31,779 --> 00:09:33,299 I know it says a minus 1, but think about it. 201 00:09:33,299 --> 00:09:38,469 When x is equal to positive 1, this is equal to 0. 202 00:09:38,470 --> 00:09:43,080 So it's going to be shifted to the right by 1, plus 1. 203 00:09:43,080 --> 00:09:43,870 We know that. 204 00:09:43,870 --> 00:09:50,600 We know that it's going to be shifted up by 1, so up plus 1. 205 00:09:50,600 --> 00:09:51,610 Then we have to think is it going to be 206 00:09:51,610 --> 00:09:53,110 opening upwards or downwards? 207 00:09:53,110 --> 00:09:57,970 Think of it this way: If this was y is equal to 2x squared 208 00:09:57,970 --> 00:10:00,830 plus 1, then this term would always be positive. 209 00:10:00,830 --> 00:10:02,780 It'll just become more and more positive as you get 210 00:10:02,779 --> 00:10:05,679 further and further away from zero, so it would open up. 211 00:10:05,679 --> 00:10:07,759 But if you put a negative number there, if you say y is 212 00:10:07,759 --> 00:10:10,629 equal to minus 2x squared plus 1, then you're 213 00:10:10,629 --> 00:10:11,470 going to open downward. 214 00:10:11,470 --> 00:10:13,759 You're just going to get more and more negative as you get 215 00:10:13,759 --> 00:10:16,279 away from your vertex. 216 00:10:16,279 --> 00:10:20,564 So we're shifted to the right by 1, we're shifted up by 1, 217 00:10:20,565 --> 00:10:22,640 and we're going to be opening downwards. 218 00:10:22,639 --> 00:10:25,049 So if we look at our choices, only these 219 00:10:25,049 --> 00:10:26,689 two are opening downwards. 220 00:10:26,690 --> 00:10:29,410 221 00:10:29,409 --> 00:10:32,539 Both of them are shifted up by 1. 222 00:10:32,539 --> 00:10:36,519 Their vertex is at y is equal to 1. 223 00:10:36,519 --> 00:10:38,740 But this is shifted 1 to the right and this is 224 00:10:38,740 --> 00:10:40,379 shifted 1 to the left. 225 00:10:40,379 --> 00:10:43,350 Remember, we said it was x minus 1 squared. 226 00:10:43,350 --> 00:10:45,370 So the vertex happens when this whole 227 00:10:45,370 --> 00:10:48,070 expression is equal to zero. 228 00:10:48,070 --> 00:10:52,370 This whole expression is equal to zero when x is equal to 229 00:10:52,370 --> 00:10:55,370 positive 1. 230 00:10:55,370 --> 00:10:57,340 So that's right here. 231 00:10:57,340 --> 00:10:59,555 So it's actually choice C. 232 00:10:59,554 --> 00:11:02,579 233 00:11:02,580 --> 00:11:04,710 When your shifting graphs, that can be one of the hardest 234 00:11:04,710 --> 00:11:05,430 things to ingrain. 235 00:11:05,429 --> 00:11:09,759 But I just really encourage you to explore graphs, 236 00:11:09,759 --> 00:11:11,960 practice with graphs with your graphing calculator and really 237 00:11:11,960 --> 00:11:15,590 try to plot points and try to get a really good grasp of why 238 00:11:15,590 --> 00:11:19,190 when you go from minus 2x squared plus 1 to minus 2 239 00:11:19,190 --> 00:11:22,830 times x minus 1 squared, why when you replace an x with an 240 00:11:22,830 --> 00:11:26,870 minus 1, why this shifts the graph to the right by 1. 241 00:11:26,870 --> 00:11:29,470 Anyway, I'll see you in the next video. 242 00:11:29,470 --> 00:11:30,500