 Quiz number one the math mouse I Thought we were ready to wrote a quiz. Yeah, I typed that quiz up later on I called it quiz zero because I didn't feel like changing the numbering system So the one you already wrote was quiz zero quiz number one the key points here And I'm going to show you two different ways two different approaches that you can get these you can do these by counting on The graph or you can do these mathematically both are valid and both are useful. I Haven't really done the mathematical approach that much with you yet, which is why I want to kind of show you today First thing I'm gonna well first thing I'm gonna do is make the graph bigger so that you could all see There's number one and I'm gonna list the transformations in the correct order Good god. Oh, she must have been in my block B. That's why I was so confused Helping somebody else this morning Order order order. I'm gonna do expansions compressions first Then I'm going to do reflections and then I'm going to do slides and I'm going to write them in that order as well Oh, and we have to do my little lame joke. Let's see here name and Sir key she gets everything right. It's amazing Tell me you'll figure it out. I stole it from another teacher But as soon as I saw that that a math seminar two years ago, and I was oh, I'm so using that my other favorite was if he's doing a fake aside like an just a fake assignment for the kids the name he puts is Check off your I'm a check off. Okay. Yeah, a bunch of stupid names like that. They were great to Vertical or horizontal. I know it's vertical instantly. How do I know because if it was horizontal It would be after the bracket in front of the X somewhere Is it next to the Y where it belongs? No, then it's not backwards. This is going to be a vertical Expansion by two any other expansions? No Reflections those are negatives. I got a negative right there vertical or horizontal. How do I know it's vertical? It was horizontal. It would be right here Vertical Reflection Now there are two ways that I can get the points on the graph or mathematically I'll do it mathematically and they'll do it on the graph the first key point is negative three comma Negative two is that right? I'm going from memory, but is the tip of the tail negative three comma negative two You see I can do it by counting so hover my pencil right here Vertical expansion instead of negative two down when I say expansion what you're really doing is multiplying by two So instead of my Y coordinate being negative two, it'll be negative four Vertical reflection positive four. I can get there that way or I could actually just look at my Y column here And I could say if I vertically expand that by two it'll be negative four and If I vertically reflect it, it'll be positive four Both of those end me up on that point The next key point that we start out with if I recall was zero zero if I vertically expand this by two Two times zero is still zero And if I vertically reflect it negative zero is still zero I probably wouldn't have actually rewritten them I would have clued in and just left the zero there But letting you know turns out this one ends up being invariant and I think when you connect them It's gonna look kind of like that My next key point is two comma two Again, I can get there in a couple of ways either by counting Vertical expansion reflection pretty sure I'm gonna end up there or I can get there by doing the math vertical expansion Reflection no matter what I end up at two comma negative four I think the next key point is four zero. I'm going from memory, but I am right, right? I hope Vertically expand that vertically reflect that zero and zero still it's going to be invariant Or I could have done it by being on the graph vertical expansion vertical Vertical reflection, why don't you be consistent? Mr. Do it can use the same color the next key point is Is what a five comma one? Is that right the top of the head? I was blank for a second there again Tyler I can do it mathematically vertically expand that by two and then vertically reflect it I'm gonna end up at five comma negative two or I can do it by hovering my pencil vertically expand it Vertically reflect it gonna end up at five negative two and The last one ends up being six comma zero now. How do I mark these for two marks? I would give you one mark if you did a vertical expansion I would give you one mark if you did a vertical reflection However, I would take a half mark off for each point that was in the wrong place Now let's talk about how they mark these on the provincial Haley and therefore how I will mark these okay? Anything anything anything that you put on your graph with a solid line is Supposed to be part of your final answer So I was helping a few kids this morning and I noticed a lot of kids did this on their graph before they started I would have to assume that that was part of your final answer and give you a zero however If you did this the rule is anything you do with a dotted line I will assume that's part of your work and I will ignore it Some of you when you're counting your points you're doing this So I end up there then I end up there then I end up there I will assume those three points are part of your final graph And I will take a half mark off for that a half mark off for that and a half mark off for that Because those are not on the graph if you want to leave yourself breadcrumb markers There's a rule we have for that instead of putting points If you put an X there, I assume that's a placeholder That's a marker, but that's not a point on the graph. Does that make sense now having said that on your test Because I typed the test up when I got better at graphics after I typed up this quiz this quiz I just did by cutting and pasting and then I learned actually how to make these funky shapes and how to type them in the Word I actually have a light gray silhouette of the original one as a dotted line You'll barely barely able to see it. I made it in a light ink. It does photocopy though, and you can see it there So on your test it will be there That makes sense that I end up That are waking you awake Okay, because you're not looking really good, right? Yes, no Okay Question two this is going to be I'm positive two left. I know because it's next to the exit backwards I gotta think about this one. It's four. Oh, it's four down because it's not next to the Y If it was Y plus four over here, that's also four down, but it's been moved over so it's no longer backwards four down and Again, I'll show you how you can get this mathematically from your key points of negative three negative two zero zero four two Five one and six zero you could go two left negative five negative two two three Four how am I calculating two left? It's really minus two from the X coordinate four down Negative six negative four negative two negative three negative four How am I calculating four down? It's really minus four from the Y coordinate so you can do it mathematically or You can just hover the points and I You need to know both. I'd use the points when there's a graph in front of me I use this method when they don't give me a graph, but they just give me a point and say where does it end up? Two left one two three four down two left one two three four down Connect them to left four down connect them to left Four down That right? Yeah, I just did a lousy job of graphing your last one. Let's try that again. That looks better and Here in here correct So again, I would give you one mark if you went to left I would give you one mark if you went four down however any point that's in the wrong place I would pick up a half mark If you went four up the best you could do is one out of two turn the page Ah the fun stuff the good stuff First thing I want to do is ask is this written in a way that's acceptable. I Look here This coefficient is Factored there are brackets So I don't need to have my little alarm bell go off here. I'm happy with this Now I'm gonna make a list in the correct order That two right there vertical or horizontal vertical Expansion by two or compression by a half. Well, if it was a two right there Everything's backwards because it's next to the Y. Ah, but it's not next to the Y So it's no longer backwards. This is an expansion by two. I see a one half Vertical or horizontal. Well, it's after the X. It's after the F. It's in front of the X It's inside the function. This is horizontal Expansion by two or compression by a half. It's next to the X. It hasn't been moved. So it is backwards compression By a half Reflections that guy right there vertical reflection Yo, it's backwards. So it should be an expansion by two Candy for you later. Sorry fight the cold or something here not working very well. No excuses. Never mind I didn't say that I should have got that right. You know what? I'm giving myself a one-hour detention today I'm gonna be here till at least four o'clock Geez stupid teacher Vertical reflection, I got that one right. Yes, Carly. Whoo-hoo, and this would be a horizontal reflection. Yes And then I think I have four right four up and Again, I can get this one of two ways. You can do it mathematically if you list the key points negative three comma negative two zero zero four two oh two two mr. Duke four zero five one and six zero You can go vertically expand by two negative four Zero times two is zero Two times two is four zero times two is still zero one times two is two Horizontal expansion by two negative three times two is negative six zero times two is zero two times two is four four times two is eight five times two is ten six times two is twelve Might not fit on my graph, but I'm not gonna panic Vertical reflection that's gonna become a positive that's gonna become a negative which is gonna make a difference That's gonna become a negative. That's gonna become a negative which is gonna make a difference That's gonna become a negative. Let's make that negative a bit clearer mr. Duke That's gonna stay where it is Horizontal reflection that'll become a positive that'll become a negative that'll become a negative that'll become a negative that'll become a negative four right Ashley what that really means is add four to all your x's because that's how you move something for right. This is going to become a ten, a four, a zero, a negative four, a negative six, and a negative eight. Four up. That's like adding four to all your y's. This is going to become running out of room here. An eight, a four, a zero, a four, a two, and a four. So I should end up at ten comma eight. Ten over eight up. Is there a dot right there? Cool. I should end up at four four. Is there a dot right there? And I think that was my tail. I'm trying to visualize it, but I think if I'm drawing the tail properly, it should look like that. Although if you drew it this way, I wouldn't really freak out because it's a bit tricky to visualize. Apparently, I should end up at zero zero. I should end up at negative four four. I should end up at negative six two. And I should end up at negative eight four. Negative eight four. Is that what you get? There it is. Now you could also do it using the graphing method. It was a bit trickier because you had to keep track when you were off in the middle of the page in No Man's Land for a split second, but then you moved four and four. And I find if I'm in No Man's Land, I just kind of call stuff out in my head. I'm at 12 high, right? Oh, what were some of them that showed up? I'm at negative 12 left, but when I move four right, negative eight, I'm back on the graph again. Three marks, there was six transformations. I would give a half mark for each transformation. But I would take off a half mark for each point that was incorrect. How many you got that one? I'm curious. I'm impressed. Number four. Inverse. This is absolutely where I would actually use the points, switching the X and Y. Negative three, negative two becomes negative two, negative three. There should be a dot right there. Zero zero is invariant, but I think your tail starts to look like this. Two two actually ends up being invariant as well, but four zero becomes zero four. Five one becomes one five and six zero becomes zero six. Yes, like that. By the way, is that a function? Does that pass the vertical line test? Does that pass the vertical line? No, shadow touches the graph in several locations at the same time. That means you couldn't graph that on a graphing calculator. In case you're wondering, by the way, the original mouse is actually sort of a semi-circle. Another semi-circle and half of a semi-circle that's been flipped if you really wanted to graph it on the graphing calculator, but I didn't care. Number five. Yeah. No, no, wait a minute. I'm a good math student. I can handle this. But the alarm bell should go off right now, Sandly. You know why? As I glanced at this, I saw there was a coefficient. I saw there was a slide, but they haven't put brackets around it. Now I had one student that I was talking to earlier this morning, and she just went, oh there. Now there's brackets. No! No, that's not the same at all because if you foil that out, you won't get the original function. You can't just add brackets. What you're really having to do is factor. I'm going to rewrite it. I'm okay with that. A little weird, but I'm okay with it. I'm okay with that. A little weird, but I'm okay with it. Factor out a negative two. I think I had an x. What number would I put right there so that when I multiply this out, I get a positive two with no brackets? I think that. Yes? Turns out, Brett, this is not two left. It's one right. Let's make my list. Vertical or horizontal? Vertical. Is it next to the y where it belongs? Well then it's backwards. Vertical or horizontal? Horizontal. Is it next to the x where it belongs? Yep. Well then it's backwards. Carly, do I get those both right this time? Vertical reflection. Horizontal reflection. Three, is it next to the y where it belongs? It's vertical and it's backwards. Three down. One right. Or you could have gone one right, three down. The order there doesn't matter. Key is to do expansions, compressions before you do slides. Alrighty, so ladies and gentlemen, start your engines. I'm going to make this one a little bit smaller so I can see everything on the screen. I'll do this one using the graphing methods. I'm going to hover here. Vertical expansion instead of negative two down, albeit negative four down. Horizontal compression, 1.5. That's not a bad decimal. I can deal with this. Vertical reflection instead of negative four down, positive four up. Horizontal reflection instead of negative 1.5, positive 1.5. Three down. One, two, three, one right. Do you guys have a point at 2.5 comma one? I get a little nervous when there's decimals, but that's not a hideous decimal, so I can tolerate it. Zero, zero was the next point. Vertically expanded. Horizontally expanded. Vertically reflected. Horizontally reflected. Three down. One right. I think we end up with that. Two, two was the next one. Vertically expanded. Four high. Horizontally compressed. One right. Vertically reflected. Four down. Horizontally reflected. One left. Three down. One, two, three, one right. Do you guys go through zero comma negative seven? No? Yes? Four zero was the next one. Vertically expanded. Zero high. Still zero high. Horizontally compressed. Four right. Why I think that's two right. Vertically reflected. Zero high. Still zero high. Horizontally reflected. Two right. Two left. Three down. Uno, dos, tres. One right. Do you also go through negative one comma negative three? Five one. I bet you's going to be a decimal again. Let's see. Vertically expanded. Five comma two. Horizontally compressed. Yeah, 2.5 comma two. Vertically reflected. Negative two. Horizontally reflected. Negative 2.5. Three down. One, two, three, one right. Do you guys end up at negative one and a half comma negative five hanging there in midair? And I'm pretty sure I'm guessing the last one's going to be right there. Let's prove it. Six zero. Vertically expanded. Still zero. Horizontally compressed. Three. Vertically reflected. Still zero. Horizontally reflected. Negative three. Three down. Three down. One right. Yes. There I am. There's my little mouth. Same as number three. Three marks. There's six transformations. So a half mark for each transformation, but a half mark off for each point that's in the wrong place. Give yourself a score, please, out of 12. How many 10 out of 12 are higher? Make sure your name is on it. Pass it inwards, please. Person in the middle, please. Put them in a nice, neat row while facing the same way. We are going to be finishing off the unit today. Yeah. Right now I'm leaning towards your test being a week from today, which means I'll be doing a big after school tutorial Friday after school, and I'll send an email out to that effect. By the way, you should have gotten an email from me this weekend. Those of you that did not, it sounds like anybody who has a telus.net email, which is not many students, but some parents, those are bouncing back and being rejected by telus right now for some strange reason. We're working on it, but y'all got an email from me in theory? If not, check your email. Check your email. Read your email. Occasionally you might want to check your email. Just be thinking out loud. Okay. Specifically in your workbook, would you be so kind as to turn to page 79? Page 79. Yo, you don't have school Monday? Is that Thanksgiving? Really? It's that fast? It's here already? Eesh. Then it's going to be Wednesday and I'll do the tutorial Tuesday because doing a tutorial Friday and then having four days will be stupid. That also gives me Friday afternoon that maybe nobody will show up and I can go home at a decent hour. My goal right now is to at least once a week drive home without my headlights, but right now I'm over. Y'all on page 79? Man, they're giving you a halt. So you guys had a pro D day week and then last week you had a half day on Friday. By the way, did you guys come to the 50th? It was bad. Some great stuff in the afternoon all set up. Quite impressive. And then you guys get next week Monday off. We need to bring in year round schooling. We're falling behind the rest of the world. On the 14th, you don't have school on Friday either? No, that's wrong. It's late. There is a pro D day later on in the month, but I can't see there being that many days off. Now you're just messing with me. I think you are. 21st is a pro D day. Nice try there, girl. Can I teach now? Excellent. Did I turn my recording back on? I think I did. I did. So reciprocal transformation. By the way, those of you that were away for that lesson last day, you want to watch it online. Very important. This is going to be two, maybe three questions on your desk minimum. Here's what we said when they say take the reciprocal, remember the symbol for reciprocal was one over F of X. That was what told you to take the reciprocal. What you were doing was taking the reciprocal of all the heights. If you were five high, Brett, your reciprocal would be one over five high, one fifth high. If you were one fifth high, your reciprocal would be five high. In fact, we simplified it. We said this, if your original graph was getting bigger, the reciprocal would get smaller. And if your original graph was getting smaller, the reciprocal would get bigger. What was the invariant heights? There was two of them, one obvious and one a little more obscure you think about. What were the invariant heights? You remember one or or negative one, anywhere one or negative one high stayed where it was. And so I said, if I'm graphing a reciprocal transformation, I put big huge dots there because those are going to stay. What was the height that we couldn't take a reciprocal of? Zero. In fact, if our original graph approached zero, our reciprocal often had a vertical asymptote right there. So let's look at example three. Example three says this, the graph of G of X equals one over F of X is shown. This is a reciprocal graph. Oh, and the maximum point, this point right here is at negative three comma negative two. Sorry, negative three comma positive two. And the Y intercept right here is at one thing. It says, given that F of X is a quadratic function, this is a reciprocal of a parabola, sketch the original graph and state the coordinates of the minimum point. Well, let's see. How high am I right there? What did they say the Y intercept was according to this question right now? How high? That's my reciprocal. How high was my original Y intercept then? Five high. My original graph must have gone through right there. And you know what else I'm going to do? I'm going to add my little axis of symmetry for the parabola, which means three over one, two, three, four, five up from the vertex. If I go three left from the vertex, I'm willing to bet there's another point right there because I know all parabolas are symmetrical. They are. Mr. Dick, how do you know this is the vertex? Is this the highest point of my reciprocal? Is the vertex the lowest point of my original? See, they become smaller. Oh, as a matter of fact, how high is this vertex right now? Sorry, let me say that again. How high is this maximum point here right now? What does the question say? What does it tell me? Negative three. You know what my vertex was? What's the reciprocal of positive two? Careful. Not negative two. A half. We're taking the reciprocal, not reflecting. In other words, my vertex must have still been at negative three, but it would have been a half high. In fact, as a parabola, which heights would have been invariant for the reciprocal, which heights would have been invariant? I think one high right there. One high right there. I think it would have done that. I see the question I'll get to you the second one. Yes, ask. The original is too high. Sorry, the reciprocal is too high. If the reciprocal is too high, what height must the original have been? What would have given you a height of two after you took the reciprocal? It must have been negative three, comma, a half, and when I took the reciprocal, I ended up at negative three, comma, two. That make sense? I think that's what the original looked like here, that blue one. And it says, state the coordinates of the minimum point, negative three, comma, one half. Oh, and the y-intercept, zero, comma, five. If my reciprocal has an intercept of one-fifth, my original must have been five high. B says, you can write this as a parabola, and we can, but that's math 11. I'm not going to freak out and I'm not going to bother. I always save this question. This is a good review of the reciprocal transformation for you guys. So having said that, the homework that I gave you, I gave you question number one, A. I'm going to add question two, and I'm going to add question six, and then I'm going to say, turn if you would, please. It's a lesson 12, but you didn't take questions about the reciprocal. I just reviewed it because I think it needs to percolate, but I will next class absolutely. Lesson 12, the final transformation, the final lesson. Now on Wednesday, I have for you one big, huge take home quiz. It's actually a unit review quiz. It covers the entire unit. I'll be going over that on Friday. And on Wednesday, your homework is going to be, start working on the great big unit review that I gave out way back when, for which the answer key showing on my steps is online at pitmath.com. Click here for today's notes. Go to block A, somewhere in block A, unit one transformations, it's there. I saw about 10 of you look blank. Go to pitmath.com. Click here to access today's notes. Go to block A, unit one transformations. And if you haven't figured it out already, if you're ever away, the easiest way to do this is sort by date modified. And the newest stuff will appear on the top. And the latest stuff will appear on the bottom. So there's lesson 11. If you were away, not as a video, but as a PDF file and transformations review. I gave you transformations review number two. And there is the answer key showing all the steps. So you can download that today. Last lesson, the absolute value transformation, the absolute value transformation. First of all, it says recall the definition of absolute value. That's actually the mathematical definition. It says if you're graphing it, graph positive X. If you're moving to the right, graph negative X. If you're moving to the left, because that's how you've taken negative of a negative and make it positive. Anyways, don't need to worry about that. What you need to know is this. Hey, what's the absolute value of five? What's the absolute value of 10? What's the absolute value of zero? Careful now. What's the absolute value of negative five? Five. What's the absolute value of negative 10? Okay, what's the absolute value of a height on a graph? If the height is positive, same answer. Ah, if the height is negative, make it. There, that's awesome. Okay, your homework, well, no, I guess I should do a few of these. So it's blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah. Warm up number two. It says we're supposed to use a graphing calculator. To you, I say no, no, we're going to use our brains. It says they've graphed f of X where f of X is X squared minus four, Hailey. And then it says, write the equation for the absolute value of f of X, the absolute value of the heights. Any heights that are positive will be invariant. That will stay where it is. That will stay where it is. Write that down. Any heights that are negative, Tyler, will become positive. Tyler, how high am I right there at the very, very bottom? I think you said it right, but I, someone moved the chair upstairs louder. What's the absolute value of negative four? That X value won't change, but it'll end up right there. In fact, you know what? It'll like this, this whole thing will flip up. And in fact, that's how I do this. Anything below the X axis, anything in the negative range, it's like you folded the paper. It's going to fold, flip right up on top of itself. That's the absolute value transformation. Blah, blah, blah, blah, blah, blah. Page 84. Here's your summary. In general, if they give you some graph and then they say, take the absolute value of all of the heights. If you're already positive, it stays the same identical fancy word invariant. If your heights are negative below the X axis, it ends up reflecting above the X axis. It's really easier to do than to explain. Here's a graph, sketch the absolute value of that graph. Okay. That's going to stay where it is, because it's positive as a height. This is going to reflect up, and this is going to reflect up. And that's the equation for a little devil's cap with little horns or marauder's cap with biking horns. V says, now sketch the reciprocal. That's overkill right now. Forget it. Example one. This is much more similar to what you'll get on the test. It says the graph of F of X is shown. Sketch that. The absolute value of this. Okay. Again, what they're really saying here, Carly, is sketch the absolute value of the heights, because F of X, that's the heights. So any heights that are already positive are going to be invariant. This ain't moving. Ah, but to the left of that, the graph is below the X axis. L of it's going to simply flip right up. It's going to look like, let's see about there. It's going to go. Yeah, the sound effects help. B says, sketch the reciprocal. Okay, I'll try this one. Sketch the reciprocal of the absolute value. Sketch the reciprocal of this here graph that I've done in red. Okay. Oh, anywhere one high is going to be invariant. So where am I one high? It looks like I'm one high right there, right there, right? And it looks like this whole line is one high. Looks like to me, it starts at about a little past just shy of negative three. So just shy of negative three, there's going to be an entire line one high. Are there any other heights on my red graph that are one high? Nope. Are there any on my red graph that are negative one high? There were on the original, but not on the absolute value. Okay. Oh, the next was anywhere my graph touches zero was going to be a vertical asymptote. There's going to be a vertical asymptote at negative 1.5. Now we do my little bug trick. Standing right here, imagine standing right there. As I move to the right, my graph is getting closer to zero. What's my reciprocal going to do? What was the reciprocal of getting closer and closer to zero? Go away from zero. Actually, I use a little phrase shoot off to actually shoot off to infinity and beyond. Oh, don't use green. Mr. Do it. Now hover right here because that point is that point right there. As I move to the left, I'm getting closer to zero. As I move to the left, I'm getting closer to zero. What's the reciprocal going to do? Shoot off to infinity. As I move to the right this way, as I move this way, Justin, what's my original graph doing? Getting bigger or getting smaller? Bigger. So my reciprocal is going to get, it's going to curve towards zero like this. Now, this is a bit of an unusual one because my original levels out at about 1.5. How high does it level out at? What's the height right here, Ellen? What's the reciprocal of three? It's actually going to, when it gets to 1.5, be a horizontal line one-third high. Instead of getting closer and closer and closer and closer and closer and closer and closer and closer to zero, it's going to level out. Now it's really tough to draw accurately, especially if you're free-handing, but I do have to say do this correctly and that's what it would look like. There's a reciprocal of an absolute value. Turn the page. Page 85 is one of those pages that you should probably doggier or put a post-it note in because there is a lovely summary of everything. What's your homework? Try number one. Also, some of the questions I signed last like 10 minutes ago from reciprocals. Try number one. Number three. Four is pretty tough. I'll try that. Six, seven and eight. Technically, that's the unit. I need to dot some i's and cross some t's next class. But that's it. Unit one, transformations. Before I turn you loose, you may recall that one of the themes I've tried to bring in throughout this unit is a certain video game, Mario Brothers I called it, because Super Mario Brothers, because all of you played it when you were young children. We talked about how that was a function. We talked about how when you moved them to the left, you were replacing all the x's with x plus one. When you moved them to the right, you were replacing all the x's with x minus one. When he jumped, you were replacing all the y's with y minus one and then y plus one moved them up and down. When he ate the mushroom, you were replacing the x's with a half x and the y's with a half y. Now, he's twice as big. So, a little walk down amnesia lane for you guys. Oh, hang on. Gotta do it this way. This was about five years ago. This was at a talent show at a university. I thought this was quite clever. Got Gordon College. So, there's your walk down amnesia lane. Remember when childhood was fun? Okay. You got about a half an hour. Classes, yours get caught up. If you finish everything, start working on the great big unit review. If you know what I'm talking about, I have extra copies. Yes, you may.