 A couple of comments. First of all, this weekend I'm going to email to you another unit review complete with a hand written answer key. It's actually the first one that I used that I made myself years ago. Yeah, man. I just got the joke. Answer key gets everything right all the time. She's great. My best student ever. Test is when? Wednesday tutorial is when? Tuesday. I'll also email you a link to my last couple of tutorial videos that I've done on this unit. I know I have at least one online, so I guarantee I'll email you a link to one and I think I have both of them online somewhere. So if you're looking to, hey, I'm bored. I got nothing better to do. I'm going to watch an hour of math just to kind of get a quick summary of everything. The only thing is last year I did logarithmic graphs and exponential graphs in this unit. Have I done logarithmic graphs and exponential graphs in this unit? So when you hear me talking about that in the tutorial videos, don't have a heart attack. It's not like, I haven't, I forgot, no, I haven't showed it to you yet. So there's probably going to be like a five minute little tangent where I go off in them. I'll try and find one from a couple of years ago where I didn't do exponential and logarithmic graphs as well if I can send you that one. I still have long enough. Here we go. Mark your own. You know what? This here means to write. Oh, no, no, no. Everything's backwards because it's next to the X. That's my fallback. If it's next to the X or if it's on this side next to the Y, everything's backwards. The Y stuff is a bit more confusing because if they've moved it over, it's no longer backwards. But since this is next to the X, it's not plus two, right? It's actually two left. So this would end up there. You know what? That, right? Correct? That would be an example of what I would consider a C-level multiple choice question. There will be about, I think there's 14 multiple choice questions on your test. I would say five of them will be of this level of difficulty or variety. What we call in the language K, which stands for knowledge questions where you don't actually have to do any real math. It's kind of a think about it, circle it. Number two, that graph is shown below. The transformation which will not change the Y intercept, what they're really saying is which transformation will leave that point there invariant. Now we have to do a little bit more thinking. Not only do we have to know what these do, we have to know how they behave on the graph. So this would be like a C plus kind of a level question. Let's see. This is a vertical stretch. You know what? That would move it to there. That's going to move it. In fact, you know what? I'm going to argue that anything that's vertical will change this. That's vertical. That's vertical. It's going to come down to the inverse or a horizontal. You know what? I think it's that horizontal reflection. I think if I spin it horizontally, it reflects about the Y axis. Zero left right stays zero left right. I think D is the correct answer. Did you notice though, I glance at the multiple choice answers for hints? That's a habit you want to develop this year. In particular, on your final exam, I can think of at least five questions where that will tell you what to do. If you glance at the answer. Oh, I know what they want me to do, where otherwise it's kind of, I'm not sure what they're going at. Okay. Number three. If the function blah, blah, blah is translated four units to the left, so the new equation would be as soon as they want me to find an equation, I fall back to listing what the replacements are, carefully cursed in substituting them in with brackets and then doing any algebra as necessary to make my answer look like their answer. Four left means I'm going to replace X with X plus four. What that would look like in my equation is Y equals three F of, the minus five would still stay, the plus two would still stay, but instead of an X, I'm going to have an X plus four. I glanced here, I don't see that answer there yet, but I'm not going to panic, it probably means that they want me to do a little bit of algebra, a little bit of simplifying, a little bit of gao. I can gather like terms, that plus four minus five because there's no coefficient in front, there's nothing right there multiplying into the bracket, this is just plain old plus four minus five. What is positive four, take away five when I simplify it? I think this simplifies to three F of X take away one plus two. Do I see, I do see that somewhere, there is D. Roxanne, math multiple choice tests are probably the toughest type of multiple choice tests because when we make up multiple choice for the wrong answers, we think about the most common mistakes and we put those answers there. In other words, if you did X minus four, because you thought four left was negative, I guarantee that's there somewhere. Number four, this would be again a good example of a K question, a C level question as far as I'm concerned. How does that compare to that? First of all, horizontal or vertical and how do I know? Steph, horizontal, I would be going, no, no, I'm not even wasting my time. X to the X, everything's backwards, it's an expansion by factor of five. And the page, okay. The graph of this is shown, what's the domain of the inverse? How are domains and ranges related for original graphs and inverses? Well, it turns out on your original graph, your domain, which is your, well, first of all, how do I find an inverse? Oh, too slow. How do I find an inverse? Okay, we're going to try that again with more of you moving your lips. You don't have to talk but move your lips and at least fake it. How do I find an inverse? Oh, you didn't even move your lips at all. Yes, I'm looking back there now. How do I find an inverse? Okay. You said switch the X and Y. I'm going to also think switch the horizontals and the verticals. See, I think the domain of your original graph will become the range of your inverse. And the range of your original graph is going to become domain of your inverse. So when they say to me, what's the domain of the inverse? It's actually going to be that right there. It's going to be from negative one to positive three, but it's domain. So I'm going to have an X. It's going to be from negative one to positive three. Less than or equal to touching, less than or equal to touching that. That's a really, really, really, really handy trick. In fact, next unit, rather than memorizing two different graphs, we're going to memorize the first graph like crazy. And then we're going to recognize that the new graph is the inverse. So when I say, hey, what's the domain of the new graph? You're just going to say, oh, it's the range of the original because you'll have that up here. We can cut our memorization in half. Oh, and you know what X intercepts become? On an inverse, they become Y intercepts. And you know what Y intercepts become? On an inverse, they become the X intercepts. Okay, man, does that make sense? Okay, because you were looking very puzzled. Good. I read your body language. Correct. How do I find an inverse? Okay. So when you switch the X and Y, what happens is your horizontals become your verticals and your verticals become your horizontals. Let's actually graph the inverse just so you can see it. I'll do it in blue. Zero negative one would become negative one zero. Sorry, let me try again, Mr. Do it zero negative when you become negative one zero, right here, the top of the circle, which is two comma three would become three comma two. The top of the circle would be right there and the bottom of this parabola circle, whatever this is for negative one would be negative one for the inverse would look if I could draw the inverse would look like that. Okay. What's the domain? Isn't it from negative one to positive three? And here's the easy trick. The domain of your original becomes the range of your inverse because you switched the X and Y and the range of your original becomes the domain of your inverse. Is that okay? Good. What is the horizontal translation of the function y equals to f of X minus 12 take away one. F of four X minus 12 in brackets. Take away one. Well alarm bell, Mr. Book that's wrong. This has not been factored out. I need to rewrite this as two f of bracket for bracket X minus when I factor a four out of a 12. Oh, it's going to be a three right there. You know what? I think the answer is none of these. I think it's three right. Is that okay? Number seven, the function f of X is transformed to that thing there. Five comma five is on the original. What's the new point? I'm going to list my transformations in the proper order. First, are there any expansion compressions at all? Are there any expansion compressions at all? No. What are you really looking for when I say expansion compressions? Coefficients, right? Numbers in front of the variables. Are there any reflections at all? Yeah? What? I'm looking for a word like vertical or horizontal because I've never ever said to you, what's the reflection? On the f vertical or horizontal? Okay, there's a vertical reflection. Are there any slides at all? To what? To what? Okay. By the way, you're going to have to play this game with me and participate, boys and girls. So wake up, sit up, do whatever you need to do, but give your brains a shake. Right now, I'm seeing drool for about eight faces. Come on, drink some coffee. Do what it takes. I heard two right. What about this one? Oh, it's on the Y side, so it's backwards. It's not four down. What is it? Four up. And if you don't believe me, if you plus it over, it would be a plus four. So four up. Now I'm going to list the point. I said to you the other day, when they give me a point, I write it, I list the first one, I list the transformations. Then I write the point and I'm just going to cross stuff out moving outwards. Vertical reflection, vertical is Y's. That's going to be a negative five. Vertical, it's the Y coordinate. Two right. What's two right from five? Seven. Four up. What's four up from negative five? Negative one. I think the point is seven common negative one, which is none. Oh, no, it is there. A. This is so much fun. I want to turn the page. The graph of f of X is shown. Which of the following could be the graph of the reciprocal? Okay. Reciprocal would have a vertical asymptote right there. Correct? That's wrong. That's wrong. Got rid of two answers already. Oh, and anywhere one high and negative one high would be invariant. That's wrong. Now, what's the best answer? B is wrong technically. I'll tell you why. My friend who typed this, he used a computer program to graph this. That should be a dotted line and his software put a solid line in for the asymptote. I don't like that, but the rest of this was so good that I said I'm going to keep it and I haven't got around to retyping it or fixing it or anything like that. When you get your graphing calculators, you're going to find software has a real tough time with vertical asymptotes. It doesn't know what to do. It wants to try and connect the dots and so it will often put a vertical line there. That's not supposed to be there like this software did, but clearly B is the best answer. So I'll go with it. Okay. Scott takes the graph of Oh, the parabola, my favorite graph expands it vertically by a factor of three, then translates it four units to the right. Ian takes my favorite graph, translates it for right and then expands it vertically by a factor of three. The vertices of the two graphs are Oh. Hmm. Hmm. Well, since they're only asking about the vertex, why don't I move the vertex around and the vertex of the original is zero zero. If I follow Scott, he expands that vertically by a factor of three. If I expand zero zero by a factor of three vertically, how high am I now? Still zero and then he moves it for right. You know what Scott ends up right there. Ian moves it for right and then he expands it vertically by a factor of three. If I expand four comma zero vertically by a factor of three, where will I end up? You know what? Still here. In fact, I think Scott and Ian end up in the same place. The graphs of this, the original and this, the new one are shown if the original equation is Oh, wounded seagull square root moved three left. Then an equation for this new one could be. What do you see? Have I stretched it vertically at all? No. Have I stretched it horizontally at all? In fact, they look the same size. Have I reflected it at all? Which way step horizontally or vertically? Okay, there's a horizontal reflection, which means I've replaced X with negative X so far. So good. If I replace X with negative X, let's see if I take this equation and I replace X with negative X, the equation would look like this, which is sitting there. That would end up there. That would end up there. That would. Oh, you know what? I think that's all they've done. I think they've just replaced the X with negative X. Now, I don't like the way that's written. If I was graphing it from scratch, I would factor out a negative and do it in the proper order. But looking at the order that they've done here, Andrew, I have to say I think all they did was that to a graph that had been moved sideways first deep. By the way, hopefully you went, it's not vertical, it's not vertical, and at least got it down to a true false question. No way is it a vertical reflection because that would be flipping it this way. I'm not even going to waste my time. Oh, and then hopefully you said, hey, square root graph move three right would point that way. You could also get there by a process of elimination. Turn the page. Ooh, what's the horizontal translation? Do you know what I see, Steph? What do I see? I see a little alarm bell ringing at me right now because I noticed that that B there is in front of the X and there's a horizontal slide, but that horizontal slide is not in brackets with the B factored out. Little alarm bell would go off. You see, Steph, don't even try and avoid it. I'll get you whenever I want to. I'm that good. I've been doing this for a long time. Good try. But now you got your adrenaline rush even better than that coffee there, right? Okay, let's rewrite this as a absolute value of B. The real question is what goes right there? Now, if I had given you this with numbers, almost every one of you would have got it. And so one of the things you're going to find on your tests because this is what they did on the provincial is if we want to make a question just a little bit tougher, we'll replace everything with what are called literals variables. We'll put letters there. It's going to be the same math, Shannon, but most of you aren't as comfortable doing math with letters as you are with numbers. Shannon, what if it had said this 2x minus 4? How would you have rewritten that puppy? Mathematically, what do you do to turn a 4 into a 2 using a 2? I think you divide. Oh, so if I hear you correctly, I think it's going to be minus C divided by B, isn't it? See how I reasoned my way there with an easier example, and that's what I often do if they give me a nasty one. I'll say, well, it's got to work with the numbers and I'll just do the same math with the letters. I'm pretty sure it's right C over B units. Okay, now we come to the written section. On your test, I won't ask you to graph more than one graph on one graph. I'll waste paper on the test, but on the quiz, I wanted to save a bit of paper. Let's see. They want me to do this one, which is three right, comma, five up. Okay, one, two, three, one, two, three, four, five. One, two, three, one, two, three, four, five. One, two, three, one, two, three, four, five. Graph A looks like this. Is that correct? One more. One more. One more each. Graph B. Hey, that's the inverse. Switch the X and Y around. Instead of negative one, negative three, negative three, negative one. And instead of three over one up, one over three up. And instead of seven over, negative three down, negative three, seven up. There's graph B, I think. By the way, half mark off for each point. That's wrong. Katie, what's this one? Hear me? Absolute value. How do I find an absolute value, kiddo? Anything that's below the X is going to... Brilliant! Brilliant! I couldn't have explained it better myself. Anything that's above is going to stay. So you would have to trace over this to show the marker. Hey, I know this is part of the graph. If you left that blank, I would take a half mark up. That's part of the graph. And then instead of negative three, it's going to be positive three. And I think you end up with that. And instead of negative three positive, you end up with a little Martian hat. Or little ears or ogre cap or something like that. That's graph C. Leon graphs the function. The point seven comma four is a point on this function. What are the coordinates of this point after that transformation there? A little alarm bell would go. Got you with your head down this time. You see, I would have to rewrite this as negative F of two bracket X minus four. Turns out this is going to be horizontal. Compression by a half vertical reflection for right in the correct order. And once I factor, I'll just list my point and we'll work our way outwards. Horizontal, okay. Horizontal compression by a half instead of seven. It's going to be a 3.5, which isn't the hideous decimal. I'll survive with that. Vertical reflection, vertical means Y's instead of positive four, negative four. Four right, right is horizontal Kirsten. I'm going to four right from here. You know what? It's going to be 7.5. I think the point is, correct me if I'm wrong, 7.5 comma negative four. Is that right? Yep. Half mark for the X coordinate, half mark for the Y coordinate would make good sense for a one mark question like, this is so much fun. I'm just getting all these right. Oh, so happy. Getting my little nerdy math adrenaline rush as opposed to my yelling alarm bell adrenaline rush. Okay. Number three, give them that graph. Write the transformations for A, B and C in terms of G of X, one mark each. Hmm. What did they do for graph A? Ah, that's an absolute value because what's below the X axis, Katie has simply flipped right up. I think what they did for graph A is Y equals G of X absolute value like that. By the way, if you wrote F of X because you're so used to putting F of X, I'll live with it, but I think they use the letter G. Did they not? So be consistent. What about for graph B? Oh, that looks an awful lot like a reciprocal kind of a thing. Let's see. Yeah, one stayed invariant. One stayed and you know what? I'm pretty sure graph B is one over F of X, G of X. See, I do it too. Be consistent, Mr. Do it, be consistent. You're right. C, you can get there by flipping it. That's one way. Now there's actually two answers for C. I think the easiest one for me to spot was this one, Y equals G of, I think if you replace the X's with negative X's, it flips, that gets you one mark or if you put that answer down here, that would get you one mark because the other possibility instead of flipping from here to here, I think we could have just slid, slid, slid, moved. I think if we moved seven to the left, we'd also get this same graph. So Y equals G of X plus seven. Now I didn't specifically say what's another equation. That's the equation I asked for the transformation. If you had just said seven left, I'd give you one mark as well because this question wasn't completely clear, but probably most of you continued with the equation theme or if you put that there and you put that there and you said horizontal reflection. Did anybody do seven left here because that's what they saw first? I'm just curious how your brains work. You did? Okay. The rest of you all said, oh, it's just flipped. Point being, there's now more, there's often more than one way to get to the same graph, especially you can do a vertical stretch or a horizontal compression because when you stretch rubber this way, it does get thinner this way. Often you'll end up with the same graph if you're clever. Number five, this is much more like what your written section is going to look like your written section. I'm going to tell you it's going to have I think six graphs. Maybe seven. Can't remember. I'll give you some kind of generic shape like this and now you can see what I meant when I said on the new graph, I'll put the original as kind of a grayscale dotted line. You can see it there very faintly. I hope the photocopied okay for you guys, right? And the way to do these is to make your list. Horizontal compression by a half. Horizontal reflection, horizontal compression by a half negative three compression positive three horizontal compression by a half negative four goes to negative two reflection positive two connect them. Horizontal compression negative one reflection positive one connect them zero zero doesn't are sorry zero one is invariant. I would probably do this point next. So it's to write compress it reflected ends up there and I think the next point that I would pick is four comma three because that way I can still do nice math. If I compress for right by a factor of a half. I'll end up that where was I to write and if I reflect it. I'll end up there. I think this ends up doing this now here's the fussy marking. First of all, if you did that and that and got the same graph you get one mark if you did a horizontal expansion by two but you did do a horizontal reflection sorry if you did that and you got the exact same graph as me you get two marks you get full marks if you did a horizontal expansion by two which is the most common error but you still do the horizontal reflection I give you one mark here is where I'd be nitpicky if you didn't clearly stop right there if you put an arrow there take a half mark off or if you didn't clearly keep going there if you didn't put an arrow there I'd take a half mark off unless you went all the way to the end of the graph and stopped we decided that if you go the edge of the graph paper that implies that you're continuing that implies an arrow but most of you are too lazy to draw that much I certainly am I'll just put an arrow there okay so now we're going to be a bit fuss here on whether it's continuing or fancy word discreet coming to a stop B what number is in front of the X right there it's invisible Amanda little alarm bell would go off as a negative one in front of the X and a slide it's not factored I need to rewrite this oh yes I do I need to rewrite this as Y equals F of I'm going to factor out that negative one which will give me a positive X I'm going to write the positive X first when I factor a negative one out of a positive one I think I get a negative one boys and girls this graph is one right not one left the curveball there this graph is one right not one left by the way don't believe me multiply that out Amanda what's a negative X times or what's a negative one times an X I get a negative X say yes and what's a negative one times a negative one will I get a pot in other words I would get a positive one and a negative X in a different order I also made this I was mean by changing the order welcome to the big leagues this is going to be then horizontal reflection one right horizontal reflection boom one right there horizontal reflection one right there connect them horizontal reflection one right there connect them horizontal reflection one right there connect them and then since I'm not dividing by a number so I'm not worried about decimals I probably pick this point here if I horizontal reflect it instead of 2 over 3 right it'll be 2 over 3 left and then one right and I probably pick this point there horizontal reflected one oh you know what this is going like that one mark for the horizontal reflection one mark for one right however if you did any points wrong you would lose a half mark for each point that was wrong how many you got that tricked a bunch of you oh that's good then because you just learned from your mistakes yes woohoo a point of this turn the page it's so much so much fun so much fun okay ooh I see three things here I see absolute value of the twist I see absolute value I see vertical expansion by two I see three up what's the correct order if I wanted you to do the absolute value last here's how I would write it rocks and don't write this down I would write it as why equals that's if I wanted you to do the absolute value last what if I want you to do the absolute value in the middle that would be vertical stretch first then absolute value then one up three up here I think I want you to absolute value first I said to you we treat it like a bracket like the bed mass rules so this is going to be absolute value vertical expansion by two three up here we go absolute value Katie instead of below the x-axis by four I'm above the x-axis by four vertical expansion by two eight three up one two three well I'm off my page but only by one I won't freak out just yet but I'm off by six or seven I'll get nervous because I probably then made a mistake let's try this one here this is another nice point absolute value of zero is zero vertical expansion of zero is still zero three up where they connected and connect them how about this one absolute value of one stays one vertical expansion by a factor of two three up let's do this point by the way I think I've made a mistake here and I'll show you go back and show you the subtle mistake in a second this is a weird one and I think I took too many shortcuts absolute value one vertical expansion three up ends up being a lovely horizontal line this point absolute value of two is two vertical expansion one up let's do this point absolute value of three is three vertical expansion one up okay this I'm good with I'm a little nervous because this look like it had a corner jagged part and this didn't I'm actually going to temporarily nuke that stretch right there I'm going to do this point let's see absolute value of negative two is two vertical expanded for one up okay I'm also going to do this point right here which is one high hanging in midair absolute value sorry what I'm supposed to want what am I going one up and I did that the whole time like an idiot that's why it looks so stupid candy why am I going one up I don't know should we try this again scene one act one take two those you watching online let's pretend that never happened just fast for anyways absolute value vertical expansion I did go three up on the first one I got that one right woohoo I'm going to do this guy because this is a little weird one and I said the absolute value ones I can't predict as much absolute value double it one two three up I'll do this one absolute value doesn't want to wear double yet doesn't go anywhere three up goes there okay that looks like kind of a little line right there how about right here absolute value one double it two one two three up I didn't go like this yes I knew there was something weird something was bugging me at least I knew something was bugging me partway through you heard me I've done something wrong here how about right here absolute value stays one double it three up all how about we move over here absolute value stays double it two three up let's go here absolute value stays double it there three up one two three let's go there doesn't do that is that what it looks like people are nodding now okay first of all if you got that three marks woohoo otherwise I would mark this if I was marking yours I would say did you do a vertical expansion can I see that things have been stretched that would get you one mark I would try and see if you got three up that would get you one mark I would try and see if you had done absolute value if everything below the x-axis had flipped I'd be looking for this pointy part up that would get you one mark but then I take a half mark off for each point that was wrong excuse me how I mark these D whoa whoa well I think I see three things going on here I think I see reciprocal vert expansion by two and a vertical reflection how would I do this I think I would do the reciprocal as dotted lines and then once I've done the reciprocal completely I'll vertically expand it by two and vertically reflected I think that's the safest way to get there I could probably do it all in one step this is a quiz or a test I'm going to hedge my bets so I'm going to change colors I'll go with blue reciprocal would be right there right there those would be invariant to be a vertical asymptote right there and then this would shoot off to infinity this would get lower not solid line Mr. Doe this would get lower and come to a stop right there because it doesn't go any further to the left from this invariant point right here I'm getting closer to zero I think I would shoot off to infinity this would be a horizontal line one high and now I'm getting bigger and bigger I'm going to curve closer and closer and closer to zero so the reciprocal would look like then I would take everything double the heights and flip it now I'm going to do the whole thing in one fell swoop just to show you that that approach also does work and because I find it easier are you ready reciprocal what was the first thing that I always looked for on reciprocal points the invariance what were the invariant points anywhere how high one and negative one so this is going to be an invariant point right here then I would vertically expand it by two and vertically reflected it's going to end up right there negative one high is going to end up right there this whole line is one high so this whole line would be invariant but then when I vertically expand it by two instead of this whole line being one high how high will it be too high and then if I reflect it said this whole line being positive too high how high will it be this will end up down except let's just stress straight line that'll end up right there the asymptote won't move at all because the asymptote if you stretch it or flip it verdict if you stretch it or flip it you're still going to end up with stretch it or flip it it's going to still be a vertical line where will this point end up let's see take the reciprocal so instead of negative four what's the reciprocal of negative four meant to be really easy what's the reciprocal of negative four negative a quarter right reciprocal are you doing is going one over the number so what's the typical of negative four one over negative four or negative a quarter so I'd be right there expand that by two what's twice as big as a quarter a half reflected this point would end up right there and I think I would curve closer and closer towards it like that see my little bug trick originally we're moving closer does I do the asymptote in the wrong place no wonder you guys are confused good gosh do it I'm not on a roll today am I how about I put the asymptote right here where it belongs is that what you guys are saying caught my own mistake at least wow remember that adrenaline coffee joke apparently you know I know you are but what am I apparently that right okay Steph I noticed that I'm going closer to zero that would shoot off to infinity but negative but when I reflect and stretch it shoot off to infinity but positive that okay here closer to zero shoot off to infinity but positive shoot off to infinity but negative and faster and don't touch the asymptote here we're getting bigger what's the reciprocal of getting bigger getting closer to zero but we'd also stretch and flip I think you would get closer to zero like that that's probably just above how hard I'm willing to ask you questions that's nasty that's an A plus or a very high a level now how would I mark this if unlike me if you got the asymptote in one in the right place that gets you one mark if unlike me you got the invariant points correct in other words if you have a horizontal line at negative two right there and you have a dot at negative four and a half comma positive two that gets you one mark if you got this shape correct that gets you a half mark if you got this shape correct that gets you a half mark so one mark for the asymptote one mark for the invariant points and one mark for the shape that's how I in other words I hope it's pretty tough to get zero on these because I'm hoping all of you can find the asymptote you couldn't Mr. Shut up let's pretend when we're wide awake and paying attention all of us can find the asymptote wherever the graph touches the x-axis or is zero number six find the inverse how do I find an inverse oh let's do that first of all x equals five y all over y plus one that gets you a half mark but it also says you must isolate the y variable okay we're going to cross multiply we're going to get x times y plus one equals five y by the way what's on the bottom of this fraction it's invisible one so it's five y times one just five y I see all sorts of things now people are like hey maybe I want to divide by the extra no get rid of brackets trust me x y plus x equals five y now I see kids they want to divide by five or divide by look how many wise do I have to how many would I prefer one get the wise to the same side how many wise do I have to how many wise would I prefer one it would be wonderful if there was some kind of a little grade nine mathematical operation I could pull out of my back pocket that would somehow turn this from two wise into one why and there is what GCF I can factor this is actually x equals y bracket five minus x and I missed it I missed it Amanda I apologize because I was supposed to say to you what is the GCF and you were supposed to say back to me because I asked you to why I think we had that little dance last day you and I didn't we yes the dance goes on the dance goes on Amanda final step how would I get the y by itself now what's happening between the y and the bracket so how would I get the y by itself divide by the whole bracket it really is an all honesty math eight and math nine at a math 12 level but there's not a single skill in there that we don't teach you in math eight and math nine final answer y is going to be x divided by five minus x there is another possible answer it all depends on which side you move stuff to you could also have gotten y equals negative x all over x minus five that's the same answer it just depends when you cross multiplied which side you moved stuff to when I cross multiplied I moved those to that side if you if you had the five y here and the x y there and instead of getting the instead you move the five y over anyways there is that how would I mark this how many marks is this one worth to I would give you point five if I saw that I would give you point five if I saw that I would give you point five if you've got the wise all to the same size and outside and I would give you point five for your final answer is you will find it is 11 plus three four five six seven eight nine ten eleven twelve fifteen eighteen twenty I think it's eleven plus twenty is the quiz out of thirty one I hope it is woohoo quiz thirty one sort of thirty one marks is way too much for a quiz that would make this by far the biggest quiz of the year so here's what we're going to do take your score and divide it by two and we're going to make it out of fifteen now technically I should make it out of fifteen point five every one of you just got a free half mark because I don't feel like typing decimals into my marks program I don't want to test ever to be out of a decimal number so give yourself a score out of fifteen that makes sense just divide your score by five now normally I would collect these except I want you to have these to study from so here's what we're going to do first over any pause so let's continue with all I have done is I've given you a handout last day of really I've gone through the provincial I found some basic ones just to show you what they look like but then I started looking at old exams and trying to find weird curveball questions if you're wondering the handout initially I think it said something like more transformations questions on the very very front of it and we've got to the very very very very last question yes okay and what I'm trying to say is what kind of by the way again let's be very clear there's not going to be twenty of these on your test but absolutely will it be a few curveballs yes okay guarantee there's going to be a question that you haven't quite seen before but I've talked about this part of it and I've talked about this part of it and I've talked about this part of it and one of the skills you have to learn is to put those three parts together on the fly the phrase you're going to hear me use all year is stubborn cleverness I'm stubborn I'm clever I will never ever quit now what I will do because it's almost always going to be multiple choice questions Cassandra as soon as I come to a question that I can't get 30 seconds I circle it and I come back to it because I'm a good test writer I won't get bogged down in it that's back so what kind of extension questions can they ask you we did a bunch but you know one of the ones they we love to do is we like to give you questions and ask how is the domain or the range been affected so I said to you the range of this function is negative eight to six and I drew the simplest possible example I said you know what just so I can kind of visualize what's going on the negative got chopped off somehow there's negative eight there's positive six I'm just going to draw a slanty line that's an example of a graph that has a range of between negative eight and positive six what would the range of this be well what's going on here what's this to doing vertical or horizontal vertical expansion by two what's this mean one up so vertical and is one up also vertical okay so if I write my original range negative y less or negative eight less than or equal to y less than or equal to six range is vertical that means this is going to affect the range and this is going to affect the range how is this going to affect the range well instead of a negative eight it's going to be a negative 16 instead of a six it's going to be a 12 how is this going to affect the range everything now is going to move one up so instead of negative 16 you'll be at negative 15 and instead of negative 12 you'll be at 13 I'll write my final answer neatly. The final range here, negative 15 less than or equal to y, less than or equal to 13. In fact, it's a moved one up because a, which we did last day, was a vertical expansion by 2. All I did for part b is say, ah, I've added a slide. What if I'd added a left-right slide? Would that change the range at all? What would that change? The domain, the domain, because domain is horizontal, right? See, what's the range of that? Now, at one point I saw Steph pants up. She got ready because she thought I was going to do an alarm bell thing and I'm not because that would be a waste of time. You know why? Is that horizontal or vertical? I don't care what it is then. Is that horizontal or vertical? I don't care what it is then because can a horizontal ever affect your range? I didn't even bother doing the factor. I didn't need to because I said there's nothing vertical going on here. How possibly can the range change? I'm going to add one more here. D, E, let's put a little f right here. What if they had done this? Negative f of x. What's your range? I've got to be careful because you said negative six except we don't write it in that order. Let's get the whole thing. What's the range? This whole thing would flip so the lowest point would now be, no the lowest point would not be positive six. The lowest point would now be negative six and the highest point would now be eight. This would go negative six less than or equal to y less than or equal to eight, right? This one is surprisingly tricky. D, what's the range of the absolute value of that graph? I'm going to re-sketch it down here because I've scrolled down so much. I'm going to say here's my simple version of the graph where this is negative eight right here and this is six right here. There's my simple version. What does the absolute value transformation do? Anything that's above the x-axis is what? Ooh, you can use the math word. That would stay as is. Anything below the x-axis is what? Reflected. This is going to end up where here, what's my new range? Careful. What's the highest I go? What's the lowest I go? Zero. This scares kids because when they're glancing at their answers, they would be saying, where is the zero? Yeah, zero is the lowest and the highest is eight and there, where'd the six go? It turns out it got dominated by the eight. By far the toughest one is E. E is a nice question and you know what? It's tough enough that once again, I'm going to re-sketch the graph here and I'm going to put it way over here in the positive area kind of by itself so I can see what's going on. I'm going to say there's negative eight right there. There's positive six right there. I'm going to make it nice and slanty. What would the range of the reciprocal be? Well, how would I graph the reciprocal? The first thing that I would do is I would find any invariant heights. Invariant heights would be one high and negative one high, about there, correct? Correct? Then I would find any vertical asymptotes. Now the way I've drawn this, there's a vertical asymptote right there. Doesn't matter how you draw it. I do know there would be at least one though if it goes from negative eight to positive six. It's going through the x-axis somewhere. Could be more than one. It could be bouncing up. I don't care. I'm looking at simplest possible case right now. Then I would do my little bug trick. As I walk to the right, I'm getting closer to zero from below. I would shoot off to negative infinity. As I walk to the left, I'm going to get closer to zero because my original is getting bigger. But how low does my original go? How low? Negative eight. You know what? This would stop right there and the height right there would be negative one over eight. Would it not? It's a reciprocal because if it was too high, became a half high, three high. You know what? If the highest, if the furthest from zero you go is negative eight, the closest to zero you'll get is negative one over eight. The same thing over here by the way. Moving to my left, it touches zero so it does shoot off to infinity. But moving to my right, Amanda, what's the highest this graph got? Amanda Ward. What's the highest this graph got? What's the reciprocal of six? You know what? The lowest it's going to get is right there and that's going to be a height of one over six. What actually happens is instead of having one range, you have two completely separate graphs, completely separate ranges. Yeah. On the test, if I gave you this, I wouldn't give you any graph at all but I would draw one to make my life easier to figure this out. Maybe you're clever enough to puzzle this out yourself. Great. Can I go back to my question? Can you see here there are two separate graphs, this one, jump, and this one. They don't connect. The two red graphs don't connect. That means I'm going to have to list two separate ranges. How high does this graph go? It's a trick question. Infinity. How low? Touching? Yes. This has a range of Y greater than or equal to one-sixth, comma, that's the top part. The bottom part, how low does this graph go? Negative infinity. How high does it go? Negative one-eighth, touching. Everything below and touching. Everything below and touching, negative one-eighth. That's a tough question. Let you think or ponder that a little bit. I know on one of my tests, I have a couple of different versions on one of my tests, I like this question. I like this question. I think I ask you to take the range either of a reciprocal or an absolute value. I can't remember which one. Let it pause and percolate and think. When's your test? I think I also see you Friday, I would imagine, of next week. That's when we're going to be starting the next unit. You want to try and, if you're planning on buying a graphing calculator, have one by then. Otherwise, on Friday, I'll hand one out to everybody. If you're planning on renting one, I'll let you keep it and I'll start hounding you for the deposit check. If you're planning on buying one, I'll just say give it back at the other class. I haven't looked yet, but you have agendas that probably say on there. I'm assuming with no Monday that they would move Monday to Friday? Just be thinking. That would make sense. Not that the calendar has to make sense, but that's what you're thinking, okay? Yeah, I agree with a lot of things. There it is.