 Reciprocal transformation. Before I talk about the reciprocal transformation, I need to ask, what the heck is a reciprocal? So, don't write this down, but what's the reciprocal of 3? 1 over 3. What's the reciprocal of 2 over 5? 5 over 2. What's the reciprocal of 1 over 3? What's the reciprocal of negative 1 over 3? Negative 3. Reciprocal means flip the number, flip the fraction. That's not a fraction. Can we decide and agree right now for the rest of our math lives that everything is a fraction, that all numbers are over what? 1. I'm not going to have, please don't make me have to repeat that one. Grade 8, I would, but let's remember that technically anything is a fraction. There are a couple of interesting reciprocals. What's the reciprocal of 1? 1 is its own reciprocal. There is another number that's its own reciprocal. Can you think of it? Not only 1, but negative 1 is also its own reciprocal. Now, I love the fact that I heard somebody back there whisper the other guest, which is actually a bad guest, so I'll show you why in just a second. What's the reciprocal of 0? Well, technically Ryan, it's that, but what's wrong with that? Ellen, you can't divide by 0. Why? Because you can't. Thank you. That's called circular reasoning. You can't because you can't, like saying I'm stupid because I'm dumb. Yes, it's circular, fine. Put your pencils down, don't write this down. Here's why. Ellen, when I say 6 divided by 3 equals 2, I'm also saying that 2 times 3 equals 6. If I was able to go 6 divided by 0 and get an answer, I'll call the answer x because I don't know what it is. I would also have to be able to say that 0 times x had to be 6. What's the problem with that statement? Yeah, we can have our 0 times table, but we can't have our 0 times table and have a 0 division table. Timesing by 0 has no inverse. And we use this all over the place, not just because it makes sense, but when you're factoring and finding roots, what you were technically doing was using your 0 times table. It's called the 0 principle in a clever way. So that's why. Nobody ever showed me that? It's like great five math. I always show my kids that when they say that. So with the reciprocal, what you're doing is you're flipping the number. What we're going to talk about is taking the reciprocal of a function, taking the reciprocal of a function, except all along we've been saying this, write this down over here. We've been saying that y equals f of x, that's been our generic function notation. We've said y is f of x, y is f of x, y is f of x. But when you hear the word y, I think the height, the height. What this is really saying, this particular transformation right there is really saying that. It's really saying one over the height. And height is EI, not IE for those of you who are spelling it correctly at home. Mitsu's figured out what page we're on, we're less than 11. So put your pencils down. Let's see if we can think about how this should work. If my original graph is one high, what's the reciprocal of one? One, my new graph should also be one high right there. The fancy word for a point that doesn't move invariant. Oh, and you know what other heights will be invariant? Anywhere on my original graph that's negative one high will stay where they are. In fact, the way I graph these is the first thing I do is they give me a graph and say, graph the reciprocal. I find anywhere that's one high or negative one high, I put huge dots there to say. They're not going anywhere. Which height could I not take a reciprocal of? X intercepts are going to be a problem. In fact, anywhere there's an X intercept, we're going to draw a vertical dotted line. There's going to be a hole in the graph there. Boy, the award ceremony could be interesting on Friday. Good afternoon, ladies and gentlemen. Welcome to the award ceremony. What's the reciprocal of 10? One over 10. What's the reciprocal of 20? What's the reciprocal of 30? What's the reciprocal of a really big number? A really small number, a small fraction. One over that big number. In other words, as your original gets bigger, you know what your reciprocal is going to do? It's going to get, I mean, everything's back. Yeah. And as your original gets smaller, closer to zero, closer to zero, you know what your reciprocal graph is going to do? In fact, special numbers. As your original touches zero, the smallest possible height, what's the biggest possible number? You all know it. You all watch Toy Story. What's the biggest possible number? Your reciprocal is going to shoot off to infinity. Oh, and if your original graph shoots off to infinity, you know what the reciprocal of infinity is? Your reciprocal is actually going to have an x-intercept and it's both going to go zero high level. That's how we're going to do this. Bigger becomes smaller. Smaller becomes bigger. Ones and negative ones are invariant. Zeroes are asymptotes. Watch. They gave you two graphs in woman number one. They graphed a line and then using a graphing calculator, they graphed its reciprocal to the right. I'm actually going to zoom in to this graph on the left to prove to you that I don't need the graph on the right. I can recreate it using Emily, the rules that I just stated to you. So let's see, watch to see if I can sketch and end up with what you guys have on the right of your page there. The first thing I would say is anywhere one high or negative one high is invariant. I would say that's one high right there. That's negative one high right there. I would put big dots there saying those are going to be on my reciprocal too. Then I would look for anywhere zero high. I see it. And I would say, you know what? There is going to be a hole in the graph right there. I draw a vertical line. Is there kind of an invisible vertical line in the middle of the graph to the right? See? And by doing that, if you look very close, I've divided this graph into one, two, three, four sections. If I use my invariant points and my vertical lines as boundary markers. I've done the hard work. Now all I have to do is go Sesame Street on you. I'm going to use my imagination. So Sesame Street is brought to you today by your imagination. Here's what we're going to imagine, Ashley. We're going to imagine that we're a little tiny bug walking on the original graph. And as a bug, I always start out on the invariant point because I know those are right. So Brett, I'll pick on you. Pretend you're a little bug right here. As you walk along your original graph heading left, your original graph is getting further from zero and negative. You know what the reciprocal of further from zero is? Take a guess. And now here's what I need to teach you as it turns out straight lines become curves. It's going to curve closer to zero and stay negative because the original was negative. Because we said that the signs don't change when you take reciprocals. This is so much fun in my imagination. Maria, let's go with you next. So stand right here. As I move to the right as I walk along this black graph, can you see I'm getting closer and closer to zero but negative? You know what the reciprocal of closer to zero is? Shoots off to infinity. Is that what the graph to the right does? That's so much fun. This is my imagination. Dancing and stories in one class. Wow. Sorry. Let's look at the book. So pretend you're here. Imagine you're standing here. Walk to your right. My original graph is getting further from zero but positive. You know what the reciprocal of further from zero is? It's going to curve closer and closer and closer to zero. Stand here. As I walk to the left, as I follow my graph to the left, it's getting closer to zero but positive. You know what the reciprocal of closer to zero but positive is? It's going to shoot off to infinity. I'm guessing my red graph looks like the same shape at least as the black graph to the right does it not? Let's find out. Was I right? Was I right? Was I right? Now I'm sketching. I'm not that concerned about getting every single point in the right location. I want the shape and I'm right on the shape. So what's our approach going to be? Invariate points plus or minus one. Assum totes zero high. And then pretend you're a little bug. However many sections you divided your graph into, we'll be doing one later where we divide our graph into nine sections. No problem. It's just a little bit more work but it's the same imagination. I should have had a little children's song with imagination in it next year. It says complete blah, blah, blah. Turn the page. Number 76, or number 76, page 76, what they've done is the big thick graph is a parabola. In fact, it's this equation right here. It's a parabola moved forward out. And then they've done the reciprocal of it. They didn't need to give me the reciprocal, that thin graph. I'll show you. I could have got it myself. If I was graphing this myself, I would look at the big thick graph and I would have gone as follows. I would have said anywhere one high or negative one high is invariant. I would have said, hey, that's one high, that's one high, that's negative one high, that's negative one high. And you notice both graphs go through those four points because they stayed where they are. Then I would have said anywhere zero high is going to be a vertical asymptote, that's zero high right there. It's going to be a vertical asymptote through here. That's zero high right there. It's going to be a vertical asymptote through there. I would have done that. And by the way, always draw your vertical asymptote as a dotted line because it's not actually part of your graph. It should technically be invisible, but as math nerds we've said, if you do it as a dotted line, we know that we're just supposed to ignore it. But if you draw it as a solid line, you're telling us that that would appear when you were graphing it. No, it doesn't. I've done invariant points, Ryan. I've done zeros. Now I've divided this graph into one, two, three, four, five, six, seven sections. It's time for my imagination. Ryan, I'd like you to pretend you're a little bug. What kind of a bug would you like to be, Ryan? A ladybug. Really. I'm not judging. So, we're a little ladybug. I'll do it in red then. Little red little ladybug. Stand right here, Ryan. As the ladybug walks to the left, can you see my original graph is getting further from zero? You know what the reciprocal would do? Sure enough, it would curve closer to zero. They didn't need to give me that. And this little short chunk right here, very short, as I walk to the right, it's getting closer to zero quickly. Shoots off to infinity quickly, steeply. Now your ladybug, we're going to ignore this because the ladybug jumps over here because this also, by the way, my original graph is symmetrical. I'm willing to bet this will be symmetrical. This kind of shortcuts, Brett, you can take along the way somewhere. And sure enough, if I move to the right, my original is getting further from zero. Reciprocal curves closer to zero. Oh. And as I move to the left, this little chunk right here, Ryan, is getting closer to zero quickly. Shoots off to infinity quickly. This is so much fun, Ryan. Let's keep going. Stand right here. Now this little chunk of graph, right there, it looks like the original gets closer to zero but negative. What's the reciprocal of closer to zero but negative? Shoots off to infinity but negative. Does that make sense? Okay. Closer to zero from below, shoot off to infinity below. They didn't need to give me. I knew that. And so this is what's bugging people right now. Let's come back to that. Let's do this little section right here. This little chunk right here is getting closer to zero but negative. It would shoot off to infinity but negative. How would I do this section? I would look at this point right here. Ryan, what's the height of this point right here right now? What's the reciprocal of negative four? Flip it. Or just negative a quarter. So this will become negative a quarter high. About there. Yes? And then I would say this. My original graph is getting bigger so my reciprocal should get closer to zero. But the highest my original graph gets is negative four. The lowest my reciprocal should get should be negative four. You know what? It should curve closer and closer and go right through there. And it should curve closer and closer and go right through there. That's where that little section came from. That's how this process works. I could have got that graph without them giving it to me. Just with my imagination. Let's actually write some of this down. Turn the page. Properties of reciprocal transformations. And we're going to use the symbol f of x for the original graph and one over f of x for the reciprocal graph. If my original graph is zero high, my reciprocal graph may have a blank blank. We said it's going to have a vertical asymptote. Now I've used that word, but you might not have seen that word before. An asymptote on a graph is an invisible line that the graph gets closer and closer and closer and closer and closer and closer and closer to, but never touches. Kind of like a guide rail, which is why whenever I can I'll graph asymptotes first because it's giving me railings. Helps to graph the rest of it. Oh, if my original graph is positive above the x-axis, my reciprocal graph is... When I take the reciprocal, does the sign change? No? So if my original is positive, my reciprocal is positive. And if my original is negative, my reciprocal is negative. Ooh! When f of x, when my original graph is one high, how high was my reciprocal? What was the reciprocal of one? One. Oh, and you know what the reciprocal of negative one high was? Those were our invariant points. Now the author says the invariant points can be found where the line's y equals plus or minus one. You know what? I don't actually sketch lines. I just slide across one high, negative one high. I can do the undergraphing. Then, here's our bug trick. When my original graph increases, gets bigger, my reciprocal does what? Gets smaller. Or since they use the word increases, what would be a clever word to use? Decreases. Oh! And when my original graph decreases, gets smaller. Alex, you know what my reciprocal graph does? Increases, absolutely. Oh! And remember the little bug? When my original graph gets closer and closer to zero, the reciprocal shoots off to positive or negative infinity. Positive infinity, if your original was positive, negative infinity, if your original was negative. And the graph approaches a vertical asymptote. And the last one, we're just going to see if we can figure out. If zero high on my original gives me a vertical asymptote on my reciprocal, you know what a vertical asymptote on your original will give you on your reciprocal? I heard it. I think I heard it. If zero gives you an asymptote, you know what an asymptote gives you? Zero high. In other words, if they give you a weird graph and it has a big vertical asymptote, your reciprocal goes through the zero x-axis right there. So if my original approaches infinity, my new one approaches zero. And then he says, and the graph approaches a horizontal asymptote, I'm just going to scribble that out. He's sort of right, but only for weird graphs, and we're not going to look at too many of those this year, so I'm not going to freak out. You know what the most common mistake on this is? It's not the actual technique. Almost all of my kids get the technique. The most common mistake is this. On the test, I ask them to graph this, and on that graph, they graph this. So the author has written here, and I'm going to highlight it. Remember, that does not mean reciprocal. This means inverse. Reciprocal means reciprocal. Now, it's not totally your fault. I've gone on a rant about how bad the notation is. Who's in Chem 12 or Physics 12? You know what we call this in Chem 12 or Physics 12? Inverse. We say they're related inversely. We honestly should say they're related reciprocally, but they don't. So they reinforce actually a mistake that's wrong. Sorry, not totally your fault. Oh, so far, Trevor, every time I've introduced a new transformation to you, I've then said, here's the replacement. Here's how you would get the equation. For reciprocal, what you're really doing is you're replacing y with 1 over y, which is what we did to graph it. That's the reciprocal transformation. Yeah, let's try some. Through the page. Yeah, yeah, yeah. Through the page. Skip example one. Let's go to example two. Example two. It says, the graph of y equals f of x has been given. Sketch the graph of 1 over f of x. And then b says, in each case, write the equation. Forget it. Let's do this first one. I'll zoom in on this first one. First one looks like a parabola that's been moved to right. I don't care. I can graph this. Let's see. First thing I said we're going to do is invariant points. What are invariant points? Which heights don't change when you take the reciprocals? Yeah, I don't think there is a height of negative one at all on this graph, but there is definitely heights of positive one. That's going to stay. That's going to stay. The next thing I looked for was vertical asymptotes. Now vertical asymptotes showed up where my graph was how high? Zero. You know what? There's going to be a little vertical asymptote right here. And I've now divided this graph into one, two, three, four sections. So far, so good, Sandley. Happy joy. Okay. Time to use our imagination, Sandley. You ready? Here we go. Oh, this is so much fun. Pretend you're a little ladybug standing right there. As I move to the left, what's my original graph doing? Getting closer to zero or further from zero? You know what the reciprocal of further from zero is? It's going to go like that. By the way, if I want it to be even more accurate, say, well, right here, you're just below three high right here. You should be just above one-third high. I'm not that worried. We're sketching. Oh, let's keep going. So much fun. Back here a little visible bug. Ladybug, as I walk to the right, my reciprocal is getting closer to zero. Sorry, my original is getting closer to zero. What's the reciprocal going to do? It's going to shoot off to infinity. Like that. By the way, I'm willing to bet this is going to be symmetrical and look like this. Let's prove it. Ready, Ashley? It's so much fun. Let's use our imagination. Here we go. Ready, Ashley? Stand here. Walk to the right. As I move to the right, my original graph is getting further from zero. You know what the reciprocal of further from zero is? Closer to zero. Further from zero is going to become closer to zero. Is that all right? Yeah. And Mitsu, as I walk to my left, I'm getting closer to zero. You know what the reciprocal of closer to zero is going to be? Infinity and beyond. What's kind of like a mustache when you're making little happy faces? I don't know. It looks like, you know what? That's the reciprocal of a parabola. Now there, they used an actual graph. That's a parabola. For this next one, they just did a eh function. That's not an equation. That's just some shape. No problem. No confidence. I can do this. Invariant heights first. Which reciprocal heights are invariant, Brett? I'm going to put a, there's one high. See, he draws this line in. I really think that's overkill. I can usually just move sideways by counting and just fill the dots in. Whatever. Asymptotes. Anywhere zero high is going to be right there and right there. And it looks like this time, Brett, I've divided this graph into one, two, three, four, five, six, seven, eight. It looks like kind of nine chunks. No one. Let's use your imagination. Excellent. I'll go with red just cause, oh, cause it's ladybugs. Standing right here, as I move to the left, my originals getting further from zero, reciprocal is going to curve closer to zero. Spends as I move to the right, getting closer to zero, shoot off to infinity. This is a weird portion here. I'm going to avoid it right now. I'm going to do this side here cause it's kind of similar to this side here. As I move to the right over here, it's shooting off to infinity, closer to zero. And as I move to the left, it's getting closer to zero, shoot off to infinity. See, Ellen, if we can logically reason our way to the bottom section. Well, Ellen, if I stand here, and if I move to my left, my original graph is getting closer to zero, but negative, what's the reciprocal going to do? Negative infinity. And Ellen, I think over here too. Caitlyn, how high am I right here? That'll be fussy. Negative two. What's the reciprocal of negative two? How high am I right here? What's the reciprocal of negative two? How high am I right here? What's the reciprocal of negative two? Negative a half. You know what, I think a horizontal line ends up being a horizontal line still, but at the reciprocal height. This whole line that's negative too high is going to become a horizontal line negative a half high. Halfway between zero and one. I can probably do a bit of a better horizontal line than that. Yes, as it turns out, horizontal lines become horizontal lines. A horizontal line three high would end up being a horizontal line one third high. Ah, what about a horizontal line one high? And being a horizontal line one high, you have an invariant line. Caitlyn, let's see if we can figure out what's going to happen here and here. Well, my original is getting bigger. My reciprocal will get smaller. And my original gets up to two. So my reciprocal will get down to negative a half. You know what, I think it'll curve in tufts there. And I think it'll also curve and connect right there. Is that okay? Yeah, yeah. Maybe, sure. That's the plan stand. That's the reciprocal transformation. Now, when they first started Math 12, they would just ask reciprocal transformations. When they first started Math 12, in fact, I think the first provincial exam when they brought in this new course back in 2001 for four marks, don't write this down, had this. That's too easy now. Now they would do, what do you think that does? Take the reciprocal and then what? Move the whole graph one up or one left? And how do you know? It's one up. If I wanted to move it one left, where would I put that X? Okay. Or I've seen them do this. Take the reciprocal and then what? Flip the whole thing vertically. I've even seen them do this. Take the reciprocal and then stretch every point by two and reflect it vertically. So instead of one highs being invariant, two highs end up being invariant. We're not going to do those today. We will eventually practice a few of those. I'll put one like that on your test. Turn the page. I'm going to sort of pause here in that I'm going to do this question next class and your homework. One A. I would like you to try doing the reciprocal of those four graphs. B says write the equation. I don't care about the equation. A, just find the reciprocal of those four graphs. Your other homework is the take home quiz. You've got about 20 minutes to work on all of this stuff. Peace go man.