 So, in theory, we're running. Here we go. Tutorials. Math 12. New page. Folks, as you come in, grab one of those, please. Transformations. Tutorial. The basic, if I was going to summarize the entire unit in two words, well, three words, everything is backwards is what I would summarize the entire unit with. In other words, if you're not sure what to do, do the opposite of what the equation seems to suggest. So, we defined function notation. We said we're going to write all of our equations as y equals a bracket f minus, or f of x minus b. Oh, hang on, Mr. DeWitt, get this right. A little rusty clearly. b bracket x minus c close bracket close bracket plus d. We said every single function. Pardon me? No, don't write on them, please. Sorry, good question. I need to reuse those, and we don't have extras. So, don't write in the little test booklets, but we're going to use them for some sample questions later on. We wrote most of our equations in this function notation. Now, remember, for what it's worth, you could actually take the time to rewrite this, grab one of those, as a bracket y minus d close bracket equals, and technically we won over a f of b bracket x minus c. Now, if you took the time to rewrite everything like this, where you got all the y stuff next to the y, and all the x stuff next to the x, then my everything is backwards rule held. Then it was just fine. So, we looked at stuff like this. If you want to get out that little booklet that I gave you, and if you'd be so kind as to open to the first test, I think it's like page four. This question here is the one you're looking for. Now remember, your provincial and your mock provincials are going to have a non-calculator section, and it's going to be 16 questions. A lot of it will be transformations, because that lends itself to non-calculator questions. So, it says the graph of y equals f of x is translated one unit to the left. Let's copy that into here, Mr. Dewick, so that you can print the question. Which equation represents this translation? Now, probably most of you are going, yeah, I can do this already, as is no problem. But here's what we said. We thought in terms of replacements. We said, first of all, if you're translating to the left, is that horizontal or vertical? Left, is that a horizontal or a vertical? Horizontal. Horizontal. That means you're replacing x with something. And what are you replacing? Well, if you want to move one left, what you're really doing, Matt, is you're replacing x with x plus one. Everything is backwards. One left in the negative direction meant replacing x with x plus one. So if my original equation was y equals f of x, that would become y equals f of replace every x with an x plus one, and now the answer is clearly c. Everything is backwards. Right? Let's look at number two. Which of the following represents the transformation of y equals g of x if it's horizontally expanded by a factor of three over two and then horizontally translated c units? I'm just going to work through some of these questions kind of as a refresh our memory gig, and then we'll take specifics and every so often go off on a tangent. See what is now, Matt? Good. Horizontally expanded by a factor of three over two, that means you're going to replace x with what? Defining silence. Everything's backwards. Two over three x. Then you're going to horizontally translate at c units. That means you're going to replace x with x minus c if you want to move at c units to the right. So if I walk through the transformations, I would have this y equals g of, first of all, I would replace the x with a two over three x, and then I would replace this x with that right there. Everyone grab the little test there, please. The test thingy booklet. So when I do that, I'll get y equals g of two thirds, and I guess I'll have to put it in brackets. x minus c. What's the correct answer here? Don't all shout it out at once. D. One of my friends who typed up a review, he summarized everything this way, and I'll put this one online under transformations review as well, eventually. He said, here's what you want to remember. Anything here is a vertical stretch, and it's not backwards because it's already been moved over. If it was on this side, it would be backwards. So a one-half over here is a vertical expansion by a factor of two, because a one-half becomes a two on this side anyhow. Anything next to the x is horizontal. This would be a horizontal expansion of compression. If the absolute value of b is bigger than zero, in other words, if b is a fraction, sorry, if b is a bigger than zero, that should say bigger than one, I think, and bigger than one, then it's going to be a compression to fix that typo a bit later. Remember your reflections, a negative there reflected vertically, a negative in front of the x reflected horizontally. You could also reflect about the line y equals x by replacing the x and y with each other. We called that taking the inverse, and we had all our translations. Then we had to remember the order. The order that we did everything was, we said factor first, then expansions, compressions, then reflections, then slides. Let's look at a few more. Hello, this thing. So, number three. Now, there's only so many questions we can ask you that say, find the point or tell me the transformation. Eventually, we'll try and mix and match a little bit. We'll start asking questions about things like the domain and stuff like that. So, looking at number three, turn that off, I remember the screencast. Graph of y equals f of x is given below. The function is transformed to that. What's its domain? Now, when I look at this, y equals f of 2x minus 6, the alarm bell goes off right now because it's not factored. Anytime they give me one that looks like this that's not factored, the first thing I have to do is factor it. I'm going to rewrite this as y equals f of 2 bracket x minus 3 because Jeff, this has not been moved 6 to the right. How far has it really been moved? 3 to the right. Now, let's see if we can't answer the question. Now, this is already a bit of a more difficult one because they haven't even given me a specific point. They've given me a point p, q. Now, if you don't like that, feel free to temporarily make up nice numbers. I could say I'm going to pretend it's 4,8 and I'll see what happens to the 4 and the 8 and see if I can translate that there. But we're going to try and do this one algebraically. So, right now, what's the domain of this function before I do anything to it? Well, the domain is everything left and right. How far to the right does this function go? Forever. How far left does this function go? So, right now, the domain is everything to the right and touching p. Now, domain, is that horizontal or is that vertical? Horizontal, because as a trick question, they could have given me a bunch of vertical stuff and the answer would have been, you know what, no change if it's domain. The vertical stuff doesn't affect the domain anyways. So, they could have put a plus 1 here or a negative 1 half here. Who cares? That's not going to affect the horizontal. This is definitely horizontal. What's that do? Horizontal or vertical? Horizontal. Expansion or compression? Is that two times as big or half as big? Half as big. Why? If you're not sure, everything's backwards and it is next to the X. So, first of all, my domain would become that. I'm not done yet. What's that mean? I'll give you a hint. Three. Three right. I would take this number and I would add three to it. Do I see that sitting there anywhere? Yeah. Is that okay? Next one. Don't want that. Want that guy. Number five, how does that graph compare with that graph? Yeah, you're going to need to turn the page. Those of you who score are keeping it home. What I'm going to do is I'm going to go through the test. That's going to jog most of the cobwebs off your brain and then I'll do specific weird ones. I'll give you a couple of provincial exams to look at as well. And I'll give you a few hints about the mock and things that, you know, preview of coming attraction type deals. Well, is this in okay form? Am I okay with this? Is there anything I need to factor? I'm okay with this. Normally, on our graphing calculators, we move that one quarter over to this side and it became a four. But it's here. If it's next to the Y, everything's going to be backwards. So, I'm going to look at this one quarter. It's vertical or horizontal and how do I know? Vertical? Why? Because next to the Y, that often gets rid of half-year-long answers. It didn't in this case. Expansion by four or compression by one quarter? And look, can you grab one of those practice tests there, please? Sorry? Expansion by four. So, already I would cross out B and I would cross out D. X plus two means what? Two left. The answer, expand by four. Two left. The answer would be A. Number six. Can I get this all in one page? I can if I shrink it. Fifth page. Make it a little larger, Mr. Dewick. Not too big. Try that. Click. This is a good one. Come on. Okay. Says this graph has transformed four different ways to two graphs, one, two, three, and four. Okay. What's this one look like? Sorry? Inverse? Now, let's convince myself this point right here looks like it's around, let's say, negative three, comma, negative one-ish. Does this point here look like it's around negative one, comma, negative three-ish? Oh, did the X and Ys switch around? I think it's inverse. So what's the symbol for inverse? Little negative one, not one over f of X. Y equals, if I'm picking answers, I'm going, haha, no, no. What do you think they did with those two graphs? Looks like the simplest would be a vertical reflection. So I think this is Y equals negative f of X. Put a negative outside the brackets, vertical. Now, technically, I'm done because I just eliminated C. And if you're a good multiple-choice test writer, I would say stop. But since we're doing review, let's keep going. How do I know that this here is a reciprocal? And where do the asymptotes occur? Wherever my original graph has what? Zeros. Wherever my graph is zero high, my reciprocal will have asymptotes. Which points won't change? What was the fancy word for points that didn't change? Do you remember? Yep. Invariant. Which points would be invariant anywhere, how high or how high? One or negative one high. So if you're doing your reciprocal graph, what I always do is I put big, thick dots anywhere one high or negative one high. Then I draw in the asymptotes. And then I think bigger becomes smaller, smaller becomes bigger, because they're reciprocals of each other. So this would be Y equals one over f of X. And I think it looks like this one is a horizontal reflection. Y equals f of negative X. We're good. Let's keep going. Let's make it back to page width. We did number six, right? Let's look at number seven. And again, I got to shrink this down a little bit. One more, but I think that'll fit. Okay. The graph of that thing is shown below. Which of the following is the graph of Y equals absolute value of f of X minus two? Well, there's two things going on here. Absolute value. And what would a minus two outside the brackets do? I heard both answers. It's two down or two right. Which one? I heard them both again. Someone convinced me that they're right or someone else is wrong. If it was two to the right, where would that X, sorry, where would that two have to be? Inside right next to the X. Is it inside right next to the X? Okay, so I think it's two down. Now one question students have is, what's the correct order? Do I do absolute value first and down first? This one I think I would go down first and then absolute value. I think if it said this, that's how I would write do the absolute value first and then go down. Which you're going to treat the absolute value sort of like brackets in our bed mass order of operations scale. So two down would look kind of like this. And then what would the absolute value transformation do? Flip what up? Anything below the X axis, that section, will flip above the X axis. I can get a specific point there. Yeah, it would look something, boy, my graph is bad. Let's try this arm a little better. Mr. Duk. It would look something like that. And I'd do it with graph paper a little bit better. Am I totally off, Mr. Kamosi? No, this one should go through three if it's symmetrical. But let's just pretend I can't draw. Let's pretend all of you that have had me this year know that I can't draw to save my life. The graph paper that this thing has built in has been a godsend. Anyways, which one of these looks right? I think C. Sesame Street is brought to you today by the letter W. Turns out the letter W is an absolute value graph that was moved down first. Who knew? I thought the letter W was just two Vs next to it. It always bothered me, Mr. Duk. Why do they call it W and not WV? It's a series of life that we just have to live with. I know. Am I the only one that bugged when I was growing up? Yeah, but a V is still a V. I don't think girls do a V like that. And if they do, I'll never understand. Okay. The partial graphs of two functions are given below. Hey, I can clip this one. It'll fit. Which equation for Y equals G of X, that's this one here, represents the transformation of that? Well, I do notice they gave me a key point. Zero comma negative two became zero comma negative a half. What have they done? I heard someone say inverse. And I heard someone say reciprocal. Please tell me which is it. Inverse or reciprocal? What does inverse mean? When you see... Yeah, the symbol is f little negative one of X. That's inverse. That's reciprocal. How do you find an inverse? Switch the X and Y around. In other words, this would become two comma... Sorry, negative two comma zero. Did it become negative two comma zero? What does reciprocal mean? Take your heights and make them one over. The height. Your X's don't change because we're saying if f of X is actually the same as Y, this is saying graph one over Y. It's your heights that change. Your X's stayed where they were. In other words, if your original graph had a point there, your new graph had a point there, the only exception was zeros became asymptotes and your graph was undefined there. But everywhere else, your X's just became new points, but they didn't change. Anyways, I'm pretty sure this is a reciprocal, not an inverse. Inverse would go through negative two comma zero. The inverse would look something like that. And you could always check for the inverse because there are always reflections about the line Y equals X, except of course I have my inverse pointing the wrong way, Mr. Kamosy. The inverse would look something like that. For those of you who are scorekeeping at home, multiple choice, select the best answer, an approved calculator may be required. I'm going to try and do this without a calculator. Oh, here's another range question. Domain and range in many ways, although they look nastier in the transformation section, they're actually easier because if they ask me about the range, Jeff, I'm already ignoring all the horizontal garbage. I don't even care. Because you know what? That plus four can't change the range, Megan. It can't. So who cares? And if they had a two in there, who cares? I've eliminated a lot of my thinking. It's actually harder to find the point than it is to find the domain and the range. All I have to do is write down what's the range of the original function. So here's my original function. What's its range before I do anything? Why? Less than or equal to one. Now I just walk through all the vertical stuff that's going on here. Horizontal or vertical? Horizontal, that's not going to change the range. What's that going to do? One down. Oh, no, wait a minute. That's not one down. One of the most common mistakes. I don't remember when we went over the transformations test. Most you were kicking yourselves for sloppy math. There it is. While we're looking at this question that they gave us, what if they had done this? Now what's the range? Piece of cake. Easy. First of all, all that garbage isn't going to change the range, so who cares? My original range was why is less than or equal to one. Is this factored already? Well, in the middle there it's not, but that's all domain garbage. So I'm not going to waste my time. This and this are nice factored separate. I'm happy with this. So correct order. Expansions, compressions. What's that going to do? I'll give you a hint. It's either expand or compress. Math. Expand because over here it would be a one half and everything's backwards. When you move it to this side it's a two, no longer backwards. What's that factor of? So that's going to become a two. Expansions, reflections. Check. Sorry, expansions, compressions. Check. The next thing we do is reflections. What's that going to do? Vertical reflection. How's it going to affect my range? Negative. What's that? Pour down. There you go. They could even be really snarky and give me the range algebraically. In other words, they could have said something like this. I'll do a domain question instead. Sure. While picking up where we left off, Mr. Dewick made a mistake. That's the correct answer right there. If you're not sure why that's the correct answer right there, see Mr. Dewick or ask Mr. Kamosy because he's better at math than me. There. Save that one. No one's going to notice online. That's the difficulty with doing this for posterity. Number 10. Which transformation to y equals g of x might change both the domain and the range? Okay. Well, what will that do? Vertical or horizontal? Vertical. Can that possibly change the domain then? No. How about that? That's a vertical. No. How about that? That's a horizontal. Nope. What's this a symbol for? That's not an inverse. What's the x and y? Does that mean you would just switch the domain and range around too? If they weren't equal to begin with, they would definitely change. They would become each other. That was actually a nice little shortcut we often use. I use that quite often if I had to find the domain of a log graph. I'm not great at log graphs. Exponentials I'm good at. I would find the range of the exponential. That was the domain of the log graph and the range. On one of your mocks, you are going to have a before and after graph. Something like this. However, when I look at this question, first thing I would do, the alarm bell would go off with my students. My students don't even twitch anymore. I got you there. Okay. This one here? Not factored. These ones are all okay, but if I got a coefficient inside the brackets and a slide inside the brackets, I need to rewrite this one. This is actually f of, I'm going to factor out the negative 2. When I do that, I recognize ooh, this is one right, not two left. Now let's see what's going on here. First of all, how wide is this graph in total? One, two, three, four, five squares. How wide is my new graph in total? One, two, three, four, five squares. Do you think we've done a horizontal stretch or expansion at all? Compression or expansion? No, you know what? I don't think that can be right because that's got a horizontal expansion compression. I don't think that can be right because that's got a horizontal expansion compression and that would change how wide this graph was. No, no. I noticed that the long side and the V is at the bottom. Here the long side and the V is at the top. I think there's a vertical reflection that's occurred. It doesn't help me at all. You know what, there's been a vertical reflection there too. In fact, there's also been a vertical expansion and both are remaining answers. What I need to do is figure out have I gone two left or two up? Well, this point right here, how high is it originally? Zero high. If I did a two left, how high would it end up? Zero high. I think it corresponds to that one. Is that one zero high? I think this one has been moved two up. I think it's this one here. I usually do use a process of elimination for certain key points. I'll look at V shapes right now. It's three high. Negative three high right there becomes positive eight high. I think they doubled it and reflected it. Sorry, I think they reflected, doubled and moved it two up. Double it, becomes negative six, reflected becomes positive six, move it two up, becomes positive eight. We're blazing. It says the given quadratic function has a positive X intercept. Check. A negative X intercept. Check. And a negative Y intercept. Check. Which of the following statements are true? The inverse will have a positive Y intercept, a negative Y intercept, and a negative X intercept. The graph of the reciprocal will have no X intercepts and the positive Y intercept. Hmm, I don't know. But they seem to think I can figure it out, so let's see. Let's answer the inverse part first. A symbol for inverse. How do I find an inverse? Why? So I guess that means that these two will become Y intercepts instead of X intercepts, and this will become an X intercept instead of a Y intercept. So read that carefully. Is statement one true or false? If I switch the X and Ys around, that's what my original would become, would it not? I think true. Statement one. I'm happy with that. Statement two. Sketch the reciprocal. Now what will the reciprocal look like? It will be an asymptote right there. An asymptote right there. Let's say one high is right about there. Let's say one high, negative one high, negative one high, I'm just eyeballing it, making up numbers. One high is right about there. Have an example clip, please, right there, Jessica. This would look like this. Bigger becomes smaller. Smaller becomes bigger. Gets closer and closer to the X axis. What about here? It gets further and further from the X axis, gets further and further from the X axis, and it would kind of do something like this. The Y intercept would become a reciprocal. I'm probably a little bit off. If this is 1.5, it should be the reciprocal of 1.5, but it would look something like that. Does that match this description? This graph will have no X intercepts. That is true. And the positive Y intercept. Coming back to you folks, Coleman's nodding, others aren't. I like number 13. I like number 13. Number 13 is a nice question, except it's a different variation that I also like. Some of you haven't had me before. Helen, why would I say that I like a question? Helen, why would I say that I like a question? You didn't hear me say that. Are you just really a good student and you pay attention to these things when you're studying? Let's look at number 13. Now, here instead of finding an inverse graph, it wants me to find an inverse equation. Well, how do I find an inverse? Switch the X and Y. This is one of the few times to me function notation is more confusing because there's an X sitting in there. I don't like that. I replace the F of X with a Y just so I have a little bit less to keep track of. And now I'll do the inverse. How? Switch the X and Y around. Absolutely. Equals negative 2 bracket Y plus 6 all squared plus 3. And now I stop and I say, oh, jeez. I guess they went and they got the Y by itself. Sometimes they don't. Sometimes they're okay with that. I've seen that as the correct choice before. They want me to get the Y by itself. How would I get the Y by itself? That's what I said. Thank you. Sorry. How would I get the Y by itself? From both sides, I would go X minus 3 equals negative 2 bracket Y minus 6 all squared divide by negative 2 and I'll get X minus 3 over negative 2. Now, by the way, I'm noticing there are no negative 2s here. We may have to work a little magic in just a second. Yeah, I agree. I'm going to square root and then plus 6. I think what they did with this negative is they divided the negative into the top and the bottom. I think they rewrote this as 3 minus X over 2 equals Y minus 6 all squared. How did I pick up on that? I didn't see any negatives kicking around down here and I saw a couple of 3 minus Xs. In fact, I'm kind of already thinking no and no. Now what? Square root isn't quite right. It looks like it's going to be plus or minus square root. When you square root both sides of a polynomial, plus or minus. So I'll have Y minus 6 equals plus or minus the square root of 3 minus X over 2. And last step, how would I get the Y by itself? Plus 6 inside the square root or outside the square root? Outside. That ends right there. Looks like the answer is A. But here's the question that I really have some strong affection for. Given Y equals 2X all over 5X plus 3 do I want an X on the top and on the bottom or is that going to get yucky? Nah, that'll work. Find the inverse. How do I find an inverse? Switch the X and Y around. If this shows up, it'll probably be a written question. And I would probably give you a mark for doing that. Now what? This seems to be where students freeze and it drives me crazy. Ready to snap? Well, little Miss Mander who has me in physics, yes? Cross-multiply. That is one fraction equals one fraction. This is really X bracket 5Y plus 3 equals 2Y. And now I'm going to get rid of the brackets. I'll get 5XY plus 3X equals 2Y. Now what? I want to get the Y by itself. We're going to move this over to this side. Get all my Ys to one side. So I'm going to go 3X equals 2Y minus 5XY. Now this is yucky because how many Ys do I have here? Two. How many would I prefer to have? How? Facture it. Absolutely. This is going to become 3X equals Y bracket 2 minus 5X. And now I can get that Y by itself really easy. Divide. I'll get Y equals 3X all over 2 minus 5X. That one has popped up on a provincial every so often as well. This is a 5 mark written question. To me, that's getting 5 marks for really doing math 10. Cross-multiplying in factory. The only math 12 is how do you find an inverse? Although actually you did it in math 11. So whether it shows up again, I don't know. They might have complained about that being too easy a question for 5 marks, but there you go. Facture a little handy-dandy test. Let's see. An invariant point is a point that remains unchanged using the graph of Y equals G of X to determine the number of asymptotes and the number of invariant points for the, well, for that thing. How about just to practice maybe we graph this? Again, by the way, don't write on these test booklets. Right? Is that a note? Okay. You're passing no notes to each other? Love notes to each other. I did play a romantic math song before we came in. I know. It's what happens when you're late. Yeah, I did. You were here at about 4 after 3. You ready? If they gave me this and they just said, hey, graph that. Let's do it. First thing I would do is invariant points are going to be anywhere 1 high or negative 1 high. That's invariant. That's invariant. And that's invariant. By the way, how many invariant points are there for this multiple choice question? 5. Wrong, wrong. Then it's going to have asymptotes. It's going to have asymptotes wherever the original graph has a root has an X intercept. It touches the X axis. It's going to be a vertical asymptote right there. Vertical asymptote right there. And a vertical asymptote right about there. If I eyeball it. How many asymptotes are there going to be? 3. Matter of fact, the correct answer is that guy. But let's graph this. Now, what I taught my students was once you get the invariance and the asymptotes, imagine you're a little bug walking on the graph and start up by standing on an invariant point. This guy here. As I walk this way, my original graph gets closer to the X axis. What's the reciprocal of getting closer to the X axis? It's going to shoot off to infinity. Well, negative infinity, fine. As I walk, go back to my invariant point, as I walk this way I get closer and closer to a height of what? Negative 2. What's the reciprocal of negative 2? Negative a half. I'm going to get closer and closer to a height of about negative a half. I'll just slide ball it, but I'm pretty close. Oh, and it looks like if I stand on this invariant point as I move to my left, I also get closer and closer to a, well, a reciprocal to a height of negative 2, reciprocal negative a half. What about as I move to the right? Well, my original graph gets closer and closer to the X axis must shoot off to infinity. What about this middle chunk right here? Well, there's a point right there. As I move to the left, gets closer to the X axis, shoots off to positive infinity, shoots off to positive infinity. The highest this gets is 2. The reciprocal of 2 is a half. I think right there would be one half high. And over on this point, as I move to the right, there's no graph, so I'm not going to put anything there. As I move to the left, it's getting closer to the X axis, and it must shoot off to infinity. It would look something like that. We're okay? Nothing? I'm going to put my microphone out on my little software program here. Graph of Y equals H of X is shown below. Which graph represents X equals H of Y? What have they done here? When they say X equals H of Y? Switch the X and Y. What's another fancy word for that? Inverse. So really, all I'm looking for is a graph that seems to be a reflection about the line Y equals X. You know what? This is 0, 4. Those two will just flip over each other. So I'm looking for a graph that still goes through 4, 0 and 0, 4. What about this endpoint right here? What's its coordinates? 6, 6, 6. What's the inverse of 6, 6? So I'm looking for a graph that still goes through 4, 0, because that would become 4, 0. And still goes through 0, 4. Because this would become 0, 4. And goes through 6, 6. I think there's only one graph that fits that, isn't it? I didn't print that one in my tutorial, sorry. Turn the page. Because when you switch 6 with a 6, but you still get a 6, 6. Does the graph not go through 6, 6? I think it does. 6 over 6 up. What's the inverse of 6, 6? 6 because you switch the X and Y around. The graph itself will be on the line Y equals X, and it's automatically its own inverse. So 5, 5, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 0, 0, negative 1, negative 1. Those are all inverses of each other. Mr. Kamosy, have I made any mistakes lately or not? Still not right? Okay. I'm going to throw this online. So we're just going to scroll back. This one here. I need to draw it out. We're not going to throw one like that. Here we go. These are the following questions. So I will definitely on your mock give you some points to do some math with. I find for me the easiest thing to do is list the transformations in the proper order. And then I usually just write the point down, work my way outwards, left and right. So I glance. I'll show you what I mean. I glance at this. Do I need to factor this or is this okay? It's okay. Let's list what's going on. What's that one half? Vertical or horizontal? Vertical by two or by a half? What's that? What type of reflection? Now remember, the phrase they'll use is a vertical of reflection about the x-axis. I always just think vertical reflection because as soon as I mention x, I start thinking horizontal right away. But in the terminology, they'll say it's a vertical reflection about the x-axis. And yeah, it's true. That's what you're flipping around. I just choose to say vertical reflection. Vert reflection. What's that? Four. It's next to the x, so it's got to be a horizontal. It's right next to the x backwards. Yeah, four right. Then what I do is I write down the x-coordinate and the y-coordinate. And I'm just going to work my way outwards. Vertical by a half? That's going to become a three. Vertical reflection? That's going to become a negative three. Four right? That's going to become a negative eight. I find that works for me just kind of working outwards. Let's do a couple of yuckier ones. Let's see what they got here. Let's do a ball. Okay. This one here needs to be factored at all. Nope. Okay, let's list the transformations then. Expansions, compressions, nothing vertical. Oh, horizontal. Compression. By one third. Because it's next to the x and everything backwards. Reflections, no. Oh, horizontal reflection. Then what's that minus eight mean? Eight down. And then I'll write my original point again, which was negative 12, six. And once again, I'll just work my way outwards. Horizontal compression by one third. That's going to become negative four. Horizontal reflection, that's going to become positive four. Eight down, that's going to become negative two. I see a reciprocal and I see a plus six. A six up. Do I do reciprocal first and then go six up? Or do I go six up first and then do reciprocal? If they wanted me to go six up first and then do reciprocal, I think they would have done one over g of x plus six. I think that's the head. Do that first and then take the reciprocal of your answer. I don't recall ever seeing one that yucky. And even reciprocals with a twist, don't happen all that often. They've only started showing up a little bit lately the last couple years. There will be a reciprocal with a twist on your mock though. What do I mean by with a twist? It won't just be a straight reciprocal. It'll be a reciprocal with a reflection maybe or with a stretch. But I'm not going to give you a reciprocal with a vertical stretch, a horizontal stretch, a horizontal reflection on a slide. I'm not going to hammer you. So we have reciprocal and six up. Negative 12,6. Reciprocal. Sam, this is what I meant earlier when I said your x's don't change, only your height changes. But your x stays the same. Six up! Add six to it. Reciprocal of six is one sixth. And then six up. I guess the answer. We pause. Is that okay? Sure. I know sometimes a little hamster on the wheel. Just give me a little kick. There we go. What about this one? Well, what's that to? Vertical or horizontal? Vertical what? Expansion. And absolute value. Now, look carefully. Do they want me to do absolute value first and then expansion or do they want me to do vertical expansion and then absolute value? Which one do they want me to do first? And I'll compare it with this. One of these things is not like the other. One of these things just doesn't belong. No, that's sorry. One of these things has expand first, then absolute value. One of these has absolute value and expand first. Think about the bed mass rules. Here I think absolute value first, then double your heights. Here, double your heights first, then absolute value. So here we have absolute value and then vertical expansion. We're starting out at negative 12 comma 6 when I take the absolute value of the height. What's the absolute value of 6? 6, vertical expansion by 2 is going to become a 12. Nearly there. I think. Determine the image of point A. So now they're going to give me a different order, perhaps. Every once in a while they'll walk you through the wrong order just to see if you can follow the instructions, make the substitutions, and come up with the equation that they want. Hey, no laughing. We're learning. I find math hysterical, Mr. Doock. Okay, fine. So, determine the image of point A. What was point A? It was negative, I forgot. Negative 12 comma 6. So there's negative 12 comma 6, and now we're going to go in the following order. Reflect about the line y equals x. What? Oh, yeah, that was another way of saying inverse. So it's going to become 6 comma negative 12, which you have to lie around. Horizontally expand by a factor of 2. One left. Let's make that an 11 and not a smear. Now there is a written section. I'm actually not going to walk through a whole bunch of these. Instead, I'm going to say, are there any specific weird ones that you are wondering about, either from your provincial exam review or your homework. Remember, you guys all have four exams, right? And if you don't, they're online. I think if not, they will be shortly at thepitmath.com. Click here for Mr. Doock stuff. Principles of Math 12. I'll set up a provincial exam, or an exam review folder or a year end review folder. Something like that. I'll put stuff in there by unit, and I'll put blank exams in there as well. I just wanted to show you a couple of exams. So here is the August 08 I think, or sorry, August 04 exam. And just letting you know kind of some of the stuff that you're going to see. That's trig, trig, trig, trig, trig, oh that's log. This was conics. We don't do that anymore. But here's one number 25. This equation represents the graph of y equals g of x after it's reflected in the line y equals x. C? What's C? Reciprocal. Is that the same as inverse? No. So we have three ways, well two ways we can write inverse. The negative one, or x equals f of y. Switch the x and y around. How is that graph related to that graph? Vertical or horizontal? Vertical. So right away you're getting crossing off b and d. Now it's a true or false question. You've got a 50-50 chance of guessing, right? By a half or by two? By a half. Here's a bit of a range twist. So if my original range is from negative one to positive two, what's my new range? When I take the, is that inverse or is that reciprocal? Okay. Here, I would probably just make a sketch. What's my original range? Well, my original range it says is from negative one to positive two. The simplest possible graph would be something, I'll just pretend it looks something like that originally. If I take the reciprocal, oh, invariant points, negative one high, one high. Vertical asymptote. What will this line do? Shoot off to infinity. Technically negative, so technically small, but I just say it shoots off to negative infinity. Here, as I move to the left, it's getting closer to zero. Get further away from the x-axis. Here, well, it's going to get closer to the x-axis. What's the highest this line goes? The lowest we go is to one half high. Okay. There's my range. There's my picture. Now what's the range? Everything below negative one, so that's wrong and that's wrong. Because that's missing y is less than or equal to negative one. How would I write this? Everything above or touching. There you go. That was number 27 on this exam. I'm willing to bet that the next question is not a transformations and I lose that bet. Okay. I thought this was the nasty one that year. Hit page. Hit width, Mr. Duke. Yep. That's fine. Not a detention. Just put the test back there, please, when you're done. The function y equals f of x is graphed to the left below. Determine an equation of the function shown on the right. Okay. Key points. Well, first thing I would say is how wide is my original function? 24. How wide is my new function? 12. We've definitely had a horizontal compression by that's a waste of my time. Shoot. They all have a horizontal compression. Nothing vertical looks like it's occurred. It went from 0 to 8 high. It's still 0 to 8 high. Now, looking at these, the alarm bell would go because these three aren't factored. Got you one last time. One for the road, so to speak. You feel good with that adrenaline rush now, though, don't you? Oh, yeah. I would rewrite this as f of 2 bracket x plus 3. f of 2 bracket x minus 3. f of 2 bracket x plus 6. Now, this graph here says it's been moved 3 in which direction? 3 right? No way! That's wrong and that's wrong. How far to the left has it been moved? Darn right. It's that one. That one. A lot of kids that year picked this one because, oh, that's 6 to the left. No, it's not. It's 3 to the left. Nearly done. Almost there. Well, this here isn't transformations, but you just wrote a test on this. How many terms are there in that expansion? Give me a hint. Not 7. Right? When you foil a plus b all squared, you get the trinomial. When you cube it, you get a fourth or a quadnomial. Whatever you want to call it. Written section. So you're going to get something like this. That graph is graph below. Graph that. What's that negative they're going to do? Vertical reflection and 3 left. Are you okay on these? Hopefully not. Any specific questions you want me to talk about now is your chance. Otherwise, I think I've gone through the whole unit. I've done reciprocals. Done absolute values. Done inverse. I've yelled at you and scared you. So, any others you're thinking about? Best advice I have over the next few weeks start working through those provincial exams. If you run out and you need more, email me. I'll happily get you more. You have more. If you run out and you need more, email us. We have more. We have lots of review stuff. Here's the worst advice that I can give you. Put everything off and try cramming this in two days before the exam. Not the best way to learn. Not mentioning any names here. Are we done then? If you feel we're done, could you please turn in my little exam booklets there? At this table. We're going a week and a day from today. Tuesday after the May long weekend. We're going logs. That one I suspect will be a little more intense.