 Real quick review, don't write this down, but just to jog your memory. Remember good old math 11, when you had something like this, they gave you a piece of graph paper, like that, and then they said sketch y equals x minus 3 all squared plus 4. You learned last year that you could sketch the parabola by cheating, by only focusing on the key points, and the key points for the parabola were the vertex, and then I memorized with my students 1, 1, 2, 4, 3, 9 different teachers of different variations. Everyone, everyone at least finds the vertex. How has this parabola been moved? What's the vertex of this particular parabola? Looking at this equation, do you remember from math 11, what comma what? Plus 3 plus 4. Three right, 1, 2, 3, 4, 1, 2, 3, 4 up. The vertex would be there, and then you would graph it from there. I don't know if it ever bugged you, Joel, why a minus here actually meant plus 3 to the right, but a plus here didn't mean minus 4. Where was the consistency? The reason is this is a specific form for a parabola, but if we were to generalize this, this 3, did it move it left or right or up or down? You told me it moved it 3 to the right. What letter is this 3 next to? X is left, right. Really, if I wanted to write this properly in our generalized form, since this moves up and down, I should really move it to this side next to the Y, because Spencer isn't Y up and down. How would I move a plus 4 over to this side? Really that equation is that. Really Alex, when we're talking about transformation, which is the fancy-smancy word, Ashley, for moving graphs around, if you write everything, the numbers next to the letters that they belong to, if you put the vertical stuff on the same side as the Y and the horizontal stuff on the same side as the X, I can summarize this whole unit right now, and the answer is everything's backwards. Everything's backwards. Speaking of backwards, so let me say that again, Mitsu. This year, because we want to generalize, take what you did with the parabola, but make it work for any function, the rule is this. Put the vertical next to the vertical stuff, the horizontal things next to the horizontal stuff, and I can summarize this entire unit with the words. Backwards. You're going to hear me saying that phrase over and over. In fact, I'm going to teach you subpoena. When in doubt, if you're not sure, everything's backwards. That's our fallback. Let's look at specific graphs then. Delete. So now, here we go. What we're going to graph in function notation is Y equals, a generic function looks like this, don't write this down just yet. There's our generic, there's any function, don't write that down. What we're going to ask ourselves is, what happens if we put something next to the X that's a subtraction or an addition, and we put something next to the Y? In fact, I wrote it here. The only problem is our graphing calculators and most software wants the Y by itself. So even though we're going to think this, Brett, we're often going to plus the D over here like we did for the parabola last year, and we're going to write it this way. Just because the software kind of forces us to. I actually kind of prefer this, because then I can just go backwards, but you'll flip flop back and forth. What are the following equations? X squared plus Y squared equals nine. It says, graph the equation below. Can you find for me just by trying, guessing, and checking, can you find for me some numbers that actually work in this equation? I'll give you a hint. Try letting X be three. If X is three, what's three squared? What would Y have to be so that this works out to nine? Okay, three comma zero works, and I gave you a bit of a hint, by the way, this graph is a circle. This graph is a circle. We don't look at it in math 12 anymore. We used to. This graph is a circle. You know what else would work? Negative three for X. Oh, not only that. Put a zero in for X. What's zero squared? What does Y have to be then? I'm going to be fussy. Three or? Okay. This is a circle. Yes, the tongue helps of radius three. Where is the center of this circle? Zero, zero. Radius is three. Oh, and good practice. What's the domain of this particular circle? How far left? How far right? Oh, this is the in-between one, and I told you then Courtney, uh, sorry, Ellen Courtney's pardon me, your sister's Courtney, and that's how I'm remembering you. Thank you. Ellen. I was doing good. Hey, I was doing pretty good. I'm looking at the seating plan. Okay. Bring up the crutch, Mr. Dewick. You're going to need it. This one has a domain negative three, and we said less than X, less than three. Is it touching negative three and three? Okay. Or equal to or equal to. And I think the range is the same thing, but with a Y. Negative three, less than or equal to Y, less than or equal to three. The reason I'm looking at this is a couple. First of all, I'm a nerd. I like to show you new equations. That's what a circle looks like. Something squared plus something squared equals something squared. Three squared. Oh, by the way, can you see the radius hidden in the actual equation? What if I wanted radius four, put a 16 there instead of a nine. What we're going to start to do is as we generalize our parabola rules, we're going to think about what we're doing in terms of replacing X with something else or replacing Y with something else. And as soon as they give me a curveball kind of a question, that's my fallback position, especially if they want me to find an equation. If they want me to find an equation, I think, well, what would I replace X with? What would I replace Y with? And the question we're going to ask ourselves today is what happens if I replace Y with Y minus two? Oh, wait a minute. I got the fancy-spanty software. What happens if I replace Y with Y minus two? Ooh, highlighting. Let's see if we can guess. First of all, if I'm replacing Y with Y minus two, do you think it's going to be a vertical or a horizontal effect? Let me say that again. Do you think it's going to be a vertical or a horizontal effect? Replacing Y with Y stuff always is going to be vertical. That part won't be backwards, sorry. Replacing X with X stuff is also going to be always horizontal. Come in. So I think I was saying, sorry for the internet, we had a little bit of an eruption, I think I was saying we're going to talk about replacements. So we just said that replacing Y with something is going to be vertical. Replacing Y with Y minus two does not move the graph two down. You know why? Because everything's backwards. It moves the graph. Take a guess, Eric. Two up. Two up. So you know what I would do to graph this one? I would say, get me my color, draw a little arrow here. Moves now our abbreviation for the graph or the equation. This year is where function notation is going to be handy. We'll use the phrase F of X as just whatever graph we're talking about. This one in this case. Moves the function to up. And the easiest way to do that for this graph is to move the center. Instead of the center being at zero, zero, it's going to be right there at zero, comma two. I'll put an X there, not a dot because the graph doesn't exist there. If you put a dot there that tells somebody the graph is there, if you put a little X, they know I'm just using that as a placeholder. What was the radius of this circle? From here, go one, two, three up, one, two, three right, one, two, three down, one, two, three left and draw your circle. So this graph is a circle with center zero, comma two, radius three. Has the domain changed? No, and you know what? It shouldn't have because we did something vertical and domain is horizontal. That shouldn't have changed at all. One of the things we're going to be doing during this unit is breaking everything up. Is it horizontal or is it vertical? A vertical will not affect a horizontal. A horizontal will not affect a vertical. So the domain is still going to be negative three less than or equal to X less than or equal to three. But the range should have changed because we did a vertical transformation. What's the new range? Negative one less than or equal to Y less than or equal to five. By the way, Dominic, just to convince ourselves that this works, I can plug in three, comma two. If I put a three right there, I get three squared nine. If I put a two right there, two minus two, zero. Oh, there's that nine zero that we did in the previous one. That's why everything's backwards. Really what's happening is, if you make this too smaller, the Y is going to get too bigger to compensate and keep the same equation working. Everything's backwards. The way they'll phrase this, the way they phrased this on the provincial, and since I'm teaching this course like there was a provincial, the way I'll phrase this on a test, this is a vertical translation to units up. Before you turn the page, here would be a good multiple choice question. Given this graph, compare this graph, what transformation has occurred, and I'll even tell you what the multiple choice answers would be. Two up, two down, two left, two right. Now turn the page. What if we replace X with X plus one? Or replacing X or are we replacing Y? X horizontal. It guarantee it. It's going to be a horizontal effect. You know what? Everything's going to be backwards. Replacing X with X plus one is actually going to move one left, not one right. And again, you can convince yourself by putting a three right here. If you go three squared, what's three squared? Nine for this to work out to zero, instead of putting a zero right there, I need a negative one right there to get that zero. This is going to move one left. The center is going to be at negative one comma zero, and it's still going to be a circle of radius three. So from there, it's going to be one, two, three up, one, two, three right, one, two, three down, one, two, it's going to look like that. This has a center of negative one comma zero. Radius is three. I'm going to skip the domain. I'm going to jump right down to the range. The range won't have changed because we did a horizontal. This should be negative three less than or equal to Y, less than or equal to three. The range won't have changed at all. Ah, but what's my domain? My whole domain has moved, yes, yes. Brett, if I gave you this and that, could you've got the domain without graphing? Move that one left. Move that one left. You'll reach the point where you'll start to do a lot of these intuitively without having to bother to pull out the graph paper, and that's also our goal. This is a horizontal translation, one unit left. Here is how we write this in function notation. In any graph, replacing every Y with Y minus K, why do we use the letter K? Because they do for some strange reason. Results in it. First of all, are we replacing X or are we replacing Y? Because I asked you to. Are we replacing X or are we replacing Y? Why? Because I asked you to. I'll never get tired of that joke. Sorry. Especially this unit. If it's the letter Y, vertical or horizontal, results in a vertical translation, K units right. Right? Mr. Dewick? Good gosh. Vertical. How about let's try up. Sesame Street is brought to you by the word up. Now, here's the only, in my mind, one of the tricky parts, Z-fang, although I like that better because graphing calculators require you to get the Y by itself, often they will plus the K over to this side, and once it's on this side no longer next to the Y, it's no longer backwards because you changed the sign when you moved it over. That's why the parabola worked the way it did last year. That's why in the brackets next to the X was backwards, outside the brackets wasn't. It was really because it was backwards if you put it on the Y side. In a function Y equals F of X, replacing X with X minus H results in vertical or horizontal. Horizontal. Translation H units right. In function notation, we would write Y equals F of, and then in brackets X minus H, we put the H in the brackets next to the X, and that tells anybody, oh, that belongs to the X. It's a horizontal. We say that H and K are parameters that affect the original function through a translation. We're going to try a few. It says there is a lovely summary of this whole topic on page 15 in your workbook. I haven't given you your workbook yet, but I photocopied all those pages. We'll look at it in a second. Who on the page? This is the kind of question I'm going to give you on the written section of your test. I'm going to give you some kind of a generic function. I'm not even going to try an equation. We did a circle, which was an actual equation. That's boring. We're just going to make it for shape. There it is. Upside-down hockey stick, sort of, and I'm going to call it F of X. Here's what I want you to notice. It has key points. I'm not going to move every single point on this graph. I'm going to move this one, this one, and this one and connect the dots. Graph that. Here's what I think to myself. Eventually, you'll probably do most of this in your head, but in our notes, we'll write everything down. What have we replaced what with? What's the difference between this and this? We've replaced X with what? X plus 1. Ellen, vertical or horizontal. How do you know without even thinking about it? X. Everything's backwards. This is going to move one left or one right. One left. I'm going to do a better capital L. What letter will I use for right, do you think, R? What letter will I use for up, do you think, U? What letter will I use for down, do you think? I'm totally good with that. If we move everything one left, this point here, which was once negative four, negative three, it's going to be negative five, negative three. I'll put a dot right there. I could graph this point. Forget it. Next easy key point. I'm going to move this one, so I'll start my pen right there, and I'll move it one left right there. We're the first two points connected, then I better connect these ones. This one is a two comma zero, one left, right there, done. Fancy terminology now, and I'm not going to ask you the fancy terminology, but I'll use it so you should recognize it. We call this graph here the image of the original under the transformation Y equals F of X plus one. For some reason I did B first, don't ask me why. Let's do A now. How has this graph been moved? Well, first of all, we've replaced what with what? Y with Y minus two, because we're always comparing it to our generic Y equals F of X. By the way, I probably will not give it to you in this format. I'll probably write it this way. How can I tell that that plus two belongs to the Y and not to the X? Not in brackets. I'm going to assume in grade 12, you can do the equation solving move in your head and minus the two over in your head and clue it. Everything's backwards. Or to be honest, a lot of my kids just memorized, well, if it's not next to the Y, it's not backwards anymore, which is technically what you did last year with the parabola. This moves the graph two up. So back to my original. Instead of negative four, negative three, negative four, negative one, right there. And instead of zero comma two, zero comma four. And they were connected on the original, I may as well connect them now. And instead of two comma zero, two, two, graph that there. Okay. So what are the transformations? We're replacing X with something and we're replacing Y with something. What are we replacing X with? X plus three. What are we replacing Y with? Careful. What are we replacing Y with? Y, you're saying X each time. What are we playing? So three, what? Left right up or down. Three left. Two. Try it on your own. I'll do it up here. I think you end up with that. I always reference this back to computer video games and the classic one is Super Mario Brothers. So in Super Mario Brothers, when you were playing it, when you were a child, when you were moving the controller to the right, Mario was a function. A very complicated function, much more complicated than this hockey stick, but Mario was a function. And to get them to move once one square right, all they were doing is they were replacing every single X value with X minus one because minus one would mean one right. And the X values are pixels because every computer screen is a graph, is called pixels. And when you were moving Mario to the left, they were replacing every single X value with X plus one. And when you press the jump button, when he was going up, they were replacing the Ys with Y minus one. And when he was going down, they were replacing the Ys with Y plus one. We'll talk about how they got him to turn around. That's another transformation called a reflection. And we'll talk about when he ate the mushroom, how they made him instantly bigger. But it's very, very simple math as it turns out, which I'm guessing at least partly influenced how they did the graphics for the page. We can also use this instead of just for a generic graph. We can use this for specific graphs as well. So take a look at this graph here. What was this original graph before I did some replacements? What was it based on? I see a squared. I think it's a parabola. This was originally Y equals X squared. But we've replaced the X with what? X minus two. And we've replaced the Y with, careful, Y plus three. This is going to take our original parabola and move it how? Two, three, down. Like to show you a slightly different way of approaching this. Last year, you did this by memorizing. Remember last day, I gave you key points. I said the key points for the parabola were zero, zero, one, one, two, four, three, nine. Remember that? Two right. Isn't that the same as adding two to your X values? Write those down and do this. That's two right mathematically. I added two to all my X values. And three down, I think that's the same as subtracting three from all my Y values. Negative three, negative two, one, six. Let's graph those and connect them. Two, negative three, three, negative two, four, one, five, one, two, three, four, five, six. There's half the parabola breadth, and then we could just flip the other points over. I'm pretty sure this one goes with this one, this one goes with this one, and that one there because it's, oh, the axis of symmetry. Now, I probably wouldn't use this method for a parabola because we've memorized that one so much that we probably don't need to fall back to the key points method. I would probably use it, though, for graph number three. Sorry, people are still writing, oh, wait. Those of you that are done, go look at graph three and see if you can figure out which of those seven basic functions that I gave you last day, this one originally was. Well, I see a square root. Is it a wounded seagull? Square root graph? Oh, I also see a squared in, what is this graph based on? Which of those seven basic functions had a square root and a squared in it? Okay, let's see. Let's see if that works. If this was a semi-circle, there would be a big square root. Is there a big square root? Check. There would be a number minus something. Is there a number minus something? Check. X would be squared. Is the X squared? Check. In fact, I think that this graph is based on this. This was a semi-circle, radius three, center, zero, zero. Is that okay, Brett? See it? Oh, but I've added some stuff. First of all, it looks like I replaced the X with what, Brett? Sorry? Yeah, did you say two or X plus two? X plus two? Yeah. People want to say you replaced it with a two and I always freak out, no, no, because then the X would have vanished. I need to replace an X with an X plus something. The X has to stay. X plus two, Brett, from what you know, replacing X with X plus two moves the graph how? Two to the, yes. And I'm looking at this bad boy here. I think I've replaced the Y with something. Brett, what have I replaced the Y with? I disagree both times. Did we see it? Y plus four. I think it was originally replacing Y with Y plus four right here and I'm glad you did that. I was hoping somebody would because that's the most common mistake, Brett. I think we had a Y plus four here and then what did I do with the plus four? Moved it over to get the Y by itself. Okay? Regardless, this is going to mean four, so I'm not going to plug in points for this. Instead, I'm going to figure out where my new center was. If my original center was zero, zero and I've moved two left and four down, isn't it reasonable to assume two left, one, two, three, four down? There's my new center and then Spencer, why don't you tell me the radius of this was? This is a semi, oh, one, two, three dot, one, two, three dot, one, two, three you know what? That's the graph. No, must, no, fuss. Homework, ignore this temporarily. We're not done because I gave you this thing here. I hope I did. Yes. Except yours looks like this. Haha. Try that. This thing here. I know I did. Yes. So this is the workbook lesson three and four. Yes, I've done two lessons in one day. This first bit is exactly repeating what I talked about. What I'd like you to do is find page 15 where I said there was a handy-dandy summary of this whole thing. Now, we call moving a graft—Found it, Eric? Okay. Help me out. We call moving a graft sliding it around. The fancy word for that is a translation. We say that we have been translating grafts from one location to another. Will I ask you for that term? No. Will I use that term and expect you to know what it means? Yes. So, know what it means? A translation is a transformation which slides each point of the figure the same distance in the same direction. And then here's what we've said. Replacing y with y minus k describes a vertical translation. Replacing x with x minus h describes a horizontal translation. In general, if y minus k equals that or they get the k over on this side, if k is positive, it moves up. If k is negative, it moves down and left and right. Then it talks about mapping notation. We're not going to worry too much about mapping notation, so you can cross that little paragraph out. Very quickly, by using a number and a letter, how has this graph been moved? I'll give you a hint, three up, left, right, down, which, let's write down here, three r. How has this graph been moved? I don't care. I don't even know what the graph is, but I can tell you how it's been moved. How has it been moved? I'll give you a hint, four. I heard four down. I heard four up. Which one? Four up. Because it will be a minus four next to the y-writ belongs, and then everything's backwards. Here, 10 and 1, 10 what? One what? 10 left, one, which one? Up or down? What you really want to ask yourself is, is the one next to the y-writ belongs? If it is, everything's backwards, so it's going to be one up. Example two. The point 2, 3 is on my original graph. I don't know what the graph is, but I know it goes through two comments, sorry, I said 2, 3. 2, negative 3. What if I asked you to graph this? Where would this point end up? Well, let's see. This graph has been moved 8 down. So let's take this point 2, negative 3, and let's move it 8 down. Where would it end up? I'll give you a hint. The 2's not going to change because that's horizontal. Move this 8 down. Where do we end up? Yeah, negative 11. B. Now, usually when they give me more than one operation, here's what I do. I make a list first of all of what's going on, so 7, 7 what? I heard up. I heard right. Which one? 7 right, 5, and then here's my system. I go off a little ways, I write the original point, and then I just start working my way outwards. I'll show you what I mean. 7 right, vertical or horizontal, the word right, is that a vertical or horizontal translation? So I'm looking at the X's. If I move that 7 right, that's going to become a 9, and 5 up, vertical or horizontal, vertical, so I'm looking at my Y's. If I move that 5 up, where's it going to end up at? And then I can see, oh, the point's 9, 2. I write the point, and I just work my way outwards in either direction, because that way, even if I have, eventually you'll have like 5 or 6 different operations, but you can just keep working your way outwards, and then this is the X and this is the Y. C. That's mapping notation, which we don't do. Turn the page. Says, write the equation of the image of Y equals F of X after each transformation. So now, instead of giving you the equation and saying, tell me about the points, now they're saying, tell me what the equation would be. A horizontal translation of 5 units left. Horizontal. X or Y. I would replace X with, with what? If I wanted to move 5 left. X plus 5. So take this equation here, replace every X with X plus 5. That's a Y, that's an equals, that's an F, that's a bracket. I'll replace it with X plus 5, and that's a bracket. A translation of 3 units up. Again, we're based on this up. Vertical or horizontal, Trevor? Vertical. X or Y, because I asked you to. X or Y, because I'll never get tired of that. So replace Y, now careful boys and girls. Replace Y with what? The replacement method, everything's backwards. So if I want to go 3 up, I'm going to replace Y with what? Y minus 3. And that's going to give me the following equation. Y minus 3 equals F of X. Or Y equals F of X plus 3. Those two are interchangeable. And that second notation actually is the most common. So Kara, you got to get used to, if it's outside the brackets, it's a Y and it's no longer backwards if it's not on the Y side anymore because it would have changed the sign when they moved it over. Does that make sense? You're going to make it? No? Don't shake your hand. Is he going to make it too? Kara, you might need to, you know, every so often. C, ooh! Here's an algebraic one. Translation of M units right and P units down. No, D is though. That's mapping notation. C, algebraic, I don't know, you're looking at D. C, they love to do the algebraic ones. And the reason is, back when the provincial was around, there was all sorts of legal programs that you could download on the graphing calculators. So they said, fine, we'll do algebraic ones that the calculators can't do. First of all, right means horizontal. So I'm going to replace X with, if I want to move M right, I'm going to replace X with X minus M. If they wanted me to move M left, I would have replaced X with X plus M. And I'm going to replace Y with Y, down everything's backwards, Y plus P. And my equation's going to look like this. Y plus P equals F of X minus M. Or Y equals F of X minus M minus P if they chose to get the Y by itself. Glad I agree, D is mapping notation. Very quickly, how has this graph been moved? I got the two part, thank you. Two what? Two right. How has this graph been moved? Two down, three left. So you'd go two down, three left, two down, three left, connect them. Two down, three left, two down, three left, and you could connect them. We're not going to because we didn't want like this already. Okay. And then I believe if you turn, you also have stuff on page 19. Yes? Once again, a lovely summary. Really quickly, we're just going to do some of these orally. How does graph, the second graph, compare to the first graph? Find it? Okay. How does second graph compare to the first graph? I'll give you a hint. Three. Three up, yeah. How do I know it's not an X? How do I know it's not horizontal? It would have to be in brackets with the four outside, like you did with the parabola. Oh, yeah. How has this graph been moved? Now, there's a minus three, but it's still there. So even though there's a minus three, that hasn't changed. The only thing that's changed, it seems to me, is they replaced X with what? X minus one, which moves it? One right. How about here? Oh, this one I know. This is the absolute value graph. I can't remember a way to remember what the absolute value graph looks like, but I think it looks something like a V or something like that, I think. How has it been moved? Six, right? And two up. I would move the vertex, six right, two up, and then I think the key points were one, one, two, two, three, three. Here's a reciprocal of a square root. I have no idea what this looks like, but I can tell you how this graph compares to this one. It's been moved. One left. Because you've replaced every X with an X minus one. It's a very nice generic way to graph without graph. It's a great shortcut. This is asking you to practice writing the equation. You're going to do that in your homework. This is asking you if the thin line is the original, what's the thick line? And what you need to do is figure out how has the thick line been moved. We'll do a together. How has a been moved? I'll give you a hint. Look at the vertex. That's the easiest one to follow. So the thick line is the new one. Here's my original. How has it been moved? I'm going to write this down. Two right. Now that would mean replacing X with what? X minus two. How else has it been moved? Four down. That's replacing Y plus four. Now here's my original equation right here. I see it. So the new equation would be Y plus four equals absolute value of X minus two. There. That's that equation. It has to be. Although Alex, they would probably minus the four over the both sides. Fine. That's a real quick covering of what we've done today. In this package here can you turn back to where the homework was? Find page sixteen. So part of your homework, number one. And you've seen that I don't write out the words. I just use a number and a letter. So for example, for one A I would write nine L. It is nine left. Some of them they're doing more like F looks like they've done a few things. You might have to tidy it up a little bit to see what's really going on. Two is good. Four is good. Five C and D. Six is fine. Seven, eight, skip nine. And then if that would be so kind as to find page 21. I'm going to give you the yucky ones. So one D, E and F, skip A, B and C. Two is fine. No, no, no, no, no. Skip, skip, skip, skip, skip. Six is good. There you go. I tried to finish this with 20 minutes. Yeah, you got 20 minutes. I gave you, if you're fast, actually only about 20 minutes worth of homework. You'll find it kind of starts to plug and chug. So I think you can possibly get it all done in class. If you, so remember the third assignment that I gave you and for Eric, the third one, that's your great big unit review. I'll tell you next class which questions you're capable of doing or you can hang on to it and save it for the end of the unit in the next class.