 We'll upload this for posterity answers to quiz number. Actually, I called it quiz zero in my handout because quiz one is a quiz That comes later the transformations quiz y equals the square root of x. Well, let's see We did do that on the day. We did the basic functions lesson. I called it the wounded seagull. I don't have it memorized I just derive it. I know that the square root of zero is zero The square root of one is one the square root of four is two and the square root of nine one two three four five six seven eight nine is three I know that it looks like and then I say to myself self. How has this graph been moved? Well in terms of replacement Replacements they've replaced an X with an X minus two and that moves the graph to What to write? And in terms of replay well, what's that plus five mean? That means five up it means they replaced the Y with the Y Minus five, but when they plus the five over to the other side it became a five plus five five up so To write one two three four five up I had a student in my other class by the way say mr. Do it I'm glad said that I thought that was five high if you use this number to use as your five high I'll give you full marks because the graph paper is a bit wonky. You have no idea how tough it is to type graph paper in word I didn't actually type this a friend of mine did it's a bit of a pain getting numbers lined up is a yucky One two right one two three four five up to write one two three four five up To write is off my graph paper. Is it a long ways off my graph paper or one square off my graph paper? Then I won't freak on my test if you're doing a question and you end up a long ways off my graph paper You've made a mistake. I made sure all the questions Don't do that and in fact if you end up anything off my graph paper on a test Be a little nervous because I think I made sure that almost all of them fit on the graph paper I might have been able to go one square over so to write One two three four. I think it looks like this five up One mark for the blue graph one mark for the red graph half mark off for each point. That's in the wrong place Two wrong points. Sorry zero example two whoo What's that a graph of I? Know what that is? Square no, it's not square root. It's got a square root, but there's also that something minus x that was a semi circle What's the radius of this semi circle? Sorry three where is it centered? Semi circle with the radius of three Centered at zero zero. So I said there's my center three left three up Three right. There's my semi circle one more. How has this graph been moved? Well, what does that mean two to the left? What does that mean four? So The easy way to do this the quickest way to do this is actually not to move these three points you could to move the center To left one two three four up my new center is there radius three radius three radius three That's the quickest way to do it, but you could have just moved each point as well and so this is a semi circle By the way, you'll notice for the center I put a little x not a dot if I put a dot that tells the marker that's on the graph if I put a x that tells the marker I'm just using that as a placeholder. It's not part of my graph, but I'm using that to get the actual graph for what it's worth Same deal one mark for each graph half mark off free point. That's wrong. Yes Right now today on this quiz. No on a test. Yes, because I just told you now She said if you put a dot at the center on the quiz here Well, I take marks off not on the quiz but on your test if you put a dot you're telling me that's part of my graph That would appear if I graft it and it's not I'll get fuss here and give you better explanations as the unit progresses. Hey from the page Sample three f of x equals absolutely that oh, I know what that looks like It goes through zero zero one one negative one one two two negative two two three three negative three fact it looks like That and then they want me to graph y equals f a little negative one that looks like an exponent bracket x close bracket That was that oh that was a stupid symbol. I remember mr. Do it one of a big rant about that he hates it. Ah, but he's stuck with it because he's not in charge of the world yet What did that symbol mean ah? This is the inverse How do I find an inverse? That's my abbreviation for switching the x and y around you can come up with your own if you want to so Zero zero will become oh zero zero it stays where it is. What was the fancy word for a point that stayed where it was Invariant oh, and one one will become oh one one stranger. Oh, you know what this whole arm will be invariant But this arm here instead of negative one positive one positive one negative one and Instead of negative two positive two positive two negative two. Oh, you know what that arm is going to end up Like that There was a way to test that you've done the inverse correctly if you drew the dotted line y equals x it should be symmetrical Except there's a problem the grid paper that my friend made up. You'll notice each square is actually not a square What shape is that? It's a rectangle. These are kind of distorted a bit So they don't quite you know look as pretty as you might Although that's okay, too Because Eric on the graph and calculators that you'll get someday if you don't have one already the screens are rectangular as well They actually distort the graphs as well. We'll talk about how actually that graph isn't quite so slanty in real life If you use square graph paper Same idea one mark for each graph half mark off for each point graph number four How many marks is number four worth? Three how many graphs as they ask you to do? Three. Oh, how about one mark for each graph? How about a half mark off for each point? That's wrong. That would make sense to me Let's see. Oh And it says label each graph in other words make sure you know which one is which I'm gonna use colors So I'm gonna start out by going Blue with this guy as Soon as I see a negative I know it's a reflection the real question is is it a vertical or horizontal reflection? Well, where is the negative? Is it right next to the x inside the function? Then it's horror that is vertical. This is a vertical reflection Which means that all of my heights are gonna become their opposites Spencer, how high am I right here? It's a trick question. What's the opposite of zero? It's a trick question invariant But positive one high is gonna become negative one high connect them Positive four high is gonna become negative four high connect them Positive four high is gonna become negative four high connect them Positive three high is gonna become negative three high connect them Zero high is gonna become oh invariant Negative three is gonna become positive three this here is Y equals negative X Looks like a fish shut up But by the way though, you may have noticed most life forms us included are symmetrical So right now when they're trying to look at how cells develop they believe that genetically Mathematically what's going on here replacing y with negative y replacing x with negative x that's going on in your DNA Genetic code the mathematicians are starting to look at that with the hope that some day perhaps If you lose an arm in an accident, we can bring the symmetry back for a new one That would be nearly cool. They can figure that out What was that Brett can they grow an extra pair of arms? You have four. No, I would be spider-man shut up. Okay This one's gonna be red Although then you could give yourself like a nice big hug twice Someone would finally hug you Sorry besides your mom let's see red Negative oh, oh, it's a reflection. Oh, it's in front of the X. This is a horizontal reflection So now instead of asking how high how high how high I'm asking how far left right and taking the opposite Instead of negative 4 to the left positive 4 to the right Instead of negative 3 to the left positive 3 to the right connect them Instead of negative 2 left positive 2 right connect them will better than that, mr. Duk Connect them. Instead of negative one, positive one, connect them. Instead of zero left right, oh, trick question, still stays zero left right, invariant. Instead of one right, one left. And instead of two right, two left. I think that red graph is the image. This here is y equals f of negative x. Am I right? I think, yep. And then, sorry, what does this one mean? x equals f of y. Oh, that's the, what I think is a better way to write inverse because it stands out. Oh, they got the x where the y is. They got the y where the x is. It looks like they've switched the x and y around. I like that much better. Unfortunately, I'm going to tell you right now, on your test, I will almost certainly use the stupid little negative one because that is the convention by far the most popular one out there in the math nerd world. I'll do this one in green. So how do I find an inverse? So I'm going to say this to myself very carefully because this is where I make sloppy mistakes. Negative four zero is going to become zero negative four. Negative three, positive one, positive one, negative three, connect them. Negative two, positive four, positive four, negative two. Am I wrong? I counted wrong. No I am right. Connect them. Negative one, positive four, positive four, negative one, connect them. Zero comma three, three comma zero, connect them. One comma zero, zero comma one, connect them. And positive two, negative three, how about negative three, positive two, I think this green graph is x equals f of y. The inverse. Yep. On a test, I'll almost certainly only have you do one graph per question. The question was on a test if you don't have a, and honestly, yeah, I'd probably have, if I told you to label them, I'd have to get, and get more than one graph. I'd have to get grumpy because how would I know that you knew which was which? The most common mistake is kids get horizontals and verticals mixed up. So you might have graphed the vertical reflection thinking it was horizontal. The horizontal thing it was vertical, but I couldn't tell. Because the two graphs are right. So yes, in real life, today I won't be freaky. Give yourself a score out of nine, please. We looked last day at expansions and compressions. We threw a lot at you. So can you get out first of all lesson seven and the homework? And I'll start out by asking, hey, any questions from here you would like me to go over now is your chance to ask. Make it as well. Make it. Any from lesson seven that you would like me to go over. Harley, number five, any before, what was number five? Any before number five that you'd like me to go over. So number one, getting the equations using their placement method, that went okay. It's substitution, but it's careful substitution, right? Number five, I would love to do number five. By the way, number three, did you notice that for a parabola, which is a bit of a unique shape, if you expand it vertically by a factor of four, you get this. If you compress it horizontally by a factor of a half, you get this, which simplifies to this. The parabola is so symmetrical, Eric, that stretching it vertically. Imagine if you had like, you've all played with a piece of rubber from a balloon or something like that. If you stretch the rubber vertically, it does get skinnier. And what you're recognizing is, as it turns out, a horizontal compression can give you the same effect as a vertical expansion on a graph. Okay, I thought that was kind of nerdly cool. Number five, pick one. I'd love to do C. Okay, so here is my original F of X, and I think, Brett, my key points are going to be here, here, here, and here. The peaks in the middles. What does that do? It's next to the X, so it's horizontal. Horizontal, what? Well, because it's next to the X, it is backwards, so it's going to be expansion by two or compression by a half. Absolutely, expansion by two. What does this do? It's vertical. It's no longer next to the Y where it belongs, so it's no longer backwards. It is what it is. It's a vertical compression by a half. Mr. Dewick. Yes, Trevor. How do you know when to use the word expansion and when to use the word compression? It all depends on the factor. In fact, what I do first is I find the factor a half, a third, three, and then if the factor is a number bigger than one, we use the word expansion. If the factor is a number less than one, like a fraction, we use the word compression. Will I take marks off if you call an expansion a compression or a compression an expansion? Probably not, as long as you have the factor correct, but for Pete's sake, let's use the correct language. So you ready? Brett, I'll do the new one in red. No, I'll do it in blue, because I use blue here. That makes sense. Right now I'm at zero, zero. If I horizontally expand zero by two, you know what I get? It's still zero. If I vertically compress zero high by two, you know what I get? Fancy word invariant, and that answers part of the question. That's part three. Now, this point right here, horizontally expanded by two. How far to the right are we right now? One, expand that by a factor of two, please. Two, and I'm going to hover there. How high am I right now? Compress that by a half. That point ends up there. I'll do this point next. You know what, Mr. Dewick? Use the technology so they can see better. Keep forgetting to do that. I'll do this point next right here, this x-intercept. Horizontally expand it by two. Instead of two to the right, you know what's going to end up? Instead of two to the right, you know what's going to end up? Expand it horizontally by two. So instead of two right, you know what's going to end up? Four to the right. Vertical compression. How high is it right now? Compress that by a half. Still zero. Not invariant, because it did move sideways, but its height ended up being invariant. So right now I have this. I think the next point I would move would be this one down here. So if I horizontally expand it by two, right now it's three to the right. Horizontally expand that by two, it's going to end up six to the right. Right now it's negative two high. Compress that by a half. Negative one high. It's going to end up right there. The last point I'll move is this one right here. Horizontally by the spot in the pattern now, I think it's going to be instead of four to the right, eight. And instead of zero high, oh, instead of zero high, zero high. That blue graph is what this black graph becomes underneath that image. So far so good? Yeah, yeah? What's the domain? How far left does the blue graph go? How far left does the blue graph go? Zero. How far right? So this domain would be zero less than or equal to x, less than or equal to eight. Or equal to because it is touching. What's my range? How low does this new graph go? Negative one. How high? Range is going to be negative one less than or equal to y, less than or equal to positive one. And invariant points, the only one that was invariant was that guy there. Is that okay? So they're going to mix and match for the rest of this question. Oh, they've added a negative. It means there's going to be a reflection in some of these two. I'll just go step by step, make a list of your transformations, hover your pencil above the point, and walk through each one until you get to the end of the list, and then plunk your pencil down and put the dot there. Is that okay? Is that okay? Whoever else asked? I don't mind doing more if I have to, but is that all right? Any others on this page that, including number five, that you would like me to do? Any others on this set of homework here? What else did I assign? Oh, number six? Okay. Then we also did some questions from lesson two cubed, lesson eight. Are there any from here that you would like me to go over? This is your chance to ask. Three? Love two. Three is tough. Can I do what? Can I do B? I don't mind doing B and then you can try A and C on your own. In fact, I was going to suggest that because that way you can still practice it. These are tough. I find if they give me the graph trying to figure out the equation, probably the trickiest concept for me to wrap my brain around. I'll give you some hints for how I'd approach A. We're going to do B. Glancing at A, here's what I notice. Those two points stayed invariant. You know how I know? Because both graphs go through those same points. You know what that tells me? It tells me we didn't do anything horizontally. Because if we'd done something horizontally, the thick graph would have been thinner or thicker. We might have done something vertically. To check that, I would look at my Y intercept and I would say, oh yeah, instead of too high, it became ten high. I think A is a vertical expansion by five. Let's do B. B looks like it has two different transformations. How could you figure that out so fast? Because my Y intercept changed and my X intercepts changed. That must mean there was a horizontal and that must mean there must have been a vertical. Two of them. Let's see. I like using X and Y intercepts, Trevor, because they give me good reference points. Alex, how high is my Y intercept originally? Good. Thank you for noticing the scale. How high is my new one? What would turn a four and two and eight? I think vertical expansion by two. Let's look at my X intercepts. How far to the right Mitsu was this original X intercept over here? Positive what? Where did it end up? Positive. I can't hear you. Positive three. What? I think we've had a horizontal compression by half. Must have. They want the equation. Trevor, anytime they want the equation, I fall back on my replacement method. Vertical expansion by two. That means we're going to replace Y with two Y or half Y. Half Y. It's always backwards in the replacements. Horizontal compression by a half. We're going to replace X with two X or half X. I've got to be fussy, not two. Thank you. For some reason, instead of using the letter F, I'm using the letter P, but whatever, F of X, P of X, I'll just put a P where the F normally goes. I think it's going to look like this. Half Y equals P of two X. Is that okay, Ellen? Well, not really, because you know what? They get the Y by itself. So instead of putting a one half right here, you know what they put right there instead, which is why I left a little space there. I'll bet you that's what it says in the back. I'm pretty sure. That's the equation of that thick graph. These are tough. So the trick is look at your X intercepts, look at your Y intercepts, any other key points that you can as well. Did I answer your question? Sorry, who asked number three? Did I answer your question? Yes? I'll go over the Y three A and C and I'll still go over that more if you want me to. Any others? Yo. For which one? B or A or C? B, yeah. My key points, when in doubt, I use intercepts. See, the reason I like intercepts is on your X intercept, if they do something vertical, will it change at all? So that means I can ignore the vertical if I don't know what it is still. I can figure out the horizontal. And on your Y intercept, if you do something horizontal, will it change at all? No, so that means I can ignore that and just focus on what's going on vertically. So X intercepts and Y intercepts, usually though, they'll have clear key points. These are also tricky graphs because they're curvy ones. It's a parabola of some type. Is that okay? Any others? Did I assign any others? Oh, I assigned number six. Okay, we're good. Okay, I'm not going to ask you to hand those in. You can hang on to those. Now you have your workbooks so we can actually write stuff in there. And we have a problem. This is the problem we're going to have to deal with. Oh, I'm not going to pause yet. Can you turn, please, in your workbooks to lesson nine? It's called Combining Transformations Part One. And it's on page 59, page 59. And once you've turned there, look up. Supposing I want to do the following two transformations. Supposing I want to move three right and I want to do a horizontal expansion by two. I want to move three right and I want to do a horizontal. So lesson nine says Combining Transformations Part One. In previous lessons, we've learned the replacement method and what replacements do to graphs, how they affect the functions. But the issue is going to be combining transformations. Warmup number one wants us to walk through what we just did with Trevor and Mitsue. So turn the page. And on page 60, they want to walk us through the same thing, but with vertical, I can't do vertical transformations in real life. I guess I can have Mitsue get on Trevor's show. We don't want to give Mitsue a little headache. So turn the page. Page 61, here is the key. Order of transformations. It says we've seen that when two transformations are applied to the graph, the order in which the transformations are performed may or may not make a difference to the final graph. You know what? What we're going to do is we're going to always assume it does and play it safe and just always do it in the correct order. Even if we could have done it in the wrong order and gotten the right answer, we're just going to say, let's always do it right. When does the order not matter if you're doing two slides or two stretches or you're doing one vertical and one horizontal? When does the order matter if you're doing a horizontal stretcher slide and a horizontal stretcher slide, two horizontals or two verticals? But you know what, Trevor? Don't bother memorizing that. What I just told you is we're always going to assume it matters and just always going to do it in the correct order. What is that order, you ask? Here it is. We're always going to do the expansions, compressions first, reflections, and then translations, or I've been calling them slides. So, translations slash slides. Wait a minute, I'm just going to do it. That can be really clever here, I think. I think. Yeah, slash slides. Oh, cool. How can you remember it? It's alphabetical. E-R-T. Or if you use the word slides, it still works. E-R-S. Or if you use the word compressions instead of expansions, it still works. Anyways, it's alphabetical. Expansions, compressions, reflections, slides. That's the order that I'm going to pound into your head. And I've just lied to you. As it turns out, actually, there is more than one type of bed mass. There is more than one order that you can do these things in, and it depends on how they're written. When I first started teaching this, Haley, I had the kids memorize both orders and how they're written. And then I said, wait a minute, that's dumb. Why don't we memorize one order, and if it's not written the way we like it, take one line to rewrite it the way we like it, and memorize one order. So what's the order? Expansions, compressions, reflections, slides. Expansions, compressions, reflections, slides. Example one, it says, describe a series of transformations required to transform graph A to graph B. And I'm going to look for them in the correct order as well. I'm going to start training my mind now. Look at graph A and graph B. Do you think any stretching has gone on? Are they different sizes from each other? I don't think so. I don't think we've done a horizontal stretch. I don't think we've done a vertical stretch. Those look like they're the same size. So expansions, compressions, check none. Reflections, which way does your original graph A point to the left or to the right? To the left, which way does graph B point to the left or to the right? To the right, we've had a horizontal reflection. They've replaced x with negative x. What about a vertical reflection? The answer here is I'm not sure. If they flipped it, I wouldn't be able to tell because this graph is symmetrical horizontally. So if we're not sure, we'll assume they didn't. It's like the parabola. You could say, well, you flipped it horizontally and then you flipped it horizontally again and then you did it again. There was three negatives in front of the x. I guess technically that could be right. We're always going to go with the simplest possible example. So here, because it opens this way, I mean, they might have flipped it this way. I can't tell, so we'll assume no. Expansions, compressions, check. Reflections, check there's one for sure. Slides. This goes back to Brett's great question. How can we find key points? You know what key point I'm going to look at here for this sideways parabola? I'm going to move the vertex around. I think the vertex started out here and ended up here. How has it been moved? Down to, okay, to left and and not only have they replaced x with negative x, they've replaced x with x plus two and they've replaced y with y minus three. You know what the function equation would look like? We're just going to walk right down this chain. Eventually you may get good enough to do this in one step in your head, but for now I wouldn't risk it. The first thing they did was that they replaced the x with negative x. Then you know what they did? They replaced the x with a what? I've got to be fussy. They didn't replace it with a positive two. They replaced the x with a what? The whole thing, the whole thing. This is where it's very important. Say it louder, because here's what that means. They drop the y down. They drop the equals down. They drop the f down. They drop the bracket down. They drop the negative down. If you replace the x with an x plus two, you have to put it in brackets because if your original x was all negative, your replacement has to be all negative. If you just say plus two, I guarantee you'll miss the brackets if you say that to yourself. What's the third replacement that they did? They replaced the y with a y minus three. Then Kara, I don't think they'd leave it like that. They'd get the y by itself. Get the y by itself. What would I do with that minus three? Plus it over. If I gave you this as a multiple choice question and I said, right in function notation, what's going on here? Y equals f of negative x plus two plus three. That's a graph that has had the following done in this order. A horizontal reflection because the negative is inside the function next to the f. Two left. Three up. Example two. It says, describe the series of transformations required to transform graph A to graph B. Okay. We won't do these ones in function notation. We're just going to list the transformations. Oh, let's look for stretches first. How wide is graph A to begin with? How many squares count four? How wide is graph B? How many squares? Eight. What would make a four turn into an eight? Horizontal what? Ellen. Bye. How tall is my graph originally? Four. How tall is my new graph? Four. Did they stretch it vertically at all? Check. Reflections. You know what? They might have flipped this graph around horizontally. Is it symmetrical? I couldn't tell if they flipped it. So I'll assume they didn't. And they might have flipped it vertically, but it's symmetrical. I couldn't tell if they flipped it. So you know what? Reflections. None I'm going to assume. Always go with the simplest case. Slides. What would be the easiest point to figure out here? I'll give you a hint. Not the corners because the corners are getting stretched. The easiest one is the middle. You know why? What's the original coordinates of the middle? Zero zero, which means any slides are going to be direct one-to-one because the stretches would not have affected them to begin with. So, started out here, ended up here. How has this graph been slided? Slid. Slid. Slided. Slid. Moved. Translated. How many left? Count. How many down? Count. Look up. Now, I did that all in one step without substitutions and I'm not expecting to get anywhere near there yet. But take a look now that you see it. Is there a horizontal expansion by two? Yeah, there's a one-half in front of the X. Okay, there's brackets, but it's in there with the X. Is there a four left? Yeah. Is there a seven down? B and C is going to be the same idea, but I think you're getting the hang of it. Next page. Describe which transformations are applied to a graph of a function when the following changes are made to its equation. Does the order in which transformations are performed affect the final graph? What I said to you, Brett, was I don't care whether it affects it or not, I'm always going to do it in the correct order. The only time I won't do it in the correct order is if they specifically say to me do it in the order that we tell you because maybe they want me to get a specific shape and the only way that we can get a specific shape is to do it in their order. Okay? I'm not going to worry about does it affect it or not. I'm just going to do this orally. Replace X with 3X. What's that going to do? Vertical or horizontal? Horizontal. Expansion by 3. Compression by a third. Which one? Compression by a third. Replace Y with Y plus 4. Vertical or horizontal? Vertical. Four up or four down? Four down. By the way, the order here wouldn't matter because you have one horizontal and one vertical and they won't affect each other. Here, replace X with two-thirds X. Horizontal or vertical? Horizontal. Expansion by three-halves or compression by two-thirds because it's that thing that flips. It's an expansion by three-halves. How did you know it was an expansion because three-halves is bigger than one? Replacing Y with negative 3Y, that's two things. What's the negative do? Vertical reflection? What's the three do? Vertical compression by a third or expansion by three? Compression by a third. And, what's that mean? Two what? Two left. Here, the order would make a difference because you have more than one X and this order would be just fine because you did expansions, compressions, reflections, and so on. Example four. Now, what does it say in bold face in example four here? Okay. They want to force me to get a certain equation, I guess, and the only way that they can do that, Kara, is to say, do it in the order that we told you even though it's not the bed mass. Okay, if they say that, I'll follow their instructions. You ready? A horizontal translation to the left two units. Let's do the graph first. I'm going to hover about the vertex. Two left. Now, right there. Yes? Reflect in the X axis. Now, if I'm reflecting in the X axis, is that a vertical or a horizontal reflection if I'm reflecting in the X axis? Think carefully. It's, you know what, I'm going to write down here vert ref so that I don't do something silly. Alright, let's walk through this. Starting at the vertex. Two left. Vertical reflection, so instead of one up, where will I end up? Instead of one up, where will I end up? One down. Right, there's my vertical reflection. Vertical compression by a factor of one quarter. Yuck. Instead of negative one, you know what my height's going to become? A quarter of that. You know what? I'm going to eyeball it. It's about, hey Mr. Dewey, why don't you make it big so they can see? Yeah, I can fit this one. So what do we say? Reflect. It's about a quarter of it. I'm going to eyeball. It's right about there. Ish. Sort of. And then what's the final thing? Three down. One. Two. Three. You know what? The vertex, the new vertex is right there hanging in midair. Following their order. I'm not quite sure if I'm right, but if I do a couple more points and the parabola starts taking shape, I'll know I got the vertex in the right place. So let's move this guy here. That's a nice point. Ready? Two left. Vertically reflect instead of too high. You know what's going to become? Negative two. Vertically compress by a factor of a quarter. So instead of negative two, what's one quarter of negative two? Who's good at fractions? Negative a half. And then one. Two. Three down. The parabola is symmetrical. Brianna, I'm going to do this one. Two left. Reflect vertically. So negative two. A quarter of that. Negative a half. Three. There. That's okay. What else would I try? Would I use this point here that's hanging in midair? Probably not. Oh, you know what? This guy. You ready? Two left. Reflect instead of positive five high. You know what's going to end up? Negative five high. A quarter of that. What's a quarter of negative five? I think negative one point two five. So I'm going to eyeball it. Negative one point two five would be right about there. Three down. One. Two. Three. Two. Three. This boy. No, this boy here. Two left. Reflect. Quarter. Three down. One. Two. Three. You know what? This is kind of, is it not? Kind of. Is it not sort of? Oh, geez, I watched that. It's sort of looking like a parabola. You know what? I think I'm right. Let's see if we can get the actual equation. I'm going to go back to the small screen. Horizontal translation. What's the replacement if I want to go two left? Replace X with, so here's my original equation. Y equals X squared plus one. That's what I'm starting out with. If I replace X with X plus two, Y equals X plus two squared plus one. Vertical reflection. Replace Y with what? Negative Y. My new equation is negative Y equals bracket X plus two all squared plus one. I'm not going to get the Y by itself yet. I'll do that at the very, very, very end when I'm done. Vertical compression by a factor, vertical replace Y with what? A vertical compression by a quarter replace Y with what? Four Y. So I think it's going to be negative four Y equals bracket X plus two all squared plus one. Three down. Replace Y with three down. Replace Y with, I'll give you a hint. Y equals Y. Everything's backwards. In the replacements, everything's backwards. So it's not going to be Y minus three. Now, look up. That negative, is that a Y? Say no. Then drop it down. Is that four a letter Y? Then drop it down. Is that Y a letter Y? Say yes. What I'm going to replace the Y with is like replacing that whole thing X plus two squared plus one. I could now get the Y by itself if I wanted to. I would multiply the negative four into the brackets first to get rid of the brackets on the left side. Forget it. It says verify with a graphing calculator. Nah, turn the page. I'm going to pause here. We're going to continue part two when I see you folks again on Monday. Monday, Monday. I'm going to pause here for a few days on this because this is really the meat and potatoes. This is the whole point of this whole lesson. So what's your homework? I think I would try number one. Number two. Three is fine. Okay. I'm really not a big fan of number four. I'm going to pass on number four. I'm really not that big on you knowing whether or not you know what? I'm going to stick with one, two, and three right now. And if you're behind in any of the other homework, this is your chance to get caught up. You've got 15 minutes.