 Mark your own I didn't type this quiz up a colleague of mine did from a different high school But I thought it was a nice kind of a hey, let's figure out some of the basics here So it says sketch the basic graph and sketch the translated graph Label at least two points on the final translated graph by label I mean clearly put dots there if you actually label them with numbers. That's fine, too This is the square root graph I can figure this out in my head I think because I know the square root of zero is zero Except let's actually make that go through zero zero just a little bit more I know the square root of zero is zero. I know the square root of one is one I know oh for square root of four is two and I know the square root of nine one two three four five six seven eight nine is three. I think the original Would look something like that one mark and then the transformed graph How has this graph been moved? Let's see it looks like they've replaced the X with an X minus two It looks like they've replaced the X with an X minus two and I know that means to write and That plus five there that means five up. It means they replaced the Y With a Y minus five, but the minus five was on this side Amanda And then they plus the five over to that side. This graph is going to get moved to right and five up Now there's a bit of a glitch on the graph paper because that five right there It's tough to tell whether it goes with that or with that if you use that number as your five Give yourself full marks But as I was saying to a student this morning, believe it or not doing graph paper in word is Really a pain. It's surprisingly difficult, especially getting the numbers to line up. So I'm just gonna count Starting right here. I'll change colors One two or two to write and then one two three four five up the first point would end up right there The next point to write one two one two three four five up This point here to write one two three four five up and This point here to write off my graph one two, you know what it's gonna be kind of about there It's off my graph on your test If you go off the graph by one point that might happen in one of my questions probably not but it might if you're going off the graph by like eight you've made a mistake and Really if you going off the graph even by one double-check your work I tried to make them so they would all fit on the graph on the test I'm less fussy on the quiz. Also. I just wanted you to realize sometimes in real life They do go off the graph you need more graph paper, but on your test. I tried to keep everything nice and condensed I think it's gonna look like that How would I mark that? Those three points right there half mark off for each point that's incorrect So if you got two points wrong, sorry you can get zero out of one one mark for this graph one mark for that graph And that's how you get your score out of two number two f of x equals Oh, I know what this is This is a semi-circle with a center at zero zero And a radius of three This is a semi-circle that's centered right there and it goes three left one two three up Three right one mark. How has this graph been moved? Hmm Looks like they've replaced the x with what so if they've replaced the x with an x plus two I think that's two left and Oh, what's this plus four over here mean? four in fact they replaced the y With the y minus four and then they plus the four over to this side Now for the semi-circle I said the easiest way to graph this is To move the center and then graph the radius you could have moved all three points by the way works Just fine, but I'm gonna be clever. I'm gonna say my center is two left one two three four up There's my new center and the radius is three the radius is three the radius is three it would Look like that. I'll double check if I go two left four up. Yep, it works two left four up Yep, it works two left four up. Yep, it works once again one mark for the original blue graph and Then for the transformation if you got it right full marks Otherwise a half mark off for each point. That's wrong. So if you get two wrong points, sorry you get zero out of one Two marks grand total turn the page number three. Oh, that's the Absolute value. I remember what that looks like. I remember what that looks like this first one is gonna go through 0011 negative 1 1 2 2 negative 2 2 3 3 it's that V shape because it's the absolute Value graph You don't have to go that far Back if you want to just go that far and put arrows on the end. I'm good with that That's what I'm gonna do Shannon joy of my heart. What does this symbol mean? I want a big rant about it. I told you how much I hated it What transformation was that one? What was that the symbol for? Louder, you're right inverse. Ah, this is inverse. How do I find an inverse? Switch the X and the Y. I'm just gonna go like this to remind myself. That's that's my abbreviation for switch the X and the Y Come up with your own So I'll switch the X and the Y here zero zero is gonna. Oh zero zero Holly is gonna become zero zero What was the fancy word for a point that didn't move? Invariant and 11 is gonna stay one one and Believe it or not two two is gonna stay two two three three is gonna stay three three Here negative one positive one is gonna become positive one negative one Ah, I can kind of see what's going on now Katie Here negative two positive two is gonna become positive two negative two in fact. I think the red graph is the inverse One mark each and again a half mark off for each point you got wrong now this graph paper is a bit weird It's not perfectly square. You'll notice each Grid each it's a rectangle not a square and so it's kind of distorting things But I think if you do draw the line Y equals X which goes through one one two two three three four five five six six seven seven I think it is a reflection That was the test that was our built-in error check that we did I'll give you square graph paper on your test It's just I didn't feel like muscle mucking around with the graphics on this thing Also, when you get your graphing calculators your graphing calculators have screens that are rectangular not square And so they're going to distort your graphs as well. We'll learn to interpret them example for Three graphs three marks how about one mark for each graph Ah Says make sure to clearly label each graph in other words make sure that if I was marking this I could tell which one was which I'll go colors. I'll do this first one in blue now Soon as you see a negative That's a reflection Question is is it a vertical reflection or a horizontal reflection? It was a horizontal reflection. Where would the negative be? Right next to the X. You know what it's a vertical reflection It's a vertical reflection now careful It is a reflection about the x-axis because when you spin things vertically you are spinning them about the x-axis So if they ever say reflect about the x-axis, that's vertical even though the letter x is there everything's backwards So I'm going to look at my heights. How high is this right here? Zero high. What's the reflection of zero high still? Zero high. Hey that guy's going to be invariant, but instead of positive one high It's going to be negative one high and instead of positive four high. It's going to be negative four high and I'll connect them and Instead of positive four high, it's going to be negative four high and I'll connect them and Instead of positive three, it's going to be negative three high and I'll connect them and instead of zero high It's still going to be zero high, but instead of negative three high It's going to be positive three high and I'll connect them that right there is Y equals negative F of S, one mark. Looks like a little fish, Mr. Duke, shut up. I'll do this one in red. Now here there is also a negative, but Amanda, it's with the X, it's a horizontal reflection. So now instead of me thinking how high, I'm gonna think to myself, self, how far left, right? Negative four is gonna become positive four. Negative three, still one high, is gonna become positive three, still one high. Connect them. Negative two, four high, is gonna become positive two, four high, connect them. Negative one, four high, is gonna become positive one, four high, connect them. Zero left, right is going to, oh, invariant. Stays where it is, Reggie. Positive one to the right, zero high, is gonna become negative one to the left, zero high. And two right's gonna become two left. I think that red graph is Y equals F of negative X, one mark. And once again, half mark off for each point that's incorrect if you get too wrong, sorry. The third one, green. Shannon, what's this a symbol for? X equals F of Y. Also, and I like that one better inverse as a symbol because the negative one looks too much like an exponent, but I'll be honest, on your test, I'm gonna give you the negative one because that's by far the most standard symbol for inverse. Even though I hate it, I'll go with what the majority says. So, how do I find an inverse? Switch the X and Y. Instead of negative four, zero, zero, negative four. Instead of negative three, positive one, positive one, negative three, connect them. Instead of negative two, four, four, negative two, connect them. Instead of negative one, positive four, positive four, negative one, connect them. Instead of zero comma three, three comma zero, connect them. Instead of one comma zero, zero comma one, connect them. And instead of two comma negative three, how about negative three comma two? I think this here is the inverse. It's a cluttered diagram, but you guys drew it so you guys know what you drew. If you would be so kind as to give yourself a score out of nine at the top of the page, please. So we looked last day at, I said pause here for expansions, compressions. So we are gonna pause here, except I don't want you to get out the photocopied lesson eight. I would like you to turn in your workbooks, please, to lesson eight. It's page 51, page 51. No, you can hang on to it if you want two copies or recycle it. Page 51. Now I gave you some questions to try on expansions and compressions, but I'm gonna hold off on taking questions about the homework because since today's lesson is kind of a part two, I think maybe I might answer some of the questions about the homework as I do this anyhow. I hope. What did we say last day? We said, look, if you replace X with something times X, that will X horizontal always, it will stretch or shrink it and it's backwards. If you replace, don't write this down. If you replace X with two X, it's not gonna be twice as fat. It's gonna be half as fat. And the magic phrase for that Cassandra was horizontal compression by a half, by a factor of a half, if you really wanna use the full magic. And we said, if you replace Y with, oh heck, a half Y, replacing Y with something is vertical. Replacing Y with a half Y won't make it half as high though. It'll make it twice as high. The only problem, we would never write a one half Y. We would move the one half over to the other side of the equal sign. And when you move a one half from one side and you move it over to the other side of the equal sign, in order it becomes on the other side of the equal sign, a two, not a one half. And so once again, the Y stuff is not backwards if it's been moved over. Like we've been saying, all unit long. So in kind of a summary, they have this. We had the following note. A number there is the same as one over the number right there. If you have a generic function replacing X with BX describes a horizontal stretch. It's an expansion or a compression. Okay. Oh, if B is a fraction, it's an expansion. If B is not a fraction, it's a compression. Replacing Y, and I don't like the way they write this as one over A, one over A. So I find the way they phrase that confusing. Instead, I'm just going to jump straight to some examples to jog your memory. Let's look at example one. So example one says this. Write the equation of the image of Y equals X squared after a horizontal, you know what? Let's all underline the word horizontal because that tells me right away I'm replacing X with something. Horizontal compression about the Y axis by a factor of three quarters. Why do we call it a compression? Is three quarters bigger than one or smaller than one? Three quarters, bigger than one or smaller than one? So smaller than, number's smaller than one, compressions, number's bigger than one, expansion. The real question is, what would I have replaced X with to get a compression of three quarters? My last class, I heard crickets chirping when I asked that question. Saw people a little bit drool going down their chin. Andrew right away instantly leaps to the conclusion. He said, hey, Mr. Dewick, I think it's this because I've noticed Mr. Dewick that when you're talking about the compression factor or the expansion factor, all you're really doing is taking the reciprocal. I've noticed Mr. Dewick, don't write this down. For example, I've noticed that you replaced X with 2X. You said the compression factor was a half, which is really taking it to and flipping it. I've noticed Mr. Dewick when you said you replaced X with a half X, the expansion factor was two, which is really just taking that coefficient and flipping it. Apparently that's the pattern. And it is. What we're really saying is, what we're really saying is, where do they have the, they don't have a one over B here. The expansion factor or the compression factor is one over whatever's in front of the X. And it's one over whatever's in front of the Y, except almost always you've moved over what was in front of the Y and taken the reciprocal as you divided it anyhow. Let's see, B. Oh, let's get the equation. This would look like this then. Y equals, I'll replace the X with four thirds X. My original was X squared. My new one is something squared, the replacement squared. Although I have to admit, probably they would write this as follows. They would probably say, you know what? I know what four squared is. What is four squared? 16 you say? What is X squared? Just plain old X squared all over. What is three squared? I bet you they'd write it like that. 16 X squared over nine. But I know that that's actually a horizontal compression by a factor of three quarters. B. Y equals the square root of X minus three after a horizontal compression by a factor of four and the vertical expansion by a factor of two. Okay. Horizontal replace X with what? If I want an expansion by a factor of four, I'm gonna replace X with what? One quarter X. You could also write that as just X over four. They do that sometimes too. And I see a vertical expansion. So I'm also going to replace Y with, if I want to vertically expand by a factor of two, I'm gonna replace Y with what? So what's my equation gonna look like? Let's rewrite it. Oh, I'm not gonna write Y. What am I gonna write instead? What am I gonna write instead? A half Y equals, then I see a square root. Oh, but I'm not gonna write an X. What am I gonna write instead? Wake up boys and girls. And then the minus three would just drop down like a domino. Or to be honest, they probably would get the Y by itself. How would I get the Y by itself? How could I move this one half over? Now here's the tricky part. It would be times everything by two, including this here minus three. It would look like this. You'd get a two in front of the square root, but you would end up with a minus six. And that's gonna pose a problem that we're going to talk about later. Y equals three X plus seven after a vertical, oh, vertical Y replaced with a vertical compression by a factor of one third. If it was a vertical compression by a factor of one third, what have I replaced Y with three, I gotta be fussy, not three, three, three, three, three Y. And a reflection in the X axis. A reflection means I'm sticking a negative somewhere. Think very carefully. Here's the X axis. This is what we're reflecting in. Is that a vertical or a horizontal reflection if we're reflecting in the X axis this way? I can do it all in one line. I'm actually replacing Y with negative three Y. That would both give you a vertical compression by one third and a vertical reflection. So it would look like this. Instead of Y, negative three Y equals three X plus seven. Ah, but then they would almost certainly get the Y by itself. They would almost certainly divide everything by negative three, which would give you a positive Y, a negative X, because you'd have a three divided by a negative three, which would just give you a plain old negative one. And minus seven over three. Kinda yucky, but Amanda, I can deal with it. It's the hideous fraction. It's okay fraction. Let's all turn the page, because we can. Okay, says describe how the graph of the second function compares to the graph of the first function. Okay, how does the second graph compare to the first graph? Well, let's see. First of all, let's ask what we've replaced with what? I think we've replaced X with what? Half X, vertical or horizontal? Horizontal, expansion by two or compression by half? Expansion by two. Let's see if we can get these without writing the replacement. Let's see if we can just in our heads see the replacement. So I look at B, I look at B part two. What's the difference? What have we changed? Have we replaced Y with something or have we replaced X with something? X, what have we replaced X with? Three X, horizontal or vertical? Horizontal, expansion by three or compression by one third? Compression by a third. C, I see two things. Let's do the expansion, compression part first and then we'll do the, because a negative is a reflection, we'll do the reflection next. So first of all, that two, vertical or horizontal and how do I know vertical? Because otherwise it would be inside the absolute value next to the X. Is it next to the Y where it belongs? Then it's not backwards. So vertical, what? Expansion by two, check. I also see a negative. Is the negative vertical or horizontal? How do I know? If it was horizontal, where would it have to be? Next to the X, it's not next to the X, it's that I, this can't be horizontal. Nicole, what is this really? This is a vertical reflection. That's all I write. Now, Katie, the textbook will always add about the X axis. The problem is that X there confuses me. When I see X axis, I'm worried I might accidentally do a horizontal reflection even though it's vertical. So I only write vertical reflection when I'm doing these. If they force me to, I'll add the phrase about the X axis, but only if they make me. D, ah, let's do the two first. Vertical or horizontal, and how do I know? Horizontal, expansion by two or compression by a half. Is it right next to the X? Then everything's backwards. Compression by a half. I see a negative, Nicole, that's a reflection, vertical or horizontal. How do I know? See the difference, right? Horizontal reflection. By the way, we know what this graph looks like. What did this graph look like, the absolute value graph? V-shaped, so Kirsten, all you would do is you would take each point, and instead of, for example, the point being four to the right, you would make it two to the right, and then you would horizontally reflect it. Now, because it's V-shaped, I think when you horizontally reflect it, you won't be able to tell, because it's symmetrical, but you'd still do it. Or maybe it'd be clever and not bother because you know it'd be the same answer. Ooh, that would be good. E, ooh, I see several things going on now. Let's deal with the two. Vertical or horizontal, and how do I know? Vertical, is it next to the Y where it belongs? No, you say, then it's not backwards. Vertical, expansion by two or compression by a half, which one? Expansion by two. What about this one third, vertical or horizontal? Horizontal, expansion by three, compression by a third, which one? Expansion by three. So don't write this down, just watch. Remember on the absolute value graph, one of the points was four comma four. It would become an eight and a 12. That's where it would end up. I horizontally expanded it by a factor of three, and I vertically expanded it by a factor of two. And I could do that for each point. And if they're in nice numbers, you can do it almost in your head just while you're graphing on the fly. If they're yuckier numbers, Amanda, you make a list of the points and you kind of do them this way. But either approach, totally valid, totally fine. This is so much fun, let's do one more. In fact, you know what? It's all, this is my birthday present for you, F. Here you go, let's see. Vertical or horizontal, and how do you know? Absolutely, is it next to the Y where it belongs? Then it's backwards, expansion by three or compression by a third? Absolutely. I think I said to you last class, and if I didn't already, if you're a little bit confused still, worry but don't panic. I have found, next class, what we're gonna be doing is equations that have everything in them, slides, flips and stretches. And I've found that actually the tougher equations are much clearer to the kids because now they can see where everything goes at the same time. Oh, that's the flip. Oh, and that's the stretch. Oh, and that's the slide when you see it all in context. So I had a few people come in this morning and I had a few people in the yesterday after school. Don't panic, trust me, it'll quick. Example three, okay, now they're giving me the graph. It says the thick line is a stretch of the original thin line. Okay, right in the equation. Take a look at A. The thin one is the original. And I think the point that's gonna be easiest to follow is that kind of vertex-y looking point. How high is the original thin vertex? Six, how high is the thick vertex? Three, I think we've undergone a vertical compression by a half. Six became three. I think we've undergone a vertical compression by a half, which means they've replaced y with what? Two y, everything's backwards in our replacement method. So I don't know if you can see in the typing, here's my original equation. My new equation would look like this. Two y equals six over x squared plus one. Although would they leave the two there? No, how could we move the two over, Cassandra? We divide by two. Now watch what happens here. I'm gonna get this. Y equals, there's gonna be a six on top. There's gonna be an x squared plus one on the bottom. And when you're dividing by two, that's the same as just sticking an extra two on the bottom there. Although, do I have a six on top? Say yes. Do I have a two on the bottom? Say yes. You know what six over two really is? What on top? A three. That's what they would write. I wouldn't give you one that tricky to spot. I'll be honest, if you gave me that and then you gave me this, I don't know if I would spot that there has been a compression vertically by a factor of a half. I might. This, absolutely. And from the graph, I can spot it right. B, on your test, I probably won't give you an actual equation like A or an actual equation like C. In fact, I'm gonna give you a generic f of x like B. Some kind of weird shape with lots of points and curves. You have lots of key points to move around. So B is a question that I like. How has it been moved? How has it been changed? Let's deal with any stretches or shrinks first. Let's see. The thin graph is the original. From the top to the bottom, how high is the thin graph originally? Count. Is it six? I don't think so. Five, I think, yes. Right, count squares, not lines. Five squares. How tall is the thick graph? Count. Weird here. I'm gonna guess 10, but is it? Ah, we definitely have had a vertical expansion by two. See this horizontal line here? How long is it? How many squares horizontally long is that horizontal line originally? Count. Three. Here's the same horizontal line, I'm pretty sure. How long is it? How many squares long is the horizontal line? The new one. Have they changed the stretch or shrink horizontal? See how you can kind of look at sections of the graph and figure out what they've done. So nothing horizontal as a stretch. I do think they've done some other stuff, though. Reflections. Which way is this horizontal line pointing originally to the left or to the right? To the right? Which way is it pointing now? A right became a left. Ah, that's a horizontal reflection. And this curvy shape originally ended up heading downwards. Which way is this curvy shape heading now? Upwards? Oh, you're saying a down became an up? That sounds to me like a vertical reflection. Let's see if we can go straight to the equation without actually listing the replacements. Let's see how good we are. So my original was y equals f of x. A vertical expansion by a factor of two. What would I replace the y with? Are you all right, I think? A half y, yes. Horizontal reflection. What would I replace horizontal x? What would I replace the x with? I gotta be fussy, not a negative x. Vertical reflection. What would I replace vertical y? What would I replace the y with? I think that's the equation. See how I put that all together? A vertical reflection, a horizontal reflection, and a vertical expansion by a factor of two. Replace y with a half y. Although they would write it this way. They would move the negative and the two over to here. That's also a vertical expansion by a factor of two. A vertical reflection and a horizontal reflection. C. Hmm. Hmm, hmm. Well, let's look at the heights. I don't think we've had a vertical stretch because it looks like that's about the same height and that's about the same height right there. I don't think we've stretched it vertically. In my original, I had an uphill and then a downhill. In my new graph, I have an uphill and a downhill. I don't think they flipped it this way at all or this way at all. You know what I think they've done? I think they've taken that thin graph and done that. Stretched it, I think. Let's see if I can convince myself of that. I think that this point is the same as which point on my new graph. Ah, nothing. You needed it. This point is the same as which point on my new graph. I think this point ended up there. I'm trying to find stuff I can spot easily, so I'm trying to use intercepts. I think that this point ended up there. I think we've undergone a horizontal expansion. There's another reason I know that as well. Steph, do you see that this point right here remained invariant, the original and the new one both go through there? How far left-right is that point? When I stretch zero, what do I get? See, I think this is a horizontal stretch. If it was a vertical stretch, this would have changed heights. A negative four became a negative eight. A two became a four. A six became a 12. You know what I think this is? I think this is horizontal expansion by two. Yeah, yeah. Which means replace every single x with what? Half x. So, Carson, see this equation right here? Replace every x with a half x. It's gonna look like this. Y equals a half x plus four. A half x minus two. A half x minus six. There, that's the new equation. I probably won't give you one like C. I might in the homework, but not on the test. It's a good thinking question, but. I find these weird shapes are far easier to figure out what's going on. You've got more corners and key points to move. Hey, let's go to the next page over. Example four says this. A polynomial function has the equation, instead of f of x, they're gonna call it p of x because p for polynomial instead of f for function. Fair enough. p of x equals, and they've given it to me factored. Lovely. Says determine the zeros. Hey, what are the zeros of this right now? What are the roots of this right now? What are the roots? So originally, right now, the roots are four, yeah. Negative three and negative six, yeah. And what's the y-intercept right now? To find the y-intercept, put a zero in for x because if you're on the y-axis, you're zero. So it's gonna be zero minus four times zero plus three times zero plus six. In fact, it's gonna be negative four times positive three times positive six. What is negative four times positive three times positive six? 72? I think negative 72 is a not because it's gonna be positive times negative. Oh, never mind, I stand corrected. Yeah, negative four times positive times positive. You're putting in zeros in for the x-axis. Yeah, I'm right. Okay, are you ready? A, how has this graph been moved? Let's do the three first. Vertical or horizontal and how do I know? Sorry, bear with me for one moment. I need to go have a chat with some kitties in the hallway. By the way, another word for roots, let's write this down for a second. Another word for roots is x-intercepts, right? Your roots are your x-intercepts and we got your y-intercepts. So let's go back and let's answer this question right here. This three, vertical or horizontal and how do I know vertical? Expansion by three, compression by one-third, which one? Okay, so we have here a vertical expansion by three and then Jessica, we got a negative, that's a reflection. Vertical or horizontal, how do I know? Okay, vertical reflection. The question says, find the new zeros and the new y-intercepts. Steph, what word is right there? What word is right there? Your x-intercepts won't be changed because you know what x-intercepts are? Horizontal. My x-intercepts are still gonna be four, negative three and negative six. And I don't need to graph it. There's no way a vertical can affect an x-horizontal. Ah, but you know what my y-intercept is gonna be? Expand it by three and someone times that by, oh, I can do it in my head. 72 times three is gonna be 216, negative 216, reflected. It's gonna become y-intercept is gonna become positive, 216, I'm gonna get rid of the plus and the right positive in front of the numbers. B, what's going on here? I got a one-half vertical or horizontal and how do I know? Horizontal y by the side of the x. Is it, you said the side of the x is backwards. Okay, so that means this is going to be a horizontal expansion by two or compression by a half. Which one? Expansion by two, comma. I got a negative vertical or horizontal and how do I know? Horizontal y and horizontal reflection because negatives are reflections. Carson, what word is that? What word is that? Will that change the y-intercept? Can't. The y-intercept is still gonna be negative 72. Ah, but my x-intercept, which are horizontal, are going to expand by two and reflect. Expand by two and reflect. Nicole, what's this one gonna end up as? Expand by two and reflect. Yes, you're back here. I don't think you're elsewhere at reflecting. I know, Sabrina, expand by two and reflect. So expand it by two, times it by two. I think you're right and then reflect it. Instead of negative six, I think you're right but I'm reading your lips, I need a bit more. Yes, there you go. Be positive on positive six. Roxanne, expand by two and reflect. So negative 12, expand by two, reflect positive 12. Expansions, compressions. So I gave you some homework from the last day on expansions, compressions. Few more. Oh, no yawning yet. Number one, practicing writing equations. Number two. So here all you're gonna say is, well, let's do two A. Look at two A. Horizontal or vertical? Horizontal expansion by three, compression by a third. So horizontal, X three. Three A and three B, three A and three B and how about G? I'll go four A and four B and number six.