 So what we're going to ask ourselves today is What if you put something in front here? What if you put something in front of the x sorry in front of the f of x in front of the function? I'd like you to look at this graph here. Now this graph here is a semicircle. It's a semicircle of radius What? It's a semicircle of radius two there is Part a asks or says write an equation which represents that there Well as an equation it would look like this Y equals if f of x is this big square root three f of x would be three times This big square root in terms of my replacements What I've really done is I've replaced y with one-third y I thought you replaced it with a three. No, no, no I replaced the y with a one-third y but Nicole a one-third right here would become what when I moved it over a three Okay What letter did we replace step? Because I asked you to what letter do we replace I'll never get tired of that joke What letter do we replace yet? We replace the y that means this is going to be a vertical something as it turns out This ends up being vertical Amanda look at the replacement We replaced y with one-third y Everything's backwards if you try and make y one-third as big It ends up being a vertical Expansion by a factor of three It ends up being a vertical expansion by a factor of three Holly what that means is all of your heights are going to become three times as high I'll show you what this would look like it says use a graphing calculator We're going to do this one by hand because I haven't handed out the graphing calculators yet Ready Amanda How high am I right there? It's a trick question. How high am I right there? Zero, you know what if I go three times as high as zero. How high am I still? What's three times zero? This is invariant Cassandra, how high am I right here trick question? And you know what a vertical expansion by a factor of three of zero is zero But Shannon here's the more interesting one How high am I right here? If I vertically expand that by a factor of three, you know how high it's going to end up That point moves to there This graph would look like this and you now know the equation for the pope's hat. It's true I thought it was a rope. No, it's an equation. By the way Want you to notice what happened to the x-intercepts? Did they move at all fancy word? Let's write that down What happened to the y-intercepts? It did expand by a factor of three. You know what I'm gonna go It did that I'm not gonna rewrite the whole phrase It's so replacing y with one-third y Which becomes a three over here? Everything's backwards replacing y with one-third y makes it three times as high not one-third is high It's a vertical expansion by a factor of three D says Write an equation which represents y equals one-half of f of x. Well, that would look like this Y equals one-half great big square root of four minus x squared If I think about it in terms of a replacement We've replaced the y with a 2y and Then Carson we divided by two to move it over and that's where the one-half came from What did we replace y with? Everything's backwards. This is going to end up still being vertical But instead of an expansion by a factor of two it ends up being what we call a compression by a factor of One-half all Of your heights end up half as high What would that look like? Well, Nicole, how high am I right here? If I compress that by a factor of a half How high am I now? Still zero Here's the more interesting one Nicole. How high am I right here? If I compress that by a factor of half of a half or how high will I end up being? Compress it by a factor of a half Yep half as high How about this one Nicole while I'm zero high compress it by a factor of a half still zero high fancy word invariant that red graph Is the image of the black one? Compressed by a factor of a half Once again, Amanda because it was vertical, you know what happened to the x intercepts They were invariant and the y intercepts underwent a vertical compression by a factor of a half Says compared to the original the graph of this now I Don't quite like the way the workbook does it I don't like that they start out with the a right here to me It should be here and then we divide to move it over But because they're starting out with the a right here in this notation step Things are no longer backwards because we've moved a over and now it's no longer backwards So here's what it says compared to this this results in a vertical stretch About the Now think about it if you're stretching a graph vertically, which axis are you stretching it about? If you're stretching it about the y-axis, which axis is not moving at all You're stretching it about the x-axis the x-axis is sitting still and you're stretching like a big piece of rubber the graph So it turns out a vertical stretch is about the x-axis just like a vertical reflection was about the x-axis And if this number here is bigger than one for example y equals three f of x then you have an expansion and If the number sitting where the a is is between zero and one a fraction for example Ryan y equals one half f of x Then you have a Compression Cassandra replacing y with something is always vertical replace y with five y one-fifth as high It's a vertical compression by one over five Replacing y with one-fifth y five times as high all of your heights will be multiplied by what about if we replace x With bx, what if we put something in front of the x another coefficient there It says the graph of y equals f of x equals that there's our semicircle of radius two and Then there's a bit of a typo here. I think look up Everywhere in part a and b where there's a f of 4x. I'd like you to instead make it an f of 2x an f of 2x and f of 2x oh And then in C where it says describe how the number for it's actually described how the number two You'll see why I did that just saying Katie. I'm gonna argue that to turn That Into that We've replaced the x With a 2x in my equation it's all it would look like this Y equals great big square root. That's not an x. That's not an x. Oh, that's an x I'll replace the whole thing with a 2x all squared See first in our place the x with a 2x By the way, probably they would tidy this up. They would probably choose to write it this way For minus 4x squared. They'd probably go two squards for I don't like this as much This is much clearer to me. What's going on? I can see the substitution I can see Matthias that I've replaced x with 2x. You know what that does first of all x vertical or horizontal X vertical or horizontal X vertical or horizontal X vertical or horizontal It's absolutely always absolutely horizontal. So when it says state what's going on. I'm gonna write here horizontal and Once again here Amanda everything's backwards It's not gonna get twice as fat. It's gonna be a horizontal Compression by a factor of one half Horizontal compression by a factor of one half in our notes will abbreviate this in our homework But if I gave this to you on a test as a multiple-choice question, that's how I write the phrase So I want you just to be familiar with it. So when you see it, you don't go like what the heck is that? Katie, how would I graph this? You ready? You ready? You ready? Let's start right here Katie Now horizontal now. I'm thinking x coordinates x coordinates x coordinates horizontal. What's my x coordinate right now on this point Katie? Compress that by a factor of a half instead of two to the right, you know, it's gonna end up One to the right same height though. Katie this point right here trick question What is my x coordinate right here? How far left and right am I? You know what if I can press that by a factor of a half, you know where I end up in variant But this left-hand graph here instead of negative two, you know, it's gonna become I agree totally negative one That red graph is the image of the original With a horizontal compression by factor that Replacing x trying to make it bigger. No, it's gonna get fatter to compensate. Everything's backwards. Sorry It's gonna get skinnier to compensate. Let's say it's gonna get better. It's gonna get skinnier to compensate Now you got me yawning What happened to the y intercept? Well the y intercept this time was Invariant why because if you're a y intercept, what's your x coordinate zero? And if you stretch zero, it's still zero What happened to the x intercepts though? Did they stay where they were? No, Jessica, you know what happened to them They underwent a horizontal compression by a factor of a half now. Let's compare that With D D says write the equation which represents f of one-third x What have we replaced x with here meant to be really obvious yep, we have replaced x With one-third x Andrew is that gonna be vertical or horizontal? How do I know instantly without to think about it x? x and Everything's backwards as it turns out this is going to be a horizontal it won't be a compression You know it's gonna end up being an expansion by factor of three What does that mean? Andrew it means that all of my x coordinates are gonna be three times bigger Andrew what's my x coordinate right here right now? What's gonna become yep? What's my x coordinate right here right now? You know it's gonna become Invariant and this one's gonna become positive six that red graph image of the original Under the transformation y equals f of one-third x a horizontal expansion by factor of two How many played Super Mario's when you were a kid? Okay, when he ate the mushroom all they were doing was because remember Mario is a function It's a weird complicated one, but it's a function to make them twice as big all they did They replaced all the y's with a half y and all of the x's with a half x and that would give you a Horizontal expansion by two and a vertical expansion by two Very easy to do Is it is he twice as big or three times as big? I've never actually measured you might be three times as big in which case they've replaced y with one-third y and X with one-third x three times very simple Once again, we notice the y-intercept is invariant in Let's try that again. Mr. Duk invariant Stop yawning. Mr. Duk knows for me didn't work rats The x-intercepts those are the ones that underwent the horizontal compression by a factor of or sorry horizontal expansion by a factor of three When is it a compression when it's a fraction less than one? When is it an expansion when it's a number bigger than one? Well, you take marks off we call expansion a compression device first Probably not but on the multiple choice section of your test. I will totally use that so Compared to the graph of f of x f of bx results in a horizontal stretch and it turns out You're stretching about the y-axis This is what you're stretching about because the y-axis is standing still and you're stretching around it What's it say right there by a factor of what what's it say right there Amanda by a factor of one over b in other words? Replacing x with 2x gave you a compression factor of one over two Replacing x with one third x gave you a compression factor of one over one over three, which is just three you're taking the reciprocal Duk. Yes, Katie. How come it's one over b here? But back here it's not one over a because they've already moved it over so it's already become backwards So you don't need to take the reciprocal anymore in this case That's why I wish they'd written it the same way both times I wish they'd kind of said look it's always the reciprocal as long as the y-thing is next to the y-thing So if b is greater than one for example Y equals f of 3x That's going to be a compression by a factor of one over b one over three in this case if b is a fraction y equals f of One half x that's going to be an Expansion turn the page page 41 is kind of nice. It asks Okay, what if there's also a negative in front? What if a is less than zero well if there's a negative right here? This is a negative right here. What else happens to your graph? vertical or horizontal Vertical and what a negative do flip Skip it page 42 Now page 42 is asking. What if you have a negative right here? What did a negative do right there? Horizontal reflection good. Thank you. You're learning that Page 43 is what I want to get to So we're looking at the a being right here if a is a fraction for example Y equals one half f of x That's the same as replacing y with 2y And when you look at the replacements everything's backwards you place y with 2y it doesn't get twice as high. You know what it's a vertical Compression what was the vertical compression here factor here being the cold What would the vertical compression factor here be? It's whatever number sitting there because you've already made it backwards moving it over so it's no longer backwards And it is about the x-axis. So as an example thin graph thick graph Then graph is my original thick graph. You can see has been compressed does not go as high does not go as high greater than one like Y equals 3f of x which you may recall would have come from this Replacing y with one third y This was the format that you were more used to the parabola Yeah, I really started something with Nicole. I'm yawning now too and I'm passing everybody else is terrible This this was what you did last you at the parabola. This will be a vertical expansion By a factor of a and you're expanding about the x-axis vertical expansion. Oh last thing For what it's worth Steph if a is less than zero. That's the fancy way in math of writing if a is negative It's also gonna be a vertical flip it will be reflected in the x axis and Compressed or expanded vertically in the x-axis, so you have more than one thing going on It's gonna leave the problems down the road, but right now it'll work. It's terrible. Hey, let's look at x's if we replace f of x if we put a b there, that's a fraction for example y equals f of One-half x where we've replaced x With one half x vertical or horizontal How do I know instantly? horizontal everything's backwards Putting a one-half x would mean expansion factor by two. This will be a horizontal Expansion by a factor of What would the expansion factor here be what did I say in my example if I generalize that one over b Take the reciprocal of whatever numbers there. That's your factor and if you're stretching horizontally you're stretching about the y-axis What if b is bigger than one would of b is not a fraction for example y equals f of three x Replacing x with three x vertical horizontal Horizontal, I don't know next to the x horizontal what well Replacing x with three x everything's backwards. This is going to be a horizontal Compression this would be a compression by a factor of one-third It's always a compression by a factor of one over b and it's always about the y-axis. Oh And and and if b is also negative There's going to be a horizontal reflection It's going to be reflected in the y-axis and compressed or Expanded about the y-axis Let's try some oh No, first let's summarize page 44 Lovely summary now. Let's try some says example one write the replacement for x or y and Then write the equation of the image of y equals f of x after each transformation a horizontal expansion by a factor of six horizontal Gonna replace x with I want to expand it by a factor of six. What am I going to replace x with? one-sixth x So it would look like this y equals f of X over six or one-sixth x they're interchangeable B sorry B a vertical expansion vertical Replace y with something factor of one-fifth Everything's backwards five y in scientific notation five y equals f of x Although they would almost certainly move the five to this side And say that looks better Okay, see a reflection. Okay reflections are negatives We're reflecting in the x-axis that means Gotta think careful. We're reflecting in the x-axis. What kind of reflection is that? Reflecting in the x-axis. What kind of vertical? We're gonna replace y With negative y There's also a vertical expansion about the x-axis We're also going to replace y with we want to vertically expand by a factor of three. We want one-third y There's our vertical expansion and we'll get this negative one-third y equals f of x although I Don't think they'd leave a negative one-third on that side Cassandra They move it right to here right to here and you know our negative one-third on this side would become Got to be fussy not a three a Negative three skip D next page Yo, why is it a negative three? Because you divide by negative one the negative wouldn't cancel Right Are you asking where the negative come from in the first place or you're asking? Why is it still negative afterwards? Okay, so how would you move this over? First you divide by negative one which would give you negative there and then you times by three Which would give you a three there, but the negative would still be there Is that okay? example to We're not gonna write this down. We're just gonna do this orally How does 3 y equals f of x compare with y equals f of x how do these two graphs compare? Looks like they've replaced y with 3 y vertical or horizontal vertical vertical or horizontal for replacing y with 3 y vertical Expansion by 3 or compression by a third compression by a third All your heights are one-third as high replace x with 4x vertical or horizontal Horizontal four times as wide expansion by four or compression by a quarter everything's backwards compression by a quarter vertical or horizontal Vertical expansion by three or compression by a third expansion by three vertical or horizontal vertical compression by one-sixth Horizontal expansion by three Let's try a graph Example four It says the graph of y equals at blah blah blah. Take a look at this What do you see going on here on the right page good I see a 2 is the 2 from the y or is the 2 next to the x x vertical or horizontal horizontal That 2 there is that an expansion by 2 or a compression by a half that 2 there Expansion by 2 or compression by a half compression By a half that's probably what I wouldn't write out Horizontal compression by a factor of a half. That's probably what I would write out in my notes. Oh I see one more thing Holly negative It's the negative outside where the y is or the negative inside next to the x It's a horizontal or vertical negative outside in front of you what where the y is or is it next to the x? Horizontal or vertical horizontal next to the x horizontal This is a what does a negative do? horizontal reflection in The y-axis, but I'm not going to write that because that confuses me So Kirsten I got two things going on here compressing horizontally Reflecting horizontally you ready? We're gonna do this together you and I this point. We're gonna do the key points and connect them Don't panic. It's gonna be easier than you think What's the x-coordinate Kirsten of this point right here? You're like you're right say it louder When I said say it louder you say it same ball doesn't help at all Negative six compress that by a half. So instead instead of negative six. It's gonna be Negative three hover your pen above negative three. Don't put a dot there. Just hover there Reflect it horizontally instead of negative three. It's gonna become positive three That's where you end up Negative six negative three reflect. Oh, that was so much fun. Chris. Let's do the next one What's the x-coordinate of this guy right here negative? What? compress that by a half Reflect we're the first two points connected then connect them Jessica, what's the x-coordinate here? compress that by a factor of one half Still zero reflect it still zero fancy word Turns out that guy ends up being invariant underneath this transformation Andrew, what's this one? compress You'll notice. I don't even care about the wise. I'm just staying on the same height You know why I don't care about the wise Do you see anything vertical written here at all? So I'm just making sure I move sideways But keep the same height everywhere connected connect them. What about this point here? a little bit trickier Holly, what's the x-coordinate right now? Compress it by a half. Yes, it's a decimal, but what do you get? 2.5 reflected You know what I can draw negative 2.5. That's not a hideous decimal. Sure fair enough about there Connect them Cassandra. What's the x-coordinate of this last one here? compress it Reflect it Got right there Connect That red graph is the image of the black graph Skinnier and flipped. It's your homework. Try number one Number one has fractions shut up and deal with it. Number two is fairly similar to number one three, I'd like you to do a b and c But not d. Skip four Five is good All of five all of five six is good Pass on seven Very quickly look at lesson eight, which I also hand it out to you, which is a continuation of the same topic Once again a lovely summary Time I got I know I've been talking a while. How far am I going to get here? Sorry what? I'm gonna pause here for now. I'll pick up the rest of this next class. I've talked too much