 So this is from an old provincial exam question. It says the graph of this function is shown below and they were even kind Enough to label the points for you. Oh, how nice. It's a little small on your copy. Sorry Trying to save paper Number one says so on your test the first half or so maybe the first 60% is going to be multiple choice questions. I think there's about 14 multiple choice questions Strangely multiple choice questions in math 12 or with 1.5 marks each Why because that's what they did on the provincial why because they're stupid But I've done that and I'm not gonna just this is the last year that principles of math 12 is running as the current course I'm not gonna suddenly change and go to the more sensible one mark for multiple choice because that would skew your grades Compared to all my other students that have come here in the past. So anyways for some reason multiple choice is worth 1.5 marks a piece Then in this final 40% I'm gonna give you some kind of generic shape Some f of x and I'm gonna give you a bunch of transformations. So here is great practice Tell me what's going on here. It says on the graph provided sketch this graph. What's going on here? List the transformations in the proper order. What do you see? Andrew what do you see here in number one? What does this here mean? Okay, let's write that down Burt X three uh-huh to Right You guys want to try it on your own and I'll do it up here and let's see if we end up with the same thing Or I can walk through it. You know what most of you nodded and said you want to try it on your own I'll freeze the screen try this yourself. So vertically expand your how it heights by three and then to right Screen is frozen So negative four negative over there And I ended up with that and if you're wondering how long it should take you You should be done in about another 20 seconds if you're not you're gonna have a tough time finishing the test I'm trying to face things like you should have really been done sooner than that Okay, it's not okay Holly we're good Watch your head. Happy joy. I would consider this Holly See plus level question Okay, it's got two transformations. I Think C level questions are with the transformations where there's two just with slides because I think the slides you guys Okay, I think that's like that, you know the parabola. I the stretches are a bit trickier. I agree Question number two Okay, sketch the reciprocal of this Step is that okay? You're looking a little question mark. Are you good? We're good Question number two sketch now, that's the reciprocal that is not the inverse That's the reciprocal inverses that stupid little negative one switch the X and Y around reciprocal means take the reciprocal of the heights Oh, okay, what was our procedure for reciprocal? Maybe we'll write this down the first thing that we did was we looked for what I called The invariant points which numbers were their own reciprocals Okay, plus or minus one high. So let's go back to my original graph You know what right there and right there? Right there and right there are going to be invariant Then the second thing I looked for was vertical asymptotes And where did the vertical asymptotes occur anywhere that we were how high? zero Hi, and Ryan if you're zero high what you real are is an X Intercept I've scrolled down. What's the X intercept Ryan on my original graph? Okay, so there's gonna be a vertical asymptote I'll do it. No black is dumb. Let's do it do it in red. So it stands out There's gonna be a vertical asymptote Right there and then I said to you we use our Sesame Street imagination. We pretend we're a little bug I'll start out standing on this point right here this positive one as I move to my left What's my original graph? Oh hang on write this down three? bug bigger becomes smaller and Smaller becomes bigger I'll show that with two arrows pointing in either direction and We remembered a few specifics Cassandra remembered when they would shoot off to infinity or what let's see as I move to my left Trevor, what's my original graph doing if I recall is it getting closer and closer to zero you guys have it in front of You but I've scrolled down my original graph as I moved to the left right from here getting closer to zero So what's it? What's the reciprocal gonna do? Shoot off to infinity Now I just lost the mark look up We're fussy when we're drawing these if you touch an asymptote with your graph That's a no-no. And can you see I just ran into an asymptote So when you're drawing these be a little bit careful you get closer and closer But you never touch an asymptote you can with the arrowhead. That's not part of the actual graph I do have to put an arrowhead there because it's shooting off to infinity as I move to my left As I move to my left Trevor, what's the graph doing getting bigger or getting smaller the original one is it? I don't know if you have the graph in front of you. I don't The original one is getting smaller the original one the original graph is getting closer to zero Okay, oh I apologize Trevor. You're right and you clearly haven't watched the video Ah Okay, let's move over to know he was late. Let's move over to Tyson Tyson my original graph it as I move this way I think it's getting a little bit bigger. Yes, but not very much bigger. In fact, what's the highest it gets is it? What's the highest it gets? To so what's the lowest I'm gonna get? What's the reciprocal of two a half and Does it keep going forever and ever and ever or does it stop right there? It's not meant to be a trick question It's meant to be really easy does this graph keep going forever and ever or did they put a big dot there to show that it stops It stops my reciprocal is gonna stop as well my reciprocal is gonna go to positive four and quit in fact It's gonna do this At positive four you told me I'm gonna be a half high It's gonna curve. It's gonna it's gonna stop right there though It's not gonna go any further because my original graph does not go any further a number of you have asked me There's arrows if you keep going There's no arrows if you don't keep going The textbook instead of arrows what it does is it goes right to the edge of the graph I'm guessing that's the software that they use to create those graphs, but Arrows, please let's do the left side Step I think if I stand right here and I move to my right my original graph is getting closer to zero, but negative Yes, so shoot off to infinity but negative and as I move to my left my original graph is getting further from zero In fact, I think it hits four negative two Is that right? What's the reciprocal of a height of negative two? Gotta be fussy not Negative one over two in other words. It's gonna end up right there It's gonna curve closer and closer there and it's gonna stop there because the original graph came to a stop there Now if the original graph had arrows and suddenly leveled out Well that I would continue with this way with arrows and that way with arrows, but the original one does know There's reciprocal You're much more likely to get something like numbers. Well, I'm gonna give you a reciprocal graph Probably worth three marks Usually I give one mark for the invariant points one mark for the asymptotes and one mark for the shape Number three little alarm bell going off here Why is there a little alarm bell going off here? What's wrong with this question? What don't I like about this question? Well, the x is that I don't mind, but it's the x is being multiplied There's a coefficient and there's a slide, but this slide is not in brackets next to the x all by itself It hasn't been factored Mr. Deweyck. Yes What does that really mean? This one here if you don't factor it the correct order changes the correct order if you don't factor it is actually do slides first Then expansions compressions then reflections You can memorize a second order Cassandra. I just memorized. I always want things factored. I Always want things factored. So I'm going to rewrite this as y equals negative 2 f of I'm going to factor out a negative 2 And what would I have left behind in the brackets here an x that you kept it obvious more specific What about right here? A plus 2 Because negative 2 times positive 2 is negative 4 we're foiling it in river We're factoring which is foil in river or front door bomber or distribution whatever your teacher called it in reverse Now I'm going to make my list Expansions compressions Let's walk through this one Sabrina. Do you see any expansions compressions? What what's this one? Did you say vertical? Okay by two or by a half? You're correct your throat sore me too Vert x to with authority because you're right. It's a vertical expansion by two Oh, what's this one? I totally agree by two or by a half and how can you remember? You're right. Say it louder with authority By a half and it's a compression since it's by a half How do you know it's next to the x where it belongs? It's backwards That's vertical if it was vertical and backwards it would be on this side since it's not on this side It's no longer backwards, right good Expansions reflections check we've done those What did we do after expansions compressions? Reflections negatives I see to there and there I see vert Reflection by the way a vertical reflection is about which axis If I flip something vertically, it's about which axis the x axis because I'm going backward in other words Sometimes on a multiple choice. They'll say which of the following is the reflection about the x axis And you have to clue in that's a vertical reflection. It's got a negative in the front I just don't like writing that because x axis confuses me vertical I know that means up becomes down and down becomes up There's a horizontal reflection To left One down right good try this one on your own go back to the original I'll see if I can do it here Boom a decimal. Oh well Expansion compression reflection reflection to left one down Expansion compression reflection reflection to left one down I think I end up with that weird thing. You do end up with a couple of points hovering in mid-areas But they're nice decimal. They're halves not like one seventh or something creepy. So I'm okay with that It was like a one-thirteenth. I'd get a bit nervous and think I've made a mistake someone I think he goes through zero three And you go through negative four negative five on the end points. Is that double-checking people nodding or no? People still going. Okay, Jessica's nodding You also go through negative two negative one and then you got those two little half points hanging in there This one had six transformations I would consider this a v-plus level question The plus comes from the fact that you had to factor it if it didn't have to factor if I gave it to you factored already It's a b-level question. It's a higher order. It's a tougher question. How many you got that? Okay Let me if you maybe I went to that See when you zigged if you didn't get it Okay Let's try another one turn the page now We're supposed to sketch the absolute value of the function, but you've all just turned the page So what I'd like you to do first of all is lightly sketch the original the original goes through there There there there and there just do it with a little dotted line And that's what I'll do on your test the original graph will be there as a dotted line for you It'll be like a gray dotted line now What's this transformation here? What's that mean sketch y equals what are those two vertical lines mean touchdown? No, that's sports What do I do for the absolute value stuff like that joke? Oh get used to them Steph what do I do for absolute value? It was actually I thought one of the easier ones anything above the x-axis remained fancy word in Variant so I would take out my pen and I would trace over this really or pencil trace over that and so that's gonna stay where it is Anything below the x-axis is going to fold right up. I think it's gonna look like this junk junk you get kind of a Sort of a seagully shape thingy right sort of this kind of Like a draw if I really can't to me absolute value. Yeah, I'll throw one on your test It'll probably a little two marker To me. That's about a cc plus like it's a fold the bottom half up Ah, but now we have in number five absolute value with a twist Once again, let's quickly sketch the original which goes through here here And here it goes through there there there there and there Sketch it as a dotted line remember I said yesterday on your quiz if you do it as a solid line I have to assume that that's part of your final answer There was one of you either in this class or my other class who use different colors for each step That is acceptable, but you may run out of colors And it clutters up your graph. What do you think this means here? Well, what two transformations do you see here? What's that mean? One up what's this mean? Which one do I do first? I'll give you a hint look at number five and look at number six and see if you can figure out the difference because I think number six Also has a one-up and an absolute value. Yes Which one do I do first? In terms of the order of operations Absolute value you treat like brackets like bed mass In number six is the plus one inside the absolute value I would do that first because it's inside the brackets like that mess Here is the plus one inside the absolute value. No, I'll absolute value first and then one up in fact It's gonna be number four moved one up now. I'll walk you through it I would go like this this one stays where it is. I'll change colors This one stays where it is one up this one stays where it is one up This point stays where it is one up. This arm is going to look like that This point here, I would take the absolute value first I would take the absolute value so it ends up there and then one up Connect them This point here. I would take the absolute value so it end up at positive two and then one up Don't Connect them. There's absolute value with a twist. What about number six? Okay This is one up and then absolute value once again, let's just quickly sketch our original Here I'll go point by point This one here says one up Don't then absolute value. What's the absolute value of negative one height? Positive it's gonna end up right there This point here says one up Then absolute value. What's the absolute value of zero? Oh, I'll wait. It looks like I lost a few See a racer is being pulled out Question Amanda asks it's a good question for me is The plus one inside the brackets next to the x that would be one left The plus one is after the function. It's one up then absolute value. Does that make sense? So let's do this next point with me ready Look up. This would say one up boom and then absolute value. What's the absolute value of a height of zero? Zero zero with authority. Yeah, look, this is kind of weird. It's gonna connect right there Let's do this point next One up and then what's the absolute value of a height of one? One good morning. Just give me a moment here Continue so This point here Amanda How high is it right now one one up and then absolute value. Oh, what's the absolute value of two? That's gonna stay where it is once I move it one up This would move one up an absolute value. So this is a bit of a strange one It goes from being a symmetrical graph to being this kind of weird funky thing, but that's okay There's absolute value with a twist What's the order you do what's inside the absolute value first Kirsten? It acts like a bracket It's not a reflection expansion compression there. You have to just think it depends on how it's written number seven reciprocal with a twist How do I know reciprocal because f of X is on the bottom, but what else do you see going here? What's that? Well, vertical or horizontal and how do you know whether having to think about it? It was horizontal. Where would it be? Is it inside there next to the X? So it's vertical. In fact, we have a vertical Expansion by two What about this negative vertical or horizontal and how do I know without thinking about it if it was horizontal? Where would that negative be? It was horizontal. Where would that negative be? It's at the X. So it's also a vertical Reflection, but it looks like before they did that They did that they took the reciprocal and then Vertical expansion by two and a negative Once again, let's sketch our original graph because otherwise this one's going to be really tricky So I'm going to very lightly put a little X right there But not what's a big thick felt mr. Doock the thin felt dope Little X right there X there little X there little X there In fact, well, okay, I'll do the dotted lines. I almost find the dotted lines getting away just the key points or what I'm going to move. All right First thing we're going to do is the reciprocal Ready I Would start with this invariant point right here. It would stay invariant. Ah, but what are my next two transformations? What are the next two things that we listed? We wrote them down. They're not on my screen. You have to read them to me. What are the next two things that we listed? Okay, so instead of one high too high and That invariant one high ends up right there. What about this invariant down here? Well vertical expansion by two don't vertical reflection It ends up there Will this point move at all here zero zero it's going to still be the vertical asymptote How do you know because if I well first of all let's draw it in if I vertically expand that asymptote by two isn't it still just a vertical asymptote and Kirsten if I flip it, isn't it still just a vertical asymptote? So it's going to be it's going to stay where it is Okay, as I start back here as I walk to my left I'm getting closer to zero normally. I would shoot off to infinity If I vertically expand that I'm going to shoot off to infinity faster If I reflect it though, I'm not going to shoot off to positive infinity. Where will I shoot off to? If I reflect it, I'm not going to shoot off to positive infinity. Where will I shoot off to? Cassandra Thank you for playing Now I can't really tell if you're shooting off to infinity faster or not It's very difficult for me on the shooting off to infinity to mark a vertical expansion by two In fact, I wouldn't even look for it. What I would look for is did the ones the invariants move? Now As I move to the right the highest I got was two What's the reciprocal of two a half? Don't put your pencil down. They'll hover it hover it hover it Then vertically expand that by two if I vertically expand a half by two. Where will I end up at? one Reflect it. I'm gonna curve Towards that you see how we can't put all this together. It's tougher by the way This would be an a-minus or maybe a question Reciprocals with a twist are considered higher or digit up that okay Let's try it on the left-hand side So we moved this point as I move closer Katie. I'm getting closer to zero So the reciprocal of closer to zero shooting off to negative infinity Vertical expansion would shoot off to negative infinity even faster. I can't really draw that ah, but vertical reflection Instead of shooting off to negative infinity. I'm gonna shoot off to positive infinity Now look at what I just drew look up on The provincial you might lose a half mark for what I just drew because it's starting to look like I'm curving back I was kind of sloppy. Can you see I started to tilt a little bit away? So I'm gonna redraw that and I'm gonna make sure that they can see that I know that it gets closer and closer to the asymptote. They are fussy I marked this question one year and they were that fussy the odds are I wouldn't catch that but I'll show you the correct Okay, could you do me favoritizing could you shut those doors at the end of the hallway? They're doing the drama class in there Okay, Nicole. Let's do this one. I would reciprocal. So it would end up a half high but negative If I vertically expand negative a half where will I end up one reflect? Oh, it's gonna end up there So, you know what? reciprocal with a twist We can we can get it I think We can get it. All right Number eight Number eight says the graph of y equals f of x is shown below sketch the graph of y equals one over f of x directly on the same grid This used to be worth five marks. It's not worth that anymore because kids got too good at it This was the very first year that this current course of math 12 came into existence. I think it was in 2002 So You know what? Try this on your own. This is a reciprocal graph. I maxed it Absolutely gonna give you one of these try this on your own. I'll do it up here. Let's see if we end up with the same thing Reciprocals remember the three steps in variants Asymptotes and then bigger becomes smaller smaller becomes bigger Okay, I'll talk you through it Ready? So if you had trouble, I'll talk you through it first thing I said anywhere one high in That part you just have to memorize Shannon. Oh and anywhere negative one high But by the way, that would get you one mark out of three because you'd have the invariant points correct So you're not gonna get zero. Oh and anywhere's real high asymptote. It was your high asymptote. You okay with that? All right Then I did my bug trick I started out here and I said as I move left my original graphs getting bigger reciprocal is gonna curve closer to zero Like that could even exaggerate that a bit more do a better drill Closer to zero Is that okay as I move right this little tiny chunk right here this little try here. It's getting closer to zero What's that gonna turn into? What's the reciprocal of getting closer to zero? Shoot off to infinity okay Stand right here as I move left this little chunk right here getting closer to zero you just told me it's gonna shoot off To infinity correct Now as I move this way, I'm getting higher. I'll get smaller. What's the highest I get right there? What's the reciprocal of two? What's the reciprocal what's the reciprocal of two? That point there Shannon is gonna end up right there So I said oh it's gonna curve closer to that and also this guy is gonna curve closer to that in that direction Okay Then what I did was as I move to the right it gets close it shoots off to infinity. Sorry. That was mean of me Okay, standing here again. Now I'm in the negative area though, right? So closer to zero, but negative shoots off to infinity, but negative. Yes Further from zero, but negative closer to zero, but negative. It's gonna look like that If you get the hang of reciprocals it actually becomes fairly step-by-step But I recognize it's a tricky concept How many you got that on your own? Okay. Whoo-hoo one more reciprocal. Is it not yet a question? Because it's a good if you have it probably someone else does no we're good figured it out Okay, he's not as dumb as you Sorry that came out wrong number nine Yet another reciprocal Okay, what's the first thing I'm gonna do Shannon? Darn right one high there one high there one high there. Well negative one high there negative one high there What's the second thing I'm gonna do gonna be an asymptote right there and a vertical asymptote right there And it looks like I've divided this graph into a bunch of sections Let's start here To the right is kind of easy. We've done that one so much closer to zero shoots off to infinity Now to the left I'm getting further from zero. I'm gonna get closer, but I do level out. How high am I right here Shannon What's there a reciprocal of negative two? Yes, it is all you're doing is flipping it right the sign stays the same you just it's really negative two over one negative one over two and Because this height is negative a half and this height is negative a half and this height is negative a half and this Height is negative a half really what I would be looking for would be a Horizontal line and you could freehand it. I'll use a ruler though I'd be looking for all that horizontal line becomes a horizontal line and I think it would curve like that. It would be getting closer to that. Why didn't I put an arrow on the end here? Because my original graph stops there too and I went right to there and quit Don't you dare go one further because the original graph doesn't exist there. There's no reciprocal height there I think this Shannon is almost identical to the previous one, but we were down. You know what? It's gonna go further from zero further from zero to high one half high It's gonna do that. In fact, it is an exact identical shape to the previous graph Here further from zero Closer to zero but too high. So we're gonna end up with a horizontal line coming to a stop This one would be an interesting one to do This graph But with a two right there Because then instead of a half high, you know what this horizontal line would end up as Negative one high and negative one high it'd be it'd be easier to draw almost actually because you're not having to freehand in Midair, I don't know. It'd be a vertical stretch by factor of two next one okay Absolute value with a twist. I see three things. I see absolute value. Yes, I See a two in front of an X. Is that horizontal or vertical? Horizontal by a half or by two by a half. I see horizontal compression by half and I see a plus one That's gonna be one Okay, what's the correct order? Yeah Yes compression a horizontal Compression by a half absolute value One up you treat the absolute value like it was a bracket and you do good old bed mass. What's inside the brackets first so Let's try a few of these I'll do this point here starting at negative six Horizontally compressed by a half. So instead of negative six to the left. Where will it end up? negative three to the left Absolute value what's its height right now negative two. What's the absolute value of negative two? Two and then one up. You know what? That point ends up there I think the next point I would do would be this negative four I'd skip the negative five because it's compressing it by a half is going to give me a decimal But this one here that's negative four left Compress it It's gonna end up at negative two absolute value Positive two one up. Oh, you know what it's gonna give me a horror. Those guys are the same height They end up the same height. I think the next point Tyson. I would do would be this guy Carson Compress it by a half. So instead of two left. How far will it be one left absolute value instead of negative two down? It'll be and Okay, what would be another good point to pick zero zero Horizontally compress it by a half nothing absolute value nothing one up right there. How much you want to bet it does that? I think I would pick two two Horizontally compress One two Absolute value stays where it is one up. What would be another smart point to pick four zero? Compress to zero absolute value nothing one up You know Well be another smart point to pick About six negative two compress Absolute value One up. Oh, it's weird. Suddenly it goes shooting back up there And then I would also do eight negative two compress negative four absolute value one up That w shape surprised me. I did not see that coming out of here at all When you have absolute value with a twist, it's much more difficult You don't keep the same shape often you'll get different Corners appearing that weren't there before Is that okay? Good. Thank you. Happy joy Good. Let's keep going so much fun I'm getting this is all like oh all the weird ones, but they're making sense now, Mr. Do it I'm not gonna do number 11, but can you tell me if they asked you this? What would you do? What two transformations are going on here? I'm looking at this is the plus one in the bottom of the reciprocal. I think I do that first if they want to be to do Reciprocal first and then one up they would write it this way That's well make that look like a one. That's reciprocal and then one up I think this one would be one up and then reciprocal It would be a bit interesting because This line here and this line here when they're raised one up would be at negative one They would become completely invariant lines. I would probably really lightly sketch the whole graph one up And then take the reciprocal of that But I want to do more so there's number 11 you can try that if you're bored. I don't have an answer key for this though Let's look at some multiple-choice questions if a comma b is the point on it y equals f of x Determine a point on that there What does this mean? What does this mean? What does this mean? Three up so if I have a comma b When I move to right, how do I calculate that? What do I actually do to this particular coordinate? Well, I'll give you a hint. It's one of those two Which is correct? Do I plus two or do I minus two to go to right? I Plus two on the graph. It's minus two in the function. It's plus two on the graph It's so I would do that and that right away What about to go three up? Do I minus three or do I plus three? number 13 two functions are graphed below the original and This one here where it looks like they've done a horizontal stretch or shrink and a horizontal slide Now glance at your answers first of all for a They haven't done a stretch or a shrink. What have they really done for a It's a reflection or not. Can you tell me look at this? Look at this has it been reflected or not? Hopefully you can see absolutely because this little thing was on the right now. It's on the left That means those two are wrong you want to get in the habit on multiple-choice questions You always look at the answers right away always especially in math You can almost always process of elimination Even if you don't know what to do now now you've turned it into a true false question because there's only two answers to guess from you Got a 1550 chance of guessing right Always look at the multiple-choice answers. They tell you a lot okay, so They would have reflected it this negative four would have ended up at a positive four right there But I noticed it actually ended up right there See it So did it get moved to left or to right? To left okay, so if we move it to left that means it must be that To left so here's the question careful now careful now What would I put in for B so that I ended up with an X plus two? The fact that there's a minus sitting there means I actually want to put in for B negative two because that would give me a minus minus, which is what Carson a plus a Lot of kids on this one said oh, it's too left B is negative Sorry, it's too left B is too because plus two They did that won't know they got a minus there sneaky beggars and I want a plus two B must be a negative two so I got a minus minus is a plus. It's That's wrong. It's the correct A is the correct answer tricky question sneaky question Number 14 How many questions that I put on this review could someone tell me? What's the last question number? Sorry? 21 okay. Thank you. Just kind of paste myself here Okay Whenever they want me to find an equation like in number 14 Shannon I always fall back to my listing the replacements So if I reflect in the y-axis now the y-axis is this one if I reflect in the y-axis Oh my chair is not working No We'll hire oh there we go if I reflect in the y-axis. This is a vertical or horizontal reflection if I reflect in the y-axis Replace what with what it's right that down replace x with negative x Now a lot of students picked B You're gonna see in just a second that B is incorrect because if I actually do the replacements. I'll get this I Would cross this out. It's not a reciprocal. I would cross this out. It's not an inverse How do you know that's an inverse because I'm pretty sure they switched the x and y's around now? I don't see that. Oh, they simplified they got rid of brackets. It's so what's a negative cubed? So this would simplify to a negative in front. Ah But what's a negative squared? Here for the second one the negative vanishes a lot of students picked this one because they said oh you're placed all the x's with negatives No, no, no, you replace all the x's with negative x's but you have to then do the mathematical operation turns out the correct answer is a number 15 Which of these represents a reflection of the graph of y of that thing? If you reflect in the y-axis, so we're reflecting in the y-axis Is that a horizontal or a vertical reflection? It's my favorite it's horizontal So we're replacing x with negative x. It's gonna be five minus negative x equals to y squared Plus y What's the correct answer? By the way, I would have said that's wrong. They switched the x and y around love inverse Inverse is a reflection. What line does the inverse drift about you remember? Might want to know that line y equals x it reflects about the line y equals x. Oh, yeah 16 I Guarantee on your test. I'm gonna give you a question where I give you a point and Give you a transformation and say what's the new coordinates? The way to handle that is first of all make your list so What's going on here in the correct order, please? What have they done first? What have they done first? I disagree I Think they've done absolute value first because there's nothing else in the brackets. So it's gonna be Absolute value that's the first thing they did Then what did they do now? We'll go in the correct order Andrew vertical Expansion by three then what? one up and The way I tend to do these once I've made my list Sabrina is I write the point nice and big and I'm just going to cross numbers out and work my way outwards in either direction in other words I would say absolute value that's going to become a six Vertical expansion that's going to become an 18 One up That's gonna become a 19 If they're getting a horizontal I would have just continued working this way if they've been a horizontal I didn't see a horizontal thing is negative 319 an option to pick There it is By the way, one of these would also be if you did three up expand vertical expansion by three first and then absolute value Is that okay Steph? So I'm gonna give you one where I give you a point and an equation I find doing it this way and just working your way outwards crossing out the numbers as you go Easiest way least amount of work, but most amount of checking your answer which equation. Oh, they want me to find an equation I'm gonna go replacements After it's compressed horizontally by a factor of a half Horizontally means I'm replacing X by a factor of a half. I'm replacing X with what? to X look up That's wrong and that's wrong because here they replaced X with a half X and a half X there I just got rid of half my wrong answers Even if I don't know anything else I can guess that I got a fifty fifty chance Then it translated four units right replace X with what X minus four So in function notation it would look like this Y equals F of Two X and then that would become Y equals F of two replace the X with an X minus four Do you see that answer there anywhere? Do you which one I don't see it there blatantly I got to do a bit of right it's not be as close, but not What did they do oh they got rid of brackets and if you get rid of brackets, what is to bracket X minus four the same as ah Those sneaky beggars a They wrote it in the way that I don't like it. They're not to number 18 If negative six comma twelve is a point on Y equals F of X what must be a point on that monstrosity Okay Learn Bell you see it you see it you see it I First of all Kirsten need to rewrite this This here needs to become Y equals. I'm okay with the negative a half. I'm okay with the F I'm okay with the negative two. I don't like that I need to rewrite this as X What would go here so with a negative two there I get a positive six there Minus three so when I foil it out. I'm good Normally I'd make a list we're running out of let's see if we can do this in the correct order without actually writing the list Just by writing the point so the original point is negative six comma twelve Expansion compressions. What do you see? What's this do? vertical or horizontal Y By a half or by two That 12 is going to become a six What's this do? vertical or horizontal Horizontal by a half or by two By a half that negative six is going to become a negative three Reflections. Oh Vertical reflection this six is going to become a what I'll just put a little negative in front of it. Whoo. Oh Reflections this three is going to become a what? Don't three what right? So instead of positive three where will we be? Positive six if we move three right Right five what five down so instead of negative six negative eleven this will end up at six comma negative eleven Six comma There's my new point so we can do them Mathematically without a graph paper or if we have graph paper we can do the hover pencil method You're gonna make it. I know I've been talking is this helping. I hope though I Figured this is my weird questions day If X comma Y is a point on that what must be a point on that? No Okay, so in math 12, how can we take an easier topic and make it tougher? They love to give you a question with instead of numbers. They'll just put algebraic variables in there They'll put algebraic variables in there look up vertical expansion by W Right horizontal Compression by one over V if there was a three there It'd be one third if there was a two there it'd be one over two if there's a V there It's one over V I'm gonna freak out. I'm gonna deal with this M right and Okay Up step says looks like you wrote mr. Right mr. Do it and that's what I'm dreaming about no relax so Here is the point X comma Y by the way, this will be considered an a-level quite this top Vertically expand by W. What would I do to that Y coordinate with whatever number was there? Multiply it so I'm gonna just write W times Y horizontally compressed by one over V I Think compressing it's the same as Dividing X by V. That's your horizontal compression Right, but was one half you're really dividing by two was one thirty really by by three M right X over V plus M K up W I Plus K want the coordinates the coordinates of this point would be X over V Plus M comma W I plus K now if you stop and think about it Ryan I didn't do a single new thing there. I thought outside of the box. I was stubborn and clever But if they give me letters, it's gotta be the same rules. I'll be a bit more paranoid a bit more careful Gotta follow the same rules. I like that question. I like that question. I like that question It was a great drink bottle, but it's weeks Don't ask me why but it worked great number 20 number 20 Given that find that oh, what is that stupid symbol mean reciprocal? Oh, no, what is it little negative one mean? How do I find an inverse? First thing I would do is I would go. Oh, okay X equals 2 y plus 1 over 3 y oh I like this question. I like this question. I like this question. This is a nice question Didn't say it was gonna be on the test currently your good student. We just know this is what I'm not done I need to get the Y by itself Well, I Got fractions. What grade 8 mathematical operation could I pull on my back pocket that would make this much nicer? I Can cross multiply because technically this is over one. So if I cross multiply I'll get 3 x y equals 2 y plus 1 Now I want to get the Y's by itself problem if I get this Y by itself I'll have this Y sitting on the other side if I get this Y by itself off this Y sitting on the other side Solution Get all the Y's to the same side first I'm gonna minus 2 y from both sides. Okay I'll get 3 x y minus 2 y equals 1. How many Y's do I have on the left there? To how many would I prefer? 1 It would be wonderful if there was some grade 9 Mathematical operation that I could pull out of my back pocket that would help me turn that monstrosity from 2 y's into 1 y and There is I can factor Really? Yeah You know what the first thing that I always always always always always always always always always always Always always always always look for when I factor GCF there is one. What is the greatest common factor on the left-hand side there? Because I asked you to what is the greatest common factor on the left-hand side there? Why because I asked you I'll never get tired of that joke ever Amanda what is the greatest common factor on the left-hand side? If I factor a Y out, I'll have a 3 x minus 2 left behind. Yes Katie how many Y's do I have on the left-hand side now? Oh Can I get the Y by itself? Sure. There's a big bracket next to it What's the Y doing to that great big bracket mathematically? Times things so how would I move that great big bracket over divide? Turns out this question is math 8 and math 9 Turns out kids leave this question and freak out and don't know what to do it. It drives me crazy Why ends up being 1 over? 3 x minus 2 that's the inverse of the original equation math 8 math 9 Math 12 I like number 21. I like number 21. I like number 21 Says if the range of my original function is that Between negative 8 and positive 6. You know what did they tell me what the original function was? Say no I'm gonna draw the simplest possible example. I'm gonna draw a graph and I'm gonna go there's negative 8 There's positive 6. There's the range It might have squiggles and curves and I don't care. I'm just trying to visualize. I made a little slanty lot there What's the range of that? What's going on here? What's going on here? Oh, are you saying this is a vertical? Expansion by 2 now is the word range also vertical? Ah So will a vertical expansion affect the range? Yeah how I Think it's going to be negative 16 less than or equal to x less than or equal to 12 this would double and that would double yes And everything in between would also double and since they were all connected originally ask Because I'm stupid and I missed my why cuz I Missed out on opportunity Thank you, Amanda candy for you later. Is that okay the really nasty one our BCD and E But I'm not gonna get to those right now I'm gonna talk about those. Can you remind me next class? I'm gonna stop the lesson because here's your homework