one all right so ladies and gentlemen okay so we're gonna have a massive tripping hazard and that's just gonna have to be that all right so ladies and gentlemen uh we're gonna be covering transformations and polynomials in a summary so just blank paper okay now if you're like no no i understand everything i just want to work on my own and feel free to work on your teamwork exam review package but i will be doing this for most of the units just to be on the safe side okay yes sir sorry oh yeah they're quizzes you keep them okay i mean i'm not gonna recognize it otherwise okay so ladies and gentlemen you got this okay to start with transformations we're going to do a very quick diploma style review for it basically what i do i do this exceptionally class so all transformations came down to this formula y equals a f of big square brackets b x minus h plus k all right now let's see what everyone remembers okay especially sleepyheads everybody ready okay here we go what is a control what is a controller oh you can show me you can see it what do you think there it is it is your vertical stretch okay so that means your y coordinates can get larger or they can get smaller right larger or smaller everybody good now let's throw an extra detail in there before i start throwing a pencil in the sleepy head okay so what if this was negative what if that a value was negative show me or tell me show me or tell me there it is what do we call that we call that reflectionware nailed it reflection x-axis okay all right now next component e okay what the heck is b what hack is b it's a horizontal stretch there you go horizontal stretch okay what if the b was negative what if the b was negative come on show me beyonce show me beyonce put a ring on it there it is reflection where the white nailed it reflection holds reflection y axis okay everybody good it's a joy all right now let's get into the one which you guys all just seem to slow up on for some reason h what does h do hey oh this class was quick the other ones have to think that's nice this is a horizontal translation easy it's recording okay they're that good all right horizontal translations okay all right so it moves it left or right and what does the k do there it is show me your tell me there you go k is your vertical translation okay so this is basically uh one two three four five six seven eight nine ten of the dirty dozen rules okay ten of the dirty dozen rules now there's a couple of details that i do want to bring up notice that these elements were found where inside of brackets or outside of brackets inside and so what's the rule about everything inside brackets not only lie but it controls your yes it's your horizontal line it controls your x and brackets lies so if this says x minus 2 it's actually going to be that's right 2 to the right and if this said 2 for your b value it would actually be one half 1 over 2. everyone remembers that good talk okay versus notice how i have an a and a k on the outside everything found on the outside of brackets is vertical y value okay so which is why we always get down to the point where we say is it in brackets out of brackets vertical or horizontal okay does that make sense okay so i'd love to say that there was more to this but there isn't so let's put you guys through the ringer by giving you a lovely little example so i do this all the time during preps and you guys gonna see it again when you do the pull press oh by the way everyone knows that pulling preps in this school are mandatory yeah they are no no lizzy that doesn't mean you're paying for it it's free here because this is public education and apparently i'm free here i still have a serious issue with that the joker said if you're good at something you never do it for free so ladies and gentlemen what we're going to do is we're going to deal with truth versus math okay truth versus now now what you're going to do is you're going to create a lovely little chart like so okay and hang on in this chart there's going to be subcategories for each one so there and there you guys are going to do a little activity okay it's small and i'm not going to count the grades and i'm not going to call on anyone you're still going to do this right okay now there is two sections of mouth or two sections of transformation we call it truth and we call it math now in the truth section we have descriptions okay so horizontal translation of two units right that's a description that is telling you the truth it is not some sort of crazy thing the guy says walk four steps that way and then takes two steps to the right you're good right like if i said right now luke take four steps to the left or sorry four steps to the right no take four steps to the right well i want you out of the building i thought you were taking a hint you're gonna go right through the wall let's go i thought you were a football player just go right through it okay so ladies and gentlemen we do that description is telling you the truth now what else is telling you the truth the whole truth and nothing but the truth mapping brilliant okay mapping is always written as a coordinate not just x not just y the pair okay it's like a pair of inseparable twins that just hate each other but somehow are always together you guys know the tips okay all right versus math now math is not necessarily telling you the truth but it is being applied mathematically so something for this would be replacement notation okay replacement notation is now okay another thing would be our formula now are there details in our formula that are telling us the truth yes but are there details in our formula that are lying yes but that's why it's under math because we can prove it mathematically okay so what we're going to do is we're going to do a little activity so i'm going to show you how this is set up and then you're going to try out three on your own everybody ready all right ready so first things first description let's say we had a yeah vertical stretch of four okay vertical stretch of four now again on the deployment we would say about the x-axis i'm doing shorthand today because i just don't have time okay what would that look like in mapping notation vertical stretch of four would anything happen to the x no would anything happen to the y yeah it would become c 4 y perfect does that make sense okay all right replacement notation we're gonna deal with that in just a second actually we'll deal with that now okay so replacement do you guys remember what replacement was okay replacement is literally we are taking our formula y equals f of x and we are replacing something in it like a true replacement i cannot put 4y in there because this is the truth so i would put y changes to 1 over 4y and let me show you why just wait okay so this is what i advise people have an issue with replacement notation i advise that you write it in the formula first so in our formula we know if we were doing a vertical stretch of four it would be 4 f x but in mapping or sorry replacement notation we would put the 4 over with the y but what would it look like if i moved it mathematically over to the other side 1 4 y so that's why this is actually what it would be in our formula and then i always ask you guys do you like the way it looks and what's the response you always say no so then you change it to this format okay does that make sense so this is math it may not always tell you the truth but it is the math component okay so you're ready to try three of these on your own so i'm going to fill in certain categories and you're going to fill in the rest and then we'll take it up are we ready okay here we go so let's go with under mapping let's do one half x plus two and then y minus four okay um next category we'll put it under replacement replace it we'll have it as x change to x minus three and y change to y plus five okay and then for our last one just for giggles um gamble you shouldn't laugh like that that's your response so this last one here's our formula y equals negative f of negative bracket x minus six closed bracket close bracket okay so you have to fill in the other details question yes i can move this back okay sorry go ahead so above the y to y plus five this is above it one over what does that say no that's an arrow i will calibrate the board okay all right so you're gonna fill in all the missing categories for those good right give it a shot give it a shot okay everyone has those written because i gotta calibrate the board real quick somehow it won't yes foreign permission to start on the first line just the first one you okay with that if you're still working keep working and i just want to do the first one okay so this is we're really time crunched today so i'm gonna use shorthand so we'll get a horizontal stretch factor of one half we had a horizontal translation of two right and a vertical translation of four down okay i'm just using shorthand that's it okay um replacement notation wise uh x changed to 2x x also changed to x minus two and then y changed to y plus four so this is actually what it should look like y plus four equals f of two x minus two i don't like the way that looks so i rewrite it as y equals f of 2 x minus 2 minus 4. now again i'm not too terribly worried about the replacement notation i'm mostly concerned about the formula in proper form so people get the formula in proper form great that's what i'm mostly concerned about okay all right now we're gonna go to this one here go to the next line okay so x changed to x minus three which means that we had a horizontal translation of three right we also had a vertical translation of five down okay because this is what it looked like in the formula y plus five equals f of x minus three i don't like the way that looks so y equals f of x minus three minus five okay everybody good so the mapping is what i care about the most brackets uh since we had three right it's gonna be x plus three five down y minus five okay good the last one was just a feast of pain so um we'll deal with the replacement last just because i really don't want to deal with it now so we had a reflection we had a reflection and we had a horizontal translation okay so for the last one we had reflection in the x-axis we had a reflection y-axis what else do we have x minus six so we had a horizontal translation of six right good all right so mapping notation that means we had negative x and then plus six and then we just had negative y yeah okay so the replacement on this y changes to negative y x changes to negative x and then x also changes to x minus six okay so again to me the priorities are formula mapping description did we have any hiccups in those ones okay if you do get a replacement notation thing it would say okay x change to this everybody sees that in which case you just put it into your formula and then you can start working with it does that make sense okay questions all right so that basically covers the bulk of the unit but now we're going to get into the fringe elements there was three fringe elements are we ready good talk okay and then there's one which we absolutely have to cover at the bearing okay um first one order of transformation all right so i want you to be asked a diploma style question first okay here's our equation our original y equals f of x okay all right so ladies and gentlemen boys and girls here we go if we had this y equals 3 f of x minus two all right where's the three found outside of brackets right so is that horizontal or vertical vertical x minus two where's that found in brackets was that horizontal or vertical so here's my question does order really count on that one does it count no no here's the reason why one's affecting the y value the other one's affecting the so does order really matter no okay it doesn't really i mean you could still you'll still go in order because you guys are just trained that way but it doesn't really matter versus this one y equals five f of x plus two where are both numbers found outside the brackets so are they affecting the same letter does order count on that one there you go okay and they have asked this before where they said which one of these would the order of transformations not matter so i just want you to be aware of that if one's operating on a vertical and the other one's operating on horizontal or it doesn't really matter okay but this one order counts and yeah that means you're always going to go on a certain order you remember the order srt did anybody ever go rst okay so we're going to write them down so rst or srt okay either way what has to come last translation you got does that make sense good talk okay so now that we have these two done okay that's order of transformations there's really not much else i can talk to in regards to that so our next element of key importance is this what are these these are inverse now notice i did not say inverse i just showed the symbols okay ladies and gentlemen symbols x sorry inverse means here's x here's why what do they do they switch x and y just switch okay so this is the key here since x and y essentially switch i'll move this up so x and y becomes y and x okay and that means your x and y intercepts also switch your domain and range also okay so let's play a little again i'm going to put down a graph let's give you a diploma style graph one two three it's not proportional i know campbell's gonna have a fit but you know what it's not okay proportional turn it right first coordinate down here is negative two two i'm gonna call that coordinate one okay next coordinate over here we're gonna say is negative two positive one let's call that coordinate two last coordinate over here we're gonna say is three positive one and that's coordinate three okay everybody sees it okay ladies and gentlemen nope wrong one sorry sorry okay i want these done to those coordinates what's coordinate one going to look like what's coordinate one gonna look like yeah negative two negative two that didn't change what do we call there it is it's an invariant point okay what does coordinate two look like one negative two perfect and what does coordinate three look like one three okay everybody good now where is this going to be something that's used potentially for diplomas right here okay let's start with the original domain okay what is our original domain where does it go from and two where does it go from and two goes from negative two up to go screen three there we go and our original range goes from negative two up to so the question is gonna say give me the inverse domain and range go you got five seconds to think about it okay give me the inverse domain and range time all right ladies and gentlemen the inverse domain and range without graphing what's our new domain going to be negative two and one what's our new range gonna be negative two three so what did you do with the original ones you just switched them so did we even need to draw the new graph or get the new coordinates no that's the point okay take it from a 20-minute question to less than a 30-second question all right so the last one that we're going to talk about before we go to polynomials is this this equation is y equals x squared plus three what restriction on the original domain must be used to make the inverse a there we go x would have to be greater than or equal to zero or x would have to be less than or equal to zero does everyone understand why be honest okay let's just swap where this thing is you see this coordinates that's 0 3 right so the new chord is going to be at 3 0 and it's going to open like this is that a function no the only way i could make it a function was if i canceled half of it okay now here's my question if i cancel half of that and i take a restriction on this to cancel half of it did i cancel a domain or did i cancel a range on this one here on the red on the red i canceled eight range part i said y must be greater than or equal to zero agreed but that's on my inverse what would range become on my original the domain x is greater than or equal to zero that's it does that make sense people if that's not in your diploma i'm gonna be shocked absolutely shocked okay like literally aghast like walking in and seeing campbell in class on time shocked okay come on man i've been a great 12 son yeah everybody good all righty calm down mlb you're having so much fun with that okay all righty so polynomials yes no yes no any lingering questions on some of the transformation stuff like that yeah remember the written response i remember everything unfortunately the one you're ever how many people care not one that's awful um so rosie that was a standard wrestler's question it has never been asked on a diploma will we see it again most likely not but can i can you do me a favor can you wait until monday okay now the reason i'm saying wait until monday if it is coming up again on the queue then that means it's fresh okay but the other reason i don't have the time to drink today okay so i do appreciate the patience on that so yes we can cover that one um i'll make one up that's decent okay and we'll go from there okay all right holly wow no yes i did spell it right the secondary i almost have it okay polynomials you guys know this is just split into two parts right there's the algebra yeah that's it it's just the algebra and the graphs okay that's it algebra and the graphs so what do you want to start with first the algebra the graphs graphs sure okay all right so the graphs have three families or three parts to it okay so let's start with the graphs the first part well let's just do an arrow diagram you guys are still getting used to it the first part was all the vocab all the vocab and the references that we have the second part was everything that had to do with expanded form and the third part was everything that had to deal with factored form and that's quite literally all the graphs so let's go through the details so we're going to start with the one that counts the most actually we're going to start with the one that counts the least but it's essential vocab okay because i know you people like writing down vocabulary yes that's right okay pretty sure she didn't fall asleep all right so vocab here we go first thing the degree does everyone remember how to get the degree of a graph it is the highest total power when expanded right that's it the next thing we had was thank you what is another word we could use for constants oh you guys are good the class blanked out on that a little disappointed okay now we also have another key one and it's not something that's going to be directly tested but something which is going to have an impact on your graphs and that is the lead coefficient right because that makes a big difference if the graph is positive or if the graph is negative right that does make a big difference all right so let's make sure that you guys have this down so i'm going to come up with an equation real quick uh let's go with 3x squared minus 6x cubed plus 4x minus 16. okay so yeah it's not written in order and i'm never going to give it to you in order if i'm asking a vocab question so ladies and gentlemen right away what is the degree of this graph what's the degree it is 3. everyone sees that what is our constant for this graph negative 16 very good and i don't know if i'm going to say constant or y-intercept okay and what's the leading coefficient it is negative negative 6. now ladies and gentlemen is this an even or an odd degree this is an odd degree and the coefficient was negative now if it's a negative leading coefficient that's going to control which arm if it's a negative leading coefficient the right arm if it's negative that means it's doing what it's going down if it's odd that means the arms go in opposite directions which brings up our last vocab point and behavior on this particular graph because it was negative this arm went down this arm went up so we went from quadrant two to quadrant make sense okay that's the reason why we dance despite aiden's protests yeah that's the reason we dance everybody good okay all right expanded form well this is expanded form what are some of the benefits of expanded form what are some of the benefits of expanded form just go for it yeah the degree you can easily see what the degree is what else y intercept clearly see what the y intercept or constant is what else leading coefficient spectacular okay well one of the disadvantages of it is can you see the true shapes of all the x-intercepts on can you see all the x-intercepts on no so that brings up factored form okay so factored form and drop that right down is something that looks like this uh let's go with x squared times x minus two times x minus four cubed times x minus two plus two squared okay all right so this brings up another key vocabulary term called blessing multiplicity do you guys remember what multiplicity is it is factored form the power that's attached to each one of the factors okay so this very first one is multiplicity one two now we're gonna draw the shapes of what this is gonna look like you guys remember what the name of that is it is a tangent yeah and you could say parabola shape that's all right okay they might use tangent just in case so it's just that touch and go all right next one x minus two what's the multiplicity just one and that means it's going to go right through just like that okay all right next one x minus four cubed what's that going to be yeah multiplicity three so it's going to do that little panic that little stall do you guys remember what we call that point of inflection yes okay point of inflection and then the last one x minus x plus two there you go just that shape okay okay so everyone remembers those now again these are the lowest multiplicities this is odd this is even okay now a great question came up from the other class i don't understand the wording when it said describe the graph with the lowest possible multiplicity okay so i'm going to draw an example here real quick pretty good okay so lowest possible multiplicity so let's deal with this and it touched okay everybody sees it what's the lowest multiplicity for that one what's the lowest multiplicity there so this might say the lowest degree of this equation could be five right because you add them up does that make sense so that's what they mean because remember in 30-1 we don't really go past two and we don't really go past three i do instruct this that this is even and this one is odd multiples right so that's all i wanted to bring up in regards to that if it says the lowest possible that's just diploma terminology okay all right so that was basically the graphing elements we talked about leading coefficients and everything else the only thing we did not discuss is what if i gave you this and i told you the x intercept here was negative four and the x intercept here was five and then i told you i gave you a point here of zero 25 remember what to do okay so you see how i gave you two forms factored we write this as y equals a x plus four squared x minus 5 cubed and then let's do a constant check what's 4 squared 16. what's negative 5 cubed like 125 125 times 16 is that going to give me 25 that's why we do a constant check we're going to need to put an a value in here so do you remember what we plug in the zero goes where the x x's and the 25 goes where the y's okay so you guys are actually really good at doing this the only thing that caught few people up is they forgot the multiplicity's attached okay does that make sense so i then had a request in the other class to go over the last little bit of vocab what are all the names for all the powers that i can ask you okay so do you guys want that as well okay x to the power of one is linear x to the power of 2 is quadratic okay x to the power of 3 is cubic x to the power of 4 is cortic x to the power of 5 is quintic x to the power of 6 is hexagon that is as far as we can go okay we can go further but i can't assess you on that okay so that was the graphing component okay all right so the last thing we're gonna do in the last like five minutes is quickly the algebra component all right already good wait let me just non-real zeros if i could test you on it i would go over it but i can't test you so a non-real zero is something that takes on this so you see how it creates that churning shape i cannot test you on that all you can say is that's a non-real zero that's it okay that's all we can do in regards to a good question all right so ladies and gentlemen let's deal with the algebra i don't know if i can actually come up with an equation it's going to be decent all right so the algebra basically had uh two parts let's be honest division and remainder theorem or factor two okay so division do you need to do long division on your test diplomas no can you just do synthetic division i'm encouraging that so let's say you have an equation i don't know if this is going to work x cubed minus 5x squared minus 14x plus 10. actually no that's wrong shouldn't it minus 10. there we go okay 99 certain this is going to work okay all right so your very first thing you want to do make sure your x's are in order and that they're all there are they all there no what's missing 0x so let's rewrite that x cubed minus 5x squared plus 0x minus 10. we good okay so ladies and gentlemen we're going to write down this little synthetic division line okay so we're going to put in our coefficients 1 negative 5 zero negative ten now we are always going to ask for this algebra always sometimes okay three minutes okay we always make sure that we fill in all the details here and to make sure that we do this algebraically look at your constant for the close yeah the constant and we want to find the potential there it is potential zeros and that's okay that's all right okay so we just list them real quick positive negative one positive negative two positive negative five positive negative ten good okay once we have that then you can graph it no one's gonna try ah you know what that's fine that's okay let's just try out two we'll just try out two because that'll actually help us to another question okay all right so here we go oh so two times nothing is nothing one and plus nothing one plus nothing is one two times one is two negative five and three negative five and two is negative three two times negative three negative six zero negative six negative six two times negative six negative ten plus negative twelve okay so basically this brings us down to 1 x squared because it was cubed minus 3x minus 6. what's this negative 22 a remainder now that remainder is telling me that it is or isn't that's right because it's not zero it is not a factor which brings us to our next component remainder theorem is going to have a remainder of some type of number the factor theorem is always always always going to have a remainder of zero so just to wrap this up how we use this is if i said here's your equation x cubed minus six x plus four and i said the room if i said it's being divided by x plus two what's the remainder what would i plug into all of these x's negative two two cubed minus six times negative two plus four no freaking one no matter actually worked okay no it didn't i killed that okay negative eight uh minus no plus 12 plus four okay good so that's going to give us eight is that a factor why is it not a factor it's eight okay it's not zero that was polynomials okay so we did not have time for radicals uh if you do need a recap of radicals i can do a quick on monday but we're gonna have to learn the next lesson almost okay all right everybody good okay were there any questions online questions from people watching online what happened we're starting a new lesson on monday okay so people just tuning in or people who are streaming live we're starting a new lesson on monday i will post the notes on d2l you can print them off or if you know someone at the school or have someone who could drop by the school they will pick they will give you the little package okay all right so people online are you good any last questions online you guys can unmute yourself all righty okay so i'm gonna start