which is graphs and transformations on your screen you can see a person and I'm sure you know him he is Mr. Albert Einstein okay and this great scientist of all time says one picture is worth a thousand words and when Mr. Einstein says something definitely it has to be carrying a lot of weightage so the reason why I kept this quotation of Mr. Albert Einstein is because graph is nothing but a picture and indeed that picture will tell you a thousand words or in fact will tell you everything about the characteristic of whatever relation it is trying to depict okay so graphs and this transformation just to highlight graphs and its transformations how useful a chapter it is now one thing I would like to tell you here this is not a direct chapter from your textbooks okay if you let's say you know open a Adi Sharma book or let's say any other Jee book and if you turn around you will not find this chapter you know as it is graph is an integral part of you know almost all the chapters of mathematics if you open like mostly 90% of the topics that you open you'll find that there is a presence of graph somewhere most of us think that graph is an optional way to solve questions no sometimes it can be the only way to solve questions okay so there are some chapters which are heavily dependent upon you knowing the graphs so if you know the graph then only you'll be able to solve those chapters so I'll list down the chapters you know I will not talk much about them because there's something that you're going to see in your two years of study with us so the very first chapter that we are going to use graphs is relations and functions relations and functions so if you turn your any standard class 11th book this is normally the second chapter after sets relations and functions is a chapter where you will be using graphs to depict relations and a certain type of relations which we call functions so relations and functions will be the first chapter where you'll be seeing it okay next you'll be seeing this in a very very big chapter called trigonometry let me tell you trigonometry is a huge chapter for you in class 11th when I say huge it almost takes me two months to complete trigonometry okay because signometry is just not about identities there are so many things associated there will be trigonometric equations there will be properties and solutions of triangles there would be you know a lot of concepts which are basically you know based on trigonometry in class 12th also you will learn inverse trigonometric functions so all of them would require you the knowledge of graphs okay apart from this just to speak in you can say a nutshell calculus what is calculus calculus is basically a vertical of mathematics where we deal with small changes or what we call as infinitesimal changes okay now this changes happens where this changes happens in the relationship between two or more variables so in order to know what is the relationship between the changes in these two variables you should know how they are linked graphically to each other also okay so calculus there are several topics I would not like to name them be it you know limits be it continuity be it differentiability be it area under curves all of them would require you to know your graph very very well okay that's the main purpose why I'm picking this chapter and not to say the least one of the very interesting chapters called theory of equations theory of equations so where you need to know how many solutions are possible to a certain equation okay graphs can be very very handy almost in let's say I talk about J main and J advance exam there are almost 60 questions which actually you know coming from theory of equations where you can apply graphs okay so with this I would just begin the chapter so this is just to tell you the utility of the concept that we are going to study so these are this concept of graph and its transformation will be useful in these chapters okay I don't know that I know you to note it down it's just for your understanding how important is this chapter for us so we are going to start with what is a graph first of all okay let's let's define graph in a you know very very generic way what is the meaning of a graph can anybody you know explain what is the meaning of a graph in your own words let it be right let it be wrong no problem try it out what do you think is a graph Rangini would you like to try representation of data very good okay anybody else good try Ria so I don't want to name people around so Adya is there Arun Dutty Avani Janya Karthik okay Siddharth very good okay in a nutshell just to define it graph is nothing but it's a pictorial representation or you can say a geometrical representation a pictorial representation of a relation between a relation between or a relation among several variables several variables okay so we know in our physical world every quantity is somehow linked to the other isn't it your marks is directly linked to how many as you are studying isn't it the price of food is directly linked to the price of fuel okay so lately you would be hearing a plot of protests going against the government for you know increasing the price of you know the fuel why because the farmers they use fuel in operating the machinery for cultivating their crops and if you increase the price of fuel obviously the cost you know incurred by them to cultivate the crop goes high and if the crop the price to cultivate the crop becomes higher even the price of the food material that we are consuming will go higher correct so things are linked to each other okay so when several parameters or several variables are linked to each other to show the pictorial representation of how they are linked we would require the use of graph okay so this is what I have written relation okay now for our study in this chapter graph in his transformation we will be restricting ourself only to relation between two variables okay so let's say a variable y is related to a variable x by any relation or this relation can be anything okay it could be a polynomial relation it could be a trigonometric relation it could be an exponential relation it could be a logarithmic relation it could be some special type of relation also okay so we'll come across those relations you know soon one by one in our discussion of this chapter so this basically are two variables which are related to each other so when you start you know plotting or you can say start sketching the x comma y points which are related to each other by this relation then you obtain the graph of that relation okay then you obtain the graph of that relation this is very obvious to all of us okay I'll just give you a simple example just a simple example to illustrate this let's say I have y is equal to x plus 2 right we all know that this is a straight line right this is the equation of a straight line so what is happening here y and x are related to each other by what relation that y is always two more than x okay so if x is one y will be three if x is zero y will be two if x is five y will be seven right so if you start you know putting some values of x that's a zero then y will automatically become two if you put one y will become three if you put x has two y will become four da da da da so if you start plotting them if you start plotting them you would obtain the pictorial representation of that relation which I'm sure all of you know is going to be a straight line like this right everybody knows it's going to be a straight line anybody who doesn't know even if you don't know don't worry about it there is an official chapter called straight lines coming your way in class 11 okay so let's say I just x and y okay now this relation need not be restricted only two two variables it can be three variables for that we need a 3d graph okay and multiple variables can also be involved so the dimension of the graph will accordingly increase okay but for our study we will be restricting ourselves only to only to show those pictorial representations where only two variables are involved now normally again stressing on the word normally normally the variable which we normally put without thinking anything we call that as an independent variable so here x would be called as an independent variable why we call it as an independent variable because here see while you were choosing x value as zero one two did anything govern your choice no right you just picked it up zero so let's put zero right let's put one let's put two let's put five let's put hundred so nothing was governing your choice of the value of x so that is why that variable is actually called an independent variable where you are not deciding anything prior to choose the value of that variable but y value will actually depend upon what you have chosen for x so this will become a dependent variable okay this will become a dependent variable okay so this is an independent variable this is a dependent variable because it depends upon what you have chosen for for x okay so now many people will say sir what if i express x in terms of y so y will become your independent variable and x will become dependent variable okay so they may change their positions so it's not a hard and fast rule that hey x is always independent and y is always dependent no it depends upon the choice of which variable or the choice of the value of which variable is independent that means you're not basing you're not you know deciding it based on some factor okay so is the definition of a graph clear to you what's a graph any question regarding what's a graph any any concerns you can always speak out you can unmute yourself you can talk you can write it on the chat box okay so any mode is fine with me okay so somebody has written something what if four variables yeah that's that's a subject matter that we have so there are very four dimensions also right there are four dimension graphs also available okay so you would be coming across something called quaternions quaternions are basically in four dimensions and those are very much a subject matter of study in the field of cryptography which is used in the field of computer science okay for data encryption so maths is not closed-ended my dear students maths is open-ended probably by the time your children or your grandchildren would be studying mathematics they would be studying a completely different types of topics which you would not have even heard of okay so yes there can be four variables there can be multiple variables involved fine very interesting i am sure some of you would also be interested in doing some kind of a research work in this field okay all right so what are we going to learn in this chapter we are going to learn how to draw the graph of certain well-known relations okay certain well-known relations so we'll be starting with polynomial relations or what we just called polynomials okay so let's start with polynomials now i would like to know from you since you have studied this in class 10th and since you are studying it in class 10th what do you understand from a polynomial what's a polynomial let's hear it out from you what do you know about polynomial yes anybody it's an algebraic expression so any algebraic expression arun nathi can qualify to call a polynomial no yes with at most two zeroes okay i'm not saying right or wrong to your response but good try i'm happy to see people trying see serious students let me tell you i'm a different type of a teacher i i do not judge anybody from the answer that i that you give okay for me your answer is your attempt to solve the question okay so as long as you're attempting i'm happy but if you keep quiet saying nothing then you are not doing yourself any good okay so not sure it's a function of x where x power is whole number okay algebraic expression with positive whole variable powers okay very good i'm very happy to see most of you trying your best to solve the question so a polynomial is basically an expression of this nature and it could be in any variable i'm using x for the sake of an illustration over here so a polynomial is an expression mark my words what i'm trying to say a polynomial is an expression in a single variable where where the exponent where the exponent of the variable here it is x where the exponent of the variable here it is x must be must be a whole number yes some of you have mentioned this in your definition this is very important so normally we represent polynomial with p with a bracket where we write down the variable that we have used in writing that expression so what is it it's an expression okay it's an algebraic expression in a single variable it cannot have multiple variables let me tell you that okay a polynomial you will only be in a single variable whether it is x whether it is t whether it is u whether it is w whatever variable you want to choose but it should be strictly of this nature okay as you can see it is strictly of this nature this should be the format of a polynomial where these exponents on these variables must all be whole numbers like in fact if you want i can write x to the power zero here for the sake of consistency they must all be whole numbers that means under no circumstance a polynomial should have any other structure than this and under no circumstance should any exponent on the variable should be fraction or negative integer okay that is strictly strictly prohibited in a polynomial now that everybody is consistent with the definition see why i'm illustrating so much upon it is because tomorrow you guys are going to face kvpy interview right and there you would be grilled by you know professors who have been in this field for the last 30 years and let me tell you these professors are very very meticulous about the choice of words you make so when you say something be very sure about what you're saying okay many of you use different varieties of you know words to explain things which may not be the right description of it so while you're learning be very very careful right from pronunciation to whatever expression you are you know talking about be very careful about it now this this number that you see in front of x to the power n okay this number is actually called lc okay what is lc lc means leading coefficient okay leading coefficient so this number this coefficient is a very important one and it is called the leading coefficient why does called leading coefficient for the simple reason that it leads the entire expression it is the the very first you know term that you see in that x entire expression but on a this is on a lighter note but on a serious note it is called leading coefficient because this fellow decides the way the graph would be leading or the way the graph will be heading towards anything okay so that is a very important you know you can say coefficient which decides which way the the graph should be heading towards so that's a leader just like a leader leads the way right so that leads the no way of the graph okay we'll soon realize it when we are trying to plot it so any question about polynomial so let me ask you few questions if if you don't have any so let's let me ask you this question okay let me not write example so few important questions i would like to ask so tell me let's say if i write a polynomial if i write an expression like this x to the power five minus three x to the power four plus six x minus one is this a polynomial first of all yes okay then tell me the degree of the polynomial i hope you know the degree of the polynomial just to revise for you degree of the polynomial is the exponent of or you can say it is the highest exponent of the variable present in that algebraic expression or present in that polynomial so absolutely correct yes it is a polynomial and its degree is five very good okay let me ask you another question let's say p2x is a polynomial given by one by x to the power five minus three x to the power four plus six six minus one okay so i just reciprocated whatever term you saw in p1 is it a polynomial is it a polynomial absolutely good you guys know your stuff it is not a polynomial because its structure doesn't match with this okay awesome very good okay next question next question very simple one x to the power three by two oh sorry x to the power three plus root two x plus one now this root is only on two is it on for is it a polynomial is it a polynomial very good yes it is a polynomial actually try to confuse you with this root two okay but you are you guys are smart enough okay try this out uh x to the power five by two minus three x plus five is this a polynomial is this a polynomial no it is not very good why because there is a presence of a fractional power on x that is not that is not a part of a polynomial okay actually tell me what is the degree of find the degree of find the degree of the polynomial six simple question what do you think is the degree of the polynomial six very good zero zero the answer is degree here is zero okay degree here is zero okay uh let me ask you this find the degree of the polynomial zero itself zero itself okay so i'm getting different varieties of answers half the janta is saying zero half the janta is saying not defined or undefined okay now here is the i know verdict here degree here is undefined okay why does undefined because you can write zero as zero x to the power of anything over here it doesn't matter whether you write zero one two three hundred four hundred five hundred so degree for this particular polynomial zero is not defined okay remember you have done a chapter call and in fact you are doing a chapter with here is sir vectors there is something called null vector what is the direction of a null vector can anyway tell me what is the direction of a null vector undefined okay its direction is not defined but it's still a vector okay so you must be wondering sir vector should have direction okay why is zero doesn't have a direction so these are some special cases you will always find where one of the parameters which is an integral part of it will not be defined so zero polynomial degree is not defined null vector direction is not defined later on you learn so many things related to it in complex numbers also okay so now that i'm convinced you know your polynomial stuff we will be starting with the simplest of all polynomials which is a constant polynomial okay which is a constant polynomial so a constant polynomial is basically where you have y equal to c okay or y equal to some constant okay the c can be a positive negative number anything it could be so this is a constant polynomial here you can see that there is no term in x or if you want to write it basically all the coefficients in the x are actually zero right so if you really want to write it you can write it as zero x to the power n zero x to the power n minus one da da da da and only a constant term c will be left off okay so this is called a constant polynomial so how would the graph of a constant polynomial look like very simple it just says that it doesn't matter whatever is your x your y will always be fixed at c so if you put x as one y will be still c if you put x as two y will be still c if you put x as hundred y will be still c so it basically says that no matter whatever choice of x you make okay by the way i forgot to tell you this we are going to draw these graphs on a plane which we call as the cartesian plane oh what is happening yeah so i'm going to draw all these graphs going forward on a plane which i will be calling as the cartesian plane okay so as you can see i have drawn two reference lines x axis and y axis and this plane will be called as a cartesian plane sometimes you also call it as an r square plane r square plane why it is called r square plane let me you know tell you a few things about it basically this r means real okay so whatever relations we are going to discuss or whatever graph of relations we are going to discuss those relations would be all real relations what are the what are the meaning of real relation in maths real relation means you can actually pictorially see the relation are you getting my point that means you can see the point on the graph so what type of relations we cannot see relations which involve complex quantities those kind of relations you will not be able to see on a real graph for that we need something called the non-real graph or the gossian plane so gossian plane argon plane whatever is the name given to it is something which we will talk little later on much later on in fact when we do complex numbers but as of now we will be restricting our discussion only with respect to real quantities of x and real quantities of y that is why we call such a plane as r square plane because your x is also real y is also real if you're if you had if you if you were using three reference axes that's x y and z then that plane would be called or that 3d plane would 3d space would be called as r cube space okay so we'll be restricting only to r square plane for the purpose of our bridge course chapter now can somebody tell me why it is called the cartesian plane why cartesian plane why not just in a simple plane 2d plane anybody knows after whom it has been named scientist guy named cartesian right shardili absolutely there was a famous french mathematician called rune decart note down the pronunciation rune decart okay i know in school and in so many places people mispronounce this guy named french mathematician rune decart okay so this is a french guy and he was very instrumental towards his work in analytical geometry and in order to honor his uh and honor this guy this particular r square plane was named as the cartesian plane now is this only way to represent a point in a plane okay no there are several other ways to do so you will be learning many of them in your due course of study there was there's one called the polar coordinates which was given by newton and in fact in your engineering and pure sciences course you will be coming across furthermore in fact there are more than 10 coordinate systems available to represent the position or location of a point in 2d space okay 2d 3d space okay so now coming to this particular constant polynomial as i told you whatever is the x value my y value will always be c let's say x is 0 y will be c now let's say i'm assuming my let let my c be positive okay some positive quantities in your mind think of c being some positive value so if your x is 0 y will be c if your x is 1 y will be c x is 2 y will be c x is 3 y will be c x is minus 1 y will be c so all of you by this time you would have already known that this would be nothing but a parallel line or you can say a horizontal line okay parallel to the x axis okay so just put arrows at the end of it because it keeps on going so such a polynomial is basically called a constant polynomial and graph of a constant polynomial would just be a line parallel to the x axis okay so depending upon c value it could either be above the origin this is the origin or below the origin okay now some people relate this to x equal to constant now this is what is x equal to constant x equal to constant how would this look like any idea anybody any idea anybody yeah everybody please try very good Ranjini says parallel to y axis parallel to y axis parallel to y axis okay now i'll try to confuse you a bit most of you are saying parallel to y axis so let's say my c is positive i'm just assuming it okay it could be anything positive negative it could be zero also so my graph of x equal to c would be a line parallel to the y axis absolutely correct okay now there was one student who asked me sir why x equal to c won't it be like and make a number line okay okay make a number line for x and just put a point x equal to c so won't this be the graph of x equal to c so why are you drawing us you know a line like this won't it be just a point this is a question to you from my side how will you answer this how will you justify that student what is happening is my question clear to everybody won't x equal to c be just a point on the real number line of x janta is confused okay so what happens shall we so y equal to zero for all values of x y is equal to zero for all values of x that is not justified because here let's say i can have a point of y equal to one also so c comma one can be a point on this line c comma two can be there c comma three can be there c comma minus one can be there y only c comma zero right siddharth karthik has actually given very close to the correct answer dear all the representation of a particular you can see an equation like this depends upon how many dimensions you are using to represent it okay so right now i'm just using one dimension one dimension is used here to represent the point and for that this point this line will actually become a point see all of you all of you just imagine i'm rotating my y axis like this i hope let's say look at the camera so let's say this this line this finger of mine is the x axis okay and this marker here is the y axis okay this marker here is the y axis let's say i'm rotating the marker like this okay till the marker goes inside the laptop screen or desktop screen whatever you are using okay so when you rotate your y axis let's say inside the plane of this laptop you realize that even this line will rotate and what you will see is like a dot over here so the reason why you can see a dot in case of a one dimension and a line in case of two dimensions is because it depends upon how many dimensions you are using to represent it now most of you would be surprised that what you actually thought to be a line is actually a plane don't believe me you know okay i'll show you i'll show you let me just open one of the uh graphing software that i normally use geo zebra i'll i'll talk about it geo zebra 3d let me show you okay so if i say y equal to x plus one i'm sure most of you will say said this is the line right okay it is a line because you're seeing it in two dimension the same y equal to x plus one if i type it in three dimension it will actually look like a plane so i'm showing you y equal to x plus one graph on this all of you please see very carefully i'm just putting a color to it okay so all of you please note down or please look at this particular graph if you see y equal to x plus one it has plotted a plane for it see i'm making it dance around do you see it okay it is because my dear you're using three dimension to show it in two dimension one dimension is always curved it is always killed so you only see for example let's say if i start seeing it from top there you go you see i'm seeing a line i see a line out of it because the z axis has been reduced to a point from top if you see z axis is like a point see this you know blue point over here so one dimension has completely been eliminated and that's why you see a line there but in reality if you provided it one more dimension it would have become a plane like this okay so i'm allowing you to rotate for you to get an idea right isn't it very interesting so your representation of a given curve actually depends upon how many dimensions you are exposed to isn't it interesting right yeah it will x equal to c will also be a plane okay let me show you let me go back so x equal to let's say five okay i hope you can see let me just switch off this yeah you see this it's actually a plane right here so if you allowed one dimension to it it will be a point if you allow two dimension to it it will be a line if you damage allow three dimension to it it will become a plane right so when we start looking at things from a very limited perspective we always think we always see from one dimension less right that's why we say never you know try to judge anything by looking at it from one dimension okay anything let's say your laptop laptop screen if you see from the front you will think that okay it is a two-dimensional structure the moment you start you know turning your head here and get oh it is it has got some thickness also so it's a three-dimensional structure correct all right so several questions i'm getting uh vashna we said in 2d it becomes a line at yes vashna sorry vashna v i said vashna say to what will it be in 4d oh my god now people are trying to explore more dimensions so it is our inability to actually you know imagine something in 4d but i will definitely leave it up to your you know research work to figure it out okay okay so no no cube is also a three-dimensional structure my dear friend so in 4d 5d in multiple these it's up to your imagination to think it out because it's very difficult to represent those things okay anyways so with this one thing is clear that y equal to constant is like a horizontal line if there is two dimension okay and if you are using only one real number line for y it will be like a point on it and x equal to a constant is a vertical line parallel to the y axis and again if you're using only one number line to represent it it will be just like a point okay all right so let's have a small question based on this 4d what is 4d what is the fourth dimension that you're trying to take it all depends on that no okay so if you're trying to represent a position of something in terms of time okay then probably 4d will be your time but how will it look like in 4d is something very difficult to know comment right now okay all right let's have a question on this let's not digress on that topic because there are so many things that we can actually explore ourselves also so here is a very small question okay and I will put a poll for this okay you need to choose the right option from the poll okay here is the poll you only have 30 seconds to answer this question is which of the following is the graph of y equal to minus 3 on your screen you would be seeing you know cross lines which are actually your coordinate axes and the actual graph will be slightly solid and it won't have numbering on it okay I'm getting the response very good just now we did this very small question based on those that concept y equal to negative 3 my dear very good so in case you are not able to see the poll on your screen you can always put your response on the chat box even that is fine for me but if you're watching it on a mobile phone please let me discourage you from doing it because the moment the poll is launched it will block your whole screen okay so please do not watch out these classes on a mobile phone better to have a desktop laptop okay all right so at the count of five we'll stop five four three two one go let's go let's go three people are waiting for my instructions okay anyway let's stop so most of you 79 percent of you said option b and let me tell you my dear students b is absolutely correct b is absolutely correct okay so y equal to something as you know it's a horizontal line okay minus 3 means it should pass through minus 3 comma 0 it should pass through sorry it should pass through 0 comma minus 3 so 0 comma minus 3 is only happening in b it is not happening in a c and d okay by the way just to add on this would represent which graph graph of which equation y equal to 3 yes this would be x equal to 3 this would be x equal to minus 3 okay we will talk more about it in our discussion of straight lines which is going to happen in a couple of months time after your board exam is over okay very good now we'll quickly move on to linear polynomials linear polynomial graph okay linear polynomial basically is a polynomial where you have the highest exponent of the variable only as one so a x plus b type so normally we plot the value of this whole thing or we plot the value of p of x along y axis hence we can also write it like this okay by the way it is stupidity to write one so we will discourage writing one over there so this kind of a polynomial will be called as a linear polynomial why it is called a linear polynomial because because the graph of it comes out to be a comes out to be a line in two dimension again please be very careful line in two dimension okay but actually speaking it's a plane okay as we already discussed a little bit okay line into ds absolutely now how do we sketch the graph or how do we make the graph for such kind of a polynomial now if you put your x value as zero if you put your x value as zero you know that the value of y will be equal to b okay so if you put if you put x as zero your y will actually be b okay that means this particular graph that you're going to sketch a job what are the difference between plot and sketch what's the English questions you're asking in a maths class what are the difference between plot and sketch very good rational okay so plot is an exact representation of the graph sketch is a rough you know representation of the graph right so when we say you know sketch this portrait or sketch this scenery you do not sketch it like you know point by point okay you just make a rough idea or rough visual representation of that particular scenery right so that is called the sketch so when when the question says sketch means roughly okay plot means you need a graph paper for that because you need to do it point by point okay anyways so how many you sketch this so one thing is very clear that whatever line we are going to get that line is going to pass through zero comma b so let's say i consider b to be my positive value so it'll pass through zero comma b okay so this point zero comma b now how will it pass how will it pass will it pass like this will it pass like this will it pass like this how do i know how do i know how will it pass will it be option a will it be option b will it be option c so from this particular expression how do we come to know about it okay so let's discuss it through an example let us say i want to plot the graph of y equal to three x plus two okay so this point is zero comma two fine that is absolutely fine with everybody now what role this guy three plays in deciding what type of graph will this line have okay now we have already seen that when it was a constant polynomial it was parallel to the x axis so a cannot be my option because a is basically meant for constant polynomials correct but this is not a constant polynomial it has got three x term also with it okay now whether it is b or whether it is c how do we figure that out okay now let's analyze it through a simple logical you know uh logical move so let's say let us say whether a or whether b or c is completely out of the picture so i let me remove it also because i don't want to keep this even in my options so i have to decide between b and c okay so let's say i take a point on this particular line x1 y1 okay and i take another point on this line x2 y2 okay so what i do i take two points on this line x1 y1 and x2 y2 so let's say these are these are points on points on this line let's say i call it as one okay so it's very obvious that they will both satisfy these two points will satisfy this line so y1 will be equal to 3x1 plus 2 and y2 will be 3x2 plus 2 correct everybody happy with whatever i have done so far any question any concern please raise the alarm okay no problem anybody okay now let us say x2 is obtained by adding a change of delta x in x1 okay let's say the difference between x2 and x1 is delta x now as you all know delta x is represented is used to represent a change in x if you don't know it then start you know making these kind of you know small small you know notes in somewhere that delta x or delta of anything means a change in that particular quantity i'm sure you would have done in physics also what is velocity of a body delta s by delta t right change in displacement by change in time right so delta s means change in that particular quantity so here let's say x2 minus x1 is delta x also as to say x2 is x1 plus delta x okay let me just do the same for y2 also so let's say y2 is y1 plus delta y now let us try to understand it on this graph now as of now i do not know which is my answer whether it is b or whether this c i do not know so let's say i take you know b to be one of my cases and i take a point here x1 y1 okay and i take another point here x2 y2 so delta x means what change did you see when you are going from x1 to x2 so this guy is your delta x okay and delta y is what change in y happened when you went from y1 to y2 this is delta y now i have no intention in taking delta x as positive or negative i do not know the sign of it right now it may be negative quantity right why if x2 y2 where let's say on the other side of x1 y1 then probably my delta x would be negative delta y would also be negative as of now i do not know what is the sign of delta x and delta y so i do not know which is my answer also whether it is b or whether it is c i do not know okay so we are trying to figure that out so just i have taken two points x1 y1 x2 y2 on that line which is supposed to be my answer and these two points x1 y1 and x2 y2 are basically related to each other by this relation okay now what i'm going to do is i'm going to change this y2 with y1 plus delta y and i'm going to change this x2 with x1 plus delta x okay this two i'm keeping at such so what i did i changed my y2 like this i changed my x2 like this and my two remains two over here okay any question so far absolutely an infinite number of lines can pass through it so that's why siddharth we are trying to decide what line is possible okay i agree to you through 0 comma b infinitely many lines can pass what is my attempt here my attempt here is to figure out which line will it be that will be satisfying this given equation y equal to 3x plus 2 wait wait wait let let the discussion be over siddharth then you can be you can ask your question so now what i'm going to do is i'm going to replace this y1 again with 3x1 plus 2 let delta y be delta y and this is again if you open it it will become 3x1 plus 3 delta x plus 2 okay so 3x1 3x1 cancels out 2 2 cancels out that leaves me with delta y is equal to 3 delta x okay that means your delta y by delta x actually becomes a 3 now this is a very important piece of information for us how it is very important let us see it says that if you move between a point x1 to x2 sorry if you move between a point x1 y1 to x2 y2 the change in the y with respect to change in the x should be 3 that means change in the y is three times the change in the x okay now let us come back to this given pictorial representation if i go from x x1 y1 to x2 y2 such that the change in delta x is let's say one unit then the change in delta y should actually be three units and that is happening only in b that is not happening in c why does not happening in c because let's say i move from this point x1 y1 to this point x2 y2 let's say one unit then here there is a drop in the value of y probably this would be minus three units okay so c cannot be my answer are you getting my point so this tells you that this tells you that first of all the ratio of the change in y to the change in x is a positive quantity means if you increase your x there would be an increase in y correct so that the ratio is still positive if you decrease your x there will be a decrease in y so that the ratio is still positive and what is the magnitude of this ratio the magnitude of this ratio is 3 is to 1 3 is to 1 means if i move 1 kilometers along let's say the x axis i should reach a point which is 3 kilometers on the y axis means the change in the y should be three times the change in the x and that is only happening in case of b so just to answer your question siddharth you will end up making that line through 0 comma b where if you move from one point to the other on that line the change in the y that should happen normally we call this as rise we call this as a run okay i'm sure you would have heard of this term in your straight line chapter so the ratio of the rise to the run should be three so is that happening in c definitely not so if i move one kilometers in this direction i actually fall my three kilometers down it is not it is not a plus three change or you can say it is not a positive change it is a negative change so dear all welcome to coordinate geometry in coordinate geometry your distances will be directed distances what are the meaning of directed distances that means they would have a sign so if i'm moving in this way along the exact positive x axis that would be a positive distance if i'm moving in the negative x axis it would be a negative distance for me if i'm moving down it is negative if i'm moving up it is positive so in which of the two cases when i'm moving one in a positive direction there is a rise of three or there is an increment of three in the positive y axis it is only happening in b not in c right and no line can actually they cannot be multiple lines stating the same siddharth just to answer your question can there be multiple lines satisfying the same criteria so you said they can be infinitely many lines doing it right no they can only be one line doing it okay so they can only be one line which passes through let's say zero comma two and it is such that if i move between any two points the ratio of rise to run will be three so in that case that line is only the line b by the way later on we will learn that this particular term is actually called this ratio is actually called or the rise by run is actually called the slope of the line okay or the gradient of the line okay so now what are the moral of this entire discussion the moral of this entire discussion is just to summarize just to summarize if you have been given up linear polynomial like this okay so first this decides the y intercept y intercept means where it is cutting the y axis okay so y intercept would be zero comma b in this case okay so first make sure your point is cutting the y axis at zero comma b okay now this a if it is positive that means your line will be oriented in such a way your line will be oriented in such a way that the ratio of the change in y to change in x as you move between any two points should be a positive quantity now depending upon the magnitude of that quantity this can actually become more steeper okay i'm just drawing few cases of okay so depending upon its magnitude it can become steeper but it will always be oriented in this fashion please note down the orientation of it please note down the orientation of this line you can think as if it is coming from left down to right up even though there's no direction of the line so you can think it to be oriented in the direction left down to right up okay and if your a is negative if your a is negative then your orientation of the line would be like this your orientation of the line would be somewhat like this okay and depending upon what is the magnitude it can become either more negatively sloped or less negatively sloped is it giving you a clear cut idea how do we actually figure out how to sketch the graph of a line depending upon my a and b values okay let's take few examples to understand it a little further okay very good so let's take one simple example let's say somebody says sketch sketch y is equal to minus 2x minus three okay so everybody first write everybody first write on your respective notebooks then we'll discuss it i'm giving you 40 seconds for it rough rough i do not want an exact sketch okay so in class 11th and 12th nobody will give you a graph sheet for making it so it's just a rough sketch that you should be knowing done everybody just write done if you're done okay good good three people are done done done done done okay so let's let's analyze this so first thing that you would see is this minus 3 this minus 3 tells you that the graph is passing through 0 comma minus 3 so whatever line you're going to get that line has to cross 0 comma minus 3 for sure okay but this minus 2 tells you that it's a negatively sloped line negatively sloped line means that ratio of rise to run should be negative 2 what are the meaning of negative 2 let's understand it once again it means that if you go one unit along the positive x direction you should drop by two units drop by two units because there's a minus sign are you getting my point or other way to understand it if you move one unit towards the negative x axis you should rise by two units so when can this happen this can only happen my dear friends when your graph is oriented like this okay so let's say choose this point and start moving one unit to the right okay if I move one into the right see I am dropping by two units and this is just a sketch so I'm just assuming this is two are you getting my phone have I said going one unit to the left let's say then I will rise by two units so this is a rising so this is the way your graph will be oriented okay make sure you put arrows at the end okay so any question regarding how do we actually figure out the graph of a linear polynomial depending upon the constants given to us so this minus 2 as you can see it actually leads the way of the graph it actually changes the whole orientation of the graph okay let's have one or two questions so here is a question coming your way which of the following is the graph of minus half x plus three which of the following is the graph of minus half x plus three your pole is right there in front of your screen pole is on easy question easy easy guys easy what is this tree line doing already very good almost 10 of you 11 of you 12 of you have responded very good others I will be giving you 20 more seconds dear let's finish it off simple question you can take a hint from these numbers the markings on the x and the y so normally I've not shown which is x which is y but it is very obvious this is x this is y kind of a thing okay all right five four three two one go okay almost 60 percent people say option number c c for Chennai okay let's check see first of all this number three tells you that the graph should cut the y axis at zero comma three so in option a and b it doesn't happen only in c indeed is happening when it is cutting it at zero comma three correct yes or no so a and b cannot happen okay now minus half tells you that the graph has a negative slope that means ratio of rise to run ratio of rise to run is negative half negative half what does it mean it means if you go two units along the positive x you should drop by one unit on the y right so in which of option c and d happens let's say I start my journey from here I move two units along oh sorry two units along the positive x axis am I dropping by one seems to be so because I think let's let's see the ratio so let's say I start from here okay and I move six units I'm dropping by three yes so c is my right answer obviously b cannot be because in d if I'm moving let's say two units along the positive x axis I'm going one unit up that means the ratio of rise to run is one is to two not minus one is two yes I'll repeat once again say two see this says that if I move along if I move two units along positive x axis okay I should drop by oh my camera is coming in between I should drop by one unit along the y axis okay along the y axis so say to tell me in which option it is happening is it happening in d is it happening in d no because when I move two units like this the line takes me up okay so to reach another point on the line I have to go one unit up yes or no so it is only happening in c option when I'm moving two units to the right and I have to fall down by one unit to hit the line are you getting my point here what does minus half indicate it tells you the ratio of rise to run minus sign just tells you whether there is a drop or whether there is an increment okay so negative sign as I told you it actually tells you the direction minus one means one unit down got the point okay so now you may say yes sir what if I move two units to the left that means this is the same thing can be written as one is to minus two so if I move two units left I should go up by one unit is it happening in d no it is happening in c only see is my answer got the point got the point got the point okay I will have one more question because some of you made mistakes so let's have one more question just a small one which of the following is the graph of 2x minus 3 I'm sure everybody would get this correct 100% correct answer I would get I'm launching the poll now y is equal to 2x minus 3 y is equal to 2x minus 3 multiple answers people are giving means some of you may be not correct okay sharduli very good in case you're not able to see the poll you can always put your response on the chat people are giving wrong responses okay last 20 seconds 20 seconds last 20 seconds five four three two one go everybody please vote please vote two of you are still waiting for my instructions okay never mind so I got 89% of the votes out of his 76% people say option number d okay d for deli and two two people have voted for a and b okay again let's start our analysis with this number minus three okay this number minus three says it should cut the y axis at 0 comma minus three so zero comma minus three is only happening in b and in d so a and c are out of the picture they cannot be my answer is that fine everybody okay now look at this number two two says the ratio of rise to run is two okay two means two by one two can be written as two by one that means if I move one unit along the positive x I should go up by two units okay so in option b and d where is it happening let's say if I move two units here I have to fall down to reach okay so this ratio is minus two is two one actually are you getting my point so in option d if I move one unit here then only I have to go up by two units to hit the line so option number d is my right option not b so janta thumbs up to you you are absolutely correct sir what is this janta sir the janta is normally a lingo that we use in IITs so I come from IIT Kharakpur so there we normally use janta for addressing people okay so don't get me wrong sir how do we know we are going hey setu when you hit the line you have to go up or down you just try to figure it out let's say if I look at b if I move one unit in order to hit the line I have to go up or down setu this is a question for you if I move one unit to the right let's say along the positive x axis and if you want to immediately come and hit the line from there you have to go down from there right so when you go down means there is a drop so it's like your x1 y1 point and x2 y2 point correct so your delta x this is what is delta x this is what is your delta y so here in this case your delta y by delta x is actually negative 2 by 1 right which doesn't match with this it is it says plus 2 by 1 okay so that is happening in option d now that point is clear setu okay good so please keep asking your doubt so we will now move our discussion to quadratic polynomials quadratic polynomials polynomials okay so from degree zero that is constant we went to degree one now we are going to degree two so strygeny has a question said if it was x equal to 2 y minus 3 would it be minus 3 comma 0 say it's very obvious strygeny so if you put minus 3 comma 0 does it satisfy your equation very simple let's say if i put my y as 0 does my x come out to be minus 3 yes it does come out so we'll pass through minus 3 comma 0 absolutely right okay okay all right now quadratic polynomial what's a quadratic polynomial so any polynomial where the highest exponent on the variable is 2 okay that means your a cannot be 0 because if a is 0 then the highest exponent becomes 1 and not 2 okay so such kind of an expression okay is called a quadratic polynomial okay so we will replace our p of x with y actually because we draw p of x along the y axis so this kind of a polynomial is called a quadratic polynomial now please do not get confused between quadratic polynomial and quadratic equations okay there are different things the other day i'll tell you a story a very very interesting story because this is the mistake which many people do i asked one of the i was interviewing some teachers in one of the nps schools and i asked the teacher that draw the graph of this okay draw the graph of this i just gave a question like this draw a graph of this now let's say you are that teacher what would you do what would you do i'm sure you are exposed to quadratic polynomials and quadratic equations both in class 10 right okay so what will you do very good char dhuli and i want to see other other people's response okay prisham am i pronouncing your name correctly prisham right okay okay good good good good good okay so let me tell you the zeros of this polynomial or in fact the roots of this quadratic now i'm using two words zeros and roots so you should be knowing when to use the word zero and when to use the word roots right now it's an equation so its equation will have roots and those roots will be two and three okay then what then what okay now some of you have done mistake while telling me the answer you have told me it's going to be a parabola okay and a parabola which is basically cutting the x axis at two and three okay let me tell you this is a wrong answer why it is wrong answer why it is a wrong answer again here your basic concept has been tested i never gave you to plot y is equal to x square minus 5x plus 6 right i what did i give you to plot i gave you to plot this particular equation okay this equation is just corresponding to two points two and three there is no y here are you getting my point are you getting my point there's no y over here so if at all you're using a y it will be like it will be like two parallel lines like this x equal to two and x equal to three are you getting my point okay and let's say if you're not using y it will be just a number line which will show you the position of two points two and three that's it you just have to make a presentation of two points are you getting this don't worry even teachers are making this mistake even teachers are making this mistake my dear students she made a parabola actually shall believe okay so don't get confused this is where your kbpy interviewers will you know catch you off guard okay this graph is a parabola i agree but this graph is either this or this depending upon what how many dimensions you are choosing to represent it are you getting my point very small things my dear but people do not take cognizance of these things and as a result blunders happen okay anyways coming back to our discussion all right so now just to give you an idea there are two types of quadratic that we normally deal with there is something called pure quadratic pure quadratic in pure quadratic your a is definitely not zero and your b is zero okay and c could be any real number okay it could be zero it could be non-zero but b is definitely zero for a pure quadratic for example something like this this is a pure quadratic expression okay so b is zero obviously a cannot be zero because if a is zero it will no longer be a quadratic it will actually become linear or a constant depending upon b okay and there's something called adfected quadratic ad effected quadratic this is for your you know general knowledge there's nothing like you know very hard and fast so oh my god what is this ad effected and all i've never heard of it don't worry too much about it it's just for your knowledge in ad effected a and b cannot be zero c can be zero or non-zero quantity so i'm just writing it as a real number so a simple expression for ad effected quadratic is anything like 2x square plus 3x plus 2 okay this is an ad effected quadratic just for you to know the name so that when you start attending your school and your teacher says hey there's a pure quadratic then you should know oh yes in sanctum akhil sir told about it okay or this is an ad effected quadratic okay fine now we are talking about graphs so let's talk about graphs only so can anybody tell me in a quadratic expression what role does a play in deciding the graph we all know it's a parabola is there anybody who doesn't know it's a parabola by the way pronunciation is parabola i have seen people pronouncing it as parabola and all those things it's not parabola okay parabola is basically a path traced by an object when it travels under gravity so let's say if i toss a piece of stone at you it'll trace a parabolic path okay yeah so my question to all of you is how are these coefficients a b and c going to affect the nature of that parabola can anybody you know tell me about that if you have studied about it okay so prisham says upward downward vabha vashna also says opening of the parabola a governs the opening of the parabola i'm sure we are talking about a so once again there was a question sir can you show a question on ad effect is that what you meant seto example you wanted i gave an example here that example of 2x square plus 3x plus 2 it's an affected quadratic oh i didn't know i'll write it down for you here instead of going back this is an example of an affected quadratic plus x minus 5 okay this is an affected quadratic okay so sharduli will come to see a little later on here let's talk about a first what role does a play in deciding the graph of a of the quadratic polynomial so for that i will go take you to a tool which is called the geojibra tool by the way i did not tell you uh this in the beginning of the class there are graphing tools that we are going to use so please make a note of it so there are two graphing tools that we are going to use one is geojibra okay and this geojibra is very very uh you can say uh robust graphing tool and it's an open software uh many many of the uh mathematicians and coders are contributing contributing to it it's an open license software so you can always download it uh i would request you to download uh the latest version i think it's classic six okay geojibra classic six you can get it from geojibra.org you can download it from downloaded from geojibra geojibra.org so some of you who were already symptom students before you would be knowing it because uh mr tushar uses it very very frequently so this is normally useful for your desktop okay so desktop or laptop whatever you are using for the classes it is good for that the second software that i normally recommend is desmos desmos okay you can get it for your phone okay so whatever mobile phones you're using you can download desmos on your phone so that let's say tomorrow we are you know meeting in an offline mode and we do not have these uh laptops and uh you know desktops in front of us we can always open it on our phone and plot the required graph so today itself you download it i think in google play store you will find it okay and apple store also it should be available so desmos and geojibra is what we recommend for graphing purposes uh do not try to install geojibra on your phone because it's a heavy software your phone may get hung up so do not use geojibra for your phone best is desmos for your phone and for your laptop you use or desktop you use geojibra so i will show you my screen where i'll show you my geojibra yeah so i have already downloaded i normally use a mac so i have installed it as you can see here this is the app this is the logo of the app geojibra classic 6 okay huh you can use it online also but online normally becomes slow sometimes so if you have the app on your system the uh the the performance is much faster okay so now what i'm going to do is i'm going to just draw a generic you know quadratic polynomial over here so y is equal to ax square plus bx plus c okay now what does this software actually do see if you see i've plotted ax square plus bx plus c i never told the uh the tool what is a b and c value so the moment i i write ax square plus bx plus c this tool comes to know that uh the user is not sure about the value of a b and c okay so it does it gives me a leeway to choose from it gives me some liberty to choose uh you know in the value of a b and c so as you can see it says that you can choose anywhere from minus five to five for a b and c respectively and it uses one of the values to actually plot this graph so as you can see right now it has used one for a i'll show you on the screen here it has used one for a one for b and one for c as of now but at the same time it knows that i am not very sure about a b c values as of now so it gives me an option from minus five to five to choose from okay it says go and choose any value from minus five to five now if i play with my a how would this graph change is what we are going to discuss now so most of you said it depends it basically tells you where the graph opens now the correct word is actually a decides the concavity of this parabola concavity means how much are the arms bend yes right how much more to put through it can become okay so it depends it also decides that factor as well okay so let's see right now a is at one so i'll increase it you know towards five so let's say i go to i go to higher value if you see as i'm going to a higher value two point five two point seven two point nine three point two three point three three five okay let's say this allows me to go to five it starts shrinking okay so it is changing the concavity of the graph that's the word rational concavity i'll write it down for you over here so a decides the concavity of the parabolic curve how much concave it is okay similarly if i decide if i start if i decide start taking my a you know towards negative side let's say i bring it back to one okay now all of you please pay attention from here on now i'm going slowly towards zero do you see the graph is becoming fatter you see that fatter fatter fatter fatter and near to one it actually not near to one exactly to sorry near to zero it actually loses its quadratic nature it actually becomes a linear graph as you can see it becomes a line x plus one correct now what happens if i take it further to the negative side if i take it to the further to the negative side as you can see it has made a bent downwards means earlier it was concave upwards now it has become concave downwards are you getting my point so a basically governs the concavity of the curve so what did we learn in this entire exercise let me plot it down over here so if a is a positive quantity the graph the parabola will actually the parabola actually opens upward opens upward okay the mathematical term for it is it is concave upward but i don't want you to get confused in the concave word right now it opens upward and if you increase your value further the the parabola becomes thinner okay the parabola becomes the parabola becomes thinner or you can say it shrinks along along the y-axis or you can say along the x-axis okay so it shrinks along the x-axis getting the point so if a is positive it will open upward and if it is a very high positive value the the higher the value of a the more shrink or the more shrunk the parabola will become along the x-axis are you getting my point okay so just to give you an idea just a small question for all of you okay if let's say this is the graph of y is equal to x square okay actually it will be yeah passing through origin over here y equal to x square okay now tell me which of the following will be the graph of y equal to x square by 2 so let's say this is one option let's say option number a and another option i'll put it in let's say blue color okay so out of option a and b which of them which of them is the graph of y equal to x square by 2 just say a or b on your chat box a or b a or b a or b which option brilliant guys very good i think almost everybody apart from i think few of them they're given the incorrect answer yes the answer is option number a okay because you are multiplying it with half okay so y equal to x square is like y equal to 1 into x square so you're reducing your leading coefficient from one to half so it will open up more it will become flatter it will become fatter so when it becomes fatter it can only be option a not b got the point b probably could be the graph of y equal to 2x square okay and just taking some in a random example and if a is negative as you can as you could see from your geojibra if a is negative your parabola would open downwards your parabola will open downwards it's a downward opening parabola and again the same if you keep increasing the magnitude of a it will become thinner magnitude only so let's say let's say i want to just give you a comparison let's say i have y equal to negative x square like this okay let's say this is y is equal to negative x square okay and if i want to plot the graph of y is equal to negative 2x square it will be downward opening only but it will be thinner like this okay i hope you can figure it out is this fine is this idea okay all right now going back to the geojibra a role is very much obvious to us tell me how does b how does b influence the nature of this graph can anybody tell about how b influences the nature of this graph a is pretty clear right a no problem now my question is how does b change the nature of this graph anybody any guesses so if i change my b if i play with my b what will happen to this graph any idea okay some people are saying open left right move laterally maybe it goes size to side okay good most of you have tried your best but let me tell you its motion is a parabolic motion okay it will move if you put your a and c fixed and if you change your c sorry if you put your a and c fixed and if you change your b this entire parabola will actually dance on a parabola okay let me show you by the way i forgot to tell you the lowest point of this parabola is actually called the vertex okay i hope everybody knows it okay vertex of the parabola now let me show you how does the vertex dance when this b value is changed thankfully in case of geojibra you can always locate the vertex of any parabola by using this command vertex and in the in place of this input you have to give the you know the conic which is basically involved over here which is f okay so as you can see the moment i write vertex bracket and the name of the conic which is f in this case it automatically points towards the vertex position which is a in this case now all of you please watch i am now making my b change okay so i'm playing with my b do you see what is happening to the parabola okay at least you're seeing what is happening to the vertex what do you see you see that your parabola is self-dancing on a parabola okay you know why does it happen why does it happen and let me give you a question based on this little later on see all of you it'll not be exactly mirror image i'll tell you what is it uh shardili okay see i'm sure all of you would have learned that your vertex coordinate your vertex coordinate is minus b by 2a comma minus d by 4a i hope everybody remembers this where d is the discriminant of the parabola anybody who doesn't know this formula was was there anybody in this class who was not aware of the vertex coordinates of a parabola i'm sure everybody knows it right and this is important for your board exam oh you didn't know okay no problem no problem now you know that okay we will derive it officially also when we are doing the chapter quadratic equations complex numbers and quadratic equations in class 11 oh no wonder they deleted it this year okay we should not have done this you know why because quadratic equation comes in at so many places in 11th and 12th you cannot do without quality equation okay that's the most in fact quadratic polynomials are the most commonly seen polynomials while you are solving maths questions anyways never mind so vertex coordinate so basically if you have a quadratic polynomial like this the vertex coordinate is given by minus b by 2a comma minus d by 4a uh setu i didn't get your question can you show the shrinking again for negative for negative i didn't get your point which negative idea for negative a value okay can you just wait for some time once i'm done with the discussion of this can i can i go back okay thanks thanks setu so now see it as you can see the vertex has b in it right in fact d also has a b in it so if i write it properly if i write it properly it is minus b by 2a and this term is actually 4ac minus b square by 2a oh sorry 4a okay so if you keep your a and c fixed that means a and c are not changing only b is changing as you can see this is changing and this is changing as a result both the x and y coordinates of the vertex is changing and because they are changing this parabolas vertex will start dancing on another curve which actually you can see is another parabola okay are you getting what i'm trying to say i'm trying to say here that if you keep your a and c fixed a and c you are keeping it fixed fixed means you are not changing its value but if you are varying your b if you are varying your b what will happen b is involved over here b is also involved in d so as a result the vertex starts moving when the vertex moves it moves in such a fashion that it is always on a curve which happens to be this dotted line as you can see over here okay now it is very easy to figure it out that it is also another parabola okay and now my question here comes for all of you okay try this out for homework yes sharduli i'm going to ask that question what is the equation of this parabola this is a homework question for all of you just to tell you sharduli that is not correct answer by the way your answer will only be in terms of a and c so the answer will be in terms of answer is in terms of a and c only okay because b is changing right so b and b cannot appear in your answer so give me the equation of this parabola in terms of a and c only of course x and y will be there that goes without saying also okay so try this out for homework try this out for homework in case you're not uh uh setu has got it well done setu okay don't tell your answer to anybody okay let not people you know call you and ask you hey what was the answer what was the answer okay everybody please give it an honest try and feel free to uh you know send me your workings personally to me okay try it out do i should know try it out okay so question is i'll again repeat my question question is there is a uh there is a quadratic expression where a and c are fixed and b is basically changed okay what is the path which is faced by the vertex or what is the equation of the path which is faced by the vertex of this parabola your answer should be in terms of a and c okay try it out one of you have actually given the answer good okay now so b-roll is clear to everybody now what about c-roll what about c-roll if i change my c what happens to the parabola anybody what happens to the parabola if i change my c values how will it dance left right up down or goes in a very complicated path very good pressure it goes up and down as you can see here i'm just making the change in the value of c just watch the motion of this graph it goes up and down in fact vertex you can see it is tracing a straight line okay guys why am i telling you all these things why am i telling you all these things of course yes sharduli the point where it is cutting the y-axis keeps changing so as a result that you know the point keeps going up and down now why i'm telling you all these things is because they are the formative part of your understanding of transformations so very soon we are going to talk about some transformations as of now we're just talking about graph no transformation was discussed so far but i'm going to talk about transformations which is going to be universally applicable okay so please focus on these small small things because they are going to be a part of your basic understanding of transformations okay anyways so going back to our going back to our sheet so a basically as i told you decides your con cavity okay so b basically you know moves the graph or moves the parabola parabola in a parabolic path in a parabolic path and this path equation is what i have asked you for homework and see basically and see basically makes the parabola move up and down moves the parabola up and down okay up and down okay now let us start understanding a bit of transformations from here on so uh please note this down because i'll be going to the next page and anybody has any question do let me know uh the class will exactly end at six o'clock so we have roughly 25 minutes so the class to end okay oh yes yes yes good good you reminded me say to us i would have moved down yeah so see uh say to just focus on okay i'll just reopen the graph once again that was too muddled up so i thought i would start with a new one okay so first of all i will keep a parameter over here let's say i keep a parameter a which goes from minus 10 to let's say plus 10 okay and i write a quadratic y is equal to ax square plus x plus 1 okay just a simple quadratic i have okay now see uh say to if i take it if i if i start with taking my a to the negative side okay first of all it inverts it becomes a downward opening parabola if i keep going more negative see start sinking start sinking sinking sinking sinking sinking you see that higher the negative value that means higher in magnitude i make the value of a more shrunk is the graph but it is opening downward because it has a negative value attached to it right is this what you wanted to see does this uh satisfy your question okay all right so now we are heading towards some transformations so let's talk about transformations by the way i'm making a para i'm basically using our parabola to showcase you these transformations but these are universally applicable transformations okay adya has a question oh very good question when does the parabola open leftward or rightward okay we'll talk about this adya but just to give you an idea if you replace your positions of x and y the parabola will start opening leftward or rightward for example let's say this was the parabola which you show which you just saw it was opening upwards right okay if you wanted to open rightward just swap the positions of x and y this parabola will open rightwards like this okay i'll talk about i'll talk about this also because this is also one of the transformations that has required for us why does it happen that's a story which is to be taken you know in in after few classes okay so just to answer your question when does the parabola open rightwards or leftwards i'll talk about leftward you know leftward basically it opens when you are changing your x sign to minus x okay so if you make it as minus x equal to y square it would become a parabola which is opening leftward like this okay why does it happen we'll talk about it some other day all right so let us talk about let us talk about few transformations few transformations so the first transformation that i would like you to understand is by the simple example let's say let's say i have a very basic standard parabola y equal to x square okay so such parabolas are mainly called standard parabola why they are called standard parabola because they are very very simplistic case of a parabola parabola can have a very complicated equation also i will not go into details of that because anyways a chapter is going to come up for you in class 11 so as of now i'm going to start with a simplistic case of a parabola which we call as a standard parabola we already have seen this type of parabola in our geojibra graph it's a upward opening parabola like this okay i've tried my best to draw a smooth one okay now my question to all of you is instead of this if i ask you to draw the graph of y equal to x minus one whole square how would you draw it of course you say that i'll sit and plot okay i'll put some value of x and i'll see the value of y and that x comma y i will plot okay now that is fine for the initial you know learning but soon you will see a pattern developing in these kind of questions so see everybody please pay attention so in order to know the graph of this let's do some plotting if i put my value of x as one my y value will be zero correct so this graph will be passing through one comma zero let's say it is over here okay if i put my value of x as two my y value will be one so it will be passing through two comma one and at the same time if i put minus or if i put zero also it is again going to pass through one so it is going to pass through zero comma one also okay so this is my x axis this is my y axis even though i've not written it okay let's further pick up few more points let's say if i put three it'll pass through three comma four so three comma four will be somewhere over here and if i put minus one also it'll pass through four so minus one comma four will oh sorry minus one comma four will also lie over here okay now let us try to join these points and see what has happened to the final graph there you go now can somebody tell me what happened to the white parabola that it became the yellow parabola in one statement if you see what happened to the white parabola that it became the yellow parabola absolutely correct the parabola vertex in fact the entire parabola my dear not only the vertex entire parabola of course the vertex shifted every point started shifting one one unit to the right as you can see every point started making a migration of one one unit to the right okay so this is a very interesting phenomena which we call as the horizontal shift okay so this is a phenomena or this is a rule which we call as the rule of horizontal shift so what does this rule this rule says that if you are basically given any graph let's say you are given a graph of this so let's say i write it down graph is given graph is given okay and let's say somebody asks you and somebody asks you the graph of y is equal to r x minus h h being some positive quantity let's say for the sake of discussion over here so some something like here what happened so if you have been given the graph of y is equal to x square so compare this with this guy and you're asked the graph of this okay compare this with this guy okay then you don't have to reinvent the wheel you don't have to sit and plot you know like the way i did initially you just have to take this graph okay so this graph can be obtained by shifting the graph of this by h units to the right by h units to the right or let me make a arrow diagram so that it is clear in your memory are you getting my point and let me tell you it's a general rule it's a universal rule it is not just you know a subject matter of a parabola it you can apply to straight lines also right you can apply to trigonometric curves also trigonometric graphs also you can apply to exponential graphs also logarithmic graphs also any special function graph also okay so this is something which is a rule which can be followed by any relation graph any relation graph okay that's why i wrote an r over here just to show that it is applicable for any relation okay not not primarily or not particularly a parabola only are you getting my point so let's say if i extend this further and ask you if i want to know the graph of y is equal to x minus two the whole square what will happen now you are very clear that will further shift one unit like this so it'll become like this okay so now its vertex will come to two comma zero position so earlier it was one comma zero and initially the beginning of the graph was at zero comma zero okay are you getting this point okay adya has a question sorry nidhi has a question so does doesn't negative mean left okay yes now this is a confusion that people have you're very correct nidhi negative means left but actually what is happening here the origin gets shifted to the left that is why the graph is the graph seems to move to the right so there is a small story behind it which i have not told you right now family because you people are not that mature enough to take it right now there's a concept called shift of origin okay in shift of origin when you replace your x with x minus h origin gets shifted actually to the left now because origin gets shifted to the left the graph appears to shift to the right are you getting a point that's why these curves they appear to move to the right but it is actually not the graph which moves my dear is the origin which is moving to the left see it's like how can i make you away from me either i push you or i go myself back then only you will be away from me correct so here origin is what doing origin is stepping back yeah so to answer your question adya origin is moving and nidhi as well the origin is moving at units to the left and hence the graph appears to move at units to the right but my serious suggestion to you would be do not as of now mix these two concepts because you will get confused okay so follow one simple rule the rule is in your graph if you are replacing your x mark my words in any fun in any relation equation if you are replacing your x with x minus h h being a positive quantity you are going to shift the graph h units to the right period okay yes so shardali if your h was a negative quantity your graph will shift to the left oh sorry if h was a positive quantity your graph so let's let me write that down also so i don't want to just speak in the air i want to just write it down and if you want the graph of y is equal to a relation in x plus h that means you are replacing this x with x plus h now the graph will be shifting the graph will be shifting shifting so shift this graph by h units to the left okay let me show you how so i'll again restart the whole thing so all of you please pay attention okay so y is equal to x minus i'm just taking a parameter a which i'm going to change so as of now it has taken a value as one and they are plotted y equal to x minus one the whole square as you can see x minus one the whole square is shifted one unit to the right from this position so let's say i i put it zero okay so if i keep increasing this let's say i take x minus one it has moved one unit to the right minus two again moving moving moving okay minus three again moving minus four okay so as i keep changing this value are you why this coming yeah as i keep changing this value of a it starts moving to the right because i have kept kept it as x minus something x minus a okay if i change it to let's say a value like this x plus one okay let's say this is your x square okay now x plus one you see it is moving left x plus two further left x plus three further left x plus four further left like that okay so this is the two rules which we call as the horizontal shift rule and the main reason behind these rules is the shifting of the origin which is something which i'm going to talk about in your coordinate geometry chapter not right now is the idea clear okay now just to give you a clearer idea in that case you can just see think of it like this if you have a curve y is equal to r minus h okay and if h is positive the graph will shift to the right and if h is negative the graph will shift to the left does it now does it now satisfy your concerns say too okay this is just for your you can note this down for your overall understanding sir are you going to give questions about these horizontal shift rule yes of course you are going to take a question a very small question i will be taking up so first everybody make a note of this rule any question any concerns please feel free to highlight here also i'm assuming h to be positive guys just keep a simple thing in your mind if x minus something is happening take the graph to the right x plus something is happening i mean x plus a positive number is happening take the graph to the left so as somebody rightly said it is working in reverse psychology it is working so minus it is going right positive it is going left simple as that okay so very small question we'll take up on this very very small question which of the following is the graph of y is equal to x plus to the whole square which of the following is the graph of y equal to x plus to the whole square please put your response on the poll which has been launched very good i've started getting the right answer awesome i hope nobody gets confused here awesome very good good shall be good okay almost everybody has responded so let me do a countdown five four three two one go everybody please vote please vote come on guys and girls you can do it okay so i received votes from 14 of you and this is what 14 of you have to say option a so everybody is concurrent on the fact that it has to be option a which is absolutely correct okay so as i told you x plus two means two units to the left okay now just to write down what are these graphs okay this guy is actually y equal to x minus to the whole square as you can see it has moved two units to the right okay we'll talk about c and d now actually because c and d basically involve vertical shifts so let's talk about vertical shifts now okay now the very same example i will take now start with now y equal to x square y equal to x square is a graph like this okay everybody knows it okay please forgive me if i don't write x and y because it's understood more or less okay now my question is how would i draw the graph of y equal to x square or let me let me write it like this how would you draw the graph of y minus one is equal to x square or in other words how would you draw the graph of y is equal to x square plus one okay now initially when we do not know we will actually try to plot so if i put x is zero y will be one so it will pass through zero comma one when i put x as one y will be two so y will be here zero one comma two when i put x as two y will be five so we'll go jump almost here okay similarly for minus one also will give me two okay and for minus two will give me a five okay now what do you see here if you start connecting these dots they tell you something very interesting they tell you that the graph has actually moved one unit up that means every point on the graph has been kicked up okay you take a point and you just kick it up so every point has been kicked up by you know one one unit up okay not not this one this is too big a kick okay so what has happened the value of y for every corresponding point has become more than one or become incremented by one so this is what we call as a vertical shift are you getting my point this is called a vertical shift okay so here is the rule that comes which everybody should note down okay vertical shift rule vertical shift rule says if you have been given the graph of a relation like this okay let's say this graph is given okay this graph is given okay and somebody comes to you and say hey can you draw the graph of can you draw the graph of this kind of a equation then all you have to do is you have to shift the graph of this by k units upwards again the reverse psychology is working y minus something means upward okay and if somebody says draw the graph of y plus k equal to something then you have to shift this graph shift this graph k units downwards are you getting my point okay so this is what we call as the rule of vertical shift okay so we'll go back we'll go back to that question which we had you know taken a little while ago and we'll try to see the ones which we did not mark which graphs where they actually okay and then I'll give you a question to end the day so please note this down and if you have any question please shoot please fire your questions at me done okay let's move on to the previous question just for a couple of seconds we'll discuss about it in previous question if you see option number c and d can somebody tell me what is this graph c representing or whose graph is c actually which equation's graph is c yes k is positive mithy sorry I forgot to tell you that k is positive k is positive yeah please write down on your chat box whose graph is c you cannot see the numbers clearly okay my dear this is minus 2 this is minus 2 okay and assume that it was coming from x square graph very good y plus 2 equal to x square or you can also say y equal to x square minus 2 brilliant you guys are very sharp okay now d1 d d d d for d tell me for d very good Nikita very good everybody has given the right answer god bless you guys very good all right now can there be a question where there is both vertical and horizontal shift happening yes it is quite possible that we can get a function or get a relation where both the shifts are being on taking place so to wrap up this session I would like you to solve this question for me a very simple one which of the following is the graph of which of the following is the graph of y is equal to x minus 1 whole square plus 3 okay think carefully I'm giving you two minutes we have two minutes to six so think carefully and then answer don't rush I'm launching the poll and once you're convinced press on the poll button whichever option you think is correct our arms say no need to hurry don't rush these are just know our initial understandings of the concepts so here and there if something goes wrong you will keep on making mistakes so here is a mix of the two shift things here's a mix of the two shift things very good janta janta is smart nice nice nice why are you smiling so much that are we going wrong somewhere okay so one person has uh is differing is differing from other people let's see let's see five of you still to vote come on guys and girls last 30 seconds so for the three people who have still not voted 30 seconds is all I can give you okay five four three two one go okay all of all 17 of you have voted very good maximum janta goes for C C for Chennai and you are absolutely right I'll tell you that setu in some time okay now see here if you start building this graph you start from the standard one okay so normally when we are going to a complicated expression we start our journey from a simpler one it's like you know you are adding a flesh to the skeleton and then you are adding let's say organs to it and then you are adding skin to it and finally you are making a human body so you start with the skeleton and see what all changes can you make in order to finally reach this stage okay so first as you can see from your observation it is x minus one the whole square so you are replacing your x with x minus one right the moment you do that I'll just uh you know take you through the stages so the moment you do that initially it was like this the moment you replace your x with x minus one the whole square I should have written it in white actually but anyways your graph gets shifted one unit to the right like this okay just one unit yeah to the right okay now I want to get this three over here so how will I get that three uses are simple change your y with y minus three okay the moment I do that automatically this minus three will come on the other side to become a plus three right so when you change in your any equation if you change your y with y minus three it basically goes three units up isn't it so this graph white graph will now go three units up that means it will be like this in the air okay in other words the vertex will come at one comma three so whichever option whichever option says your vertex is at one comma three that will be your right option and of course it should be opening upwards this cannot be my option because the vertex seems to be at minus one comma three this cannot be your option because the vertex seems to be at one comma minus three this is your option my dfn which all of you have said in fact most of you have said so c is the right option and by the way this also cannot be because here your vertex is at minus one comma minus three okay so option number c is the right option well done okay so with this we are closed the today's session I think some of you have a doubt what are the difference between a quadratic polynomial and a quadratic equation it's a very big difference equation is where you are trying to solve for that particular quadratic polynomial let's say I basically give you this okay and I basically give you this okay so normally this is a quadratic polynomial okay where you are basically replacing this with a y because you want to plot it along the y axis okay so this entire value of x square minus so see every point here is what it is x comma let's say I'm assuming that this is this equation only which is not the case in this case okay so you are basically plotting x comma x square minus 3x plus 2 let's say I'm just assuming that this graph is the graph of this quadratic polynomial but when you say quadratic equation my dear basically it is the meeting of this graph so this thing is basically where do these two graphs actually meet this and y equal to 0 are you getting my point so meeting point of the quadratic polynomial with y equal to 0 basically makes a quadratic equation okay and you're trying to solve this now this can be y equal to 0 this can be anything that depends upon whatever number is over here or whatever term is over here so here you are solving for x to get the value of x for which this meeting will happen but this is not a meeting kind of a thing this is basically a relationship between x and y that's the difference between a polynomial and an equation got it setu is it clear okay a sharduli says y is negative which is not possible okay that's also a very smart way to address the problem okay all right