 So graph is nothing, but it's a relationship between a Independent variable which we call as x With a dependent variable which we normally call as y Now many people say sir if I write x in terms of y then x will become independent and y will become independent See, it's just a name given to it. Don't be like, you know too much into the the fact that Which is the subject of the formula over here? Why we call it as an independent variable is because many a times This kind of relation even though I've written it as an F read this not as a function read it as a relation or Read it in your mind as if there is a machine mathematical machine Okay, think as if there's a mathematical machine, okay like this which I'm calling as Machine F Okay, this machine You feed in x as an input Okay, and depending upon what is this input mark the word Depending upon what is this input you get an output? Correct. That is why we call y as a dependent variable and x as an independent variable Yes, yes, yes, no Shashwath See you may write some of the points that is fine But don't be like, you know busy writing busy writing and you know you're missing out what I'm saying Okay, it is put on record and PDF will also be shared with you. Okay, so pay attention to whatever is going on Okay, so what we do is when we are plotting such kind of a relation or we are plotting some kind of the dependency of these two variables with each other on A graph that is what we call as the on on the x y axis That is what we call as a graph. So what we do is We normally plot the output Okay along the y axis and The input along the x axis. Okay, and this plane is called a R square plane Okay Now what is this R written with a double stroke? Many people are getting confused. Okay. This actually is you know a Representation for real numbers Okay, normally there is a single stroke R also But we use this for relation Okay, you'll soon study a chapter in 11th called relations and functions So there also will come across R. So in order to distinguish between the two Rs I have put a double stroke for this to signify real and One single stroke R will be for relation. What is an R square plane? It's a two-dimension plane Which says along both the dimensions Okay along both the dimensions you will have real numbers So basically x and y both will be real in character That means I'm only learning to plot those relations Where my input and output both are real numbers Now many people ask me said there are complex numbers also We can also put complex numbers into some kind of machine and get some output Definitely you can but those are not going to be plotted on this plane, which we call as the Cartesian plane Okay Cartesian plane is a name which is given after the famous French mathematician René Descartes The pronunciation is this The spelling is René It is written as Descartes, but pronounced as Descartes. Guys be very very, you know Spot on with your pronunciation. Tomorrow you are going to appear for an interview in KBPY Right, I'm sure you are going to clear the written down So there when the professor is talking to you, you cannot use a wrong Pronunciation they they will get completely pissed off with the wrong pronunciation, right? There's something called, you know Some people in yeah, how do you pronounce this? How do you pronounce this? People call it format, format, format. It's not format. It's phoma, phoma There's something called you'll come across this soon in your limits chapter How do you pronounce it? It's not L hospital. It's l'opital. Yeah, it's French. It's French. It's French. Yes There's something called. Yeah, René Descartes Okay, so these are the pronunciations that you will be you know, you know people will Try making opinion about you. Okay in the interview. So don't don't I know mispronounce all these things Yeah, so basically I'll be plotting only those functions Which will take real input and Which will give you real output Those functions I'm going to plot on this Cartesian plane and Those functions will be called as the real valued functions. I will write it down. They are called the real valued functions Okay, so whatever function chapter you will be having, you know in your class 11 That's matter in class 12. Also, you have functions chapter. We'll be mostly talking about real valued functions What are real valued functions? I'm again repeating the functions which take in real input and Give out real output. Is that fine? By the way, we'll come to know little later on in our math curriculum. This input belong to a set which we call as domain Okay, so if you hear this word from me accidentally, don't be like, oh my god, what did sir say? I don't understand what he said Domain is a set from where you choose your inputs Okay, that means this machine may not work for every real number Right, there may be some inputs which will not work in this machine. For example, let's say I make a machine called 1 by x Right. See just for you to understand in layman terms. I'm using this word machine Well, you don't find this you'll not find this word in books. Okay So as a did you see the first code that I have put on my the first slide? Maths is to make things easy for you not difficult. Oh No, no, no, no, it's not to do with ordered pairs. I'll come to ordered pairs little later on in our course Think as if you are choosing your inputs from a given basket That basket is called the domain basket. Okay, for example in this I can put everything, but I cannot put zero I cannot put zero. It will become undefined. So that basket will contain all real numbers except zero Okay, so this becomes the domain of the function Right from a given set where you choose your inputs that is called the domain in plain and simple words So when you're drawing a function Let me tell you all the x values that you are having in the span of that function That will be a part of the domain. For example, let's say I end up getting I may end up getting a graph of some function like this Let's say hypothetically hypothetically. There's a function. So this function I'm drawing it only between a and b. So your domain is all real numbers between a and b That means left to a right to be you cannot put anything in the machine The machine doesn't know how to process those inputs Correct. Will you ever put your shoes inside a microwave? Microwave no, right? It is not meant to process and it is not meant to heat up your shoes, right? It's only meant to heat up food items. So shoes is not a part of the input to your microwave So microwave is a machine right What are we going to put inside? That is the domain inputs Getting my point. Whatever you get is called the range Okay, as a question. Yes, sir. Tell me. Yeah, so is it is it the inputs? Which are individual inputs which are called the domain or the full set which is called the full set is called the domain Okay, okay. So input is chosen from that basket which you call as a domain or the set which you call as the domain Okay, so this is a domain where all real numbers Except zero are invited to be put in the machine Correct, and whatever you achieve out of it. That is part of another basket and that is called the Range that is called the range. Now there are other things also involved co-domain and all but right now Don't worry about it. The bridge course agenda is only to make you aware of all the tools I forgot one important thing. I think I skip that slide Agenda for the bridge course. Okay, so this is the agenda for the bridge course. Sorry. I missed this out I'll just take two minutes and explain this. I'm sorry to Put that topic and the whole see bridge course is basically designed for a month so we are closing on May 1st to May 14 and Till that we'll run this bridge course The idea is to basically equip you with all the tools and skill sets which you need to immediately face your school. Okay, in fact immediately to face your You know a J level of problems or CT level of problems, whatever see what will happen when you go to school Two different teachers will be teaching you. Okay. Most probably most well it happens in NPS Hsr. So probably in Yashankul also it might be happening So now they will not teach you everything, but they will expect you to know these concepts In fact, I do not blame anything anybody for it. It's actually The magnitude of the syllabus that they will not teach you about very basics of things. Okay, so I have Ensured that before the school reopens you are Introduced to these chapters. What is the graph which I'm doing today and probably I'll do it the next class I'll try to see whether I'm able to finish it I plan for seven hours under this, but I don't think so Seven hours will do justice. It may run to more. Okay It will include graphs of various types of functions algebraic trigonometric logarithmic Exponential modulus floor function fractional part ceiling function and not only those will also talk about Some basic transformations on these graphs. Okay, so what are the, you know, transformations these graphs undergo, right? So that we can plot a complicated graph made out of the same for the same function Most of the things would not be knowing right now. So just be patient during the course We will tell you most of the things the next topic that we are going to take is calculus Okay, you'll require calculus immediately in physics. I think you haven't had a session on Physics so far, but I'm sure the research is going to start with vectors. I guess and You'll also start learning calculus. They're applying calculus. So calculus is basically Yes, Sharon. This PRMO training will happen separately. It's not a part of this course Okay, so that that training is separate. So people who are preparing for PRMO RMO There is separate batch for that. I'll be conducting sessions in fact to shower sir and me will be conducting sessions together for that You can enroll separately You can join separately for that. Okay Yeah, so basic of calculus will be introduced to you so that you know, what is this sign a max vertical of math all about See calculus is a modern-day mathematics concept. Okay, it started with geometry Then we went to analytical geometry, which we call as the coordinate geometry Then algebra came into picture where we started dealing with the equations and slowly now We are moving towards calculus. Okay, calculus is a science or you can say a mathematical science which deals with Continuously changing quantities Okay, and most of the quantities that you will come across in your day-to-day life are continuously changing Aren't you continuously growing? Okay, am I not continuously becoming batter? Just a joke here. Okay, everything is continuously changing and how to study those Changes is what calculus teaches you? class 12 70% of the topics are calculus based 70 70% Let me write it in yellow 70% topics of class 12 is based on calculus and I don't mind mind speaking over the camera, but let me tell you CBC curriculum is in fact any curriculum in India is very badly designed Okay, let us CBC secretary or board secretary hear this There is no logical flow of topics between different verticals physics. You start vectors in Class 11 max vectors is in class 12 Physics you start using calculus math calculus comes in the last few months of 11 There is so much of disparity. There is so much of disconnection between subjects Go and see the international books. They are so beautifully designed. I'm also a teacher for a levels and I be There the way they have you know carried out the flow is superb But in India, what do we do board exam? syllabus should be light 11 syllabus should be heavy Right just to make board exam syllabus light. There has been a illogical sequencing of chapters Okay In fact, if some of you go and you know have a later on if Agupaya good position in bureaucracy, please Remember a sincere request from a teacher make some change. Okay, even we requested but nothing is happening Okay, the last chapter would be a miscellaneous content Where we'll be talking about some important tools like determinants quadratic in equations and any equations Okay, this is actually a miscellaneous topic. So let's see time permits will be able to do that One more request guys whenever you are answering to something Make sure you are using the private this thing so you can communicate only with the the faculty members So I'll do one thing. I'll make this only with the host. Okay, so if you're speaking something that is only with me Okay Well, so coming back to Yeah graphs So what do I mean by this particular thing is that this function is only allowed to accept inputs from a to b So this entire Values is called the domain Okay And the graph Span along the y-axis. Let's say from here to here Okay, this span is basically what we call as the range Okay, so somebody says what's the domain of this function graph that you have plot? You should see on the x-axis from where to where is your graph spanning? That span is called the domain If you see along the y-axis from where to where the graph is spanning that is called the range of the function Okay, don't worry too much about it separate chapter is devoted to it. I'll talk about it So what are we going to learn my dear friends here is to first to draw? Algebraic graphs or graphs of functions, which are algebraic in nature. Okay Now before I go to algebraic graphs There is a small in fact under the algebraic graph. We'll talk about first the graphs of polynomial functions Now I should not be using the word functions right now because you don't know officially what's a function Think as if it's a relation Okay, but before that I would like you all to tell me what's a polynomial? What is a polynomial? Who will unmute himself and explain what is a polynomial? so it's a algebraic expression with With a degree not in fractions not in decimals Okay, not only degree But all the all the terms in the polynomial all all the terms in the algebraic expression has to have a degree only in Only in natural Numbers whole numbers. I'm sorry. Yes. Yes. Very good. So what basically charan said that A polynomial is a expression I'll mark my words as I told you you will be judged on the way you are going to use these terms So polynomial is an expression in single variable You cannot have two variables in a polynomial So either you use x or use y or use t or use any 26 alphabets. Okay, it should be in a single variable And its structure should be like this Let's say I'm using a variable x. That's why I'm using the word p of x Just to you know write something like f of x. Okay p for polynomial x because I'm using x as my x as my variable So a polynomial basically is something which looks like this It should have The same structure that I'm writing over here Okay Where none of these powers None of these powers that you see on this variable must be Negative integers if nothing is there you can write zero also These powers should only be whole numbers. They should not be negative integers. They should not be fractions Correct. So these powers or you can say the index Index of the variables Okay, should only be belonging to whole numbers Okay, so two things to be noted it should have this structure It should not be any deviance from this structure And none of the powers should be Negative integer or fraction It should it can only be Whole numbers. It can only be non negative integers. So as to say Okay, take some example. Let us say x cube minus 3x square plus 4. Is it a polynomial? Is it a polynomial? Yes, because its structure is matching with this And moreover, none of the powers are fractions. None of the powers are are Negative integers everything is a whole number. Okay, so this is a polynomial But is this a polynomial 1 by x cube minus 3x square plus 4? Is it a polynomial? No, it's a rational function, but it is not a polynomial. You cannot call it as a polynomial Polynomials are types of rational functions. Now, what is the rational function? Somebody asked me in the previous class See rational functions are basically functions of the nature A polynomial by another polynomial Okay, so it's a ratio of two polynomials you can say Okay, so this is a rational function, but not a polynomial And all polynomials are rational functions getting my point So this is not a polynomial because its structure is not the same as what I have mentioned Okay, I tell me this one x to the power 3 by 2 plus 6x plus 5. Is it a polynomial? Is it a polynomial? No, it is not a polynomial. The reason is it has got a fractional power. Okay Now, what is the degree of a polynomial? So degree of the polynomial is defined as the highest index Or the highest index occurring in that given expression. So here your degree is n And that is subject to the fact that the leading coefficient is not zero Okay This is yes highest power present in that expression Provided this term guys, I'll be talking a lot about this term today This term is called leading coefficient What it is called Leading coefficient Because it leads the way like a leader. Okay. Now, I'm not joking Yes, it definitely leads the way the graph looks Okay, I'll explain you in some time why it is called a leading coefficient and how it leads the way Excuse me, sir one doubt. Yes Sir, yeah, this leading coefficient, sir So, uh, this leading coefficient does it only come as a coefficient for the highest number that uh, which the The variable which consists the degree of the polynomial or the first coefficient found in any quality This is the coefficient of the highest index term That is called the leading coefficient Let's say charan I write it like this Minus the x plus two x square this will be the leading It doesn't mean it is coming first or last It is the coefficient now. He used the word coefficient. I'll tell you everybody these numbers are called the coefficients Okay, out of these coefficients the first one which is present with the highest power That is called the leading coefficient Is that fine? Okay Now just a quick question for everybody Just a quick question for everybody What is the degree of? Quick question What are the degree of A polynomial six? What will you say? Type privately to me everybody please Six what is the degree? Oh my god. I'm getting various responses All the way from somebody who is saying six from one from zero Okay Some of you. Oh, it's not a polynomial. Okay. I'm getting various responses Now here comes some kind of unlearning and relearning Guys, let me tell you one important thing which I wanted to tell you There are many things which you have not learned correctly Intent and that is not because of mistake of yours Okay, you're going to unlearn them and relearn them Okay So be prepared to unlearn certain things Now this is actually a polynomial Okay, so people who are saying doubting that it's polynomial or not. Let me tell you it's a polynomial and its degree is zero Okay, because you can only express it as Six x to the power zero so degree has to be zero But now my follow-up question is what if I have zero itself? All right, is it a polynomial and what is the degree of the polynomial? Okay Hariharan, Tanishka, Madhav, Arjun, Charan, Andy, okay, not a polynomial Okay, okay Okay, now here comes the verdict my dear Yes, it is a polynomial, but its degree is not defined So Charan was absolutely correct Because when you write zero you can write it along with any power. I can write power of two I can write power of six. I can write power of hundred. I can write power of one lakh It will work with any power. That's why we say even though it's a polynomial the degree is not defined No, no, no, no infinity is a different thing altogether. That's that's something which is again Uh, you know part and parcel of this bitch goes There's a difference between the word not defined and infinity Okay, what are the difference between not defined and infinity? Not defined means That operation is not mathematically possible Right, so this is not a legal operation in maths. So example one by zero Right, you cannot Perform such an operation That's why we say this is an undefined operation in maths But when you write one divided by a quantity x Where x is very very small on the positive side We say or you can say some constant divided by x where x is Approaching towards a very very small positive value Then we say this this takes a very very large value And that large value is a value which is too big to represent. So we use infinity symbol for that But this operation is a legal operation Right, even though the result is too big to manage That is still a legal operation Okay, yes, I've used limits concept my dear So I'll be talking about this in the limits chapter, which is your basics of calculus. Don't worry. Let's have patience Okay, so I'm going to begin with The basics of all polynomial Not the constants. I know you know constants. By the way, I'll just revise this also constants How do you plot a constants? So if I ask you Represent this graph y is equal to 2. What do you do for that? What do you do for that? What does it mean actually? There is no independent variable over here. Everything is, you know Just a constant Right, what does it mean? It means that Whatever you put in this function this function is always going to throw out to at you Correct. So if you plot this or if you show a diagrammatic representation of a machine which does this activity You will represent it by a parallel horizontal line I can say a line parallel to the x-axis Which will always be having y value s2. That means it'll cut the y axis at 0 comma 2 And this is how you represent it Okay, ensure that whenever you are making a line you put arrows at the ends That so shows that the graph is moving indefinitely to both the sides Okay, so just you ask just to ask you a simple question What is the domain of this function my dear? Anybody domain of this function? How do you represent everything in maths? Didn't I tell you it's a r square plane? All right, it's all real numbers very good aditya hariharan very good Okay, one more question I'll ask what is the range of this function? Right, you'll say it's a set which only contains two Okay, don't worry sets is also a chapter. We'll come to it Okay, so I'm not going to talk about you know constant graphs. You already know it I'm going to directly jump my dear to linear graphs or graphs of linear polynomials I should not say linear polynomial. I should say linear functions. Okay Okay, linear function graphs So I'll start with the very basic one which is y is equal to let's say x plus 2 Okay, why let's say y equal to 3x plus 2 Okay Now how many of you have been introduced to lines in class 10th? Okay, I'll again run a poll poll is type Yes, if you have been introduced to lines before type. No, if you don't know lines at all At least the basic equation graph concept of slope intercept Wonderful everybody. Please vote everybody. Please vote the count of five. I'll stop the voting five four three two one Stop Okay, so 23 of you have voted out of that 18 of you have said I know it Five of you say I don't know it. Okay. Never mind. I'll I'll go by the very basics see uh This is something which we call as a linear graph. Now this name has come because of the fact that This graph the graph of such functions is a line Just like this. This is actually also a linear graph. You can attach a zero x to it. Okay So this is a line. This is a graph of a geometrical structure which we call as a line And that's why it gets its name as a linear graph. Okay now here Many people know how to plot it Okay, people who don't know how to plot it. I'll discuss with you how to plot a function in a slightly unique way Okay, you will not have heard of this method before I always start my graph with the constant term. So what I'll do is all of you please pay attention I will make a coordinate Axis so basically x in the y-axis All of you please pay attention I always start with the constant term two Okay, so I express zero comma two on this graph That means I will get a structure or I will get a curve which will be hinged to zero comma two Hinged means attached to zero comma two. Okay Next what I do is I introduce the next term three x So see I'm coming from constant side. So slowly after constant. I have brought the three x term Okay Now these three x term actually makes the graph depend on x because now x has Made its appearance Okay Now the leading coefficient here is Three three is your leading coefficient Okay, all of you please pay attention I know most of you know that three is also the slope I'll come to that little later on just listen to me So because of this three x my dear It gives a linear dimension to the graph. That means my graph will start becoming like a line Hinged to zero comma two Now this line can either be like this. This line can either be like this Right, it can be so many ways. It can be either be like this. Let me choose a different color Uh, it could be either be like this blue one Okay, it could be either be like this red one Okay, now how do I identify that because of this three x what is the direction that the line is going to take Who is going to lead the way? That's the leading coefficient. I was talking about leading coefficient leading the way So that's the leading coefficient which actually gives me an idea Where the graph is heading towards? Okay, I know for sure it is passing to zero comma two, but where is it heading to Only the leading coefficient will show me the way How let's see Now if you see here If the leading coefficient is positive If this leading coefficient is positive. So this three is a positive number, right? It tells you that If you increase your x, let's say if I increase my x Okay Constant is not going to you know contribute If x increases will y increase or decrease? simple question If x increases will y increase or decrease? No, it's not a hit-and-trial I'm going to tell you a very very in a logical way. It's going to increase. Correct very good Now it's because thanks to this leading coefficient I'm thankful to the leading coefficient for telling me that x will increase When y sorry y will increase when x will increase correct So it tells me that Boss this cannot be my answer These red graph neither this White graph these cannot be my answer why these cannot be my answer Because for these graphs as x is increasing y is falling down See it was like this, right? So if you increase along x y is falling down Are you able to see my hand on the screen? Okay guys in today's class. I'll keep on dancing on the on the camera. Okay So please excuse me for that because I always love to teach in a physical class So this technology even though it is helping me to reach out to you in this corona crisis Is actually, you know curtailing my attempts. Okay So the the blue graph and the yellow graph are the only two options which is left to me now Which of the two graphs I should choose again? That's a dilemma Now that is also going to be rectified. Thanks to this leading coefficient. See how What I'm going to do is I'm going to increase my x with a delta x Okay What is delta x? What is delta x? anybody Change in x the rate is change Don't use the word rate again the long use of technology. It's the change in x Okay, rate means divided by change in time that will be rate dx by dt if I do then you say rate of change of x Okay, just like rate of change of displacement. That's called velocity right rate of change of velocity is called acceleration anybody knows rate of change of acceleration also Jerk very good. Okay very good Now jerk may not be a very good word to use. Okay. Anyways, so this basically is uh telling you that if I Change my x by delta x. That means let's say I'm somewhere on this graph. Okay. I'm just going to you know extend this line Okay, let's say I'm somewhere on this graph with some x comma y on it What is graph graph is basically set of all points Which are satisfied by that relation. Isn't it? That's what is graph Right, nothing else. So let's say x comma y is such a point which satisfies that given equation Let me take 1 comma 5 for your convenience. Okay Now if I change my x If I change my x and come to del x plus delta x, let's say I come to x plus delta x And this will also bring a change in the y So I'll come to this position x plus delta x and y plus delta y Hope you can see this Okay, so this is what we call as change in x. This is what we call as change in y So if you move from point a to point b on the line, there would be some change in the x value To reach the point b and there will also be a change in the y value. Isn't it? Okay, so this change is what we can come to know from this leading coefficient. See how So even x plus delta x and y plus delta y will satisfy the same relation yes or no Subtract these two. So let me call it as one. Let me call this two. Okay. Do two minus one What does it give me? Tell me on the left hand side. I will get oh, I'm so sorry. This is y Sorry slip of pen. This is why So if you subtract it, you'll get a delta y Correct If you subtract 3x and one of the 3x goes you'll get 3 delta x Correct two and two will get cancelled So I'm left with this relation Which tells you a very very important thing It tells you that if you're moving along that line the ratio of the change in y To change in x is three What does it mean It means that for every one unit change in x The y is changing by three units Three times change it is experiencing That means its steepness Should be more Are you getting my point? So it's like you're walking one kilometer in this direction. It is raising you by three kilometers Right, that's why this leading coefficient three here actually based Actually works as the slope of the line Yes, so Raj it's the slope of the line. We also call it in the language of Calculus as the gradient Okay gradient and slope both are the same thing Okay, now if there is no change in y when there is a change in x what do we say gradient is Zero for example x axis if I go x axis is also a line no at the end of the day It's also a line. So let's say if I move from x comma y To a point x plus delta x there is no change in the y Isn't it flat so knows the slope is zero the gradient is zero If there is infinite change in y with respect to a small change in x We are basically walking along the vertical direction almost Okay, and that's why we say Sorry, let me write it over here. That's why we say slope of x axis is tending to zero And slope of y axis is tending to infinity How many people ask me? So why you say tending to zero and tending to infinity? See, uh, when you say slope Basically it is with respect to some reference You cannot tell the slope of Line with itself You have to treat something else as a reference So here when I'm saying I'm treating x axis as a line Normally and I'm treating x axis itself as a reference So I should say this change or this particular, uh, I would say A deviance in the choices is represented by tending to zero Okay, later on you will learn something about product of slopes being equal to minus one when they are perpendicular There you will realize why I use the word tending to zero Whereas y axis has a very very large slope Right. Yeah, actually I think that answers your question. What is tending tending means Going very close to zero, but not achieving zero even though for practical purpose While solving questions you can safely assume the slope to be zero for an x axis But in reality it is very very small Okay, so yeah, somebody's raising the hand anybody any question Okay, guys, please feel free to stop me At any time where you want some kind of clarification very good Okay, so this three that you have seen which is actually your leading coefficient was also your slope Okay, now something very interesting If you have a line in this way They will always be called as a positive slope line Because there is an increase in y with the increase in x If the line is in this way, we call it as a negative slope line because With the decrease in x there is an increase in y Okay If let's say if you increase your slope, it will become more vertical in nature. It will become more steep in nature The moment it becomes parallel to the y axis we say that the slope is infinity Right that doesn't mean that if you're coming from negative side in order to become y axis your slope has to be negative infinity No, it is still called infinity Right, so many people, you know ask me this question sir is plus infinity and minus infinity meeting each other You may say so Okay, especially for a graph which is following a linear characteristic Okay, so now here Yeah, now here try to understand when your graph becomes more negative in slope It is going to start tilting this way. Hope you are able to see the motion of my hand on your video The moment it becomes here. We still say the slope is tending to infinity Okay, we don't use the word minus infinity in this case Uh, which part you did not understand uh enough See actually what happens I'll show you in the next graph Let's say I have a line like this Let me show it in uh gray color. I have a line like this Okay Now this has a negative slope. Let's say the slope here is minus four Okay, what is the meaning of minus four slope? So if there is a one unit change in x, there would be a four units change in y Okay If I start tilting this line in this way, let's say I now tilt it like this Okay The slope will become more negative. That means let's say it would become let's say minus 10 Okay, so if I keep on more increasing, let's say I choose now this orange one like this like this Okay, here you probably will have a slope of let's say No minus thousand If you keep doing that ultimately when it becomes vertical that means like this vertical parallel to the y axis Will you say the slope has become minus infinity? Actually, we never use this word even though you are logically going more negative more negative more negative more negative But ultimately when it becomes y axis, we say it is tending towards infinity only Because there is no direction of a line, right? Even if I had gone, you know You know more positive, I would have used the same position Right, even if I would have gone negative more, I would have used the same position Okay, so we don't use the word negative infinity slope positive infinity or tending to zero is what we can use All right, I'm talking about lines not talking about quadratic equations Okay, and that too you're talking about gradient hariar Okay, yes, there are negative infinity words used. I'm not saying you can never use negative infinity But not for the slope of a line. Okay. Hope I have clarified myself So it tells you one thing that This number that you see signifies where it is basically cutting the y axis. That's why we call it as the y intercept This number basically signifies the slope which gives you a rough estimation Of how much is the line inclined and which way this way or this way? Okay, now like you to take up these two questions everybody Okay on your notebook now time to open up your notebook and Do a sketching of this graph. Now, what is the meaning of sketch? People ask me how is sketch and plotting different sketch means rough estimation Okay, you have to give a rough estimation of the graph Right Plotting means exact point by point. So for plotting you would need a graph sheet for sketching You would not need a graph sheets. So I am assuming that you are sitting with your notebook Just a pen you could require have If there's a scale it will be helpful, but even scale is not required Okay, now I would like you to sketch the following graph y is equal to minus half x plus three Y is equal to let's say five x minus two Well, let's say I keep the last term as my plus three always And I would ask you to plot y is equal to Minus six six plus three. Okay on the very same graph I would like you all to draw these three figures for me quickly Uh, can I just have 90 seconds for this not more than that times us now Let me know once you're done Guys, I want rough. Please do not sit and start making Uh Calibrations on the x-axis and the y-axis roughly roughly That's why I asked you to make it on the same graph so that I I know that you are understanding the relative slopes and all now. I may ask somebody to Scribble on the screen. Hope you know how to use your annotation tool. Okay. There is an annotation tool on the On this zoom platform by which you can also scribble on my screen. Okay, please do not misuse it Don't start scribbling on the screen because I can come to know who is scribbling So I'll name a person and that person only will scribble Yeah, there is a pen kind of a structure. I'm sure you can see it in uh, I don't know what you are seeing on your screen, but uh On my screen it is appearing like a pen. Okay Annotate it is written annotate there Akshash, did you figure that out annotate? Uh, anybody who has seen it can can you guide Akshash? Okay, let me just put it for everybody to communicate Akshash, you found it No, Anurag found it I don't think it's here. Okay. Never mind So only those people who can see the annotate they will be under scrutiny Yeah, there's a pencil button in the screen. Plot it anywhere. No, just let's say I call this a 0 comma 3 Okay, rough estimation. Okay, not very accurate. Okay, so let's say this is my 0 comma 3 Yeah, if you're using a mobile phone, then you can see it as a pen coming up on top Okay, anyways I will do one graph for you And relative to that you have to show me the other two graphs. Okay So I'll just draw the first one for you. It's a graph with a negative slope that we need to go this way And half signifies that The change in y is one half the change in x. So it is not very high negative So you can say somewhat like this Okay, by the way, guys, there's something which I wanted to discuss When you see slope Slope of a line Is basically the rise overrun. Isn't it? So slope is nothing, but it's the change in y by change in x. Okay Now this change is basically represented by an angle. So in this case the angle is basically Let me just Extend this a little bit more Basically, this is the angle by which we represent Something which we call as the angle of inclination Angle of inclination I'll talk about it in the straight line chapter What is this angle of inclination? It is basically The positive angle it is defined as the positive angle. Please make a note of it if you want It's a positive angle between the x-axis And the given line Okay, that is called the angle of inclination of a line Okay, and this angle has to be measured starting from the x-axis Angle has to be measured starting from the x-axis Okay Now what is it? I'll explain you in this diagram. So if you start from x-axis And go in a positive direction. Now, this is something which is may not be known to you There's something called positive angle What's the positive angle? Anybody knows it? Anybody knows what's the positive angle in mass? There's something called positive angle Absolutely mother absolutely angle measured in Anticlockwise is called positive angle. See what is an angle? Angle is basically a mathematical Parameter which is trapped between two rays Let's say this is a ray which is initial position and this ray moves to a terminal position like this This trap parameter here will be called as an angle Okay Now if this movement is happening anticlockwise, we say this is a positive angle So let's say my angle my ray was like this first If I move like this then this angle which is transverse by this ray will be called as a positive angle Okay, if you go this way that means from this position to this position if you go So this is your initial position And this is your terminal position Then this angle will be called as a negative angle. So this will be a negative angle in mass Okay So angle measured in anticlockwise sense is positive angle measured in clockwise sense is negative Okay, so when I'm starting from x-axis and going in an anticlockwise way Anticlockwise way To hit the line then this angle will be a positive angle which we call as the angle of inclination Okay Now this angle of inclination tan is what we call as the slope as of now is just important to know this I'll be talking more on it in the lines chapter Meanwhile people who have learned how to on annotate who is going to come up and do the second one Anybody, Charan would you like to try it here? Show on the same graph Okay, by the way, I have missed out. This is now my 0 comma 3 Okay, let me assume this to be 0 comma. Yeah, doesn't matter. Yeah, can you show me on the same graph? How would 5x plus 3 look like Vedha smutha I would request you to unmute yourself because I have not understood your question Sir, I just want to know how like you just you just extend the line line from the x point from the y From the y garden from the y point from the y-axis to any point in the x-axis A line is an infinitely extending figure. So it's not about from one point. It can pass through see What's the line? line is basically It's a infinitely extending figure Which actually Goes from One point to another without any change in the direction Okay, so you can draw a line from anywhere to anywhere Anyone you can draw it from anywhere to anywhere. So what's your actual question? No, sir to draw the slope How do you lot rise in the run and No, no, I don't draw the slope there. I don't draw the line Slope is calculated for that line Okay, for example, there is something which Charan has done Charan This is not correct dear See it is a Right, I'll tell you it's a positively sloped line That means see your line is saying that as x increases Okay, y is decreasing Is that the case with second question? No dear x is increasing Okay, so okay. Now. This is something for Vedha Please pay attention dear here. I'll give the answer to your question through this example So right now, this is not the right answer. So can you just erase it? Can you just erase it Charan? Or I will also erase it. No worries. I have the wherewithals Okay, thank you so much. Now. This is a line which will pass through 0.3 only And since this 5 so we don't plot the 5 thing it actually gives you an estimate Of what is the steepness of this line? How steep it is Okay, so plotting is only for a line only. So let's say I draw this line And I'll draw it at a very high slope of 5 climb means if there is a change of 1 Let's say I erase this part and light it down again somewhere else. Okay, it means that if I change my Change my x by 1 unit There will be a change of 5 units in this graph. This is how you figure out from the graph. Okay You don't draw a slope anywhere Okay, so a slope is a characteristic. It is not some which is not a graph Yeah So You can you can make a sketch of a person Okay, but When you are mentioning its features You show that feature in that sketch. For example, if you want to make somebody tall So you make the graph of the person or sketch of a person and show him tall in that graph So tallness is not what you are sketching. You're depicting it in the picture You're sketching the person image. So the same way line is what you are sketching And with what slope is what this number 5 tells you? So this is your slope that leading coefficient tells you okay Who will go for third one Something somebody other than charen. Just type me if you want to go for the third one anybody Okay, akshaj go for it By the way six o'clock. I'll give you a break Akshaj go for it go for it. Let's see Wonderful even your line is a little bit crooked, but I understood your intention here. It's absolutely correct It should be a negatively high slope. Okay. So see half is like this. No, so it should be somewhere More inclined in this so half is like this. The answer should be somewhat more inclined than this So he has drawn a line perfect and I'm just trying to show it on the line. So if this is one then this will be six Okay, and there is a decrease in the value of y with the increase in x and vice versa So if I'm going this side decrease in x there's an increase in y. So it is a negatively slope line Are you getting my point? So he has not drawn the slope. He has drawn the line With that given slope Okay, and never mind. It's okay. It's okay. So I will draw it for for him. So I'll just clear the annotation Thank you so much akshaj. That's absolutely correct So you are going to draw a line like this. I'm taking a different marker color. So let me take Let me take gray. Okay, so it would be a line like this passing through the very same point And ensure you are put arrows at the end Is that fine So with this, I think there is no doubt whatsoever uh regarding the The graph of the graph of a linear function. Yes, any question here? So I have a doubt about this graph. I didn't know how to draw negative. Yeah, can you just uh explain me the negative line drawing Negative it's very simple. Uh, uh, charen was the person right see charen positive so clients will be like this Hope you can see the motion of my hand. Okay negative so See, how do you identify whether it is this or whether it is this see If it is this it means as x grows y will also grow Correct, if it is like this it shows if x reduces then y will go Getting it. Okay. So if there is see what is what is the slope slope is the ratio of delta y by delta x So if this guy increases Uh, if this is for increase means positive Then this should also be positive that this slope will be positive But if ulta happens that this fellow reduces when this fellow increases then it will be a negative slope line Sir, I have one doubt. Yes If x is heading towards left side is it's increasing or decreasing It's decreasing In this side if you're going you are reducing If this side you are going delta x is increasing If you're going up y is increasing if you're going down y is decreasing Okay, sir. Thank you. What's the point. Okay. See line extends infinite is milling. Okay It goes it goes uh from you know one end to other infinitely. Okay, it doesn't stop So it doesn't matter if you're going from a to b or coming from b to a You will experience the same sign of the change So let's say if I'm going from a to b There'll be positive change here. There'll be positive change here also That means the ratio will still be a positive numbers here If I'm going from b to a there'll be decrease in y also There will be a decrease in x also That means still negative by negative will give you a positive Is that fine? Okay So what I'll do is I'll take a break All of you know need a break will resume at six Oh seven p.m. Okay. Just a 10 minutes break for everybody. Okay And before that I would like you to participate in a poll How did you like the pace of the class? So please comment on the class pace. Was it fast? Is it as desired by you or is it slow than expected? Honest feedback I don't know who's we know Giving what response? So don't think as if I'm I'll come to know by your response. Who is that one? Okay Honest response before we go on a break See, it's my first class with you. So I need to calibrate myself that okay I need to go fast or I need to go slow or I need to be With whatever pace I'm sticking Okay, it's good. What would fast 10 five seconds more five four three two one go everybody Some of you don't want to vote. Okay, never mind Okay, so 60% of you has said the pace is as desired eight of you are saying it's fast Okay, I'll try to slow down only two of you feel I'm slow. Okay So people who are thinking I'm so please bear with me, right? It'll definitely increase to a certain extent in some time. Okay So take a break. I'll mute myself and I'll shut off my camera also. Okay, stop sharing So now we're going towards Quadratic polynomial graphs that's that's very important Uh, I'm sure you have been exposed to quadratic equations. In fact quadratic polynomial also was in class 10, right? So we'll be not talking about quadratic polynomials polynomial graph Okay Now how does a quadratic polynomial actually look like so it's basically Something which has got this kind of a structure ax square plus bx plus c. Okay Where your abc are basically your real numbers It may be complex also, right? There is there are quadratic equations uh Okay, there are quadratic equations which has got complex numbers also as the coefficients But we are not going to talk about them. I will wait for complex number chapters to take that up For our uh, you know graphing purpose will assume abc to be real numbers Now this abc a should not be zero For the simple reason if a zero it will become a linear polynomial So why should I call it as a quadratic polynomial so that that uh Distinction has to be maintained between linear and quadratic and that's why a should not be zero Now uh something very uh, you know Trivial not for you to uh, read about There are two types of quadratics that we normally talk about when it's called a pure quadratic Okay, what's a pure quadratic a pure quadratic? Is one where not only a is Not zero, okay, but also b should be zero So a should not be zero, but b should be zero C may or may not be zero Right example Example something like x square minus two. It's a pure quadratic Okay, we'll be talking about pure quadratic in some of the chapters like integration next year So we'll be learning how to integrate when they are pure quadratics in the denominator Okay This is just for your knowing purpose. There's nothing to be worried about. Oh, what's a pure quadratic? Should I do more research on it? No just for your Knowledge Then there's something called adfected quadratic Adfected not affected adfected adfected quadratic Is those quadratic where a should not be zero and b also should not be zero essentially remember in a quadratic normally A b and c could be zero right But in an adfected quadratic a should not be zero and b also should not be zero The word affected means varying power of x, okay A typical example could be something like x square minus five x plus six. This is an example of an adfected quadratic Okay Yeah, c can be zero c can be zero, but a and b both should not be zero for Adfected quadratic. Okay, just for your information. So mostly we'll be talking about uh adfected quadratic But we'll start with but we'll start our discussion with The most basic of all quadratic you cannot get a quadratic simpler than this guy y is equal to x square, okay And i'm sure you know the graph of this also. What are the graph of this this graph is basically a graph which is A curve which we call as a Parabula This is a curve which we call as a parabola Okay, now again pronunciation please be watchful I've seen people saying parabola and all those things is not like that. It's a parabola. Okay. What's the parabola? It's basically a path traced by an object when thrown under gravity So let's say if this is a digital pen, which I throw at you this will come Like this to you isn't it? Okay, so make up A Part like this just like you know when Dhoni hits ball out of the cricket park It takes a parabolic part. Okay the launch of missiles. Okay, most of them will be parabolic in nature Okay, of course, there are some which you know go in a very very straight line, but they will be parabolic in nature So this is a parabola and you can see the graph. It's basically having two arms both are extending towards plus infinity That's why I have shown them with arrow sign. It is it doesn't stop anywhere So whenever you are plotting a parabola, make sure the end point you should put two arrows signifying that they are still moving upwards Okay What is this point called? Which I'm showing with a white dot. What is this point called? anybody Origin Oh, that is origin for sure. What is that of a parabola? What is that color for parabola? No, no, no hero of the pollen organ No, no, no, no It is called the vertex of the parabola Okay Now it is or it is at origin vertex is at origin, but we call that as the vertex Okay Now this is the most basic of quadratic graph that you can ever come across That's why many times I call this as the skeleton graph for my quadratic Now this skeleton word is my own point word. Don't you not don't expect to see these words in your books So why I call it as a skeleton graph because I'll be doing lots of Experiments on it just like we do it experiments on skeleton in bio lab So now I'm going to take you to a tool which we call as the geojibra tool Have I have you been exposed to geojibra tool? anybody Yes, okay. So normally there are two graphing tools my dear, which I would like you to download after this class Okay, one is called geojibra You can download it from geojibra.org. So you can just go to geojibra.org and download download Geojibra classic six Geojibra classic six Now coming to resources, this is some one of the very good resource that you can have for plotting things If you're using your phone, this is for laptop primarily laptop or desktop Okay If let's say you are using your phone, then I would recommend something called desmos Desmos, I'll show you how it looks I have it on my phone I like I like plotting graphs a lot. I can't solve any question without plotting its graph. So almost 90% of the questions I solve I plot the graph Okay, so it will look like this Okay, hope you can see on the screen. Okay Uh, I'll show you the I'll just close it and reopen it once again. I'll show you the logo also Yeah, this is the logo. Okay. This is the logo desmos you'll see Okay, and then immediately the x y coordinate system will come. Hope you are able to see Yes, yes, akshaj y equal to x square is a parabola the graph the structure is called a parabola Okay, now, where do you see parabola in your daily daily life? Can somebody tell me? Of course Paces containers Okay, you have some containers which are parabolic in nature Uh, you'll see parabolic mirrors headlights Correct in your car, especially if you see those older Toyota cars From the side you can see the mirrors are like this Is it right nowadays? We don't use spherical mirrors. Those used to be in ambassador cars. Okay, probably Uh, your parents would have seen it. My dad used to have an ambassador car So now spherical mirror is not used anymore because of the spherical aberrations We use parabolic mirrors nowadays. Okay. Now these mirrors have a characteristic see like this There is a focus over here Okay, my this hand is focused There's a bulb kept at the focus When it hits the parabolic surface it becomes parallel. That's why they're used for for Parabolic so it's also used in operation theaters to you know, uh kill stones in kidney Okay, let's say if I have a kidney stone, what we'll do doctor will do He will keep the he'll place the mirror in such a way that the kidney is at its focus Okay, there's something called focus also. I'll talk about it in the conic section chapter And you'll pass the laser beam like this it will hit the surface and pass through the Pass through the focus and it'll kill my kidney stone Yes, uh, please note one thing. There is an example which people normally cite that is wrong example They say that if you hang a wire Okay across two points it takes the shape of a parabola. This is not a parabola This is actually a catenary Okay, so don't misuse these things. This is not a parabola. It may look like this, but it is actually a catenary So if I if a wire of uniform mass per unit length is hung between two points Okay, the structure of the wire Is actually a catenary not a parabola Hyperbola is something else hyperbola is I'll show you in some time. What is a hyperbola? Catenary is a cost hyperbolic function Kosh Kosh, this is called a catenary. Okay. I'll show you on the graph. Meanwhile, let me take you to the graph so I have plotted this. Uh, I've downloaded the uh tool on my system So I'll directly open to the tool Geogebra tool This is Geogebra classic six Okay Is better to uh, yeah, it's parabolic curve symmetrical to sedan to answer that question Yes, it is symmetrical about its axis Okay, I'll tell you what is the axis Yeah, so people who want to see a parabola Our typical parabola will look like this. Y is equal to let's say x square Okay this Okay, this is a parabola Here this line is called the axis So somebody was asking what is the axis of this line here Which is dividing the parabola into exact two parts. It's called the axis of the parabola Okay This point here is called the vertex of the parabola Okay, in fact I can show you on the tool itself your tool can easily show you the vertex of any conic. So just type bar It'll show you one Option click on that It'll say which conic. So conic. This is the name of the function f. So put f there So it'll show you zero zero and it'll also show you by a Okay, so a is the vertex of this parabola Okay, you can say a vertex to be The minimum most point of the parabola in some case when the parabola is upside down, let's say When your parabola is like this, I'm just drawing another version of a parabola. Let's say your parabola is like this Then it can represent the maximum point of the parabola. So vertex is either See they are different types of parabola. You're right now seeing only two types. Your parabola can be like this also Okay, that is called slightly tilted parabola. Your parabola can be like this also Okay, there can be so many directions. So you can just think that where The axis is meeting the graph that point is called the vertex Okay So it would be wrong to say that it is always the minimum or always the maximum No, it is not like that. It's the point where the axis is meeting the parabola. That is the vertex point Okay, right now my aim is not to tell you about parabola I have a dedicated chapter to that In conics where I'll talk about it Now guys, I would like you to understand the difference between a catenary and a parabola. I'll show you the difference What I'll do, I'll just add a one here. Okay, just permit me to add a one. I'll go up. Okay No worries. I'll come to that. Why it went up and all I'll talk about that Now I'll draw a catenary. Catenary is cosh Cosh now cosh is a function Okay, which you'll see when the moment you type c os You'll see a cosh here. You can see last second last option If you plot cosh see the difference, I'll move this graph slightly down I'll zoom in also Okay, hope you're able to see the difference. Sorry. My figure is coming. My photo is coming This graph that you see which is green in color This is what a Wire will hang when it is hung between two points. Let's say I I hang it about this two points. Okay Okay, let's say these are the this is the ceiling for you This wire will take this structure. So wire will hang like this wire hanging This structure is what we call as catenary in the language of mathematics Okay, it follows cosh hyperbolic function This is something which is called hyperbolic trigonometric function. I'll not discuss right about this right now people who are willing to Study about hyperbolic function you will be Discussed about this in uh pre rmo and rmo sessions. Okay, I'll talk about it in that. Okay So do you see the difference in the shape? Very similar, isn't it? They look very similar But there is actually a difference. This is your parabola while this is your catenary Okay So please, you know make a note of this because no, I remember this very well because they asked me in one of the math quizzes I went to represent my school in a math quiz So they asked me this question and I was surprised to know I learned there that it was actually a catenary not a parabola Okay, anyways, okay, this is not my uh Discussion agenda today. I'm going to discuss with you simple parabola first of all Before that, I'll just stop a normal parabola Okay Just a normal parabola y is equal to a x square Plus b x plus c Okay Now normally when you type such a thing in your jujibra jujibra doesn't know a value, right? You're just typing a there b there c there. It doesn't know a b c. What value is it? So what it does It sells assumes some value So it'll put some value. Let's say it has put one one one as of now But at the same time it will give you some liberty to change it Okay, so it'll make it'll give you some Range in which you can change your a and b values Okay, so people who are first timers to this tool I'm just telling you a bit about how it works. So you can vary it Okay, you can change that limit also for example, you may make it from minus 15 to 15 also I can make it from minus 15 to 15 also Okay, now it is going from minus 15 to 15 Now all of you please put down your pens I don't want you to have your pen in your hand also put it down Okay, and listen to me what I'm saying listen to me what I'm saying very important thing When you write the equation of a parabola Like this Our equation of a quadratic polynomial like this. This is your leading coefficient, isn't it? This is your leading coefficient Now somebody was asking me Does leading coefficient always represent slope? No, it represents slope only for linear functions for a quadratic function like this It represents something very important which we call as the concavity of the parabola A here is responsible I'm saying responsible for the concavity of the parabola What does concavity mean? See in this case it is concave upwards. Do you see this? Okay, if the parabola was like this, let's say hypothetically let's say the six that donates It's the parabola which is concave downwards Okay Now this guy is responsible for that Do you want to see how? If you change this a The concavity of the parabola Not only upward and downward it can change the breadth Also of the parabola length and you can say the Thickness and thinness of the parabola will also be governed by a Okay, all of you see how All of you see how I'm just going to you know erase whatever I've shown on the board I'm just going to take you to this particular slide now. I'm going to vary my a All of you please pay attention, huh? This a I'm increasing from one. I'm increasing more. See is it becoming thinner? That means a is Influencing the concavity of this curve Correct. So higher values of a it is becoming more thinner Can somebody tell me why? Why does it becoming thinner? Anybody unmute yourself and say anybody Tanik Shah Arjun Madhav Harry Kiran. Hi, sorry Harry Aaron Siddharth Shashwath Shares Anybody would like to take a call Gradient increase. Okay. Let's not go to the gradient level What does common sense it common sense will answer this right? See my always my teacher always used to say Supply common sense. Okay, then comes knowledge So common sense is something which God has bested you with It is a inherited knowledge Then comes acquired knowledge which the teachers give you so first common sense. What does it say? Okay. See common sense is I'll take a very simple example if I had Y is equal to x square Okay And if I had Y equal to 2x square Okay, I'll drop both the graphs here. Okay. Just roughly. I'm right. So red one is y is equal to x square Okay now At value of let's say one For x y should also be one Right. So if you look at this when x is one y should also be one easy to calculate Now for this guy when x is one Y should be two right correct That means for the same value I should hit the next graph that is y is equal to 2x graph. Let me change my marker Let me show it with some other color blue color. I should be hitting up At the same time for every value I should be going a little up, right? Okay for two this gives me four, but this will give me eight So for two, it'll come to eight. That means my graph should be looking like this Okay, I'm sorry for this, you know Bad shape at the end, but Roughly, you understand why it should be thinner Because for the same value of one now I have to go and get a two value and can only happen when my graph is more shrunk in other words When you have a quadratic And you increase this value Y will also is going to increase For the same x For the same x Yes or no If y is going to increase that means for the same x coordinate. I have to go a higher value to meet my requirements Are you getting my point? Yes or no, so let's say if I had y is equal to x square plus three And I got y is equal to two x square plus three At zero, let's say it gives me the value of three Okay For one it's let's say it gives me a value of four Okay, but the same guy for one will give me a value of five So I have to go a higher that's why the graph must be shrink So higher positive values means my graph is going to become thinner and thinner and thinner I'll come to b and c analog in some time I'm just playing with aim idea B and c I'm assuming they are not changing Here in the graph you would have seen I was just changing my a not b and c, okay Right now See something very interesting So you saw that when I increased it from 1 to 15 this guy became thinner concavity reduced Okay If you go the other way around that means if you decrease it'll become fatter See it's opening up opening up opening up opening up opening up the moment. I've become zero It becomes a straight line. Oh my god. Did you see that? Sorry, I missed out that mark Yeah, I I can actually put the value also here Okay That means the quadratic equation has lost its parabolic property. It has become a straight line now So see that transition. Okay. Now see here. See my hands High value of a Lesser lesser lesser lesser still positive. Still positive but lesser lesser lesser lesser the moment it becomes Zero it becomes a straight line Now continuing the you know trend if you go negative it becomes the other way around that it becomes Concave downward parabola Why it happens like this again, let's reason it out very simple see When you had ax square plus bx plus c and let's say this a was positive This a was positive If your x goes to infinity What will happen to y Please note this term is much more heavier as compared to the other terms So if I'm going towards infinity This term is going even more towards a positive value. So y is also going towards infinity Correct. So this explains why I just go back to the to the graph once again this explains why When I was having a positive value of uh, when I was going towards infinity y was also going towards plus infinity. Check it out. Okay Now even when you're going towards minus infinity That means you're going towards this side Since it is a square over here and since this is positive y will still go towards infinity That means both the arms have to go up in the air Correct, right? Which means if my a is positive my curve will always be opening upwards that is concave upwards Okay, now see what will happen When my a is negative If a is negative As you go towards plus infinity Okay, this will even go towards plus infinity. This will also go towards plus infinity But because there's a minus sign along with it Why will go towards minus infinity in this case? Getting my point Even though the other terms may be positive But this is the very heavy term which dominates the other two Getting my point So the leading coefficient is taking the lead. It is dictating terms to y It is saying to y boss. You will do whatever I tell you If you if I am negative You will go down negative Are you getting my point? In the same way if x is minus negative also see Observe this graph if it is a concave down Even if you're going towards plus infinity y is going towards minus infinity Even if you're going towards negative infinity y is still going towards minus infinity Because negative infinity square will still be positive infinity Correct and still minus sign is attached to it. That means it will ask it to go till again minus infinity That's why it goes down both the hands of the graph goes down Is this idea clear? Now I have tried to explain you in a very layman language Later on I will introduce calculus to you. Basically a is related to the double derivative of this function But this is something which I'll take later on. Okay. Don't worry about it. Okay. Don't worry about calculus part Okay So as a teacher, I always want you to understand things like a layman because you'll remember it for the rest of your life Okay, I could use a lot of jargons, but that will not make, you know, your life easy Is it? Now next thing is What does a roll? What does do? What does a do? I know it now concavity decides the concavity What does b do? If I change b, what do you expect the graph to do? Changing a you know shrinking expanding going towards negative whatever if I change my b What will happen to this graph? So Hari Kiran Hari Hiran is saying it going is will go to sideways Anurag is saying it should shrink shrink Anybody else? Okay Now you'll be very surprised to see what is happening. Actually all of you. Please Focus on the vertex of this graph Okay, it is actually going to perform Dance, I'll show you how it dances. Okay, focus on the vertex of this conic Okay, focus on a okay. In fact, I will on something called show trace You can see here if I click on the point, it'll go to an option called show trace Let's click on that show trace means if I change my b, it will show what what path a or the vertex is undergoing. Okay. So see here I'm increasing my b. It's going down. I'm decreasing my a is going to the other side Do you see that? What structure do you see? Let me increase the range of it. Let me make it 25 to 25 So you'll see that as you're changing your b Your vertex is dancing. It is dancing on what it is dancing on another parabola It is dancing on another parabola No, no, no concavity is not decided. Yeah, it's dancing on a parabola, which is concave downwards But if I do the downward parabola, let's say I make the parabola down and then I dance my b Then it'll dance on an upward parabola See here You see that Basically, it is dancing on a parabola, which is of opposite concavity to it Okay, this I gave us a question to the 12th graders a few days ago Asking them to get the equation of this path raised by that, you know, vertex Okay, if somebody is interested, you can find it out as no problem. Wait for the right time to come So my dear, what is the conclusion here? The conclusion is if you are varying your b The vertex starts dancing. In fact, the parabola starts doing this And this is dancing on a parabola Okay, how many of you are aware of the coordinates of the vertex? If not, note it down. It's minus b by 2a comma minus d by 4a Okay, what is d here a b c you can understand. What is d? d is actually the discriminant I'm sure you would have heard about it in your quadratic equation chapter. Okay So because now see now see the see the relation why it is dancing If you write it in proper way, it is actually Something like this that you have written isn't it? Let's say if you fix your a and you fix your c that means a and c are fixed a and c are fixed And you are varying your b You are varying your b then This varies And this varies as a result The both the x and the y coordinate of the vertex starts varying Okay, that's why they start dancing And they will dance in such a way that they will still be on a parabola One of the Kora Mangala students of class 11th I mean same same session I had with them. He actually gave me the equation the very next day Okay, if you can do that nothing like that, but I'm not putting a compulsion on you Any question here? Any question here? How do you know him? Okay, yeah Oh, he's in class with you. Okay good Oh last year Okay Now Next is how does c change if if c change is how does the graph change? This c is just responsible for moving it up and down like this Okay, see see the c For a and b here if I just vary my c See the graph is just moving up and down see the motion of the vertex Okay, so at whatever position it is it is just moving up and down So now, you know in in a quadratic equation What role these a b and c play? Okay, this is something which is very important for you to know again. I'm repeating a decides concavity B decides Or if you change b it is going to move On a parabola that means it is going to swing Right to left left to right up to down down to up Okay, that means it's going to move like this the parabola is going to swing like this C just makes it go up or down Yes or no Okay, now guys what we are learning here is transformation along with it up till now We have not discussed transformation. So now the first time I'll be talking about transformation Now this graph We have already seen This graph you have already seen y is equal to x square now would like you to tell me What will happen to this graph if I if I Let me change the color If I ask you to do y is equal to x minus 1 whole square. How would this graph change? Will it say wherever it is or will there be some small change in the graph? Please type privately to me. What is the change that the graph will experience? I'm waiting for a response No, concavity will not change the done A value has not changed my dear x square also has a one x minus 1 whole square also if you expand a will be one there. So there's no change in the concavity Okay, akshaj is saying you'll go down. Anurag is also saying go down Hariaran says move a little up and down Arithra is saying move right by a unit Mother is also saying right by unit Vertex shifts up by one. Oh my god. So many types of answer. Good to see you all responding very good Always keep interacting. Let it be wrong guys. We are learning here, right? I'm not making any kind of a judgment about your knowledge Now whenever you're responding You should actually do a bit of analysis If I were to answer this question, I will do a small analysis. Okay. What analysis I will do See originally The vertex is at zero zero Okay, that means when x is zero y is also zero But for this function to have y as zero x should actually be one Isn't it Isn't it that means the zero of this polynomial will now be one Earlier the zero of the polynomial was zero itself Correct. So what will happen this vertex will move slightly right by one unit So well done those people who said it will move right. You are absolutely correct But there will be no change in the concavity as it does so So every point starts moving one one one one one to the right Including the vertex Do you see that? Okay, so this is the transformation which I want you to appreciate Because you you will not be able to draw all the graphs Right, you'll not be able to plot all the graphs But if you know the skeleton and if you know what all transformations require me to reach it Then you can accordingly shift those graphs and make those changes to reach your final graph See akshaj Let's say when x is one over here, what is the value of y? One only correct But to achieve the same one you have to put x value as two Are you getting my point? So let's say this one value to achieve the same thing earlier it was having it was achieved at one But now it will be achieved at two Correct So it is like saying that this point has moved one unit to the right Okay, are you getting my point now akshaj? Okay, I show you on the the tour itself all of you please come to this tool So let me erase, you know all these unwanted things. So I'll reopen the graph all of you please pay attention, huh? Now what I'll do is I'll plot y is equal to x minus k the whole square Now again since k I have not specified Uh, this function will assume a value one. Okay, so let's say I'll put it to zero So now this is your x square graph. Hope you can see it. Okay If you increase your k to one that means you make it x minus x minus one see it is starting moving starting to move right Yes, it's basically the zero of the polynomial is changing Are you getting so if you move it to two it'll move two units to the right So people who said it will move to the right by one unit, right? I can see most of you giving that answer to me absolutely correct Okay, if you do the other way around that means if you go towards if you make it x plus one See it has gone one unit to the left. Now the vertex has come to minus one comma zero this point Okay, if you still move it, let's say plus two it has gone to the left again Correct. So here There is a rule which I want all of you to note down the rule is the rule is if the graph of Y is equal to f of x Now why y equal to f of x because this rule is universal rule It is not that it is only applied to a parabola. It can be applied to straight lines to conic to exponential functions to logarithmic functions to modulus functions Whatever function you can make in your life. This rule is equally applied to all the functions. So what is this rule? Listen to this the rule says If there's a function y is equal to f of x whose graph is known to you The tick mark means known to you Okay And let's say if you want to achieve the graph of y is equal to x minus h H being some positive quantity. So see what I'm doing. This is my f of x thing like that Okay, then this becomes f of x minus one Right, that means you're replacing your input From x to x minus one Then this graph will be nothing but the original graph shifted H units to the right H unit towards right. See the arrow which I am facing This is a universal rule So if somebody says Hey, this is the graph of two x plus one Can you tell me the graph of I'm just giving a very simple example for you to relate Can you give me the graph of y is equal to two x minus three plus one You say very easy my dear. It is just going to move three units to the right It'll come over here. Okay. So this transition is basically Three units every point starts moving three three units every point starts moving three three units Getting my point See h is a number right H is a number x is a variable x is a x x is a Coordinate representation of any point on it, but h will be fixed Getting the point tanishka So so don't get confused here x represents any points x coordinate So it's a variable If it is a variable you cannot say it is greater than the variable because variables still changing, right? Okay, so the agenda is very clear that if you're replacing your x with x plus h the graph will move Sorry, if you're replacing x with x minus h the graph will move h units to the right Provided your h is positive But if you're replacing it with x plus h the graph will move at units to the left At units to the left Okay, so what happens in x minus two whole square very good question it'll move two units to the right Correct, so I'll draw that so if it is x minus two it will become like this Okay, so this orange one is the graph of x minus two whole square But does this square matter there was already a square in the function, no Akshash, can you just unmute yourself and ask that question here? Ah sir, so like over here you have given The square is there like even in this y is equal to f of x minus h The square is already there. Yeah, and this function can be anything I'm just giving an example in this case it is x square but tomorrow it can be something like e to the power x also It can be log x also f of x means a generic name I'm given some some function that is what is the meaning of f of x It doesn't mean f of x. There's no x square inside. It doesn't mean it's not a quadratic f of x is a name For example student Now this student can be Akshash also this student can be Arnab also this student can be shashpat also So f of x is like a name given in general to any function Which has x as the variable in it. It doesn't mean x if it is x square. I have to put x square here No, it could be I can call e to the power pi of x also as f of x correct f of x is just like You can say a name given to a function Right. It can be anything any function in this world Is this easy to lose clear to you all of you? Okay, if this is clear to you, I have a follow-up question to you What will happen if I do this? y minus 1 equal to x square Okay, let me first draw y is equal to x square for you So let me just go back to my skeleton graph. See I'm playing playing playing with my graph. That's why I call skeleton So let's say this is x square Okay, normally I will use the same pen color to represent the name here in the graph Now tell me if I want x minus 1 the whole square, what will happen? What will happen to this graph? Hariharan says it will move up by one unit Now don't say just shift shift in which direction Advik Shift in which direction Madhav A left right I already discussed with you. So, okay, if you're thinking it's still going to go at left and right go ahead and write it Okay, let me have a poll here question. What will happen to the graph option a? shift Up by one unit Okay option b shift down by one unit option c shift right by one unit Option d Shift left by one unit Okay, I'll run a poll. Just wait for me to run the poll So here you go Only one option is correct. Okay I'll give you 30 seconds to answer this time starts now Everybody should answer please 30 seconds only five four three two one go end of poll Just 19 of you voted. I don't know why So maximum people That means Janta is saying option a That is 54 percent of you Then the next highest vote has gone to d. Okay. Let's see. What is the answer actually? So I'll stop the sharing of it See very common sense again. There's nothing above common sense When I'm writing it like this, it means I'm doing y is equal to x square plus one, isn't it? I just brought the one to the right side. Okay. Now see the trend Earlier when x was zero here again, let me change the pen color Earlier when x was zero here y was also zero right But this time my dear When your x is zero y is actually one that means The graph is now going the vertex is going to take a jump by one unit And if vertex is taking a jump Rest all of the points will also take a jump. That means every point will start moving up up up there every every point will start getting a kick up by one unit You get this point vertex is just an example Okay, that means your graph will be appearing to be a parabola like this And people who answered with option a Thumbs up to you. You are absolutely correct Think a bit and reply. Okay. So d and all c and all this is not going to happen. Okay Is that fine? Is my logic clear why it is moving up? So it is adding a one to every y value which it was having in the previous graph So every point is getting a kick Up by one one one one one right? They're all jumping up by one. Let's say they are getting into an excited state. Okay of electrons Okay So here comes another rule my dear students for you the rule is the rule is If you have been given the graph of a function that means the graph of this function is known to you tick mark Okay Then the graph of y minus k where k is some positive value Is equal to f of x will be the same graph kicked k units up Okay And if you have y plus k equal to f of x then it is the same graph Kicked k units down. Do you see that? Let me show you in juju graph Okay, now here what I'll do is I'll remove this function Now I'll go do y Minus k is equal to x square again. They're going to assume some value of k. So they have kept it as one. Okay. See, let me bring it to zero So this was your original graph, right? So if you increase your k value, that means y minus one it'll go up by one y minus two it'll go up by two y minus three it'll go up by three like this Similarly, if you go negative side, let me y plus one Down y plus two down y plus three down like that And This is applied to any function in this world not only quadratic. You can apply it to any function in this world Now I'm going to ask you a few questions which will be a mix of these two That means I'll change x also for you and I'll change y also for you And let's see who all are able to get those questions correct. So here comes the first question for you Sketch is sketch sketch means a rough estimate. Okay sketch y minus three is equal to x plus one the whole square Okay, and in that sketch, I want you to show me two things the vertex coordinates And where it is cutting the y-axis your time starts now. Let's have 90 seconds for this time starts now If you're done type done, okay, and uh Then I'll put some options. You have to take on those options. Okay If you want to reply with your answer, you can reply privately, but don't don't put it for everyone to see. Okay Okay, I can see answers coming from Hariharan uh, sharvi uh, shashvat aditya Cut set four. Okay, so say zero comma four. Yeah All right guys time up time up time up now it's time to vote For vertex, please choose the options. Okay option one one comma three Sorry option a option b minus one comma minus three option c one comma minus three and option d We put it up option d minus one comma three your uh polling begins Okay, one second your polling begins now Uh, just 10 seconds to poll because you've already done it everybody please press in the poll button for vertex It didn't come So I didn't get the poll once again once again. I'll I'll relaunch it earlier Now is it coming? No, sir. Are you? No. No, sir. The poll Just a second. Okay. Let it be I think I think there is some problem with the poll. Okay. I'll just uh Try one more. Uh, we just got it and it is close all of a sudden But those were the end results. Yeah, those were the results Now No, sir Never mind. So which option do you think should we correct? Just type it out type it out for me There is some issue with the poll. I'll get it resolved Absolutely option d is correct. Okay. Let's check how it happens. Let's check Let's check. Okay. Now here comes a problem where I have given both the transformations together Okay Now whenever you're trying to solve such question even in the assignments that you will be getting start from the skeleton graph Okay If you do your skeleton if you take your skeleton graph And start making transformations slowly you will reach your end goal. So let's say this is my skeleton graph Y is equal to x square. Okay. Now, which is the first transformation you want to do? Do you want to change x to x plus one or do you want to change y to y minus three? Anything you can do doesn't matter doesn't matter end result will be the same So let's say I change my y to y minus three. Okay. Oh, let me write it in different ink. Let me write it in yellow If you change y with y minus three Then see what will happen? This graph is going to go three units up Remember remember the rule So it will become like this Okay now many people think that Many people think that Sir is changing the concavity of the graph. No, I'm not changing the concavity dear concavity still remains the same as the previous Even though it may look a little bit, you know thinner, but the concavity is not changing Okay Next transformation that you can do is the final transformation which will bring you to the final result. Let's say you're changing x with x plus one So in this yellow graph now, you have to make the change. You have to make the change in this graph now So in this yellow graph, you have to take it one unit to the left Right, so your end result would look like this Are you getting my point and of course it will cut the y axis somewhere Over here. So I'm just extending my y axis will come to that little later on But meanwhile what has happened to the vertex? Vertex has gone three up one left Three up one left. Where will it reach? Minus one comma three. So option D is correct Are you getting my point and where is the y intercept? It is very easy just put x as zero to get the y intercept So y minus three is equal to one square. That means y is equal to four So this point is zero comma four Any questions here? See simple trick is minus means right and up Okay It goes ultra. I know that because minus our brain thinks left and down right But in see actually what is happening? I'll tell you the inside story It's actually not the graph which is moving It is the origin which is moving But I don't want you to think like that because if you do you'll get confused Okay, so just have a image in my mind that if I'm changing my x with x minus something I have to go right If I'm changing my y with y minus some positive number I have to go up And reverse will be the opposite direction. So x x plus something go left y y plus something go down That's the way I can recommend shares for you to remember Yeah, you can say that it's it's following the opposite sign Okay, now you would be very lucky if you have to get a question like this in the exam This is like telling you the transformation bluntly. Okay, where to go? Normally the question will be somewhat like this y is equal to x square minus six x plus five like that they'll give you this question Can you all Sketch this graph and tell me where is the vertex and the y intercept. So please sketch it number one Number two, tell me the coordinates of the vertex And number three, tell me the y intercept Fast, uh, can we have just one minute to do this because you already know, okay, let's say 1.5 minutes 1.5 minutes Any question anybody? Yeah, I can see people asking questions. Take positive or negative negative. Yeah, you can do that Yeah, yeah, sure mother. We'll do more questions. We'll do more questions so that the concept sinks in Uh, in case you by mistake drop out of the session, you can click on the same link to join back again So don't panic if your internet has gone off for some time It has happened with me so many times that my internet only has gone off So you can click on the link and get back In case there's a power cut See these technical issues will happen, right? It's all beyond our, uh, uh, hands So don't get panicked And recording is always on Okay, I'm getting answers now from akshaj Uh from hariharan. Also. I got okay. Okay last 20 seconds. Then we'll start the discussion Guys in the group also, uh, if you have any doubt, please ask on the group itself Okay, that group is also meant for, you know, clearing your doubts I don't think like that the group is only meant for Uh, you know serves to send their it's timetable and schedule You can also post your questions and everybody please try to answer everybody's question. Okay Help others if you help others god will help you Yeah, so see this is not in the format which we would have liked it to be in Okay, I would actually like to see this in this format y plus minus k and x Plus minus some h whole square, right? This is the format Uh in which I would like it to see right because then only I would be able to decide whether to how much to go right or left or how much to go up and down Okay So what I'll do in order to bring it to this form. What do I have to do? I have to complete the square I have to complete the square now. I'm assuming everybody here Has done this exercise of completing the square so many times in your life Okay, so now I'll do the completion of squares here. That's a very simple What I'm going to do is I'm going to write the first two terms So see how it works people who have not done completion of squares before are not very confident about it We see See, what do we do is we make the coefficient of x square Directly or indirectly as one so in this case you don't have to worry. It is already one. Okay Then take this number along with the sign half it What is half of minus six Minus three, correct. So what I'll do I'll do x minus three whole square Okay, basically what I'm doing. I'm treating it as minus two ax Okay, there's x square minus two ax like that. So I should have a minus a over a right So now when you expand it, you realize that x minus three whole square will give you x square Minus six x but it'll give you nine also, right? But nine, I don't want nine. I don't want so I remove the nine Okay, and bring this five down over here So it actually becomes y is equal to x minus three whole square minus four. Isn't it so far so good Okay, which actually tells me that it was a case of this And now looking at it the transformation is very clear What is the image coming in your mind? three units, right Four units down Isn't it? So it is very easy there after to imagine. So let me start with the skeleton graph y is equal to x square which is like this yellow graph Okay, now which transformation you want to do it's your call. So I would let's say this time for a change I will do x minus three whole square first Now people were asking sir. How do you remember minus means right? So I'll draw it right by three units. So somewhat like this Okay, without any change in the concavity Okay, so now I'm at three comma zero And now the final nail in the coffin I have to do y plus four is equal to x minus three whole square That means it'll go four units down. So this guy will reach this, okay? Now here also, please be very very careful while sketching it because many of us will choose a wrong y intercept So, please understand y intercept is obtained by putting x as zero. So y will give you nine minus four, which is five So I'll cut at somewhere five comma zero. So come down like this. Okay, that's the best way to sketch it Okay, so cutting it at zero comma five And this point is three comma minus four that is your vertex Is that fine everybody? Is that fine everybody? Okay, let's see who all got the right answer very good Sharon Arnav So Raj now you got it dear Now you got how it works any question now I'll stop here And take your questions first then we'll I'll go to the next uh question Vedha clear Please ask please ask if you have any kind of a question here Okay Now before we move on to Another question. There is one small thing which I would like to do In fact, I would like to ask you this question Okay, I'll start with a very simple graph. Y is equal to 2x plus three. Okay, we all know this graph Okay, straight line graph having a slope of two and having a y intercept of three. So somewhat like this it'll appear Yeah Okay, so let's say this point is zero comma three Both sides it is extending indefinitely. Okay. Now my question is What will happen if I change the sign of x to minus x? I mean, I'm asking you what is the relationship between the graph of these two I know you can plot both of them. So I'll only plot it for you. So no need for you to plot So the graph will look like this, isn't it? Okay Can somebody tell me how are these two graphs related? What are the geometrical? Yeah, of course negative slope Is there any kind of reflection happening of yellow graph? Absolutely, it flips about the y axis Right, so basically what is happening? These two are like mirror images These two are like mirror images treating the mirror as your y axis. So think that the y axis is like a mirror Okay, so the point is getting reflected about every point here is getting reflected about this mirror So this point a is coming to this point a dash This point b is coming to b dash like that every point is undergoing a reflection Are you getting my point? So what I want you to understand here boys and girls listen to this carefully In any function, this is the rule which I would like you to note down If your function graph is known to you And somebody comes and says draw me this function graph All you need to do is reflect the above graph about the y axis About the y axis Getting my point. So I'll just give you a hypothetical case. Okay. Let's say there was a graph like this Okay, just hypothetically some graph I've drawn. Okay This is the graph of some function y is equal to f of x now which function is that even I don't know Okay, I've just done the graph And let's say somebody says give me the graph of y is equal to f of negative x What are you going to do? You are just going to reflect this graph about y axis treating it as a double mirror That means both the sides are silvered. So this triangle is going to go like this And this you can say a sine curve is going to come like this Simple as that Getting my point this rule is very important rule in a similar way if somebody says What is the graph of minus y is equal to f of x now? He's changing the sign of y What will you do in that case any idea anybody anybody can take a guess? It's not minus x plus one mother. It is just f y is equal to f of minus y is equal to f of minus x Absolutely Advik brilliant Harir and very good. Shashwat. It is going to reflect about the x axis Okay, why it happens is very simple. It is not a rocket science. I'm just going by the logic. Okay. What is the logic? Logic is When you're changing the sign of x so let's say there was a point one comma two If I just say change the sign of x coordinate, where will it go? Where will it go? minus one comma two Isn't it so minus one comma two If it from one comma two if it becomes minus one comma two, isn't it getting reflected about the y axis as the mirror That is what is happening in the first if I say change the sign of y Wouldn't this point become one comma minus two my dear That means we'll come directly below it in reflected about the x axis This is the reason for this particular change. That's it common sense prevails Getting my point. So I have a question because sorry. I'm so sorry. I pressed on the long Yeah, so I have a question if I say plot the graph of A three minus y is equal to let's say x plus two whole square. How will you plot it? How will you plot it? So this is the next question for you sketch this graph And secondly, please Tell me the coordinates of the vertex And please tell me the y intercept Let's see. Let's see fast 90 seconds for this Let it be wrong. I understand. You're the first time you're doing this type of concept I'm not saying right or wrong Hariharan noted your answer Let's see what others have to say I hope you have drawn the sketch also Okay, just don't mention the vertex and keep quiet draw the sketch also And guys one more thing, uh, you you now have both the graph graph tools graphic tools Graphing tools at your Disposal that doesn't mean you will solve your assignment questions by using them Right, please have an honest learning that is very important honesty is the most important thing while you are learning something Okay, please don't try to impress anybody by you know doing using those tools Your learning is important Making mistake is a ingredient. You can say important ingredient of learning So verify your answer by those tools. Don't solve your question by those tools Okay, shashwat sharbhi shares also very good anurad See, I'm not saying right or wrong to answer dear I'm just happy that you're responding. I just want that as of now first class. I don't I'm not expecting miracles from you Okay, guys 20 seconds more please wrap this up so that I can discuss it Okay, okay, okay So I really got a very good response from everybody had saw uh I'm very glad to see everybody answering It's a different thing whether it's right or wrong, but initially you have to take the sleep I know accuracy is important but uh as of now Your responding is important for me. Now. See how will I attack this question? Okay, first of all What is the skeleton graph that we are going to look at? So I'm going to start with x square Okay, so I'm going to draw it Step by step. So let's say x square graph y is equal to x square graph is a parabola like this Okay now Next transformation that I would do is I will make x as x plus 2 Oh, sorry, let me change the ink. I don't want to write with the same ink So I'll change x with x plus 2 That means I'm shifting the graph two units to the left. So it will be appearing like this Okay, so this point is minus two comma zero. Okay Now I'm going to change sine of y because I have to get Here. Okay. So now see I'll change my pen color again. Let me check take green So minus y is equal to x plus 2 whole square my changing the sine of y means what my dear Reflecting about x axis see there's a rhyming thing Change the sine of x means reflection about y change the sine of y means reflection about x Again change the sine of y reflection about x change the sine of x reflection about y Okay, so if you reflect it about the x axis, it's going to look like this Is that fine? Now Many people ask me said no, how are you going to fit this three over here? Uh, how are you going to put this three over here? Never mind. I'm going to still do it I'm going to replace my y with now y minus three Okay, if I do that that means I'm shifting this graph three units up Okay, so my final answer would look like this Getting my point So this should be a final graph with the vertex position at minus two comma three So those who said minus two comma three Absolutely correct. See I've got my final answer final, uh, uh, uh expression, which I needed Okay, I'll also verify this on the desk more geojabra tool Now coming to y intercept for y intercept. We all know put x as zero So when you put x as zero over here, it gives you three minus y is equal to four So y is equal to minus one. So yes, this point is going to be zero comma minus one A y you mirrored. Okay, so tanishka I mirrored because from here I was replacing my sine of y to be minus y Remember, what did I discuss in the rule? If you change the sine of y You have to reflect the graph about x axis. That's why I reflected it I'm just following the rule. Are you convinced? Okay Yeah, see tanishka question was, uh, why did I reflect the graph? From this white to green Because I was changing from y to minus y If I'm changing from y to minus y Now why the change in sign see ultimately I have to achieve this my dear, you know This is my end goal I have to reach this position So doesn't it require minus y here? If you don't do it, then you will always be getting y plus minus something. So y will not become minus y ever, no How else do I change it minus? How else do I change y to minus y? Tell me some other way to do it I'll do that only There's no other way Okay, yeah Now people are asking why did you change it to y minus three because I had the end vision in my mind that I have to get a plus three So if I would have written y y plus three with a negative outside see here Had I written see I'd already achieved this goal, right? If I do y plus three and when I expand it, I would have got minus y minus three is equal to x plus two whole square But did this is this what I need? No, right? I need this So I have an end goal in my mind and according to that I'm making the transformation It's not a blind thing that I'm doing. Okay. Let's Put minus three, you know, let's put plus three. Let's do plus two. No, I have this end goal. This is my end goal Okay, and I'm working according to that end goal Getting a point Okay, can you just verify this? Three minus y is equal to x plus two whole square. Let me verify this on the desmos on the geojabra tool Sorry for speaking desmos every time because I normally use my phone for plotting it. So Three minus y is equal to x plus two the whole square I your Rama what happened? Bracket I missed out Yeah, see this I can also show you the vertex Vertex of this equation one See minus two comma three. Okay. This is what we got Vertex minus two comma three See minus two. See blue graph is what is my answer. Check the blue graph. This blue graph Okay, it happens Sharon. Don't worry initial few classes. You'll be like, you know Going mad When to reflect one two, you know, take the mirror image. It will happen. Okay Let time take its own course Did you like it? Can we end the problem with a positive note one question? Will you like to try? Okay, see I I didn't get enough time to expose you. This is a very very long chapter I wanted to expose you to so many graphs. Okay one last question. I'll take sketch the graph off This time I'll give you it in a slightly different way. Uh, let's say three uh Plus eight x minus x square And uh, I would like you to Uh, show me the number one vertex And number two The wide intercept Time starts now. Let's have 90 seconds for this time starts now Oh Last question after that, I'll stop Yes, sir. So Raj, please ask what's your doubt here? You can unmute yourself. You can talk But then typing it out. Please ask There's not a very big batch. There's a batch of 25 26 students. Yeah You can unmute yourself and talk How do you find the vertex? On jujibra, you're asking No, sir On the graph itself. Yeah, okay Now wherever was the original vertex I keep applying the transformation and see where does that point reach That point becomes your vertex Okay, if you're asking in general how to find vertex You will have to wait till I do quadratic equations with you Okay, sir. Okay If you want a vertex formula directly, but don't use it right now It's minus b by 2a comma minus d by 4a. I had already given this formula while I was explaining How does the parabola jump when your b is changing, right? So this is the direct coordinate of the vertex Uh, you can use this as a verification also. Okay time up. Let's discuss time up. Let's discuss See again, this is not in the format which I would like it to be in right I don't I don't like this format because it doesn't give us an any any idea about In what direction and how much the shifting is happening so again Complete the square. How do I complete the square? See there's a way to do that also. So if you see Uh x square has a coefficient of minus one, right? So take it common from these two terms like this Now half this half this Half of that is minus four. So write x minus four whole square But remember when you were writing Uh, we'll discuss Hariaran. Uh, when you're writing x minus four whole square, you will generate x square You will generate minus 8x, but you'll also generate plus 16 But plus 16 beta. I don't want Correct. So I'll remove it Now I'll again reopen the brackets and it'll become 19 minus x minus four whole square. Am I right? Which actually means y minus 19 as negative x minus four whole square Now this is the desired form This is the desired form And I can apply my transformations here. All of you. Please watch how I'm doing this transformation So I'm going to first draw my skeleton graph. What is my skeleton graph y is equal to x square? So that's going to be like this That correct. Is that fine? Now what do you want to do next? What do you want to do next? Let us change the sign of y itself. Okay You'll be surprised why I'm doing that Let's change the sign of y. Okay. See whatever approach you take ultimately the end goal must be achieved See there can be so many paths Okay, if you change the sign of y you have to reflect this about the x axis. So basically the graph will now look like this Okay Now what I'm going to do is I'm going to Send this minus sign on the other side See like this Will the graph change? Will the graph change? No, right because equation is not changing. I'm just sending the minus sign to the other side. Okay Now next what I'll do I'll replace my x with Let me choose green color Or no brown color. Let me replace my x with x minus four. So minus is already there. Now I made x minus four That means this guy white guy Will shift four units to the right So it'll become like this Okay, so this vertex now has gone to four comma zero Understood any question here? Any question here? No question. Okay Now again, I'm going to make a final change. I'm going to put y as y minus 19 and everything will remain the same That means this green guy will now go up by 19 This will go up by 19 Getting my point. So if this goes up by 19, it will appear to be I'm just drawing a rough structure. It will appear to be Like this parabola. Okay. Sorry for this coming in between So this point will now be four comma 19 This will be four comma 19 and y intercept will be zero comma three. Check it out Okay, so this is going to be your final final graph. Who all got it correct? Give yourself a pat on the back. I'll also verify this on the tool Three plus eight x minus x square three plus eight x minus x square three plus eight x minus x square Three plus eight x minus x square Oh, oh, see we are absolutely correct Okay, and I'll also show you the vertex position vertex of this. Yeah four comma 19. You can see here four comma 19 Okay So boys and girls I'm going to stop over here I'm going to stop over here Okay