 Can you see the screen now? So actually being the first class, I'll first brief you up about the course content of class 11. And of course the pattern of the exams that you are willing to write. So that it can help you plan your entire year accordingly because there are some chapters which are very important not only with respect to class 11, but also with respect to class 12. And they require your utmost attention and seriousness. There are some chapters which you will find very difficult but may not be very useful in class 12. So I'll list down or give you an overview of those chapters along with their difficulty levels. Fine? So yeah, so class 11 mathematics. So class 11th is quite loaded. Class 12th is not as loaded as class 11th because it has been the, what to say, rationale of all the boards to keep the pre-board year heavy and the board year slightly light so that students are not under a lot of stress, okay? So when you talk about mathematics, your mathematics will start from a simple chapter called set theory, okay? This is a very, very easy chapter. I would name it as easy, E for easy, M for moderate and let's say D for difficult. This chapter has very limited application in your class 11th and 12th, but when you talk about exams like KVPY or when you talk about exams like J-Advance, they actually frame questions on set theory in probability chapter. So with school point of view, yes, yeah. So they will ask this in probability chapter. From school point of view, this chapter is very simple and don't waste too much time on this, okay? And you'll always find questions only in the two markers and there'll be no questions in coming in four and six marker. This chapter is followed by relations and functions, relations and functions. This is a very vital chapter because all the chapters in maths are actually working on functions. Functions are mathematical machines, okay? Where you put in some input and you get an output from it, okay? So yeah, whether you talk about calculus, whether you talk about trigonometry, right? They all deal with functions. So functions has to be very, very, I mean, you have to be adept in function with respect to not only knowing how it works, but also the graph of those functions because next year we'll be doing a chapter called area and the curves where you will be required to sketch the graph of functions. And any questions so far? So this topic is also easy, I would say. Basically, relations lead to functions. So you have to understand what are relations in order to understand functions. I'm sure you must have done this in bits and pieces in your ad math, okay? Yeah. Yes, yeah. Now this chapter will be followed by a very, very huge chapter. In fact, very, very, yeah, dead end chapter also trigonometry. Tignometry itself is made up of several small, small chapters. Like there are chapters of trigonometric ratios and identities. You will have chapter on trigonometric functions. You will have chapters on trigonometric equations. Equations slash in equations. In equations is not a part of school syllabus, but for KVPY and all, we would need that. And you have the last chapter, which is called solution of triangles. We can say solution slash properties of triangles. And normally we take around one, one and a half months to just do trigonometry. And despite that, I have seen people not getting enough confidence in this chapter because the problem with this chapter is the problem of plenty. There are so many formulas. There are so many roots. There are so many identities that you don't know which is the correct way to actually solve the problem. So we'll work that out through a lot of problem practicing. Now after trigonometry, there are chapters which are very, very new to you. For example, complex numbers, okay? Complex numbers, basically, it's a different type of numbers which has got real and imaginary part. And this chapter is also very new to the students. So this will also take some time. By the way, trigonometry, you are going to find this relatively difficult. So I'm just writing D over here. So this requires a lot of attention. And if your trigonometry is going to be good, you will be having good, you'll have a better time doing other chapters like integration because they'll all use trigonometric functions. Next year, inverse trigonometry also you'll find comfortable. Complex number is also difficult, relatively difficult, I would not call it difficult. It's relatively difficult with the other chapters if not understood properly. Okay, I have seen that in school. We don't, I mean, I'm not blaming any school teacher for that, but it's the CVSC curriculum or the ISC curriculum which doesn't allow too much things to be taught about complex numbers. So when the students take up their problems on complex numbers in competitive exams, they feel that they have not understood it in a better way because they just know the basics of it. So this is also relatively difficult chapter. Then it is followed by permutation and combination. Yes. You may have done this chapter. Yeah. So the scope of it will be slightly higher in the class seven syllabus. And this chapter is moderate because it's based on the art of counting. So people who are very good at counting find this chapter easy. Then you have another new chapter called binomial theorem. Expansion, yes, yes, yes. And these two chapters, you'll find there, okay. Okay. Because the binomial coefficients are essentially going to be used in your permutation combination. So one after, okay. Then there is a very easy chapter. I'll call this chapter as, then there's a very easy chapter called progressions and series. I think you would have heard of arithmetic progression. Yeah, so arithmetic progression, geometric progression, harmonic progression, arithmetic or geometric progression. And how is the connection proper? Can you hear me? Okay, on the Skype as well? Can you hear me on Skype as well? Okay. Okay, so your Skype is muted or is that's on? Okay, okay. You can do one thing. You can listen from Skype and you can keep your phone and talk so that you don't have to hold the phone or you don't have to use, you're using your phones. Any way that's convenient to you, you can use that. This chapter of progression and series is quite easy and quite scoring as well, okay. After this, we have chapters like probability and this chapter is going to be with you for at least three more years. If you're planning to do engineering, probability is there in your 11th, in your 12th, in your first year of engineering. So it's a very, very vital part, probability and statistics go hand in hand and it actually is used for a lot of calculations in data sciences. So this chapter is going to be again there for you. This is moderate chapter. In fact, this is an easy chapter in class 11 but slightly moderate chapter in class 12. Okay, then we are going to have chapters like linear inequalities or we call linear in equations. Linear equations, you would have learned how to solve any equations, right? Let's say three x plus five is greater than seven. So find the interval of x which satisfies this inequality, right? A similar thing, but in a much, a little bit advanced level would be there. Not only in one variable, but also in two variables. So you'll have linear in equation solving in one variable as well as two variable for your class 11th, okay. Then this is followed by chapters of calculus. Now, this is not necessarily the order in which we are going to go. We are going to follow the order of your school, whatever. I think Deepa ma'am would be taking or whatever somebody, anybody is taking will follow that order here itself, okay. Then you have a calculus introduction part where you have limits chapter coming up. This is a relatively difficult chapter because many people have not come across this concept before. How familiar are you with calculus? That's wonderful. That's good, that's good. That will make my life easy, okay. And then we have derivatives which is direct application of limits. And as you have already done it, you'll find this easy because derivatives are basically based on conventional way of following the rules of the derivatives, the sum rule, difference rule, product rule, quotient rule, chain rule, et cetera, okay. And derivatives is going to be the foundation for your calculus for next year as well, okay. Apart from this, we have coordinate geometry chapters which we call as conic sections. By the way, I'm assuming that you are going to attempt section B of the paper. I hope you know, I see the pattern is section A, section B and section C. Okay, section A is basically 80 marks paper which is compulsory for both science and commerce students. Section B is normally attempted by science students and section C is normally attempted by commerce students because it has a lot of statistics part, right. Maths is divided like this. At least I know of maths, okay. So maths will have section B and section C and one of the sections of B and C you have to attempt. Section A is compulsory, okay. So section B will come, yes, yes. But last year what happened, some of the students who had already taken economics, they found section C aligned to their economics syllabus. So they didn't want to study some extra things. So they just took section C for their exams. So I mean, what is your combination for ISC? Have you taken biology, right? Okay, okay, then I would advise you to go for section B only. So section B will have conics and under conics you itself is made up of chapters like, in fact, circles they say as a separate chapter but I call circles within a conic. They have circles, you have parabola, you have ellipse and you have hyperbola, okay. With a touch of pair of straight lines but that is not a chapter in itself but you should be aware of what is a pair of straight lines. So you'll have a touch of this chapter, pair of straight lines, okay. Even before this you have a chapter called straight lines itself. So straight lines would be a chapter for you. Straight lines, yeah. So straight lines, conic sections are going to make your coordinate geometry part. Okay, now apart from this, you will have, sorry. Okay, okay, yeah, difficulty level. Straight lines is, I would say moderate, okay. Circle is also moderate. Parabola, ellipse and hyperbola are relatively difficult because you need to understand the second degree equation, generalized second degree equation, how they function, all those things. So they're relatively difficult but if you understand, you'll find them very easy, okay. Apart from that, you have few small chapters like statistics. Statistics is, there's a compulsory statistics and there's a statistics which is meant for section C. So what I'm talking about is your statistics which is compulsory. And here you will have to study mean deviation and standard deviation. In fact, you're going to study something which we call as the measures of dispersant. Measures of dispersant. How is the data dispersed? That is what you have to study in statistics, okay. If you're taking section C, then you will have further things like correlations and all those stuffs. So I'm not going to put this as a syllabus here because we are mostly going to focus around section B part. Then you'll have a very easy topic called mathematical reasoning. Yes. Mathematical reasoning and there is another chapter called principle of mathematical induction. Principle of mathematical induction. There are two separate chapters. By the way, statistics is very easy and scoring as well. Mathematical reasoning is basically based on logics. So a truth table and all would be dealt in this chapter, okay. Again, it's a very easy chapter. You will find that only two marker questions or max to max of four marker can come on this. Principle of mathematical induction is basically alternative way of proving a statement is true or false by use of induction and that's also easy, okay. So more or less, this is the content of your class 11 syllabus and you will find that you have quite a lot on your plate right now. Yeah. How many chapters normally you had for AdMaths? Okay. So you'll not find much difficulty in aligning to the class 11 syllabus, okay. But our aim is not to do just class 11. We have to go beyond it because we are also targeting exams like KBPY and all JEE main exam, okay. So our complexity level will be slightly higher. Okay. So my course plan is I would first spend around three to four classes with you doing bridge course. Bridge course, okay. Bridge course is basically throughout your class 11th, you will be coming across certain important concepts which we require to introduce right before the course. For example, how to plot the graphs, right. How to see the transformation on graphs. A bit of calculus, okay. You may be aware of it, but I'll quickly brush through it so that when you're doing in physics also or chemistry also, you are familiar with those concepts. So around four class we'll spend on bridge course and under bridge course, I plan to do these chapters. First is functional graphs and transformations. So you will be coming across a lot of standard functions and some special functions also. So I'll introduce it to you. For example, you'll come across something called fraction part function, greatest integer function, least integer function, how to deal with those functions. And if at all there is a transformation or there is a change in those functions, how does the graph actually change because of that? So that is what I'm going to start today itself on this particular topic, okay. Later on, I think this will take around two classes easily, two to three classes. And then we'll talk about your basics of limits and derivatives because in physics, you will be requiring it immediately. And this is going to be the very integral part of your understanding of calculus. So you can call it as an intro to calculus, okay. Yes. After this, I'm going to start with your class 11 syllabus and I would actually start with trigonometry if at all because your school is going to reopen only in the month of May, right? May last week, correct? So before that, if you're able to finish a substantial amount of trigonometry, you will find at least your UTs and semesters will go very easy for you, okay. Good. So let me begin with graphs, functional graphs and their transformations. First of all, since you have already done functions to a certain extent in your admins, what do you understand by a function? Absolutely, correct. I mean, it's basically a special type of relation between two variables X and Y, which we normally denote like this, correct. So normally there's a way to represent relation. This is one way to represent a relation where we say, where we say X is related to Y in a particular way, okay. For example, if I say Y is equal to X plus three, correct. This is the kind of a relation which says that your second variable, which we call as normally, you know that what is the name which you assign to these variables, we call this as a independent variable. Independent variable, correct. This, these terms are not to be taken literally because many people say, what if I express X in terms of Y? They're just names assigned to it so that we are able to connect to it. So when we say Y is equal to X plus three, it's a relation which connects Y and X, right? Now, when these relations follow certain criteria, then those relations are named as functions. So when does a relation become a function? What is the criteria which makes relations as functions? By the way, functions are also called mappings. Yes, mappings or correspondence. Why does it call mapping? Because it maps X to Y, correct. So basically the mapping word has come from the arrow diagram which we normally use to represent relations. So relation what it does, your X comes from the set, let's say a set A, right? And your Y comes from the set B, right? This set has elements of X and set B contains elements of Y, right? For example, if I talk about this relation and I have an element one in A, then it would be mapped to the element four in B, right? Because when you put X as one, Y will become four. So when we have, yes, so if you have two, it'll map to five, correct? If you have six, it will map to nine, right? And it depends upon the set provided to you. So whenever a relation is defined like this, they'll always give you the relation, where is the relation defined? So a typical way of representing a relation is, let's say if I want to express this relation, I would say there's a relation R from set A to set B, right? Where X is related to Y and they put this equivalent symbol such that Y is equal to X plus three. This is one way. Another way would be there's a relation from set A to set B where R contains all points of the form X comma Y, such that Y is equal to X plus three, right? The various ways you can follow the way which you feel like, okay? That means if you look at this second way of representation, you can actually relate X and Y like coordinates. So you can actually plot the relation on a graph sheet, correct? So for example, if I have to plot this relation, I would plot one comma four, two comma five, and six comma nine, understood? So a pictorial representation of a relation will be actually the graph of that. So in this case, it would be just a straight line like this, correct? If you choose your A as, if you choose your A as set of all real numbers, by the way, in order to distinguish between relation and real numbers, we put double arrow about real numbers. So that means real numbers. I hope you're aware of the notations and means natural numbers. Yeah, Z or I is used for integers, okay? W for whole numbers, C for complex numbers, C is used for complex numbers, okay? Yeah, so if you say that your relation is from set A to set B, where A itself is set of real numbers and B is also set of real numbers, then the graph of it would be a straight line like this, Y is equal to X plus C line. Are you getting it? So you can also plot the relations on a graph like this. Now these set A and P are given special names, okay? This set A is actually called the pre-image set, okay? The set of points from where your X is picked up, it's called pre-image set in case of a relation and B is called the image set, okay? These are basics which you should know. I talk about this more in the chapter relation and functions, but right now we are just focusing on plotting these functions and looking at the transformation. So this is just an overview that I'm going, giving you. I'll go in details of this in the chapter relation and functions, okay? Now what makes a relation a function? That's very important for us to understand. Just a second, I think I got a call. Yeah, Arushi? Yes, yes, yes, I got a call from somewhere, so that's why it got disconnected. Yeah, yeah, yeah, okay, you can call on this or send. Yeah, yeah, yeah, fine. So what makes relation a function are these two criterias? So any relation which follows these two criteria will be eligible to be called a function, okay? So what are the two criterias which make a relation a function? First criteria is every element of the pre-image set, every element of the pre-image set must be mapped to, must be mapped to an element of the image set, an element of the image set. So here the word is every, okay? I'm sorry, I got every element of pre-image set must be mapped to an element of the image set. That means no element of set A must be left out. Are you getting this point? Okay. Second criteria is that, yeah, this is the first criteria which should be satisfied. Sorry? Yes, yes, yes, because in general, the relation may not satisfy this criteria. You may have a relation where I'll give you example, let's say there's a relation, I'm making up an arbitrary example. Okay, there may be a relation from set A to set B where let's say you have an element like A, B, C, and let's say one, two, three, four. A is mapping to one and C is mapping to three. So basically you can have a relation like this also. So a relation can have, a relation can have only these two elements as its point, okay? But in this case, it will be not a function. It will be not called a function then. Because the every element, that is element number B is unmapped, correct? So for a relation, you may have some elements of set A left out, that is no problem. But for a function, it should not be the case that any element of set A, that is your pre-image set is left out. Second condition is the mapping of the pre-image must be unique. It means that if let's say A is mapping to one, A cannot now map to two again. That means in this case, the mapping is not unique. So this will again not be eligible to be called a function. Understood these two criteria? So every element must be mapped and that mapping must be unique. These are the two words that you should remember, every and unique. When these two criteria are fulfilled, then the relation would be eligible to be called a function. Yes. So let me give you some arrow diagrams of relations and let me ask you whether they are functions or not. Okay, let's take this question first. This is set A, this is set B. Normally we represent this by oval structure and I have a relation R from set A to set B. Let's say this is one, two, three, four. This is one, nine, 16. Okay, this is mapping to this, three is mapping to nine, four is mapping to 16. Okay, is this relation a function? We even set must be mapped. So two is not mapped to anything. So it's not a function relation. Absolutely correct. Okay, so in this case, two is not having an image. So two is unmapped. So this will not be a function. What about this? Let's say one, two, three and I have let's say minus one, minus two, minus three, minus four. Correct, so yes. So this is also not a function because in this case, three does not have a unique image. Okay, try this one out. Let's say A, B, C, D and I have one, two, three, four, five, six, this, this, this, this. Is this a function? Absolutely, so it's a function because every element of set A is mapped, correct? And even though B and C both are mapping to three, their respective images are unique. Correct? It's not happening that one is mapping to two of the elements in set B, okay? And in case of function or relation, whatever, it is fine that some elements of B are unmapped. That is okay, but it is not fine to have an element of set A unmapped. Is that fine? Yes, yes, okay. Now, there are two terms that we normally state in case of relation and function both, okay? The two terms are domain and range. You are aware of these two terms? Yeah. Okay, so in this case, let's say I talk about the first instance that I gave you. What is the domain in this example? Example one, no, domain in case of first example will be just set of points one, three and four because they are the ones which are getting mapped. Yeah. Yeah, there's no two. What will be the range by the way? Absolutely, okay. For example, two, tell me what is the domain? Correct, what will the range? Yeah, everything in your set will be a range. What will be the domain in case of example three? A, B, C, D, correct? And range? One, three, and five, absolutely correct. This is what I wanted to know whether you are clear about the domain and range concepts. So B set normally in function, in relation it is called the image set, right? I'll tell you the nomenclature difference. Nomenclature, how it differs? In case of a relation, you call set A as the pre-image set, I told you that. And you call B as the image set. But we don't use the same nomenclature when that relation actually qualifies to be a function. In case of function, A will be called as the domain. It's obvious because there is a criteria which we just now studied that every pre-image must have an image. That means your set A completely becomes your domain, right? As you can see in your last example here, this domain is actually your complete set A. That's why instead of saying pre-image set, we actually call it as domain only, okay? B is called a word rhyming with domain, codomain. What is the difference? Yeah, no. What is the difference between codomain and range? Codomain is everything. One, two, three, four, five, six. That's your codomain. Range are the only ones which are getting mapped. That is one, two, and five. Sorry, one, three, and five. What's the difference? Five. So in future, when I am talking about functions, I would refer, I would ask you, tell me the domain, tell me the range, et cetera, okay? Now, codomain is something which is actually user-defined. It is up to the person who is mentioning the function to you to choose his or her codomain, okay? But the range is something that you can also find out. But remember, range has to be within the codomain. For example, yeah, for example, let's say, I say that there's a function whose domain is one, two, three, and four, okay? And its codomain is one, two, three, four, five, six, seven, eight, nine, okay? And this function is defined as x squared. Correct? Okay. So this is your domain, this is your codomain. Now I'm asking you, tell me the range of this function. By the way, I'll use the short form, rf for range, okay? Okay. Tell me the range of the function. Okay, so one. Okay. That's it, that's it, absolutely. So this will become your range. This is your domain. I'll write it down, domain. And this is your codomain. Now, your range was curtailed by the domain which the codomain which has specified, isn't it? So range we say is actually a subset of codomain, right? Of the codomain, it cannot exceed codomain because codomain actually restricts what values you can assign to your range. And domain actually decides what input can you put to the function. Is that correct? Now, having said this, you can actually plot functions or relations on a graph. For example, if I have to plot this, what points will I plot? On the graph? On the graph? One, two, four. One, one, two, four, three, nine. Right? Yeah. So normally I will deal with, in this chapter I'll deal with the graphs of functions and transformations. But even before that, even before that, I would like to ask you, if through graphs you have to distinguish between a relation and a function because both of them can be graphed, right? So if let's say I give you a graph of a relation, okay, let's say the relation graph is given to you. Can you identify from the graph whether that relation is qualified to be called a function or not? Can you do that? Yeah, you can. Yeah. Okay, let me give you an example. Ah, Dan, you know that. Let's say I give you a function like this, a relation graph like this, okay? Yeah. So this is your Y, this is your X. Yeah, now what, is this relation graph a function? Absolutely, absolutely. There's something called the vertical line tests. Okay. Vertical line test is a test which is actually used to show that a given pre-image should not have more than one images. In this case, it is having three images to it, right? So let's say an element, yeah, let's say an element X1 is mapping to Y1 also, Y2 also, Y3 also. That is not allowed in a relation. So this will, sorry, that is not allowed in a function. So that will not be a function. Absolutely good. Great. So what are the functions that we can, we'll come across in our entire 11, 12th curriculum. So let me run you through some standard functions and their graphs, standard functions and their graphs. Of course, I will not go in, I'm not going to talk about straight lines because you're all aware of it. Okay, so let's say I say Y equal to X. You know how to plot it. It's easy, right? You know about the concept of slope and intercepts. So if I say Y equal to MX plus C, M is the slope, C is the Y intercept. You know that? Yeah, if M is positive, your graph will be in, your line will be inclined in this fashion, right? And if your M is negative, your graph will be inclined in this fashion, correct? If C is positive, it will cut the positive Y axis and if C is negative, it will cut the negative Y axis, correct? So I'm not going to talk too much about it. I'm directly going to jump to parabola or quadratic graphs. So second one is your basic, most basic I'll start with, which I call as the skeleton graphs, okay? So skeleton because all the quadratic graphs have evolved from this simple graph, yeah? So it's basically a parabola. You have already learned about it, correct? So I'm not going to talk too much about it. So this is a parabola. Now my question to you is Harushi, if this is the graph of f of X equal to X square, then what would be the graph of X minus one square? Yes, X minus one whole square. To which direction? Up, down, right, left. Are you sure? Okay, so I'll give you options. Do you think it is this way? The white one? Or do you think it's the pink one? Which is your answer? White one, okay. Let me show you some examples. If it is the white one, this value will be what? Minus one comma zero, correct? Does minus one comma zero satisfy it? That means when you put Y as zero and X as minus one, is it satisfied? At, at, no, correct? No, correct, correct. So what are the answers then? It's the pink one actually. Yes, see it's very simple. If you look at both the graphs, X square and X minus one square, correct? Let's say when I put X as zero, your Y will be zero here. Correct. But this fellow will become zero. Y will become zero when your X is equal to one. So the same value of Y is realized little later. So another example, let's say X is one, then Y is one. But this guy will get one when X is two. Are you getting it? That means the same value, let's say zero, is realized at a lag of one. So it'll, you know, realize the same value of Y at a higher value of X, correct? So what happens is that every point here starts shifting to the right. Are you getting this? Okay, okay. So the same value of Y is now obtained one value later on in X. Okay. Understood? Yeah. In a similar way, if I ask you what is the graph of X plus one the whole square? It will be the white one, so correct. So this will be the graph of, this will be the graph of X minus one the whole square and this will be the graph of X plus one the whole square. Oh, I'm sorry, sorry. X plus one and this will be X minus one. Slip of pen, yeah. So here is a rule for you. The rule is, the rule that comes out from here is if you replace your X with X minus H where I'm assuming that your H is already a positive quantity, right? Then the graph shifts H units to the right, understood? So here, yeah. So here if you see I have replaced my X, this X with X minus one, right? Yeah. So my graph got shifted one unit to the right and it is applicable for any function graph. It is not only just parabola graph that it is applied to, it can be applied in general. That's why I'm giving you as a rule. Okay, got it, yeah. Okay, if you replace your X with X plus H your graph shifts left to H units. Is that fine? At H units to the left, right? Actually later on when you study straight lines I will tell you that this concept is actually related to shifting of origin. But right now I'm not going to confuse you with those concepts. It's just a rule that you should keep in your mind when you're dealing with a graph of functions. It's a very, very handy rule. I'll just box it up. Okay, now the very same question. Let me draw the parabola once again. By the way, you know the namings of critical, what is this point called actually in a parabola? It's called vertex. It's called vertex, absolutely. Yeah, let's say I take my old friend parabola, this one, Y equal to X square. Okay, this is Y equal to X square. Now tell me what will happen to the graph if I plot X square plus one? So will it be up or will it be down? Will it be the white one like this? Or will it be the pink one like this? Just use your logic, use your logic. Take your time, no need to hurry up. Absolutely correct. See, you reasoned it out well, good. So this is your answer. Okay, not this one. But at the same time, if I ask you X square minus one, is it a pink one? So this will be the white one, this will be the pink one. Okay, now here comes next rule. If you see this clearly, it's actually saying F of X minus one is equal to X square and this is saying F of X plus one is equal to X square, correct? So the rule says if you replace your Y, in this case your Y is your F of X, hope you are able to understand Y and F of X are same things, okay? So if you replace your Y with Y minus H, again H being a positive quantity, your graph goes H units up, up, yes. If you replace your Y with Y plus H, the graph goes H units down. Goes H units down. Down, exactly, so up is equivalent to right and down is equivalent to left. Okay, now I'll give you a mix of these two on a question. Plot, just a rough plot I want. Plot, F of X is equal to X plus two, the whole square minus five. That's the initial answer. And Y, F of X is equal to X square and it will come with its vertex at the logic, right? Right. Nothing to it, yeah? Mm-hmm. What would it note? H units to the left. Left, right. Okay, so that's... Okay. This is minus five. Mm-hmm. No, no, no, no, Naseer, that's the trick over here. If you see here, it is F of X plus five is equal to X plus two, the whole square. So it's like saying Y plus five is equal to X plus two, whole square. And remember I told you when you replace your Y with Y plus H, the graph will move down. So this may cause a confusion because I wrote it on the other side, no? Okay, fine. Yeah, so X moves to the left. Yeah, so just tell me what vertex position. Two. Mm-hmm. Minus two, minus five. Okay, so that means your graph will look like this. Kind of. So this point will be according to your minus two, minus five. Okay. That's correct. Now let's verify this on a tool called GeoGebra. I would request you to install this on your laptop. And on your phone, normally I use these two graphic softwares. One is GeoGebra. GeoGebra. You can open it online also geogebra.org. And on phone, normally I keep this tool called Desmos. Yeah, these are graphing tools. And it's very important to keep looking at the graphs while problem solving in maths. Okay, so I'll just open to the, can you see my screen? Y is equal to, Y is equal to X square, okay? You see this parabola? Yeah. Now, the moment I do Y is equal to X plus two, the whole square, see what happens? It got shifted to the left. Okay, so you see that this is the position now. So earlier it was this, now it has become this. Okay. Now, in the same graph, if I add a, did I, did I subtract a five or add a five? Subtract a five, yeah, subtract a five, see what happens, it goes down. Yes, and the vertex position has now come here. As you can see here, minus two, minus five. Is that fine? Yeah, understood. Understood, okay. Now normally the question that would be given would not be as straightforward looking like this. They may also ask you a question in this form, plot f of X is equal to X square minus three X, sorry, X square minus six X. Oh, I'll take minus three X only. Minus three X plus let's say seven. Then how do you do this? Yes, completing the square. Completing the square, absolutely. So do that. You should have got this, X minus three by two whole square, but this will generate an extra term of nine by four, so subtract nine by four and add the seven back. Yeah, yeah. So that would be X minus three by two square plus 19 by four, right? Yeah, so let this 19 by four come on the left side so as to give us a clear idea what to do with the graph. And now tell me what will happen to the graph. Correct? So where will the vertex go? Absolutely correct. So 1.5 comma 4.75, the vertex will come and the parabola would look like this. Very good. So now you have known how to use transformation in case of quadratic equation graphs. Okay. Now, going back to the same question, what will happen if I replace my X with negative X? Think properly. Will this change the graph by the way? It won't change. Yeah, yes. But let's say my graph was like this. F of X is equal to X plus one the whole square, right? And here I replaced my X with, let's say I've got a graph G of X. I replaced my X with negative X. Then how will the graph look like? So by the way, you know how to plot this one, right? Yeah. So I'll erase this one because this is not giving you a clear idea of this transformation because of some reason which I'll tell you why. But let me plot F of X first. So F of X graph will be, tell you what will I do to get the graph of X plus one whole square? Shift it which direction? Left. So it will be like this. This point vertex will be at negative 1 comma zero, right? No, this is the graph of F of X. Yeah. Tell me the graph of G of X which is negative one plus one whole square. If I bring one minus one outside, I'm just going to do the same. Okay. So what's the graph that you see finally? Do you see this graph, white graph? Yeah, yeah. Okay. Now, if you observe this closely, essentially what has happened is that, you're there, Arushi? Yeah. Yeah. So essentially what has happened is that your graph has got reflected about Y axis. You see that? Okay. So this is the rule that I'm going to tell you next. The rule is if you replace your X with negative X, then the graph gets reflected about Y axis, Y axis. And the reason why it did not work in X square graph is because it was already symmetric about Y axis. So the thing which is already symmetric if you reflect it about Y axis, will anything change in the graph? Nothing will change, right? No. It will remain the same. That's why we did not observe anything. That's why I mind it. Right. Is this rule clear? Yeah. If X is replaced with minus X, your original graph that is F of X would be reflected about Y axis to get G of X. You know how to imagine reflection, right? So the reflection of this part is your this part. And reflection of this part is your this part. Correct. Correct. I just try one more. Plot of Y is equal to three negative X whole square minus two. And this is to not have minus two. And what is happening? Okay, so X is getting replaced by X of X. So it's equal to two. And yeah, it's equal to plus. Minus three comma minus two, okay? Okay. Okay. Right. Three comma minus two, that's brilliant. Very good. See, if you look at it, I really just respect your analysis, but that would be more useful in cases where you actually cannot manipulate. Here you can see that, since there's a square over here, you can actually do this, correct? Yeah. So basically you're plotting this graph where you just have to shift your vertex C units to the right and graph two units down. So this will be your answer. And of course, you know, whatever you have done, it is going to give you the right answer, but that would be more useful had you not been able to do these kinds of manipulations. Correct. Okay. So that was actually a trick for you that I thought I would take you to do little long away, but here you can do it a shorter way also. Yeah. Great. Now, again, the same scenario, but this time I'm asking you to plot minus of x square. So if this is the graph of x square, the graph of negative x square is going to be the white one, the opposite direction. Right. So if this is f of x equal to x square, then this will be f of x is equal to negative x square. Correct? So here the rule is, if you replace your y with negative y, then the graph gets reflected about the x-axis. Is that fine? Now with all these rules, I'll give you a plot. The plot is three minus y is equal to x plus one, the whole square. Y equals x plus one, which first, so it'll be minus y equals x plus one, whole square, x plus one, whole square, minus three. Okay. Okay, okay, fine. So now I'm just going to plot this and then I'll reflect it. Let's plot this. So if I first plot this, then x is going to process x one and x is going to move one unit to the left. So if x moves one unit to the left, then what's going to happen? So if x moves one unit to the left, the vertex is going to be at minus one and this is, and y, okay, wait, wait, wait. Yeah. Okay, make this the next one. Yeah, there you go. And then if y is going to move three units, wait, it's going to be y plus three. Wait, wait. So it's going to be y plus three, and y plus x is going to move three units down. So it will be at minus one, minus three. If it's just y equals x plus one, whole square, minus three, it will just that and be minus one, minus three. That's not going to be the vertex, but now it has to reflect about the x axis. So my x value is going to be the same. So I'm going to be the same, it's going to be minus one, but my y value is going to change, okay? Okay, wait. So if it's going to change, then what's going to happen next, one second. Okay, so I have minus three, minus one, minus one, minus three here. And I have to reflect this across the x axis. Okay, how would I, oh, okay, that will simply go in the positive side. Okay, so that means that will be minus one, comma three. Yeah, is minus one, comma three going to be the vertex? Okay, let's check. So there are various ways to proceed. Let me start with plotting the graph of y is equal to x square. Okay. Let's go step by step. So y equal to x square, I'm just drawing a miniature graph, correct? Yeah. Next, what is the next step that you did? You replace your x with x plus one or you replace your y with minus y. What did you do that? First what it was, I replaced x with x plus one. Okay, so x plus one whole square will shift this entire thing, one unit to the left. Yeah. Okay, then what did you do our issue? Then minus three, I brought the minus three to the left side. So basically I shift that graph down. I shift that graph down, yeah. Down means you're plotting y minus three then. If you're shifting this graph down, if you're shifting this graph down means you're doing y plus three is equal to x plus one whole square. Yeah, I did that. Okay. Yeah. Basically it would be like this now. This point will now come at minus one comma minus three. Yeah. Then what do you do? I have to reflect it and I reflected it. Okay, now you reflected it and you got minus y plus three is equal to x plus one whole square, right? Yeah. So when you reflect it about the x-axis it will now appear to be of this nature. Yeah. And x is, you know, I used to point, so it will be like this. Yeah, is it? Yeah. So the vertex will now be at one comma minus three, right? That's what you got? You should get across the x-axis, right? Oh, I'm so sorry, I reflected about y-axis. Yeah, sorry, sorry. Yeah, what would it be? Minus one comma three? One second, yeah. You will reflect it about the x-axis, it'll go up. Yeah, it'll go up. So that will be minus one comma minus three, minus one comma three. Yeah. Absolutely correct. Yeah. Okay, now there are various ways also to draw it. I will take a different route and I'll still reach the same answer. See what route I will take. Okay. I will first plot y is equal to x plus one the whole square that you actually did in the second step. Yeah. Okay. Now here itself, I will change the sign of y. Okay. That means I will reflect it about the x-axis. So it'll become like this, correct? Yeah. And then I will replace my y with y minus three. That means I'm taking the graph three units up. Yeah. So it'll automatically open up to give you three minus y, isn't it? Correct. And we are at the same graph once again. Correct. So approaches can be various, but whatever approach you take, whether you change the sign of y first or whether you change the sign of y later on, ultimately it should not, it should bring you to the same function back. Yeah. Got the point? Yeah. Okay. But just a simple thing, what will happen if I multiply x-square function by let's say two? It will shift the graph anywhere? No, it won't. Will it shrink the graph or will it make it expand? It'll expand. Okay, let's check. Let's say originally it was x-square. And let's say at a value x equal to one, your value of y will also be one. Yeah. Okay, so originally let's say this is your x-square graph. Correct. Now when you're plotting, let's say I name it as g of x just to make things different. Now when you're plotting g of x, for the same value of x equal to one, now I should get the answer is two, right? So won't the graph be more shrink like this? Because for the same value one now, I will get a value of two. Correct. Are you getting this one? Yeah. So if you multiply your function, this is another rule, if you multiply your function, okay, with k, and k is greater than one, then the graph will shrink along the y-axis. Oh, graph will, the graph will shrink along the y-axis. Along the y-axis, okay. Okay. In fact, I should say modulus of it is greater than one because you may have a negative number also, but a negative number you'll first flip it and then shrink. For example, if I ask you dog k of x-graph, which is negative two x-square. Yeah. So in that case, you'll first flip it down and then shrink it like this. Correct, yeah. So because of this negative sign, yeah, because this negative sign will flip about the x-axis and then shrink. Okay, so if function is multiplied by k, where k is greater than zero, then the graph will shrink along the y-axis. No, greater than not greater than zero, greater than one. One, greater than one. Modulus of that number is greater than one. Okay. Modulus means I'm ignoring the sign right now. Yeah. And if it is less than one, but greater than zero. No, this will always be greater than zero because modulus function always gives you the positive of it. Yeah. Correct. It will shrink along the x-axis. Okay, yeah. In other case, you can say it will become more flat. So for example, if I want to plot, let's say a graph of p of x, which is x-square by two, then in that case, it will be more flatter like this. Okay, so if modulus of k is less than one, shrinks along the x-axis. That means starts coming closer to the x-axis. Okay, yeah. Okay, got it, yeah. Understood? Okay. Yeah. Now, with all these rules in our mind, we'll proceed on to the next function, which is a cubic function. So first of all, I would like to know from you, have you ever seen the graph of x-cube? Yeah, I think it has two, two, this one, right? Right, so that two vertices are stationary points. Two stationary points, how? I think I remember seeing that, or maybe not. It's like this. Oh, it is one, okay. Oh, no, it's stationary, okay, anyway. Okay, it has only got one stationary point here. Yeah. Okay, so I'll show you on GeoGebra as well. Y is equal to x-cube. Is that it? Oh, yeah, nice. Correct? Okay. Yeah. Now, since you have learned all your transformations, I have a question for you. Plot minus y is equal to two minus x-hole-cube plus one. Two minus x-hole-cube plus one. Oh, okay, okay, okay. So if I took the matrix, then what do I do? Hmm, interesting. Interesting. So let me go back and check what I did. Okay, so when it was minus x minus one-hole-square, then if it was minus x plus one-hole-square, then it became what? It was minus x plus one-hole-square. Then I basically made it into x minus one-hole-square. But can I do it here? It was minus x plus one-hole-square, and I made it into x minus one-hole-square. Here, if I need to make it that way, will it become, okay. Will it be, can I make it minus y equals minus, minus of x minus two-hole-cube plus one. So y will be x minus two-hole-cube minus one. Okay, does that make sense? I did, I brought that in the form of y equals x minus two-hole-cube minus one. Yeah, you can do that. Yeah, okay, now I think I can apply, I should be able to apply. Absolutely, you can do that, yeah. I should be able to apply the rule. It's always good to simplify it to the form that you can easily manage with. Yeah, so now I should, if I use a rule, then x, if it's x minus two, then it'll move hatch units to the right. So it'll move hatch units to the right. So hatch units to the right, meaning that it'll go two units to the right. Perfect. So that point where it was initially at the origin, right, that point, it'll be at the x quadrant of that point should be two. Right. Okay, and now it's y plus one, okay? If I do y plus one, then it moves one unit downwards, so it'll be two minus one. That'll be the point. Absolutely good. Yeah. So your graph will look like this. Yeah. Okay? Yeah. And you rightly said from origin, it will shift to two comma minus one. Yeah. Let's check this on GeoGebra as well. Okay. So we had y is equal to x minus two minus one. Moves to two comma minus one, yeah. So now this point has come to B position. Origin has come to B position. Great. So without much ado, we can move on to now the bi-quadratic functions. Okay. So the simplest of all the bi-quadratic function is x to the power four, which we call as a skeleton graph for bi-quadratic. Now here, you would realize that the shape is more or less like a parabola, but it is slightly more flatter. Oh, sort of like that, okay. Okay. And it's very important to realize the difference between x square and x to the power four. Okay. The difference is this. So white graph is x to the power two, and yellow graph is x to the power four. Now, where is this meeting point? Can you tell me? Where x to the power four equals x square. Yes, so what is that meeting point? You can simultaneously solve it. Yeah, x to the power four equals x square. x to the power four minus x square equals zero. So that will be in the form of a square minus b square, right? So there will be three points, right? One will be plus minus one, other will be zero, right? So these two points will actually be minus one and one. Yeah. Now see here, it's very important to note that in the interval minus one to one, you would realize that x square is the boss, that means it is greater than x square, x to the power four. As you can see, it is higher, correct? Yeah. This graph will be higher in position as compared to x to the power four. That's why white graph is above the graph of yellow in this zone, correct? But the moment you, but the moment you cross that one or you cross that minus one on both the sides of it, that means when x becomes greater than one or when x becomes lesser than minus one, x to the power four will start dominating. So as you can see, yellow graph is becoming higher than this graph, okay? So this is a small thing that you need to keep in mind when you're dealing with both the graph simultaneously. Okay, and to the power four, between x is one and x is minus one and it's also inclusive, right? So it's equal to two also. Yeah, if you say equal to, then it will become equal to here. Oh, okay. If you say equal to, then it will become equal to here. Okay, okay. Correct? Bye. Okay, I'll just show you the difference on GOG, brah. Okay. So y equal to x square I've already plotted, so I'll just click it, okay? And y is equal to x to the power four. See that? Two and black. So in the interval minus one to one, as you can see, this will be minus one comma one and this will be one comma one. So in the interval minus one to one in this zone, your x square is higher than x four. Yeah. Okay, and post one, x four dominates. Okay, yeah. Sorry, lesser than minus one and greater than one, x four dominates. Lesser than minus one and greater than one. Greater than one, x four will be higher than x square. Okay, got it. Understood? Yeah. Now, I will not ask you any question on this because you've already learned most of the transformations. So you can also deal with this. Yeah. So dealing with degree five, or what we call as the the Quenry graphs, degree five, simple example will be x to the power five. Okay. How would you plot this? The graph of this will be almost similar to the graph of x cube, but again, it will be slightly flatter over here. Oh, okay. Just like x to the power four was flatter than x square. Yes, exactly. Okay, I'll show you the difference on GeoGebra. Let's first let me plot it on the, this thing. So on the whiteboard itself. So what will happen is, x cube graph would be of this nature. It will be like, okay. Again, this meeting point is one and minus one. So see the shift in the nature of the x to the power five. Now it has become more flatter. Okay. So what I want to say is that in the interval minus one to one, okay, magnitude wise, x cube will be higher than x to the power five, magnitude wise. Okay, okay. And beyond one and beyond minus one, x to the power five will be magnitude wise greater than x cube. Okay, correct. Understood. It just like the previous one. Like the previous one, exactly. So let me show you. So x cube, x to the power of five. Just slightly expand it. It looks almost the same. Yeah, but there's a difference. The red one is your graph of x cube. And the pink one is a graph of x to the power five. Okay. Do you see that change here? Yeah. Okay. Now if I continue by the same trend, what will happen to the graph of x to the power seven? Okay, x to the power of seven. So x to the power of seven will be even more flatter. Yeah, even more flatter. Correct. Than x to the power of five. Yes. So x to the power of nine will be even more flatter. Than x to the power of seven. Yeah. So if I take a very large odd power, let's say x to the power of 101, let's say. You see, almost it's like a. Oh yeah, like a square. The square it should have become. Yeah, almost it will become squarish like this. You see that? Yeah. In a similar way, even for even powers, you'll see the same nature. For even power also, you'll see the same nature. For example, x to the power of two, you have already seen the graph. X to the power of four, you have already seen the graph. Let me just bring it down. Oh, nice. Yeah. X to the power of six will be even more flatter. Yeah, yeah. X to the power of six will be even more flatter. X to the power of eight will be still more flatter. Right? And if I take x to the power, let's say 100 and all, that means almost appear like a U. X to the power of 100, let's say. Oh, yeah, yeah, yeah, yeah. It's almost like right angle also. Almost like a right angle, yeah. So if I raise it to a further power 1000, let's say it'll become this. Wow. 10,000 again. So it'll almost become a kind of a rectangular structure. Okay. Okay. Great, so these are the graphs which will play to get the graphs of polynomials which are of degree two, degree three, degree four, degree five, et cetera. Okay. Okay. Just a quick test. Half of minus y is equal to three minus x whole to the power of five, plus one. Plus one. Okay, so now what you have to do is that, okay, let's see, one, one equals. So this would be the same as x minus three to the power of five, plus one. So I can bring it to the form x minus three to the power of five, minus one. Yeah. So now x will shift, x will shift three units to the right. So it will be the black points at the origin that we have three, three, and that will be the x square and the black point. And y plus one, right? So the y plus one, y plus one will be at one point. Y plus one will push it, yeah, to minus one. So it will be at three, minus one. So that will be at the point at the origin will move to three, minus one. Absolutely correct. Yeah. So as you rightly said, you could actually bring it to x minus three to the power of five, minus one. That means x to the power of five graph will be shifted three units to the right and one unit is down. Correct, yeah. So slightly more flat like this. And this point, put this arrows to show that it is still moving up. Okay, does not stop. Well, now we'll take a case of non-algebraic functions. So we'll start with logarithmic functions. Okay. Are you aware of logarithms? Yeah, we did that. Okay, you know the properties as well? I mean, I'll revise that with you because unfortunately you're going to need this so many places and formally neither ISC syllabus nor CBSC teaches it. But I'm sure people from ICSC have a bit of exposure to logarithms. Yeah. So what's the log function? So when you say y is log of x to the base of, let's say a. What do you mean by it? That means that a to the power y should be equal to x. Absolutely. It is equivalent to saying a to the power y is x. Okay? Yeah. Now log function is defined under certain conditions. Now when I say defined under certain condition means for log to be a real valued function. What are the meaning of real valued function? That's not imaginary. Yeah, that means it should neither take non real inputs, neither should give non real inputs. Right? That means it can only take a real inputs. That means your x has to be real and defined, of course, real defined and its output must be also real and defined. Outputs must be. And defined by defined, you mean? Defined means it should not be like infinity and all. Oh, okay. For example, one by x, this is undefined at x equal to zero. At x equal to zero, this function is undefined. So for log function to be a real valued function, these two criteria should be met. First criteria, whatever input you have must be positive. Whatever input that you have must be positive. Since I've used log x to the base, say I'm writing x, but if at all you are finding something like log of some function itself, okay, to some base itself, then your f of x must be positive. Yeah. Okay. So x should be greater than zero, cannot be even zero. No, it cannot be even zero. It has to be greater than zero. So in this case, if I say your f of x should be greater than zero. Yeah. Okay. And second criteria is your base, which is your A, should be positive and it should not be one. Positive? Positive and it should not be one. Now, there's an obvious question that will arise in your mind, why the restrictions have been imposed? See, for a simple reason that, let's say if I ask you, what is log of minus two to the base of three? So three to the power of what will give you a negative number, so that's not possible. This is not possible. Yeah. Okay. So much cases will not exist. And therefore, we said that we will never allow the input to become negative. Okay. Now, why is this restriction on the base? So let's say if I ask you this question, log of what to the base, negative one is going to give you half. Okay, so that'll be imaginary. Yes, so it'll become imaginary in nature, which we call as I, which will study in complex numbers, but remember I told you in the beginning itself, it should take real inputs. Yeah. So X cannot be imaginary. Yeah, okay, but if minus one equals two or something, minus one to the power of two equals X. Yeah, right. So in some cases, it will not, it will work. In some cases, it will not work. That means when you have fractions over here, it won't work. Okay. So normally, A should be, cannot be lesser than zero when Y is a fraction. Yes, but normally what happens in order to avoid the confusion, we say that we will never put A as negative. Okay, correct. Okay, Y is not one also because we cannot answer such questions. What is the answer for this? What power? Power, oh yeah, you can't get that. Right, so that's why. Everything is one. Exactly, so that's why we avoid taking, in fact, we do not take one also as our base. Yeah. Okay, so only under these two conditions we'll define our log functions. Now, when I say A greater than zero and not equal to one, this itself gives you two conditions when your A is between zero and one. Yeah. And when your A is greater than one. Okay. Now I will introduce you to the graph of log X function for both these scenarios, for A between zero and one, and for A greater than one. Okay. Okay. So let's talk about their graphs. Okay. So graph of log X to the base A, for situation number one, where your A is between zero and one. Okay, let me take an example. Let's say I want to plot log X to the base of half. Okay. And whenever we don't know any graph, we normally tend to choose X and see what is our Y coming out, right? Yeah. So let's make a small table so that we get the idea of how the graph is. Yeah, so let's say I make a table like this. So here I'll keep the value of X, this is the value of Y. So tell me what is the value of Y when X is going to be one? When X is going to be one, half zero, zero. Zero. When it's going to be half? Half to the power half is equal to what? Half to the power half, one by one. No, no, no, X is half. Oh wait, X is half, correct, one, one. One. One-fourth? X is, so wait, X is one-fourth, so half square, two. Two. Okay, I'll say minus half. When X is minus half. It is not possible. Yeah, that's not possible. That's why I tricked you here. Okay. What about two? Half, what's two? What gives you two? What power and half will give you a two? What power and half will give you a two? Half square will give you one by four. Half root, half will give you two. I don't know, what will give me a two? Minus one. Oh yeah, minus one, correct. Okay. What about, which power will give you a four? Minus two. Minus two, okay. So we have sufficient number of points with us to start plotting the graph. Yeah. So it's obvious that my graph will pass the X axis at one comma zero, right? This point, one comma zero. Now as we decrease the value of X, graph will become higher and higher. So let's say this is half comma one. This is one-fourth comma two, correct. Had I taken one-eighth, it would have been at three. Okay. And as we increase the value of X, let's say two, it becomes minus one. Yeah. At four, it becomes minus two. Yeah. At eight, it would have become a minus three, like this. Correct? Yeah. So if you start combining these dots, this is the graph of log function. And remember, it'll go on and on in this direction. So put an arrow. And this also goes on and on in this direction. Remember, it will never touch the Y axis. There's an SMTot. So that's why we say Y axis is an SMTot. Okay? Got it, yeah. Let me show you on the GeoGVLA as well so that you have an exact idea of how these graphs look like. Okay. This is the graph. Right? Seems to touch, but doesn't touch. Yeah, seems to touch, but doesn't touch. Actually, it touches it at infinity. As you rightly said, it would be SMTot to the graph. Okay. So a quick question. If this is the graph of log X to the base of half, then what would be the graph of log X to the base of 1 fourth? Okay, I'll give you an option. I'll give you an option. In fact, I'll give you two options. Will it be like this? The green one? Or will it be like this, the red one? That means it'll become more bent towards the X-axis or more away from the X-axis. Which one of them, red or green, will be my answer for this? Think relative to this graph. This is the graph of log X to the base half. So which of the two, that is the red or the green, will be the graph of log X to the base 1 fourth? X to the base 1 fourth. Right. So in the case, okay. So the red graph is going closer to the X-axis and the brown graph is going further away from the X-axis. Yes, so which one would be the answer? Okay, so we'll see if I take X to be, I'll try, I'll try with this one. So if I take X to be, one second, if I take X to be, wait, so it's gonna be one by four more than X. I'm gonna take to be, if I'm gonna take to be two, then I would get minus half of it. Okay, so for the other one, it was minus one, right? So for this, it is minus half. Okay, and I'll also try for four. Four, it is, let's see. One point is sufficient. You said when you take X is two. Yeah. The yellow one was giving you, the yellow one was giving you the answer as minus one, right? Yeah, so be right. But this one would give you the answer as negative half, right? Yeah, so it would be red. It would be red, absolutely, because this is going to give you, this is going to give you two, right? Yeah. Yeah, so this one would be the graph because for the same point, X equal to two, your answer should become minus half. So this would be the graph of log X to the base, one fourth. Okay, now in general, if you are between zero and one, Yeah. as you go towards zero side, the graph will bend more towards the X axis. Okay. And as you go towards one side, the graph will bend away from the X axis. Okay. So graph will bend away. Yes. Graph bends away. So I'll show you here itself. So let's say Y is equal to log of one fourth of X. You see that? Oh yeah, okay, it's a red line, yeah, yeah. And if I make it three fourth, let's say log three fourth more towards one. See that? Oh, see, it's bending away, correct. So that means that the one that you drew, the brown one was closer to one. Yeah. Closer to one, yes. Yeah. Now, another question for you. What would be the graph of log X plus two? X plus two, I should tell you like what, whether it's how it shifts, right? How it shifts, yes. Of course, this rule applies everywhere. That's a universal rule. That's why it's called a rule. So X plus H, so it moves H units to the left. So it moves two units to the left. Right. So yeah, it moves to... So when it was log of half of, oh yeah, it moves to the left by two units. Comes minus two at one point. When X is equal to zero. This graph, you just have to shoot two units to the left. Yeah, you should shoot two units to the left, correct. How will it pass, tell me? Hmm? How will it pass then? If I move it two units to the left, how will it go? If it moves two units to the left, then... Goes like this, correct? Yeah, okay. Okay, the same graph, I've shifted two units to this side. Now where will it get the X axis? Okay, at that time it cut the X axis at what point? It cut it at... One comma here. Okay, so now we'll cut it at minus one, how will it go? Absolutely, yes, yes. And what will be the asymptote line now? It will be two units to the left. So what is the equation for this line? Y equals... Not Y, X equal to negative two. Is that clear? Understood? I'll show you on GeoGebra as well. So here I'm going to change my X with X plus two. See that? And X equal to negative two, if you thought it would be a line exactly very close to this. Correct, okay. So this becomes your asymptote. This line becomes your asymptote, okay? And it cuts it here, see? Minus one comma zero. This point. Get in the point. Okay, so one more. This is a small one. Let's say three minus Y is equal to log half two minus X. Three minus Y equals to log half, not half the base half, two minus X. Okay, so now I would do minus Y equals to log half two minus X, minus three, okay? So I would do minus three, okay? So two minus X is... Okay, two minus X, okay, okay. Oh, so is this sort of like log? Okay, I can break this actually. Minus Y equals log, wait. No, I can't make it, I don't know if I can make it. Unlike the case of cubic and quadratic, you cannot send a minus sign on this side like that. Okay, yeah. So I would first have to do it without that and then reflect it. See, what steps will you follow to reach this graph? Let's say I start with this. This is my starting point, step number one. This graph, you know? Yeah. Correct? Okay, now see what I'll do. Yeah. Step number two, I change my X with negative X. Okay. No, if I do that, what should I do to the previous graph? Step one, what will I do to step one to get step two? You'd reflect it about Y-axis. Reflect one about Y-axis. Yeah. Absolutely correct, Arushi. Then step three, I replace my X with X minus two. Do you see that? That will generate a two minus X arrow. Oh, correct, correct, correct. That means what am I doing to graph in two? I'm shifting the graph in two by shifting the second graph by two. Two units where? Right. Correct? Step number four, I'll change my Y with negative Y. Okay. By the way, let me write it as two minus X now. Yeah. Now, what am I doing to the graph in step three to get a minus Y over here? You have to reflect it about the X-axis. Brilliant. Okay. Now, having got this, I can't go down actually. So step number five would be what? Replace your Y with Y minus three. That means what am I doing to the graph? I'm sorry. What am I doing to the graph in step number four? Are you are shifting it, you're replacing what? I'm replacing Y with Y minus three. Yeah, Y minus three. So you are bringing it down? No, I'm going to up, up, up, up. Yeah, I'm shifting it up, correct. So let's follow these steps. So first, you reflect it about Y-axis. So I'll be drawing it and I'll be raising it, okay? So this is my original graph, correct? This is the graph of step number one. Step number two, I have to reflect it about Y-axis, correct? So I'll draw it like this, correct? Step number two, I'm shifting it two years to the right. That means instead of this minus one comma zero, now it'll come at one comma zero like this. Correct. So I'll erase the older one. Now what am I doing next? So what is that? You finished? Two years to the right, yeah. Now minus Y. Next, you have to reflect. Reflect about X-axis, correct? What's axis? So if I reflected about X-axis, it will look like this. This part will appear like this. So it will appear like this. So let me erase this as well. Next, what I'm doing? After that, you have to shift it up by three units. Three units. So basically a final graph would look like this. Just go up by three units, okay? Yes, okay, I'll show you step by step on Jiu-Jitsu as well. So this was the graph of Y-axis. Yeah. Now what is the first step that I did? I replaced my X with negative X. Yeah. See that? I'm talking about Y-axis. Yeah. Now I replaced my X with X minus two. Correct. See that? Two units to the right? Okay, correct. Correct. So I replaced my Y with negative Y. See that? Oh, oh, interesting. Okay. The minus three. Yeah. And then what we do is we replace our Y with Y minus three. Minus three, okay, okay. See that? It seems to have stopped, but there's some graphic issues. So it'll actually go down below. This will go down below, okay? Exactly, that is what we had plotted over here as well. Sounds good? Okay, so I'll take your five minutes more. Let's look into the graph of those cases where your A is greater than one. So right now we saw A between zero and one, right? Yeah. So let's take, for example, log of X to the base of two. Okay. Okay, so without much waste of time, I mean, without plotting it actually, we can, I'll give you the graph. The graph is like this, okay? It is actually the mirror image of the previous graph about the X axis. Oh. That means it brings something very interesting in your mind that if it is a mirror image, then this should be negative of this, right? Yeah. Which actually is true. I'll show you through the properties. Okay, correct, correct. No, it's telling me that negative of that is, isn't it going to be the negative of X? No, negative of X means it would be reflection about Y axis. Okay, correct, correct. It's Y, correct, correct. So basically when you're plotting this and when you just change the sign of Y to this. Yeah. It actually becomes this graph. Okay, correct. So basically you're trying to say negative log X to the base half and log X to the base two are same graphs. Okay, yeah. That means this property is true. Okay. So basically log to the base, Oh, yeah, it makes no sense. So it's basically like log of X to the power minus one base half is same thing as log. Yes, you can generalize this. You can say log X to the base of A is negative of log X to the base of one by A. Oh. In general, you can say that. We'll prove this, we'll prove this through properties. Okay. So basically log X to the base A equals negative log X to the base one by A. One by A, okay. Now, just a quick question. How would the graph of log X to the base four look like? Okay, log X to the base four. Will it be like this? Or will it be the green one like this? Again, use your logic. Okay. So, okay, that's it. So if log is four, right? So we should see whether it bends. Towards the X axis? The yellow one is the original one. Yeah, yellow one is the original one. So if you see whether it bends again towards the closer to the Y axis or further away from the Y axis. Yes. Okay. Okay. So if I take in, so can I do it with values? Yeah, you can just take a special case and check whether the bending should be more or the bending should be less towards Y axis. Okay, so if I take the original graph of Y equal log, that was log two X, right? Yeah, log X to the base two and other is log X to the base four. Yeah, log X to the base four. Okay, so if I take X equals, okay, wait. Take X is four. Then Y equals four for the two cases. We'll see where X is. Yeah, I think that also. Yeah, so when log X to the base two equals four, then X should be 16. 16, X should be 16. But when log four again. That will four to the power four. That's 256. Okay, that should be 256. Okay, so when X is 16 for this, Y is four. But then Y is four only when X is 256. Okay, so Y is four when X is 256. So if I have to take that, so basically mean that when X is four, when X is just 16, when X is a smaller number, Y is the number. So, okay. See, Arushi, very simple. Let's say I take X as four. Yeah. Right? Yellow one would give me the answer as two, right? Yeah. When I put four over here, I should get two. But when I put four over here, I should get an answer of one only, right? One. So my answer would be below, yeah. My answer would be below. That means green one is your answer. This is your log four to the base X. I'm sorry, X to the base four, yeah. Nice, yeah. So means in general, if you go towards, let's say if you're greater than one and you go towards one and you go away from one, then this graph will be more, this will be less bend. I can say this will be bend more towards X axis. Okay, and more. And this will be bend less towards X axis. X axis, and this will be bend more towards, so, okay. So if A tends, it's closer to one, then it's bend less, no, wait. But X, oh, okay, okay, okay, okay, okay, okay, okay. So if let's say instead of four, I give you 16, it will be even bent more. Like this, it will be bent. Yeah. Correct. So I'll show you quickly on the Geo Geo tool as well. So let's say I plot Y is equal to log, log of two base X, okay, so this is the graph. And the moment I increase the value to four, it's more bend. Okay, yeah, correct. The moment I make it 16. When it's closer to infinity, I mean, it's closer to infinity, it bends more towards X axis. Yes. When it's closer to one, it bends lesser towards X axis. Yes. So if I take let's say log, who's it? See, more and more bent. Oh, wow. Okay. Great. Yeah, got it. Fine, then I will stop here. Okay. I'll see you next week. In fact, I'll give you some assignments to plot. Okay? Okay, sure. I'll send you a sheet and I'll mark the question numbers which you need to do. Don't do the entire sheet. Okay. So I'll send you by evening today. Okay. And yeah, I think next class, next time on Sunday, I don't think I'll be there because I'll be going out. Okay. I'll be going out to the station. So. No problem. We can have it on Saturday if you want. No, I'll be going between Wednesday and Sunday. So if we have like a Tuesday or something. No problem. Anyways, vacations are going to start Arushi. So we can fix up a schedule for that class. No worries. Okay, fine. Okay. Yeah, let me know. Okay, because Wednesday and Sunday, I won't be there. Sure, sure. No issues, no issues. Fine. Yeah. Okay. See you in a little bit. Thank you. Thank you.