 Quickly, first of all, you start with the graph of x cube Like this Then we make the graph of Okay, now I'll do it with a slight change Can I write this as y plus y is equal to minus x minus 1 over q Okay, now can I do one thing? Can I first, so it will look like this So this gap is 5 minutes now, so now this position is 0, negative 5 Is that right? Next, what are the next steps that I have to do to get minus x minus 1? Change x with minus x So if I change x with minus x means I am reflecting this graph about y axis Again I am repeating the rule, changing the sign of x means reflecting the axis Changing the sign of y means x axis So if you select it up, this is how you will see the graph Is that fine? Next, I want, so you have already got y plus 5 is equal to this So now I am going to change my x with x minus 1 Okay Yes, I know This is as we were saying, negative x minus 1 cube So changing x with x minus 1 means this reflected graph will shift only this to the right So now your final result would look like this 1 comma negative 5 It will come at 1 comma negative 5, so it will be like this So this portion will be 1 comma negative 5 Is that fine? Now if you understood this Moving on If I extend the concept to x to the power of 4 now So quadratic we saw, cubic we have seen just now Now let's talk about bi-quadratic In bi-quadratic, in fact you can try it by plotting some points In bi-quadratic you will realize that the graph would be of a similar shape as a parabola But slightly more flatter I'll show you on GeoG So y is equal to x to the power of 4 Okay This time we're raising up Okay I'll plot the same here also So it takes time Okay Now I will plot y equal to x square simultaneously Now you'll see something happening over here If you see in the minus 1 to 1 zone something strange is happening After 1 if you see the blue one dominates the green one After green one But between minus 1 to 1 the green one dominates the blue one Dominates means higher than the blue one Why is that so? Dominate means higher in value See if you zoom it The green one is above the graph of blue one In minus 1 to 1 domain But post 1 and pre minus 1 the blue one is higher than the green one Why is that so? Because it has a higher power So higher power means higher values Absolutely Anandya very good You're only dealing with fractions there So which is the higher value? Minus 1 half square has a So x square graph is higher than x to the power 4 graph Higher means for the same x value Y value is more for x square Yes or no? So that's why in this zone you realize that Your green graph is higher than the blue graph But the moment you cross one That means you're no longer dealing with decimals No longer dealing with fractions Then let's say I take a 2 value Then 2 square will become lesser than 2 to the power 4 And that's the reason why now blue graph becomes the boss So between minus 1 to 1 X square is the boss But after that Before minus 1 x to the power 4 domain Can you predict that trend if I do x to the power 6 Please draw in your notebook These 3 graphs Y is equal to x4 Y is equal to x square And Y is equal to x to the power 6 On the same x y axis And carefully tell me what do you observe In the graph That observation is very very essential Now when the graph comes I'm predicting it will come Least of the 3 in this interval again Are you getting that? Let's see Let's see when we plot it Y is equal to x to the power of 6 Yes Seriously Now this Dark green one You see that It's even higher than the blue one Post one And even higher than the blue one 3 minus 1 But in the interval minus 1 to 1 This is the lowest That means it will be more flatter That means if I put A is that It's like a U So let's say Y is equal to You see that It's become so flat that you almost realize It's becoming parallel to the x axis And then it's becoming flat But it's not actually flat It's actually the differentiability Which you'll lead in higher section I think your girls are doing that There is no corner actually here But you perceive it as a corner Because it's very very small I'll show you x to the power 3 first This was the graph of x to the power 3 How do you expect the x to the power 5 It would appear like this At the same interval At the same interval Absolutely At this point it will be minus 1 Plus 1 Are you getting it? So it has become more flatter In the minus 1 to 1 zone So after predominating In terms of magnitude But between minus 1 to 1 X cube is the boss Are you getting this point? So let me show you again on gg graph Let's delete all these Okay So let's say the graph of X cube first I will draw Y is equal to X cube You see that X to the power 5 dominates X to the power 3 But in the interval minus 1 to 1 X to the power 5 is having a lesser value in magnitude Than X cube graph What will happen if I put X to the power 7? Even more flatter X to the power 9 Even more flatter So X to the power any odd power Less power very Less power very Is equal to X to the power 7 They are typing in the name Oh my God X to the power It has become so flat that it is looking like a It's not exactly not flat There is a smooth turning over So with this we will move on quickly So that we can do a lot of questions from your worksheet also This nature 1 by X 1 by X square 1 by X cube and all Now Y is equal to 1 by Unic section chapter So I am not going to spend too much time talking about it It's just that You will see this in your thermodynamics In your class 10 But you will be having it in class 11 There is something called isothermal curves Where you will just be It will come out to be a shape Which is now what I am going to draw in front of you So a rectangular hyperbola looks like this Now listen to this concept very very carefully It is drawn in this way because You can see for yourself that When you put X value as What is the quantity Let's say 0.0001 Very reciprocated So when you are very close to 0 But on a positive side This value will become very very large Of course I cannot go and go there So I have just shown it by an arrow That means having a very large value And as you start increasing the value Let's say 0.0001 You make it 0.1 Then it becomes 10 So at 0.1 let's say here It is having a value of 10 Let's say 1 It will have a value of 1 It is not to this scale So as you keep on increasing Now from 0.0001 to It starts falling down and down The value of Y will start becoming smaller And smaller and smaller Now the important point here To be noted is that You can never make it reach 0 This value will never become 0 Right That means it will never touch Your X axis So we say this phenomena as Your Y axis is asymptotic For your X axis is An asymptote to this function It is felt like this Asymptote What is asymptote right now? Asymptote is basically a line Which is at a finite distance From the origin Right now definition of an asymptote A line which is at a finite distance From the origin And appears to touch the curve at infinity And it appears to touch the curve At infinity So in this case Your X axis And your Y axis Both will be asymptote Normally we call this as a horizontal asymptote We call this as a vertical asymptote People who would be writing calculus AP calculus exams This asymptote concept is very important for you There are questions where you would be Asked to plot these asymptotes Is that fine? Now what happens On slightly left to 0? Let's say I choose negative 0.000 What will I get? I get a very huge Negative value To minus infinity Here it was going to plus infinity And as you say this one Your value will of Y will come out to be At minus 1 Your value will come out to be minus 1 At minus 2 Your value will come out to be minus half So it starts decreasing down in magnitude As you go towards more Negative values of X That's why the reason for this shape Again All the rules that we have discussed Is equally valid to this as well