 So, hi I am Pratik again and now we will talk about affine transformations on on independent variables and their effects on the plots of different functions. So, here we considered t to be a real variable a comma b are reals and so, the affine transformation we define as t goes from t to a t plus b and so, when the variable goes from t to a t plus b we study the changes which is the plot of the function x t undergoes. Here in the future slides I will show you different plots plots of x t and x of a t plus b in various stages and in all the plots ahead we consider that a is greater than 1 and b is greater than 0. So, this is the function I choose x of t. So, now we move to x of minus t. So, as you can see the plot has just flipped around the y axis I will just show you the previous plot again for reference. So, this is how x of minus t looks like. So, here we have multiplied the independent variable which is t by minus 1 and plotted the function. Next we go to x of a t a as I reminded you earlier is greater than 1. As you can see the plot has compressed here and I can see that Ashwit here has a confused phase if you might want to say something. Normally when we multiply with something that is greater than 1 we expected to get bigger. Why is the graph getting smaller? Yeah, I understand that this is this might confuse a lot of students. So, we now take the example of an exponential which Ashwit talked about in a previous video. Consider e power 2 t. So, we draw e power 2 the plot of e power 2 t here and this is the function e power 2 t. Consider when t is equal to 2 the value of the function which is the function is x of t. So, the x of 2 will be equal to e power 4. So, plot t equal to 2 here this is the t axis. So, e power 4 lies somewhere here. So, note that e power 2 which is e square will lie here and e square is the value of the function say x dash of 2 and x dash of t will be equal to e power t. We plot e power t again on the same graph. So, this is the plot of e power t they build this e power t. Now, as you can see the graph of e power t has compressed when you go from t to 2 t and that is how you explain the apparently counter interactive paradox which Ashwit noted and I hope that answers Ashwit's question this time. So, now we move on to x of t by a again a is greater than 1 and as expected the graph of the function has expanded and not compressed as was in the previous case. Now, we move on to shift which is x of t plus b remember that t is positive here as you can see the graph of the function has shifted to the negative real axis remember that t is positive this might cause some confusion among the student Ashwit here has something more to say on this I hope what he says makes you understand it a bit more. So, intuitively this looks like you are looking into the future and the signal in the future would look like something, but we are in the past and we are trying to bring that signal back to the past. So, intuitively the signal will be shifted towards the negative axis. So, that is why it looks like this. Yes, that is a very intuitive explanation. So, let us do this a bit more mathematically this time around. So, consider the plot of the function x of t the original function we saw. So, that function were looked something like this. So, this is the t axis this is x of t. So, now we have changed from t to t plus b b is positive. So, consider a specific value of t say t is equal to 0. So, when t is equal to 0 t plus b actually lies somewhere here. So, when we plot it on the t plus b axis which is this plot the previous plot this t plus b value actually gets mapped to the value of 0 and similarly every value on the t plus b axis gets mapped to a value b units to the left of it. Therefore, every value on the t plus b axis gets shifted to the left by b units and hence the plot on the whole also shifts to the left as is shown in the diagram. So, that is how you explain a linear shift in the independent variable. This is similarly a graph of t minus b here as expected the entire graph shifted to the right instead of to the left. So, now we combine both the transformations we have studied so far multiplication of the independent variable and a linear shift. So, this is the graph for x of a t plus b as you can see the effect of a can be seen in the compression observed in the graph the effect of b can be seen as the graph has shifted to the left and not to the right. But if you notice carefully the shift in this case is not as much as the shift you observed in just the plot of x of t plus b I will show that graph again. So, this is x of t plus b and this is graph of x of a t plus b the shift here is noticeably less. So, how we explain that we show in the next slide. So, again we observe that in the previous plot the shift is smaller than the one in the plot for x of t plus b. So, we explain that by writing x of a t plus b as x of a into t plus b by a. So, a is non 0 and a is greater than 1. So, as a is greater than 1 b by a will be smaller than b and the shift observed will be b by a and not b as a result the shift observed in this graph is lesser as compared to the shift observed in the previous graph. So, this is the graph for x of a t minus b again the shift is smaller and it is to the right and not to the left. So, Pratik do we apply scaling first or shift first? Yes I would like. So, before answering that question I would like to show you another graph. So, this is the graph of x of minus a t plus b as you can see here b is positive as I remind you again b is positive. However, the graph has shifted to the right and not to the left. So, here I come to the question Ashwit asked the operation you apply first shift or scaling is important and if you do not take care of that you would not be able to analyze situations like this. So, I would like you students to discuss this on the discussion forums and I will be ready to help whenever you have question. So, we move on to a next last graph which is again an extension of this the same graph. So, here we just show x of minus a t minus b again as in the previous graph the shift is not as expected and this again is can be explained by the order of the operation. Thank you.