 Hi, I'm Zor. Welcome to a new Zor education. Continuing our linear function theme, we have discussed the major principles, the definition, the main, etc. We have discussed the graphs, and there is one very important property of linear function, which I would like to talk right now. This is proportionality. Now, what actually I mean in this particular case? Proportionality is a concept very much embedded in the linear function in a way that as you are proportionally increasing arguments, the function also proportionally increasing its values. Let me be a little bit more precise. And I think graph actually might help in this particular case. So if you have a straight line which represents the graph, and I will take three points a, b and c with values p, p, u and r, and the function is ax plus b, where a is not equal to zero. Now, what I mean is that these are increments of argument. These are increments of the functions, the function value. So my point is that the ratio of increment is exactly the same, arguments or the value of the function. What I would like actually to prove more precisely is that the relative increment of the value of the function in two different points. So this relative to this should actually be the same as relative increment of the argument. The geometrical standpoint, it basically means that these two triangles are similar. Since they are similar, then this catheters relates to this as this to this. So the similarity from the geometrical standpoint means this algebraically. And obviously it's all related to the fact that this is a straight line, obviously nothing connected with hope in general. So for any linear function, this is actually true. Now, how can I prove it? Quite frankly, it's very easy. Let's just think about what r, p, q and r relative in this term considering a, b and c are arguments, and we will very easily come with this particular ecology. Okay, r minus q is a, c plus b minus a, b plus b divided by q minus p. q is the value at point b which is a, b plus b minus p is value at a equals, what does it equal to? Well, obviously if I will open these parentheses, now b would be here as a plus, here would be the minus, and I will have a, c minus a, b, or a, c minus b. Now, in the denominator, I will have a similar picture that would be minus a, a minus b, so b will be reduced, so it will be a, b minus a, a, so it's a, b minus a. Since a is not equal to zero, I can reduce by a, and it's c minus b divided by b minus a. So this is an algebraic expression of similarity of these triangles from the geometrical standpoint. In any case, it signifies the proportionality of the increments of the function to the increments of the argument, that's very important. And obviously this proportionality depends, obviously on this factor a, and the fact that the whole thing is linear. That's why we can reduce all these formulas and improve this proportionality. Now, this is kind of an obvious property of linear function, but why did I, why did I ever talk about this trivial stuff? Well, there is a very important reason to this and very practical one. This is a characteristic property of the linear dependency between argument and the function. So if I have certain statistical results of some experiment where on the input to some system I give one particular parameter and derive with the value of another parameter on the output of this system, then I'm changing my input parameter. My output parameter is also changing. I can check if dependency between input parameter and output parameter is linear or is not linear based on whether this particular property is observed or not observed. And I would like actually to go to a specific example, physical example if you wish, doesn't really matter which one in this case it's physical, which would illustrate how people in science or wherever else can actually make a judgment about dependency between certain parameters, whether it's linear or not linear. All right, here is exactly what I mean. Now, let's assume you have some kind of a rigid reservoir filled with air. Now, here you have some kind of source of heat. This is supposed to represent the flames. All right, so let's consider this is some kind of a pressure cooker and this is your source of heat. Whatever, doesn't really matter. So what's important is it's rigid, which means it really has certain shape which does not change. Now, and it's completely airtight. Now, as you increase the temperature from the physical standpoint, you understand that the pressure inside would grow. Why? Because the molecules are moving much faster and that's what actually means that the pressure is growing. So people were definitely interested in how dependent the temperature of this particular container related to the pressure. So let's consider somewhere you have an instrument which measures the pressure. Here you have a thermometer which measures the temperature and here are the results. So you have a temperature and you have a pressure. Okay, in my case, I check the pressure for these four temperatures and the pressure in some units is 200, 300, 400 and 1200. So let's just completely disregard what kind of units of measurements, whether it's Celsius or Fahrenheit or Kelvin doesn't really matter. We're dealing with plain numbers. In some unit of measurements, these are temperature and these are pressure. Now, my question is, is pressure linearly dependent on the temperature? Well, let's remember that we have that property of proportionality between, if it's linear dependency, then it should be proportionality between the pressure and the temperature. So let's just check this proportionality. The change between 50 and 60 is 10 units and change between 200 and 300 is 100 units. So the ratio is 10. Now, this change between 60 and 70 is also 10 units. Between 300 and 400 is also 100. So again, ratio is between the increment of the function relative to the increment of the argument is also 100. And then the third ratio from 70 to 80 is also 10 from 400 to 500 again 100. So you have 10 again. So it looks like there is a proportionality between change of the argument and change of the function. So there is a strong indication, at least based on these four measurements, that the dependency is linear. Now, dependency is linear and this ratio actually is the slope. So if I'm looking for y is equal to a times x plus b, a is 10. That's the slope because change of the argument results in the change of the function 10 times. So the only thing which I don't have is b, right? How can we establish it? Very simply, let's just substitute x and y, let's say 50 and 200. So I will have y, well, which is 200. 200 is equal to a, which is 10, times 50 plus b. So this is 500, so b is equal to minus 300. Now, what if I will substitute these two or these two? Well, obviously I must receive exactly the same result, otherwise I did something wrong. But let's just check. For instance, I put 60 and 300. So 300 is equal to 10 times 60 plus b. Now, what is b in this case? This is 600, this is 300, so it's minus 300. The same 300 as here. Well, 70 would be, what, 400 is equal to 10 times 70 plus b, 700 minus 400, again minus 300. The same thing over there. So obviously everything is held. So my equation is minus 300. So my equation is y is equal to 10x minus 300. That's the result of my analysis. Based on all information which I have, and I have only four measurements, I can actually derive the equation which connects these in this way. But the first and most important was I checked the proportionality to make a decision whether there is a linear dependency or there is not. So now, let me make another example and then we will decide whether there is a proportionality there or not. Okay, here's an example. Let's assume I have the same table where I measure my pressure at this temperature. But instead of a rigid container, as I used before, I will use a balloon. Now, balloon is expanding as the pressure is rising, which means pressure will rise a little bit less than in case of a rigid container. So as the temperature rises, the rigid container is preserving the volume and the molecules are moving faster and that's increasing the value. The rubber balloon is expanding so the molecules have more room to grow, so the volume also is changing. And obviously the pressure depends on the volume as well. If the volume grows, pressure usually is going down. So increase in the pressure should be not exactly the same as in case of a rigid container. Now, I suggested to use the following measurements. 150, 200, 260, and 330. I promised the pressure grows not as much as in the rigid container case. Now, let's check exactly the same thing. Do we have a proportionality or not? Now, 60 minus 50 is 10. That's the change of the argument. Now, the change of this is 50 from 150 to 200. So the ratio is 5. Next, from 60 to 70 also have temperature raising by 10. But here I have by 60. So my ratio is 6, 60 to 10. Next, from 70 to 80 again, my temperature is changing by 10. My pressure is increasing by 70 from 260 to 330. So ratio of the change of the function to the change of the argument is 7. So as you see, here we have different ratio. Now, that alone, I mean, considering the numbers are pretty different. It's not like 5.001 and 5.002. They are very, very close and can be explained as some kind of errors. But this is not an error. These are really dramatically different numbers, which proves that in this particular case, there is no linear dependency between the rising temperature and rising the pressure. And the reason for this is physical, obviously, because the balloon is expanding, so volume is changing, and volume is just yet another very important factor in changing the pressure. So, well, basically, that's it. That's all I wanted to talk today about. And what's important is to understand that the characteristic properties of the linear function can actually be used in practical life, and they are actually used to get to some kind of maybe physical or whatever the practical implementation is, dependency between arguments and the functions. By the way, in this particular case, what's important is there is a real physical law which is kind of in the foundation of whatever I was just talking about. So, if you have all three variables changing the pressure, the temperature, and the volume, there is a law that this is constant. So, if V is constant, then P is proportional to T, because if P over T is a constant, then that's basically what it means, that the P is a linear function of T. But if V is involved as well, that actually changes this direct dependency and it actually introduces this new factor, the volume from which pressure depends upon it. But this is all physics, so it doesn't really matter. What's important is the characteristic properties of the linear function are very helpful to determine physical dependencies between certain things, between certain physical parameters. All right, so that's it for this particular lecture. Thank you very much. Next, probably I will talk about quadratic functions, I hope.