 Hi, I'm Zor. Welcome to the Unisor Education. Today's topic will be quadratic functions, quadratic functions. The lecture has notes which are on the Unisor.com website, so you can read the notes first or after my lecture, whatever. I would like to present quadratic functions and not as just, okay, this is piece of information we should have to remember. I would like to increase the complexity of the explanation about whatever quadratic functions are, gradually from a simple case to a more complex case. So we will all be thinking about what exactly the quadratic function is and what's its properties, et cetera, et cetera, and we will derive this information rather than I just give it to you. So let's start with the simplest quadratic function. The simplest quadratic function is this. y is equal to x square. Now, it's basically a particular case of the most general quadratic function which I can write like this. So this is the most general one, but we are not considering it right now. We are considering this function with p equals to 1 and q and r equal to 0. So this is for later. Right now, we are concentrating on the most simple one. Now, when we are talking about functions, we have to talk about, first of all, where it is defined, what's the domain, where it takes the value from, codomain, and what's its range. All right. Quadratic functions as we will be discussing in this and some future maybe lectures are always defined on a set of real numbers. Yes, quadratic function can be, in theory, considered for complex argument. Remember, the complex is something like a plus e i, where i square is equal to minus 1. We are not considering complex numbers as arguments to quadratic functions. When we will be talking about quadratic equations, we will definitely consider complex solutions. But right now, when we are talking about quadratic functions, it's only functions defined on the real argument. So x is any real number. Now, why is it any real number? Because any real number can be multiplied by itself, which is square. And as a result, you will get, again, the real number, which means the main for this function is all real numbers. Codomain is also real numbers, but now my question is the range of this function, the values which it takes, doesn't cover an entire set of real numbers? Well, the answer is now, because positive and negative numbers squared would result in the positive number. So minus 2 squared is 4 and 2 squared is 4. You cannot get a negative number by multiplying any real number by itself. So, the main is all real numbers. Codomain, well, you can say that it's all real numbers, but the range where the function really takes values is only non-negative real numbers. 0, by the way, is a real value. If you put 0 here, you will get 0 as a result. So 0 and all positive, which means non-negative real numbers are the range. Okay, we've done that. Now, how about the correspondence between the domain where the function is defined, the arguments, and the range where the function takes values? In some cases, like, for instance, in case of a linear function, we have a one-to-one correspondence between the main and the range, because for every real number of a function, the result is the real number of the argument, which if substituted into this function gives the result. It's not the case with y equals to x where y equals, you cannot talk about one-to-one correspondence if 2 will result in the value of function 4, and minus 2 also will result in the number 4. And any other pair of positive and negative numbers with the same absolute value will actually result in a similar situation. So it's not one-to-one correspondence between all the values from the domain and all the values from the range of the function. However, however, we can always restrict our function to a narrower domain. So what if I would consider this function, y is equal to x square, only on the domain of non-negative numbers? Well, in this case, we do have a one-to-one correspondence, because for every... So let's say it's a different function. I will use q is equal to p square. p is an argument, q is a function. But I will consider this function only on this domain. Now, if I have any p, I will obviously get some value of q. But if I will get any q, then I can always have a square root of q, the arithmetic value of square root of q, the positive or equal to zero, the arithmetic value. And that will be an argument, the square of which substituted into this function will give the q. And if you remember, I'm always capable to take the square root of positive or equal to zero number, because actually that's how real numbers were introduced. If you remember the lectures where I was explaining what are the real numbers, I was talking about certain short comings of rational numbers. Like, for instance, you cannot find a rational number, the square of which equal to two. There is no rational number like that. So we can introduce real numbers to expand our universe of rational numbers to be able to find such the value p, real number p, the square of which is equal to two. So that's why I can always say that in the domain of non-negative real numbers, this function does establish a one-to-one correspondence between all the domain elements and the range, because the domain will be non-negative and the range will be non-negative. And for every domain, I can always, for every element of the domain, I can find one and only one element of the function. And obviously for any way of the function, I can find one and only one element of the domain. And that's what establishes one-to-one correspondence. Okay. Let's forget about this one-to-one correspondence and let's go back to our real numbers as the full scale of all the real numbers as arguments to this function, as it is usually done basically. So what I can say about this function, that this function is even. Now, if you remember, what is even function? Function is called even if for any x, if for any x from the domain, for any argument. I can say that the value of the function at certain value of the argument is the same as the value of this function for the value of the argument being negative to this one, if the values are the same, then the function is called even. Now, is this function even? Well, obviously yes, because we have already stated that positive number being squared and the corresponding negative number with the same absolute value with a different sign being squared actually results in the same thing. So x squared is equal to minus x squared. Interesting detail. I mean, I was just explaining this, but can you prove it? Well, that's maybe a little bit less trivial, but let's just think about what is minus x? Minus x can always be represented as minus one multiplied by x, right? I mean, that's from the definition of multiplication, negative numbers, etc., etc. Now, being squared is minus one times minus one. That's what it is. Square means you multiply it by itself. Now, multiplication is commutative, which means you have a construction running. Anyway, now if you will multiply it sequentially, minus one times minus one is one times x times x, x times x squared is x squared, so this is equal to x squared. So this is actually a proof definition of what is minus x, and I'm using the commutative property of multiplication. So basically I've proved that this function is even, because x squared is the same as minus x squared. Yeah, this is a trivial proof, but look, everything should be logically solid. So if you are stating something, you have to be prepared to prove your point. So I proved my point. This is an even function. Now, since this function is even, then its graph is symmetrical relative to the y-axis. Now, I will start working with the graph right now. I will explain how the graph should go, but whatever I will use as my logic to draw the graph for the positive x, not negative rather x. Then I will use the symmetry and reflect it relatively to the y-axis to get the other part of the graph. So I will draw the graph of this function only for non-negative x, and by symmetry I will extend it to the negative part. Alright, so let's talk about how this particular graph looks like. Well, first of all, we can start from the point where argument is equal to zero and the function obviously is equal to zero as well. So the graph starts here and then we extend it to the right, increasing the argument, and then I will reflect it by symmetry relative to the y-axis. Alright, so first of all what we can say is that the graph should go upwards because for every greater value of x the value of y is also greater. If the number is multiplied by itself, which is x square, the greater the number we take, the greater the result will be. So I would like to stop at point one where the function is obviously also equal to one. So we have two points. We have point zero zero and we have point one one. These two points definitely belong to the graph. Just out of curiosity consider this function. Now you know this is the straight line and it also intersects. It also crosses these two points, zero zero and one one. Now how does this graph look relative to this one? Let's think about it. If you multiply the value which is less than one by itself, which is also less than one, you will make it smaller. For instance, one half times one half is equal to one fourth. So the result is smaller than the argument. So if I am between zero and one with my argument, the value of my function would be smaller than the argument. So in this function, the value of the function is equal to the value of the argument. This is the argument and this is the function. Now this is smaller, so the graph should be below this. So no matter where I am in this interval from zero to one, the value of my function would be smaller than the argument. So the function will go under this particular line, y is equal to x. However, at this point it intersects it because the value of the function is exactly the same. So I am assuming basically the function will probably go something like this. Now after x equals to one, if you increase the argument, so you are multiplying the number which is greater than one by itself, which is greater than one, which is increasing. So the function will be, the value of the function will be greater than the value of the argument. If the value of the argument is 1.5, for instance, squared, it will do what? 2.25, right? So I am increasing the value of the function is greater than the value of the argument. So the graph will be above this line. So it will be something like this. So I am assuming from purely x equals to y, qualitative properties that the graph will probably look like this. Now let me put a little bit more logic into this particular shape. The next consideration is the steepness of this graph. Now what is a steepness of the graph? Well, I can measure the steepness in the following way. I take two points, let's say x1 and x2, and then I take the value of the function, which is y1 and y2, and I measure this particular angle. So how much function has increased, which is y2 minus y1, relative to how much argument has increased. So I am saying is that the steepness of the function can be measured by this ratio, how much function has increased, versus how much argument has increased. Now if function is linear, y is equal to x, we know that the function and argument are exactly the same, right? Which means that this particular ratio for this function will always be 1, because y2 is equal to x2 and y1 is equal to x1, so their ratio is equal to 1. So the function, which is a straight line, y is equal to x, has a steepness of 1 always. Now how about this function? Well, that's not the same argument. If you will take y2 minus y1, which is x2 square minus x1 square divided by x2 minus x1, as we know this is equal to x1 plus x2. For those who don't believe it, cannot apply this by this, and we will get this. So this is actually not a constant like in this particular case. It's a variable which depends on x1 and x2. Now, what I'm also trying to establish is local steepness. Now local steepness means when x1 and x2 are very close to each other. So in this case, when these two values are close to each other, it's a good measure of the steepness at the point of argument x1. If x2 is very close to x1, then this particular thing would be very much a local characteristic of the curve. And now we see, okay, so this is approximately equal to 2x1 if x2 and x1 are very close to each other. Now, what I'm actually saying is that this thing is increasing with increase of the argument. So if x1 is shifting to the right, increasing basically, then the local steepness is actually increasing as well. It's equal to double x1. So the steepness must increase, which means the curve goes steeper and steeper and steeper as we are increasing argument. That's very important, actually. Now, what I did right now is I have introduced basically the beginning of analysis where you have the derivative basically from the function. I'm not using the word derivative. I'm not using the limit theory, et cetera, et cetera. I'm just trying to explain the more elementary, from the more elementary standpoint. But basically the steepness is something which is in this particular case related to somebody's phone. This is basically how the steepness can be measured. And as we see, the steepness is increasing, which means the curve goes steeper and steeper to the right. Now, obviously using the symmetry, because this function is even, I can say that the continuation of this would be symmetrical relative to the yx. So the complete graph of this function, x square, looks like this. And it's called parabola, the word which we probably all know about. But this is the beginning where the parabola actually started. This is the function, the main function, and that's how parabola has started. All right. So now, we are ready to complicate the concept of function. And we will gradually try to increase the complexity of this function from the elementary quadratic function to, basically, ultimately to the general form. And that will be the subject of the next lecture. Thank you, and good luck.