 Hi, I'm Zor. Welcome to a new Zor education. I would like to continue talking about quadratic functions. The previous lecture was dedicated to a very simple quadratic function, the very beginning. This is the elementary one, and today I would like to expand on this. So, by now, we all know that this is the function which has a domain of all real numbers, range all non-negative real numbers. We know that the graph is parabola, which has a steepness increasing as we increase the arguments. We know that the function is even. And now let's just try to generalize whatever we know about this particular function, gradually towards more general one. So, the properties of this function we will gradually transfer to this function. Okay, so how do we do it? Well, we will do it in steps. The first step is we will complicate just a little bit our function. So, the first step is we consider this one. What's the difference between these two functions? Well, not much, quite frankly. Domain is exactly the same thing, all real numbers. Now, the range, well, since you know that the x square is always zero or positive, it means that the range of this function is this. Because we are edging to see something which is non-negative, right? So, if c is positive, then the y is more positive. If c is negative, it's still less negative or a game more positive. Now, from the graphical standpoint, we know that if we will add a constant to the value of the function, this is the value of the function, we add the constant. The graph will be shifted upwards, which means that parabola will be here. This is for positive c. If c is negative, it's similar. So, all we're talking about is that this particular point, which we can call a vertex of the parabola, the vertex will be shifted from whatever it was here, and it was, vertex was at zero, zero, obviously, right? For argument equal to zero, the function equal to zero. In this case, the vertex would be, for argument zero, it will be the value of c. So, the vertex is shifted along the y axis to the position zero c. And that's the only complication of this function relative to this one. There is nothing actually more to consider here. Next, a different kind of complication. What if, instead of this function, we will consider this one? We multiply it by eight. What happens in this case? Well, first of all, the vertex of this parabola would be exactly in the same place. At zero, it's equal to zero. But then let's think about what happens if this function had this graph, one, one, and one is one, one. Now, the new function for x equal to zero is also equal to zero. But for x equal to one, it's equal to eight. Let's consider a is positive, just, you know, for argument's sake. Then what happens with the graph of the function? Well, then if this is eight, the function will go through this point. So, the parabola will look like this. So, if a is greater than one, it will be steeper. If a is less than one, if this is a, then the parabola will be less steep than the original one. So, in any case, the positive eight doesn't change the direction of these two, how should I say it, parabola horns. So, the horns will still point upwards. The vertex will still be in exactly the same place at zero, zero. But the steepness would be dependent on whether a is greater than one, in which case the steepness would be greater than the original parabola, or less than one, in which case the steepness would be less. That's for the positive eight. What if a is negative? Well, obviously, it would be symmetrical towards x-axis. So, instead of if a is negative, then it would be somewhere here, and the parabola would be correspondingly like this. So, horns will be directed downwards for a negative eight. And again, steepness depends on the absolute value of a. If it's absolute value of a is greater than one, it would be steeper. It would be closer to the vertical axis. If a is less than one, it would be less steep, it would be further from the vertical axis. But that's the only complication. Again, it would not change the character of the curve. It was a parabola, and it is still a parabola, just a different parabola. It used to be parabola with this type of steepness and the horns pointing upwards. Now, this parabola will have different steepness depending on the absolute value of a and different direction of the horns depending on the sign of the eight. Okay, so, fine. Let's complicate our story even more. So, little by little, I would like to come up with a general expression for quadratic function and its properties. But I don't want to give it to you. I would like to kind of gradually complicate the story and we will all derive to the conclusion about this generalized function. So, next complication is this. y is equals to x plus b square plus c. Again, a little complication, right? Now, from the general graph theory, which I have presented in other lectures, you can say that what happens with this particular function if we do this particular transformation? We transform argument, we add b to the argument, and then we take this function and add another constant to it. Now, you know again from the theory of graphs that if we will add to the argument the value b, then the whole graph will be shifted left. If b is positive, it will be shifted left. If b is negative, it will be shifted to the right by absolute value of b, y, for a very simple reason. If the point p and q belong to this particular graph, then the point p minus b, q belongs to this one. And, yeah, actually plus c, sorry, plus c. So, the graph of this function would be, let's start again from original one, and original one would be like this. That's our original parabola, y is equal to x squared. Now, let's consider for argument say that b is positive and c is positive for it. So, in which case this is minus b and this is c. So, this point would be the vertex, and the parabola would be like this. Well, let's just consider. If the point minus b is substituted into this equation, y would be equal to minus b plus b is zero, so y would be c. So, that's why we have this, and everything else is just the same parabola as this one. So, direction of the coordinates is the same, steepness is exactly the same. Steepness is actually determined by the coefficient here, but we didn't introduce it yet. So, we just shifted the vertex. Now, if b is negative, for instance, then it would shift to the right by an absolute value of b. If c is negative, it will shift down by the absolute value of c. So, this is basically exactly the same thing. So, we know what happens with the graph when we add something to the argument and then add something to the function. The positive addition to the argument shifts the graph to the left, negative to the right. Positive addition to the function shifts the graph upwards, negative downwards. We know that. All right. So, now I think we are ready for the last complication which we would like to introduce, and this is a x plus b square plus c. Now, what happens in this case? Well, let's just think about what we have already learned. Adding b to the argument shifts the graph to the left by absolute value of b. Let's say b is, let's consider them to be positive. It's just easier for me to explain. So, if b is positive with a and c are all positive, then this would shift the vertex of this parabola into point minus b. So, I'm assuming everything is positive in this case, but negative would be the same thing. Now, then I actually know that my vertex would be at point c here, right? Because if I will substitute minus b to x, this will be 0, 0, so it will be c. So, this is my vertex of the parabola. Now, this steepness is determined by this. So, a is positive and depending on how big a is, it will be steeper or less steep than the original parabola, y is equal to x square, but in any case it would be something like this. The greater the a, the narrower, narrower parabola would be steeper basically. The smaller a, it would be more less steep. Okay. Now, what if something is negative? Well, let's just think about it. If a is negative, the parabola changes the direction of the horns. If c is negative, it would move downwards instead of upwards. If b is negative, it would be shifted to the right. But basically, it's all manipulations based on the absolute values and signs of a, b and c. Parabola will still be parabola. However, it will differ from this parabola by number one, location of its vertex, direction of the horns depending on the sign of a, and steepness depending on the absolute value of a. So, these two parameters, b and c, determine the vertex and a determines the direction of horns and steepness of the parabola. But it's all parabola. Again, for whatever coefficients a, b and c, we choose it's still a parabola. And now, let's just do the last step which we would need to find the properties of our general function. This is our general function. What I would like to do is, since I know the angle of this, I know where the vertex of the parabola is, I know the steepness, etc., how can I derive from these a, b and c from these? Well, let's just do a very simple thing. Let's just solve this and let's just find what our a, b and c are if p, q and r are given. a, b and c, we need to find out if we know p, q and r. Now, this is supposed to be for all values of x. These are two functions which are supposed to be equal to each other for all values of x. Well, it's very easy to derive from this that basically if I will open the parenthesis here, I will have a polynomial of the second degree on the left and on the right. So, let's just do it. It's a times x square plus 2 b x plus b square plus c equals to p x square plus q x plus r. Or if I will open the parenthesis here, I will have a x square plus 2 a b x 2 a b x plus a b square plus c. So, this is my second degree member and it should be equal to this one. This is equal to this one and this is a free member without any x. It should be equal to this. Now, why if I have two polynomials equal to each other, why the corresponding coefficients must be equal to each other? It's actually very easy to prove. But in any case, let's just think about it, since it's supposed to be for all x's, right? So, we can actually choose an x equal to 0, which means this part disappears and this part disappears and that will have c is equal to r. I mean, a a b square plus c is equal to r. With x equal to 0, these two disappear and these two disappear. So, I have a b square plus c is equal to r. Now, since this is equal to this, I can basically write that a x square plus 2 a b x is equal to p x square plus q x, right? Since I have this piece is equal to e to this one, I can already use this. Now, I just drop them because they're equal from both sides of the equation and I have this one. Now, I can reduce it by x while considering the x is not equal to 0 and we can consider it because this is supposed to be for all real x's, right? So, let's take it for any x not equal to 0. I will reduce it by x and I will have a x plus 2 a b is equal to p x plus q. And now, we have a similar situation. We have equality between two polynomials and I can again, after that, just assign x is equal to 0 and have 2 a b is equal to q. So, that's another thing. 2 a b is equal to q. And finally, the same way, I can drop this and I can a x is equal to p x so I have a is equal to p. So, now, I can basically, using these, find what my a, b and c really are. Actually, I can, I found a already. I can substitute it here. So, I will have 2 p b is equal to q from which I can derive that b is equal to q divided by 2 p, right? So, I have a and I have b. Now, I substitute it here and I will get what? From a, I will put p. From b square is this one. So, it's q square divided by 4 p square plus c is equal to r from which c is equal to r minus, now, we can reduce this by p. p is not equal to 0 by the way otherwise it will not be a quadratic equation minus q square divided by 4 p. So, from this, I have derived this where a, b and c are these coefficients. I don't need this anymore. I don't need this anymore. So, I can always rewrite this particular equation, sorry, this particular function in this format. y is equal to a which is p times x plus b which is x plus this square plus c which is r minus q divided by 4 p. So, what is this? It's exactly the same function just represented in this format. Why do I need it? Because now I can definitely say the following that since I express this particular function in this format and I have already researched this format before. So, I know everything about the behavior of this function. I know that its vertex e is at point minus bc. So, this is the parabola with vertex at minus q divided by 2 p r minus q square divided by 4 p. So, I know that much. I know that this coefficient p determines the direction of the horns. If p is positive, it's upward. If p is negative, it's downwards. And it also determines the steepness. The greater absolute value of p, the greater the steepness of this parabola is. That basically concludes the research related to this function. So, I can know that any quadratic function, and again, it's quadratic so p is not equal to 0. So, for any quadratic function, I can say that it is represented graphically as a parabola with this particular vertex and the steepness determined by the coefficient p positive or negative horns up or down and steepness depending on the absolute value of p. Okay, that's it. That's all I wanted to talk about quadratic functions. So, it's always a parabola. We know how to determine where exactly the vertex of this parabola is and the steepness. We know that the function is defined domain. It's all real numbers. And considering its range, it all depends on whether the p is positive or negative and relative to this free coefficient, which is c, actually, it goes either above or below while depending on the sign of the p, whether the horns are ups or downs. Okay, that's it. Thank you very much. That concludes my theoretical part of the quadratic equation, sorry, quadratic functions. Since I mentioned quadratic equations, obviously, when I will be talking about quadratic equations, I will use this particular full square, actually it's called full square form or format. So, I will use this full square format to help you to solve the quadratic equation. What else can be said about the function? Practically, that's it. I think the theoretical part of this quadratic function is basically completely exhausted in this case. So, thanks very much for listening to me. Don't forget that unison.com contains notes for this lecture and it does contain lots of other things, including exams. Your parents or supervisors are encouraged to have a login to basically supervise the process of your education and well, good luck. Thank you very much.