 I'm Zor. Welcome to Unizor education. We will talk about graphs of specific functions today. Function looks like y equals x to the power of r. And we are not fixing r to any specific variable. We will try to examine how this function looks for any different value of r within a certain range of course. And we will assume that x and y are real numbers. Before doing that, let me just spend a couple of minutes on what is x to the power of r, where r might be different. Primarily, we all know what it is if r is integer number. Obviously, x to the power of n is x times x, et cetera, times x, n times. Everybody does know that. Now, what about rational power? What if it's x to the power of p over q? What is this? Well, let's think about it this way. Let's call it z. We know that x to the power of n is equal to x multiplied n times. So basically, what I would like to say is that these powers are edit. Like, for instance, x to the power of 3 is equal to x to the power of 2 times x to the power of 1. 2 plus 1 is equal to 3. Now, in a more general case, you can say that x to the power of n plus n is equal to x to the power of n times x to the power of n. Because this is x multiplied by itself n times. This is x multiplied by itself n times. So together, they're multiplied n plus n times. So this obviously is a property of the function of power. Now, if this is true, let's do it in this particular case. What if I will have z times z, etcetera, times z, q times? It should be equal to the z to the power of q, right? So these powers are edit together. Now, what is z? z is x to the power of p over q. Which means if x p over q is multiplied by itself q times, we will get z to the power of q, which is x to the power pq. That is another property. Now, what is this? So the first property was x to the power f plus n equals to xm times xn. Now, what is x to the power fm multiplied by n? Well, that's very simple. That's x to the power of m to the power of n. Why? This is x times itself m times, right? m times. Then we have to raise it to the power of n, which means multiplied by itself n times. So this is repeated m times. So m times within the brackets, times n brackets by itself, so we have m times n x repeated. So that's another property. Now, based on these properties, this is equal to x to the power p over q times q, which is x to the power of p, right? So what I'm talking about right now is basically the definition. What is a rational power? What does it mean to raise to a power which is a rational number p over q? As usual, how do I define it? I define it in such a way to retain the basic properties which power has for integer numbers. So these are basic properties. So I retain these properties. And if these properties are correct for my newly defined x to the power of p over q, I have to have this equation. So what does it mean if the same number repeated q times is equal to another number? Well, in a different symbolical representation, it means that x to the power of p over q is equal to root of q of x to the p. Why? Because this is repeated q times and we get x to the p, power of p, right? So what is each of these? That's the definition of the root of the q's degree. So basically what we can say is that this is the root of the degree q from the x to the power of p. That's the definition. It's not really anything which I have proved. It's the definition. But I base this definition on certain properties of integer numbers. So these properties are retained. So that's about rational. And one more thing about the power. Negative. What if I have x to the power of minus n? Let's think about what this is. Well, again, based on this property, I can say that x to the power of n times x to the power of minus n is equal to the x to the power of minus n plus n. Minus n plus n is 0. x to the power of 0 is 1. x to the power of 0 is equal to 1. Always, by definition, to retain these particular qualities. So what follows from here? That x to the minus n is equal to 1 over x to the n. Again, that's the definition of the power to the negative number of the power. So we know that x to the power of p over q is actually root of the power of q from the x to the power of p. And x of minus n is equal to 1 over x to the n. All right. So these are just properties which are supposed to be known to everybody because these are kind of definitions of the function power for any particular value of the power. What I did not actually talk about here is what if the power is any real number, including irrational. But again, you understand that every irrational number can be represented as a series of approximations with rational numbers. So basically, we can define for every irrational number a series of irrational numbers, a series of rational which are tending to this. So basically, I don't want to go into these details too deeply, but the function x to the power of r is defined for irrational number as a limit, as a result of this, as a limit of this tendency if you approximate with rational numbers. Okay. Enough of that and let's talk about the graph of this function. Well, let's start from the very, very simple thing. First, r is equal to 0. So we have y equals x to the power of 0. But we know that this is always 1. Right? x to the power of 0 is over 1. Which means that the graph will be, it will be this. It goes to the point 1. So for any x, the function takes the value of 1. So the graph will be this horizontal line. Okay. That's it. So for any other, let's say, to the first degree or to the second degree or to the power which is any rational number which is not equal to 0 or negative or whatever it is, x being equal to 0 will always result in y equals to 0. So any function which we will consider from now on will have this power not equal to 0. And if it's not equal to 0 then the function for x equals 0, y will be equal to 0. This is, for instance, for y equals x to the first degree or just plane x. But if y is equal to x then for every value of x y is equal to exactly the same thing. Right? So they're always, so the point where they're crossing will always be equally distance from both axes which means that the whole thing will be on it by sector of this angle. And everybody knows about that. Everybody saw this graph, y is equal to x. And most likely everybody saw y is equal to x square which I'm going to the next. Okay? x square. For every value of x which is, let's say this is 1 and this is function y is equal to x. All right, now you try to build the function y is equal to x square. Let's think about this way. If the value of x is greater than 1 x square is always greater than x. Right? Like 2 square is 4, greater than 2. 3 square is 9, greater than 3. Any value which is greater than 1 being squared result in the value which is bigger than the original one. So if I start from here, from any point and I will try to make the point if this is x what is y which is x square it will be much further than this y is equal to x line. So points will be somewhere there. If it's here, somewhere here. And obviously if x is equal to 1 x square will be 1 as well. So this is the point where both y is equal to x and y is equal to x square have exactly the same value for x equals to 1. But after the 1 x square grows faster than y is equal to x. So it goes something like this way. Now, before 1 from 0 to 1 any number which is from 0 to 1 if you square it it will be a smaller number. The result will be smaller. For instance, 1 half. Square of 1 half is 1 quarter smaller. 1 third. Square of 1 third is 1 9. Smaller than 1 9. So basically these values will be below y is equal to x. And at 0 they need to gain. So the basic graph is like this for x is equal to 2. Now, what will be the negative part? Well, if x square is an even function if you remember because negative number being square results in the same number as positive, the corresponding positive. So it will be symmetrical here. And this is parabola whichever goes so. Okay, great. Now, let's talk about a bigger power. What's bigger than 2? 3, obviously. If we are talking about integer numbers. We will return by the way to rational and negative etc. Now, 3 is bigger. Power is bigger than 2. Now, what does it mean? It means that for numbers which are greater than 1 they will grow faster. Again, 2 cube is bigger than 2 square. So as we raise into a bigger power the numbers which are greater than 1 are bigger. So here x cube now this is y is equal to x square. Now for cube it will be even higher. You know what, let me just take this equation out of this. I'll put it here. So we are considering x cube right now. So this is y is equal to x. This is y is equal to x square. Now y is equal to x cube. So as I said, starting from 1 for all numbers which are greater than 1 it will grow faster. So it will be even higher. That will be y is equal to x cube. And whenever we are going whenever we are drawing the graph between 0 and 1 we have an opposite effect. If you have a bigger power then the number which is in this particular range will become smaller. So x to the power of 3 4 let's say x equals to 1 half will be what? 1 eighths. Which is smaller than x square which is 1 fourth. So what I'm saying is that in this particular area the x cube will be below than x square. But again in x equals to 1 and x equals to 0 all of these functions are the same. For 0 and 0 it's this point and for 1 it's 1 it's this point. So it's like a knot. If you will consider all the different graphs for all the different powers in this particular case they are tied together. And in 0 they are tied together. So in 0 to 1 those with a bigger power go smaller and those with a bigger power they will be positioned higher. Now what about the negative x? Well why is it equal to x cube is an odd function because it changes the sign if you change the x. So this will be this way this is minus 1 Now you obviously understand that the bigger the power is the more graph will be the lower graph will be this area and the higher in this area so this will be y is equal to x and x to the fourth degree is again an even function so it goes here. S power is increasing our graph becomes smaller and smaller here but then since the graph still has to cross the 0.11 it will be steeper growing in the area which is close to 1 and then even steeper growing further up when x is greater than 1 and as far as the negative numbers if the power is integer if it's an odd integer like 3 it will be centrally symmetrical it is an even like 2 or 4 it will be symmetrical relative to y axis so these are about positive integer numbers Now what about additional numbers which are of p over q side type well first of all let's think about odd or even functions since this is a definition of x to the power of p over q obviously if p is odd then the function is odd and if p is even then the function will be even well that's almost true because now the whole thing depends on q consider if q is let's say 3 and x to the p is let's say negative number because x is a negative number well the cubital root of the negative number exists and it's negative but what if it's q equal to 2 it will be square root of something which is negative square root does not exist among the real numbers we need complex numbers to resolve it so traditionally when we are talking about general function of y is equal to x to some power and the power can be even rational number usually people do not consider these graphs in the area when x is less than 0 than negative x so usually all these graphs are considered for positive x so we don't have this problem of dealing with when the function is defined when the function is not defined etc because as I was saying if you have something like y is equal to x to the power of 3 second for instance then this is basically square root of x to the power of 3 and if x is negative you cannot really extract the square root of it so it's not defined function this type is not defined on negative x so for some kind of a simplicity purposes all these functions y is equal to x to some power usually considered as defined with positive x so we don't have these problems with negative so forget about function being odd or even it is defined when the power is integer but it's not defined when the power is rational it's not always defined so we don't consider it at all so if we have completely cut out the negative part then everything becomes actually much simpler because if you have p over q then let's say it's y is equal let's say y is equal to as I was saying 3 over 2 well this is 1 on the half it's greater than 1 but less than 2 and if you will start thinking about this graph if this is the graph of y is equal to x to the first degree and this is the graph y is equal to x square 3 second which is 1 on the half it's in between 1 and 2 so the graph will be in between so the graph will be somewhere here in between these two guys here and in between two guys here so it's still it still has this pseudo parabolic shape so it's a little bit lower than 1 than y is equal to x in this particular area and it's above y is equal to x in the area of x greater than 1 and it grows to infinity faster than y is equal to x but slower than y is equal to x square so this is basically in between these two things and that's how it is defined now the only thing which we have left out is when the power is negative and that's the last component to that let's do it for integers first it's easier and then we will say that the rationals are in between the corresponding integers so we are talking about x to the power of minus n let's say but you know that this is 1 over x to the power of n right so let's draw this graph and then we'll see what happens and I will consider our own positive numbers as I was saying it's easier so we don't have to deal with when the function is defined when it's not defined it's always defined as long as x is greater than 0 alright so first of all we will do something which I was talking about during one of the lectures on the graphs I will use I'll just try to divide one to another so I have function x to the nth degree and I will divide one into it ok we all know that this is 1 this is y is equal to x and this is y is equal to x to some to some power of n whether it's 2 or 3 it always goes below y is equal to x in this particular interval and above and faster grows when x is greater than 1 so what is 1 divided by this number so this is my number and I have to divide 1 by 8 first of all y cannot be defined when x is equal to 0 we cannot define by 0 we divide by 0 that's out however as x tends to the 0 x to the nth tends to the 0 as well so 1 over x to the nth will go to infinity so obviously the function will go to infinity as it approaches as function approaches as x approaches to 0 1 over x to the nth will go to infinity now if x is equal to 1 then 1 to the nth degrees 1 1 over 1 is 1 so the function still goes through this point you remember this is the point 1 1 which ties together all these graphs it still ties them together even these types of function with a negative n so here we have the function which goes here now what's further as if x grows from 1 to plus infinity obviously x to the nth degree also does it and 1 over x to the nth degree goes to 0 this is 1 over an infinitely increasing number so this goes to 0 so the function will go this way I'm not sure this is the best drawing but probably I should let me try again I'll just draw it a little lower scale and that would be better so 1 1 it's too big for me I will do it this way this is 1 1 this is my knot so this is y is equal to x this is y is equal to x the nth degree and here we have from plus infinity it goes to 1 1 and then goes down to 0 this is my y equals to x to the minus n so if I divide 1 by this function I will get this function again they meet in 1 1 and then the x to the minus n goes to 0 as x goes to infinity and it goes to plus infinity if x goes to 0 and by the way this in some cases is called hyperbola but there are different hyperbolas if n is equal to 2 that's a quadratic hyperbola and if n is equal to 1 it's just plain hyperbola so in this particular case it's 1 over x or 1 over x square or 1 over x cube or whatever it will be a plain hyperbola or quadratic hyperbola or cubical hyperbola etc but now let's think about how the function behaves with different negative n's what if my n is greater let's say let's compare what n is equal to 1 and n is equal to 2 so if n is equal to 1 I have y is equal to 1 over x but if n is equal to 2 it will be 1 over x square now what's faster goes to 0 if my if x goes to infinity obviously x square so 1 over x square will be faster going down to 0 so if this is x to the minus 1 for instance then x to the minus 2 will be faster going down to 0 and opposite when we are discussing this interval between 0 and 1 the function will be going to infinity which means it's going above the y is equal to x so this is y is equal to 1 over x square so the whole hyperbola I will still call it hyperbola but quadratic hyperbola in this case is turned a little bit it's growing faster to 0 and faster to infinity so if this is how the 1 over x looks then this is 1 over x square and this is 1 over x cube so it will be faster growing there and faster going down to 0 here so that's the difference and again when n is not an integer number but some kind of a rational number we always can say that the curve will be very much like those with integers just shift it a little bit or turn it a little bit if you wish depending on the value so all graphs of y is equal to x to the minus any rational or even irrational number when this number is negative they are all kind of hyperbolas so it's completely different group of curves if you wish because you can always say that the positive power they're all going this type of thing and for negative correspondingly to they're all hyperbolas they're completely different now I did not yet discuss I did actually have an example of a rational number which is greater than 1 I had 3 over 2 remember this is something which I probably have to separately maybe show if y is equal to x to 1 half well obviously it will be the function which will be in between y is equal to x to the 0 and x to the 1 but in this case it's not exactly clear so let me just do it separately you will understand what I mean first of all again using these things we start again with y is equal to x to the 1st degree and as I was saying all the x square, x cube, etc they all go this way they're below y is equal to x here and above there this function if you consider it it's symmetrical in some way the graph will be this way now with y because obviously again in 0 and 1 it takes the same values 0 and 1 correspondingly now as far as x this is basically a square root of x so if you consider something like x is equal to 1 quarter then y is equal to 1 half right? so if x is equal to 1 quarter which is somewhere here this is also 1 quarter 1 half is greater so as you see the function will have the value above this bisector so that's exactly what I meant when I draw the function this way in this interval from 0 to 1 this function will be above y is equal to x but then as the x grows square root of x is smaller than x so the function will be below it will still go to infinity but slower than y is equal to x so if you compare y is equal to x square and y is equal to x square and y is equal to x to the 1 half they are kind of symmetrical that's symmetrical relative to this bisector so all these functions with power less than 1 they will have this type of shape the smaller power it's still positive but smaller close to 0 the more in this and closer to 1 is equal closer to this particular line when x is greater than 1 basically that concludes all these functions we have considered greater than 1 powers they will be of this shape equal to 1 it's this less than 1 but still positive it's this shape of the graph and whenever we go to a negative territory the power that will be hyperbolas which are somewhere here in this particular error so these are all the graphs of x to a power r and again what's interesting is they are tied together in these two points 0 and 1 they are all the same in 0 is 0 and in 1 is 1 well except 1 when r is equal to 0 in this particular case it's completely separate and the function is horizontal 1 alright basically that concludes my problem number 4 to graph all the functions of y is equal to x to the power of r hope you enjoyed it thank you