When we talk about the end behavior of rational functions, it means that this is an end state of any of the mental processes. It is a common concept that end behavior of rational functions is very important to be understood by students. In this article, we will discuss about asymptotes, which are nothing but sharp discontinuities in rational functions, and what they imply in psychology. The discussion might be quite technical for you if you do not have adequate knowledge on concepts like asymptotes.
Symmetry is one of the symmetries of any rational function. A set of such symmetries will ensure that every point on the graph has an equal probability of being either closer to the origin or further away from it. Every point on the graph has a mean value that will be equal to the mean value of the corresponding mean value of the other future point on the graph. We can say that there are a set of asymptotes depending upon the symmetries of a given rational function f(k), where k is any real number.
For instance, let us study the behavior of rational functions as it relates to the symmetries of the first and second arrows of the graphing diagram above. As we know that every interval on the horizontal axis is equal to its mean value, so we get by interpreting the left and right arrows as x and y values.
The meaning of the end behavior of rational functions is, the probability of finding any point other than the origin on the left or right side of any interval on the horizontal axis is one. And the probability of finding any point on the x axis is one – the higher the x value, the greater is the interval on the left and the greater is the interval on the right.
Then we find that the left and right arrows of the above diagram have the same direction of symmetry and therefore, the left arrow represents the mean and the right arrow the standard deviation of the mean. Now, from these facts we can easily draw the conclusion that the end behavior of rational functions is indeed symmetrical.
This leads to the conclusion that there is no way that we can make an error while working with the basic end behavior of rational functions as it is given in this article. In other words, by following the above procedures we can be absolutely sure that our conclusions are correct.
There are a few other methods of working with asymptotes, but they prove to be rather difficult to work with in all cases. One of these methods is called as the accommodation approach where we take the mean of the data and set the end behavior of rational functions such that they lie on a very narrow range of values.
This ensures that they are extremely small and so the risk of an asymptote being discovered is very low. The other method of working with asymptotes is called the mean integral approach where we adopt the mean of the integral of the function and set it equal to zero.
The other two approaches that are not so popular are the horizontal bar approach and the vertical bar approach. The horizontal bar approach gives the end behavior of rational functions such that the x value of the mean of the function plotted on a horizontal bar is equal to the y value of the corresponding line on the vertical bar.
On the other hand, the vertical bar approach gives the end behavior of the reciprocity functions as the x value of the mean of the function plotted on a vertical bar is equal to the y value of the corresponding line on the horizontal bar. So both these methods have a certain level of accuracy associated with them and they are suitable for work with small data sets. However, if you would like to deal with more data sets, then the horizontal or the vertical bar graph is a good option.
Analyzing the End Behavior of Rational Functions – Part II
The end behavior of rational functions is a map from the inputs of a state t to the end result of that state t in a hyperreal model of time and space. The end behavior of rational functions can be described by the map obtained by the application of a calculus function on the model plane, which maps inputs of any sort that get observed in the future to the end state t.
In this sense, it is more important to say that the end behavior of rational functions is rational, rather than descriptive. That is, the model of rational functions that we use at any time can be used to compute any other model.
The end behavior of rational functions can be illustrated in the well-known game called “Tetris”. At the start of each game, there is a horizontal position called the starting block. The blocks are moved left and right by using the mouse or keyboard, so that they fall between two vertices of the original board, called the edges. When a block falls on either edge of the board, it will be turned into a piece of play currency (Zucker, Sickle, or Hammer).
Here, for simplicity’s sake, we will assume that the starting and ending values of each board are different. For simplicity’s sake also, we will assume that the board has only four edges – two top and two bottom.
We will also assume that the value of each piece of play currency is uniformly distributed over the board; that is, it lies between the x y co-ordinates, thus there are no such points on the board which lie outside the range of any piece of play currency. Finally, we will assume that the board has an even number of edges, so that each edge does not coincide with any other edge, so that the set of eights of the polynomial function f is even.