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When we study about functions and polynomial, we often come across the concept of end behavior.
End behavior is useful to examine the trend in the function value as the value of x gets larger and larger in magnitude. End behavior of a polynomial function is the behavior of the graph of y = f(x) as x approaches positive infinity or negative infinity. Now, we can find the end behavior by just knowing the values of Leading Co-efficient an and Power n of the Polynomial Equation. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
Case 1: If Leading coefficient(an) is positive and the power(n) is even, both ends of the graph go up. Case 2: If Leading coefficient(an) is negative and the power(n) is even, both ends of the graph go down. Case 3: If Leading coefficient(an) is positive and the power(n) is odd, the right hand of the graph goes up and the left hand goes down. Case 4: If Leading coefficient(an) is negative and the power(n) is odd, the right hand of the graph goes down and the left hand goes up. Asymptotic behavior of graph of a function involves limits, since limits are the situations where a function approaches a value.
The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction. An end behavior model of a polynomial uses only the leading coefficient and the variable of highest degree.
End behavior of a graph can be based on the degree and the leading coefficient of a polynomial function.
It is clear from the graph that that when x approaches to $\infty$, the right part of the graph is going upward.
From the graph, when x approaches to $\infty$, the right part of graph extends towards upward direction.



Graphs, the visual representations of functions are used to understand function behavior easily.
This is an exponential model,where the function approaches a finite value at one end and unbounded at another end. This is a logarithmic model, where the variable increaseswithout bound at only one end.The function approaches + ?or -? at this end depending upon the function definition.
We are quite familiar with the end behavior of some simple or basic functions like linear, quadratic exponential and logarithmic models.
The end behavior of Polynomials is determined by the degree and the sign of the leading term. Examples:Discuss the end behavior of the polynomialsa)  p(x) = x3 - 14x - 4       b) q(x) = - 4x4 + 16x3 + 31x2 - 49x - 30The leading term in p(x) is x3. The Horizontal asymptote of a function is found in a manner similar to the way the end behavior is discussed.
From the pattern seen in the function values, we can conclude, that f(x) > -? as x > -? and f(x) > ? as x > ?.The Graph of f(x) = $\frac{x^{2}+2}{x+1}$ shown above confirms our observations. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right as shown in the figure.
As the name suggests, "end behavior" of a function is referred to the behavior or tendency of a function or polynomial when it reaches towards its extreme points.
It helps to tell us how does f(x) behave as the value of x increases to positive infinity or decrease to negative infinity. In other words, we are interested in what is happening to the y values, as we get large x values and as we get small x values. We can determine the end behavior of any polynomial function from its degree and its leading coefficient. For the large values of x, we can model the behavior of function that behave in the same way. Similarly, when x approaches to $\infty$, then left part of the graph is also going upward.


But, when x approaches to $-\infty$, the left part of the graph extends towards downward direction.
The end behavior is an important aspect of graphs along with other features like vertices, intercepts, turning points and asymptotes. This is represented by x > ? and x > -?, the former meaning x assumes greater and greater positive values and the later tells that x becomes smaller and smaller on the negative side.
The function value becomes larger and larger as the absolute value of the variable increases. The function becomes more and more negative as theabsolute value of the variable increases. Discuss the function's end behavior as observed in the graph.It can be noted that the green graph ( f(x) ) appears to coincide with the horizontal line y = 4, which serves the the function's horizontal asymptote.
The end behavior model of a function gives the equation of the asymptotic curve to the graph of the function. Let us now learn how to determine the end behavior of a function algebraically and how end behavior is analyzed from the function graphs. It can also be directly manipulated by having a look at the graph that which part of the graph will go up and which one will go down when the function reaches nearest to the extreme points. Then both when x > ? and x > -?, f(x) > $\frac{a}{g}$Let f(x) = $\frac{4x +3}{2x-5}$.The end behavior is explained by the model y = $\frac{4}{2}$ or y = 2. End behavior may also be found by applying some rules and formulae on the equation of given function. In the page below, we shall understand about the end behavior of a function and method of its estimation.




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