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You can come up with an equation for a polynomial function if you have the graph of the function or you know the x intercepts of the function. The arrows in each section show the direction of the ends of the function, and the dotted lines in between express that what happens in between the 2 arrows is negligible; the ways that the arrows point will always be the same. As you can see from the chart, positive even functions have both arrows pointing up, meaning that the ends of a positive even function will always rise. You can use all of the skills you just learned in this section to draw the graph of a given function.
We can tell that this function is even because the highest degree in this function is 4, which is an even number. The x2 implies that the intercept at 0 is a double root, because technically the function can be completely factored out to look like (x)(x)(x − 4)(x + 2). It is a rule that the highest degree of a function has one less than that number of turns in the graph. Which of the following represents the equation of the polynomial of degree three whose zeroes are -3, 0, and 5? The graphs to the left show the end behaviors of both the positive and negative equations of p(x) = x^2 and p(x) = x^3. 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.


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. 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. 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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