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End Behavior

When we study about functions and polynomial, we often come across the concept of end behavior. 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. In other words, end behavior is meant to find what will be the characteristics of the function at extreme points, i.e. at points when independent variable (say x) approaches near to infinity on positive axis as well as negative axis.

End behavior of a given function can be estimated with the help of its graph. 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. It is also dependable upon the leading coefficient - whether it is positive or negative ; also upon the degree (highest power of the variable) of the function or polynomial - whether it is
even or odd.

By knowing about end behavior, one comes to know about if it will go in upward direction or in downward direction when function tends towards positive infinity and negative infinity. In the page below, we shall understand about the end behavior of a function and method of its estimation.

 

End Behavior Definition

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End behavior is useful to examine the trend in the function value as the value of x gets larger and larger in magnitude. It helps to tell us how does f(x) behave as the value of x increases to positive infinity or decrease to negative infinity.

End Behavior of Polynomial Functions

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End behavior of a polynomial function is the behavior of the graph of y = f(x) as x approaches positive infinity or 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.

Polynomial End Behavior


Consider a polynomial

f(x) = anxn + an - 1xn - 1 + .......... + a2x2 + a1x + a0

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.

Feasible Cases:

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.

Let us explain the end behavior of the function, f(x) = $x^3$.

Given polynomial function, f(x) = $x^3$

Degree of function = 3 = Odd

Leading coefficient = 1 = Positive

End behavior of the function:

f(x) -> -$\infty$ as x -> -$\infty$

f(x) -> + $\infty$ as x -> + $\infty$

Graph of f(x) = $x^3$


End Behavior of Polynomial Functions


End Behavior Asymptote

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Asymptotic behavior of graph of a function involves limits, since limits are the situations where a function approaches a value. A vertical asymptote will usually exist whenever the function doesn’t exist. Basically, there are two types of asymptote, Horizontal and Vertical.

Horizontal Asymptotes:


A function f(x) will have the horizontal asymptote f(x) = a, if either $\lim_{x \rightarrow \infty}$f(x) = a or $\lim_{x \to -\infty}$f(x) = a.

Vertical Asymptote

The line x = a is a vertical asymptote of the graph of a function y = f(x) if either

$\lim_{x \rightarrow a^+}f(x) = \pm \infty$ or $\lim_{x \rightarrow a^-}f(x) = \pm \infty$.

End Behavior of Rational Functions

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The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction. We can determine the end behavior of any polynomial function from its degree and its leading coefficient. Rational functions are in the form of F(x) = $\frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials.

Asymptote of Rational Function are:
  • If $\lim_{x \rightarrow \infty}$F(x) = a, then y = a is a horizontal asymptote.
  • At number where Q(x) = 0, either a hole or a vertical asymptote occurs.
Let us graph the rational function, f(x) = $\frac{x^2 - 3x + 5}{x - 2}$ and describe it's end behavior.

Given rational function, f(x) = $\frac{x^2 - 3x + 5}{x - 2}$.

For Vertical Asymptote:


For the vertical asymptote, put denominator = 0

=> x - 2 = 0

=> x = 2 is a vertical asymptote.

Behavior near Vertical Asymptote:


$\lim_{x \to 2^{-}}(x - 2)$ = -$\infty$

$\lim_{x \to 2^{+}}(x - 2)$ = $\infty$

For Horizontal Asymptote:


Since degree of numerator > degree of denominator. So, no horizontal asymptote.

End Behavior:


Dividing polynomial, we get


End Behavior of Rational Function


=> f(x) = x - 1 + $\frac{3}{x - 2}$

Now,

$\lim_{x \rightarrow \infty}$ f(x) = $\lim_{x \rightarrow \infty}$(x - 1 + $\frac{3}{x - 2}$)

= $\infty$

Similarly,

$\lim_{x \rightarrow -\infty}$ f(x) = $\lim_{x \rightarrow -\infty}$(x - 1 + $\frac{3}{x - 2}$)

= -$\infty$

Graph of Function:


End Behavior of Rational Functions

End Behavior Model

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An end behavior model of a polynomial uses only the leading coefficient and the variable of highest degree. For the large values of x, we can model the behavior of function that behave in the same way. If a function g(x) satisfied conditions given below, it is simply called an end behavior model.

The end behavior of function g is
  • A right end behavior model for f iff $\lim_{x \rightarrow \infty}$ $\frac{f(x)}{g(x)}$ = 1
  • a left end behavior model for f iff $\lim_{x \rightarrow -\infty}$ $\frac{f(x)}{g(x)}$ = 1

The leading coefficient term for a polynomial is an end behavior model for a function.
Let us find the end behavior model for f, f(x) = 2x2 - 6x + 7

Given f(x) = 2x2 - 6x + 7

Here, g(x) = 2x2

$\lim_{x \rightarrow \infty}$ $\frac{f(x)}{g(x)}$ = $\lim_{x \rightarrow \infty}$ $\frac{2x^2 - 6x + 7}{2x^2}$

= $\lim_{x \rightarrow \infty}$ $(\frac{2x^2}{2x^2} - \frac{6x}{2x^2} + \frac{7}{2x^2})$

= $\lim_{x \rightarrow \infty}$ $(1 - \frac{3}{x} + \frac{7}{2x^2})$

= 1 - $\lim_{x \rightarrow \infty}$$ \frac{3}{x}$ + $\lim_{x \rightarrow \infty}$$ \frac{7}{2x^2}$

= 1

=> 2x2 is an end behavior model of f.

End Behavior of a Graph

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End behavior of a graph can be based on the degree and the leading coefficient of a polynomial function. Given below are some examples for finding end behavior using graphical end behavior.

Solved Examples

Question 1: Describe the end behavior of the function, f(x) = -3x9 + 7
Solution:

Given polynomial function, f(x) = -3x9 + 7

Leading Co-efficient = -3 = -ve value

Degree of polynomial = 9 = ODD

So, we can conclude that the right hand of the graph goes down and the left hand goes up.

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.]

End Behavior: The right part of the graph goes down and the left hand goes up.

End Behavior of a Graph

Question 2:

Discuss the end behavior of the graph given below:

End Behavior of Graphs


Solution:
It is clear from the graph that that when x approaches to $\infty$, the right part of the graph is going upward. Similarly, when x approaches to $\infty$, then left part of the graph is also going upward.

Question 3:

Discuss the end behavior of a graph given below:

Graphical End Behavior


Solution:
From the graph, when x approaches to $\infty$, the right part of graph extends towards upward direction. But, when x approaches to $-\infty$, the left part of the graph extends towards downward direction.

Question 4: Describe the end behavior of the function, f(x)= 10x1000 + 151
Solution:

Given polynomial function is f(x)= 10x1000 + 151

Leading Co-efficient = 10 = +ve value

Degree of polynomial = 1000 = EVEN

End Behavior: Both ends go up.



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