Graphs, the visual representations of functions are used to understand function behavior easily. The end behavior is an important aspect of graphs along with other features like vertices, intercepts, turning points and asymptotes.

End behavior describes the course of the graph to the far left and far right. Let us now learn how to determine the end behavior of a function algebraically and how end behavior is analyzed from the function graphs.

The end behavior of a function f(x) describes how the function values are influenced as |x| becomes greater and greater.
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 different end behaviors of the graph can be pictorially explained as follows:

Both End Up Behavior
     Both End Down Behavior    
 One Limit Behavior
Above is shown a situation:
when the graph rises at either end.
The function value becomes
larger and larger as the absolute
value of the variable increases.
  In this case the graph falls at
either end. The function becomes
more and more negative as the
absolute value of the variable
increases.
  This is an example showing
the function approaching a
finite value at either end.

               
One Rise Behavior
                No Limit Behavior   
  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 increases
without 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. These function models are used in determining the end behavior of more complex functions.
The end behavior model of a function gives the equation of the asymptotic curve to the graph of the function. For polynomials the end behavior is determined purely by the leading terms.
Example:
Suppose f(x) is a rational function, where f(x) = $\frac{g(x)}{h(x)}$, where g(x) and h(x) are polynomials of the same degree, with corresponding leading coefficients, a and b. 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. This tells that as x assumes larger values at either end the function value approaches 2. The graph of the function appears to coincide with the horizontal line y = 2 at either end.
The end behavior of polynomials are determined by the leading term.

End behavior rules for Rational Functions:

Let f(x) = $\frac{g(x)}{h(x)}$, where h(x) and g(x) are polynomials.

Case (1): Degree of g(x) < Degree of h(x)
As both x → -∞ and x → ∞, f(x) →0 and the graph closes on to the x axis.
Case (2): Degree of g(x) = Degree of h(x)
As both x → -∞ and x → ∞, f(x) → $\frac{a}{b}$, where a and b are correspondingly the leading coefficients of g(x) and h(x).
Case (3): Degree of g(x) > Degree of h(x)
f(x) approaches +∞ at one end and -∞ at the other end.

For other functions formed by compositions of simple functions, the end behavior models are found applying suitable algebraic techniques.
The end behavior of Polynomials is determined by the degree and the sign of the leading term. The method is summarized as follows:

Odd degree Polynomial

      Leading Term Positive    
        Leading Term Negative
      Odd Degree Polynomial          Odd Degree Polynomials
The Graph falls down at the left
end and rises up at the right end.
      As x → -∞  => f(x) → -∞
      As x → ∞  =>  f(x) → ∞

The Graph rises up at the left
end and falls down at the right end.
         As x → -∞ =>  f(x) → ∞
         As x → ∞   =>  f(x) → -∞ 

Even degree Polynomial

      Leading Term Positive             Leading Term Negative   
       Even Degree Polynomial           Even Degree Polynomials
The Graph rises up at either end.
       As x → -∞   f(x) → ∞
       As x → ∞    f(x) → ∞
  The Graph falls down at either end.
       As x → -∞    f(x) → -∞
       As x → ∞     f(x) → -∞

Examples:
Discuss the end behavior of the polynomials
a)  p(x) = x3 - 14x - 4       b) q(x) = - 4x4 + 16x3 + 31x2 - 49x - 30
The leading term in p(x) is x3. Odd degree and positive.
Hence the end behavior is Down left and Up right. The graph drawn also confirms this.

 End Behavior Polynomial Example          
End Behavior Polynomial Examples


q(x) is an even degree polynomial with negative leading term.
Hence the graph falls down at either end as seen in the graph.
The Horizontal asymptote of a function is found in a manner similar to the way the end behavior is discussed.
Whenever the function approaches a finite value k, as x → -∞ or x → ∞, the graph of the function has the horizontal asymptote y = k.The graph of the function appears to coincide with the horizontal asymptote towards its end/s.
Note the function approaches the finite value only at the ends, but somewhere for the middle values, the graph of the function can cross the horizontal asymptote.

In addition to Horizontal asymptotes, slant asymptotes and asymptotic curves are also associated with the end behavior of the graph of a function.
Discuss the end behavior of the function f(x) = $\frac{x^{2}+2}{x+1}$
f(x) is a rational function with the degree of the polynomial in the numerator being greater than the degree of the polynomial in the denominator.

As per the rule, f(x) should approach ∞ at one end and -∞ at the other end.
Let us make a table to check this, for high values of x both positive and negative.

  
     f(x)  
 -10    -11.3333
 -100    -101.0303
 -1000   -1001.003
   
 10   9.2727
 100   99.0297
 1000   999.003

From the pattern seen in the function values, we can conclude, that f(x) → -∞ as x → -∞ and f(x) → ∞ as x → ∞.


End Behavior of Function

The Graph of f(x) = $\frac{x^{2}+2}{x+1}$ shown above confirms our observations. The slant asymptote y = x -1 is also shown along with. The graph of the function seems to coincide with the slant asymptote at either end.
The graph of the function f(x) is represented by the green graph below. Discuss the function's end behavior as observed in the graph.

End Behavior of a 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. On the right end, the green graph rises up without bound.

Hence the end behavior of f(x) can be summarized as follows:
As x → - ∞     f(x) → 4.
As x → ∞       f(x) → ∞.