The end behavior of a polynomial function describes how the graph behaves as x approaches +∞ or -∞. In other words, it indicates the direction in which the ‘tails’ of the graph extend as the input value (x) becomes extremely large (x → ∞) or extremely small (x → −∞).
The degree and the leading coefficient of the polynomial determine its end behavior.
The degree of a polynomial is the highest exponent (n) of x in the function. It is used in shaping the graph:
Polynomials with even degrees (e.g., x2, x4) have graphs that are symmetric at their tails.
Polynomials with odd degrees (e.g., x3, x5) have tails that extend in opposite directions.
The leading coefficient is the coefficient (a) of the term with the highest exponent of x. Its sign (+ or -) determines whether the graph rises or falls as x increases or decreases:
A positive leading coefficient means the graph rises as x → ∞.
A negative leading coefficient means the graph falls as x → ∞.
Thus, there can be 4 possible cases for finding the end behavior of a polynomial function, represented in the following chart.
For Even-Degree Polynomials
With a Positive Leading Coefficient
If an > 0, both ends of the graph rise upwards.
x → ∞, f(x) → ∞
x → -∞, f(x) → ∞
With a Negative Leading Coefficient
If an < 0, both ends of the graph fall downwards.
x → ∞, f(x) → -∞
x → -∞, f(x) → -∞
For Odd-Degree Polynomials
With a Positive Leading Coefficient
If an > 0, the graph falls to the left and rises to the right.
x → ∞, f(x) → ∞
x → -∞, f(x) → -∞
With a Negative Leading Coefficient
If an < 0, the graph rises to the left and falls to the right.
x → ∞, f(x) → -∞
x → -∞, f(x) → ∞
Here is a table that summarizes the patterns of the end behavior based on whether the degree of the polynomial (n) is even or odd and whether its leading coefficient (a) is positive or negative:
Degree (Odd or Even)
Leading Coefficient (+ or -)
End Behavior
Even
Positive
x → ∞, f(x) → ∞ and x → -∞, f(x) → ∞
Even
Negative
x → ∞, f(x) → -∞ and x → -∞, f(x) → -∞
Odd
Positive
x → ∞, f(x) → ∞ and x → -∞, f(x) → -∞
Odd
Negative
x → ∞, f(x) → -∞ and x → -∞, f(x) → ∞
Determining Algebraically
We can also find the end behavior of the polynomial function algebraically.
Let us find the end behavior of the polynomial f(x) = 6x3 – 2x2 – 5x + 3
Step 1: Identifying the Leading Term
Here are 4 terms in our given polynomial: 6x3, -2x2, -5x, and 3
Since 6x3 has the highest exponent of all terms, 6x3 is the leading term.
Step 2: Identifying the Degree and the Sign of the Leading Coefficient
Here,
The coefficient of the leading term = 6 (positive).
The exponent of the leading term (degree) = 3 (odd)
Step 3: Determining the End Behavior
According to the rules, a polynomial function with a positive leading coefficient and an odd degree will have a graph with the following end behavior:
x → ∞, f(x) → ∞
x → -∞, f(x) → -∞
The graph rises as x → ∞ and falls as x → -∞
Solved Examples
Determine the end behavior of the following polynomial functions: a) g(x) = 3x4 – 2x3 + 5 b) k(x) = -2x5 + 3x3 – 7 c) f(x) = -x3 + x2 – 6
Solution:
a) Given, g(x) = 3x4 – 2x3 + 5 Here, The terms are 3x4, -2x3, and 5 Since 3x4 has the highest exponent of all terms, 3x4 is the leading term. Now, The leading coefficient is 3, which is positive. The degree is 4, which is even. Thus, the end behavior is as follows: x → ∞, f(x) → ∞ x → -∞, f(x) → ∞ Both ends of the graph rise as x → ±∞ b) Given, k(x) = -2x5 + 3x3 – 7 Here, The terms are -2x5, 3x3, and -7 Since -2x5 has the highest exponent of all terms, -2x5 is the leading term. Now, The leading coefficient is -2, which is negative. The degree is 5, which is odd. Thus, the end behavior is as follows: x → ∞, f(x) → -∞ x → -∞, f(x) → ∞ The graph falls as x → ∞ and rises as x → -∞ c) Given, f(x) = -x3 + x2 – 6 Here, The terms are are -x3, x2, and -6 Since -x3 has the highest exponent of all terms, -x3 is the leading term. Now, The leading coefficient is -1, which is negative. The degree is 3, which is odd. Thus, the end behavior is as follows: x → ∞, f(x) → -∞ x → -∞, f(x) → ∞ The graph falls as x → ∞ and rises as x → -∞
Sketch the graph of g(x) = 4x6 – x + 2 and find its end behavior.
Solution:
The graph of g(x) = 4x6 – x + 2 is shown alongside. End Behavior: The degree of the polynomial is 6, which is an even number. The leading coefficient is 4, which is positive. This means: x → ∞, f(x) → ∞ x → -∞, f(x) → ∞ Both ends of the graph rise as x → ±∞
