End Behavior of Polynomial - Algebra-1

1. Fundamental Concepts

Definition: The end behavior of a polynomial function describes how the values of the function behave as the input values become very large or very small.
Leading Term: Determines the end behavior; it is the term with the highest degree in the polynomial.
Leading Coefficient: Influences the direction of the end behavior (upward or downward).

Dominant Role of the Leading Term: The term with the highest degree (leading term) in the polynomial governs its end behavior. Lower-degree terms have no impact on the trend of as , since their growth rate is far slower than that of the leading term.
Four Core Types of End Behavior (based on leading coefficient a and degree n):
1. When and n is even: As , ; as , (both ends of the graph extend upward).
2. When and n is even: As , ; as , (both ends of the graph extend downward).
3. When and n is odd: As , ; as , (the right end of the graph extends upward, and the left end extends downward).
4. When and n is odd: As , ; as , (the right end of the graph extends downward, and the left end extends upward).

Problem: Determine the end behavior of the polynomial $$f(x) = -4x^4 + 3x^3 - 2x + 1$$ .

Validation: The polynomial has an even degree and a negative leading coefficient, confirming that both ends go down.

Problem: Analyze the end behavior of $$g(x) = 2x^5 - 3x^4 + x^2 - 7$$ .

Validation: The polynomial has an odd degree and a positive leading coefficient, confirming that the left end goes down and the right end goes up.

Step 1: Identify the Leading Term: Ignore lower-degree terms in the polynomial and find the term with the highest degree. If the polynomial is in a factored or nested form (e.g., ), expand and combine like terms first to determine the leading term.
Step 2: Extract Key Information from the Leading Term: Clarify the sign of the leading coefficient a (positive/negative) and the parity of the degree n (odd/even).
Step 3: Match to the End Behavior Type: Refer to the four core types of end behavior (see Key Concepts) based on the sign of a and the parity of n, and directly determine the trend of as and .
Verification (Optional): For simple polynomials, substitute extremely large or small values of x (e.g., or ) to calculate the function value and verify if the trend matches the determined end behavior.