To determine the end behavior of a polynomial function, which part of the polynomial is most important to analyze?

• A. The constant term (the term with no $x$)
• B. The leading term (the term with the highest power of $x$)
• C. All terms contribute equally to the end behavior, so no single term is most important
• D. The term with the smallest power of $x$

Answer: (B)

Verified step by step guidance

1. Recall that the end behavior of a polynomial function describes how the function behaves as the input variable \(x\) approaches positive or negative infinity.
2. Understand that as \(x\) becomes very large in magnitude, the terms with lower powers of \(x\) become insignificant compared to the term with the highest power of \(x\).
3. Identify the leading term of the polynomial, which is the term with the highest exponent on \(x\), because it dominates the value of the polynomial for very large or very small \(x\).
4. Analyze the leading term's coefficient and degree to determine the end behavior: the sign of the coefficient affects whether the graph rises or falls, and whether the degree is even or odd affects the shape of the ends.
5. Conclude that the leading term is the most important part of the polynomial to analyze when determining end behavior.