Topic 1.6 Polynomial Functions and End Behavior: determine the end behavior of a polynomial from its degree and leading term, and express that behavior using limit notation.

What this topic is asking

The College Board (Topic 1.6) wants you to determine the end behavior of a polynomial: what the output does as $x$ runs off to $-\infty$ and $+\infty$. End behavior is controlled entirely by the leading term, that is, by the polynomial's degree (even or odd) and its leading coefficient (positive or negative). You must also write the behavior using limit notation.

The leading term dominates

This is why you can read end behavior at a glance: ignore everything but the highest-power term.

The four cases

A quick mnemonic: even degree gives matching ends (both up or both down, like a parabola), odd degree gives opposite ends (like a line or cubic), and the leading coefficient's sign decides which way.

Limit notation

The exam expects end behavior written with limits. The statement $\lim_{x \to +\infty} p(x) = +\infty$ reads "as $x$ increases without bound, $p(x)$ increases without bound". You give one limit for each end. This notation reappears throughout the course, including for rational functions in Topics 1.7 and 1.9.

Why end behavior matters

End behavior is more than a graphing detail; it constrains what a polynomial can do. An odd-degree polynomial runs from one infinity to the opposite one, so by continuity it must cross the $x$-axis at least once, guaranteeing a real zero. An even-degree polynomial has both ends pointing the same way, so it has a global minimum (if the ends go up) or a global maximum (if they go down). These structural facts let you reason about zeros and extrema before doing any algebra.

A useful clarification is that the leading coefficient's magnitude does not affect the direction of the ends, only the steepness. Whether the leading coefficient is $2$ or $200$, an even-degree positive-leading polynomial still has both ends going to $+\infty$; the larger coefficient simply makes the graph climb faster. So when classifying end behavior, only the sign of the leading coefficient and the parity of the degree matter, which is what makes the four-case rule so quick to apply on the no-calculator section.

Try this

Q1. State the end behavior of $p(x) = 7x^6 - x^3 + 2$ in words. [1 point]

• Cue. Even degree, positive leading coefficient: both ends rise to $+\infty$.

Q2. Write $\lim_{x \to -\infty} q(x)$ for $q(x) = -x^5 + 4x$. [1 point]

• Cue. Odd degree, negative leading coefficient: the left end rises, so $\lim_{x \to -\infty} q(x) = +\infty$.