Polynomial Functions and End Behavior — AP Precalculus
1. Core Definitions and Limit Notation ★★☆☆☆ ⏱ 5 min
The leading term is $a_n x^n$, $a_n$ is the leading coefficient, and $n$ is the degree. End behavior describes the trend of $f(x)$ as $x$ grows without bound in the positive ($x \to +\infty$) and negative ($x \to -\infty$) directions.
- $\lim_{x \to +\infty} f(x) = +\infty$: As $x$ grows without bound positive, $f(x)$ grows without bound positive
- $\lim_{x \to +\infty} f(x) = -\infty$: As $x$ grows without bound positive, $f(x)$ grows without bound negative
- Replace $x \to +\infty$ with $x \to -\infty$ for behavior as $x$ grows without bound negative
Exam tip: On the AP exam, if the question asks for end behavior in limit notation, you must write both the $x \to +\infty$ and $x \to -\infty$ limits to earn full credit; verbal descriptions alone are not accepted.
2. The Leading Term Test ★★☆☆☆ ⏱ 4 min
The leading term test organizes all possible end behavior into four cases based on two properties: parity (even/odd) of the degree $n$, and sign (positive/negative) of the leading coefficient $a_n$:
- Odd $n$: $(-x)^n = -x^n$, so end behavior is opposite on the two ends
- Even $n$: $(-x)^n = x^n$, so end behavior matches on the two ends
- The sign of $a_n$ reverses the direction of both end behaviors
Exam tip: For a polynomial in factored form, you do not need to expand it to find degree and leading coefficient: just multiply the leading terms of each factor to get the leading term, which is all you need for end behavior.
3. Constructing Polynomials from Specified End Behavior ★★★☆☆ ⏱ 5 min
A common AP exam question asks you to write the equation of a polynomial that meets given end behavior requirements, often with additional constraints like given roots or a specific degree. Since end behavior only depends on degree parity and leading coefficient sign, there are infinitely many correct answers, any of which will earn full credit if they meet requirements.
- Use the given end behavior to find required degree parity and leading coefficient sign
- Add any required roots or degree constraints to build the general form
- Choose a valid leading coefficient matching the required sign, then confirm all requirements
Exam tip: If the question does not specify a minimum degree, the simplest correct answer is just the leading term itself (a monomial) with the correct degree and leading coefficient, which will always earn full credit.
4. AP-Style Concept Check ★★★☆☆ ⏱ 4 min
Common Pitfalls
Why: Students confuse number of distinct roots with total degree, forgetting repeated roots add to the degree
Why: Students forget the negative leading coefficient cancels the negative sign from the odd power of negative $x$
Why: Students confuse coefficient size with degree when identifying the leading term
Why: Students incorrectly carry the negative sign of $x$ through to an even power
Why: Students mix up the parity rule for end behavior