Polynomial functions and end behavior — AP Precalculus Study Guide

For: AP Precalculus candidates sitting AP Precalculus.

Covers: General form of single-variable polynomial functions, limit notation for end behavior, the leading term test, relating end behavior to degree and leading coefficient sign, and constructing polynomials from specified end behavior.

You should already know: Limit notation for infinite behavior of functions. Basic polynomial algebra for simplifying and factoring. Classification of even and odd integers.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Precalculus style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Polynomial functions and end behavior?

A polynomial function of degree $n$ (where $n$ is a non-negative integer) is defined as $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$, where all $a_i$ are real constants, $a_n\ne 0$ (called the leading coefficient), and $a_n{x}^n$ is the leading term. End behavior describes the trend of the function's output $f(x)$ as the input $x$ grows without bound in the positive ($x\to +\infty$) and negative ($x\to -\infty$) directions. Unlike local behavior around intercepts or turning points, end behavior depends only on the leading term of the polynomial, not lower-degree terms. This is one of the core foundational topics in Unit 1: Polynomial and Rational Functions, and accounts for approximately 5% of the total AP Precalculus exam weight per the official CED. It appears in both multiple-choice (MCQ) and free-response (FRQ) sections, most commonly as part of graphing questions, or as a step in analyzing rational functions or real-world polynomial models.

2. Limit Notation for End Behavior

To communicate end behavior clearly on the AP exam, you must use standard infinite limit notation. For any function:

• ${\lim }_{x\to +\infty }f(x)=+\infty$ means as $x$ grows without bound in the positive direction, $f(x)$ also grows without bound in the positive direction.
• ${\lim }_{x\to +\infty }f(x)=-\infty$ means as $x$ grows without bound in the positive direction, $f(x)$ grows without bound in the negative direction.
• For the left end (large negative $x$), we replace $x\to +\infty$ with $x\to -\infty$, which means $x$ grows without bound in the negative direction.

The core principle of polynomial end behavior is: for sufficiently large values of $|x|$, the leading term dominates all lower-degree terms. For example, take $f(x)=2x^3-1000x^2$: at $x=1000$, the leading term $2x^3=2,000,000,000$, while the lower term $-1000x^2=-1,000,000,000$. At $x=10,000$, the leading term is 20 times larger than the lower term, and the gap grows as $|x|$ increases. As $|x|\to \infty$, the ratio of the leading term to any lower term approaches infinity, so we only need to analyze the leading term to find end behavior.

Worked Example

Problem: Write the end behavior of $f(x)=5x^4-3x^2+7x-12$ using correct limit notation.

1. First, identify the leading term: the highest degree term is $5x^4$, with degree 4 (even) and leading coefficient 5 (positive).
2. Analyze ${\lim }_{x\to +\infty }5x^4$: for large positive $x$, $x^4$ is positive, multiplied by positive 5, so it grows without bound. Thus ${\lim }_{x\to +\infty }f(x)=+\infty$.
3. Analyze ${\lim }_{x\to -\infty }5x^4$: any even power of a negative number is positive, so $(-k{)}^4=k^4>0$ for any positive $k$. Thus $5x^4$ is still positive and grows without bound, so ${\lim }_{x\to -\infty }f(x)=+\infty$.
4. Full end behavior: ${\lim }_{x\to +\infty }f(x)=+\infty$ and ${\lim }_{x\to -\infty }f(x)=+\infty$.

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.

3. The Leading Term Test

The leading term test codifies all possible end behavior into four cases based on two properties of the leading term: the parity (even/odd) of the degree $n$, and the sign (positive/negative) of the leading coefficient $a_n$. The intuition for this rule comes from basic properties of powers:

• For any odd $n$: $(-x{)}^n=-x^n$, so the sign of $x^n$ flips when $x$ is negative, leading to opposite end behavior on the two ends.
• For any even $n$: $(-x{)}^n=x^n$, so the sign of $x^n$ stays the same when $x$ is negative, leading to matching end behavior on the two ends.
• The sign of $a_n$ flips the sign of the entire leading term, so it reverses the direction of both end behaviors.

The four cases are:

1. Even degree, positive leading coefficient: ${\lim }_{x\to +\infty }f(x)=+\infty$, ${\lim }_{x\to -\infty }f(x)=+\infty$ (both ends up)
2. Even degree, negative leading coefficient: ${\lim }_{x\to +\infty }f(x)=-\infty$, ${\lim }_{x\to -\infty }f(x)=-\infty$ (both ends down)
3. Odd degree, positive leading coefficient: ${\lim }_{x\to +\infty }f(x)=+\infty$, ${\lim }_{x\to -\infty }f(x)=-\infty$ (left down, right up)
4. Odd degree, negative leading coefficient: ${\lim }_{x\to +\infty }f(x)=-\infty$, ${\lim }_{x\to -\infty }f(x)=+\infty$ (left up, right down)

Worked Example

Problem: Match the end behavior description "As $x\to +\infty$, $f(x)\to -\infty$, and as $x\to -\infty$, $f(x)\to +\infty$" to the correct combination of degree parity and leading coefficient sign, then write an example polynomial.

1. First, note the two ends have opposite end behavior (one goes to $+\infty$, the other to $-\infty$). Opposite end behavior always means the degree is odd.
2. Next, check the right end (as $x\to +\infty$): it goes to $-\infty$, which means the leading coefficient must be negative. For odd degree, a positive leading coefficient gives $x\to +\infty \to +\infty$, so this matches negative.
3. Confirm the left end: for odd degree negative leading coefficient, ${\lim }_{x\to -\infty }{a}_n{x}^n=(-)(-\infty )=+\infty$, which matches the given description.
4. An example polynomial is $f(x)=-2x^3+4x-7$, which satisfies all conditions.

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.

4. Constructing Polynomials from Specified End Behavior

A common AP exam question asks you to write the equation of a polynomial that meets a given end behavior requirement, 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, but any polynomial that meets the requirements will earn full credit.

The process for this is: 1) Use the given end behavior to find the required degree parity (even/odd) and leading coefficient sign (positive/negative) from the leading term test. 2) Add any required roots or degree constraints, then choose any valid leading coefficient that matches the required sign. 3) Confirm your polynomial meets all requirements. Unless the question specifies otherwise, leaving the polynomial in factored form is acceptable, but you may need to expand it if requested.

Worked Example

Problem: Write a 3rd-degree polynomial with roots at $x=1$ and $x=-4$, and end behavior ${\lim }_{x\to +\infty }f(x)=-\infty$.

1. Degree 3 is odd, which matches the opposite end behavior required. We need a negative leading coefficient to get ${\lim }_{x\to +\infty }f(x)=-\infty$.
2. Roots at $x=1$ and $x=-4$ give factors $(x-1)$ and $(x+4)$. We need one more linear factor for a 3rd-degree polynomial; we can add a root at $x=0$ for simplicity, giving the factor $x$.
3. The factored form is $ax(x-1)(x+4)$, where $a$ is the leading coefficient. We choose $a=-1$ (any negative number is valid).
4. Expand to standard form: $f(x)=-1x(x^2+3x-4)=-x^3-3x^2+4x$. This polynomial meets 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.

5. Common Pitfalls (and how to avoid them)

• Wrong move: For a polynomial in factored form, miscounting the degree parity by counting the number of distinct roots instead of summing exponents of factors, e.g., calling $(x-2{)}^2(x+3)$ degree 2 even. Why: Students confuse number of distinct roots with total degree, forgetting repeated roots add to the degree. Correct move: Always add the exponents of every factor to get total degree, then check if the sum is even or odd.
• Wrong move: For an odd degree negative leading coefficient, writing ${\lim }_{x\to -\infty }f(x)=-\infty$. Why: Students forget the negative leading coefficient cancels the negative sign from the odd power of negative $x$. Correct move: Explicitly calculate the sign: $(\text{leading coefficient sign})\times (\text{sign of }{x}^n\text{ for }x\to -\infty )$ to get the correct limit sign.
• Wrong move: Identifying the term with the largest coefficient as the leading term, e.g., calling $f(x)=100x^2-x^3$ leading term $100x^2$ with even degree end behavior. Why: Students confuse coefficient size with degree for leading term identification. Correct move: Always sort terms by descending degree and select the term with the largest exponent as the leading term, regardless of coefficient size.
• Wrong move: For an even degree positive leading coefficient, writing ${\lim }_{x\to -\infty }f(x)=-\infty$. Why: Students incorrectly carry the negative sign of $x$ through to an even power. Correct move: Memorize that any even power of a non-zero real number is positive, so the sign of $x^n$ for even $n$ is always positive, regardless of $x$'s sign.
• Wrong move: When constructing a polynomial for opposite end behavior, using an even degree. Why: Students mix up the parity rule for end behavior. Correct move: Write the rule down first: same end behavior both sides → even degree; opposite end behavior → odd degree, before writing your polynomial.

6. Practice Questions (AP Precalculus Style)

Question 1 (Multiple Choice)

Which of the following correctly describes the end behavior of $f(x)=-3x^5+7x^3-12x+2$? A) ${\lim }_{x\to +\infty }f(x)=+\infty$, ${\lim }_{x\to -\infty }f(x)=+\infty$ B) ${\lim }_{x\to +\infty }f(x)=-\infty$, ${\lim }_{x\to -\infty }f(x)=+\infty$ C) ${\lim }_{x\to +\infty }f(x)=+\infty$, ${\lim }_{x\to -\infty }f(x)=-\infty$ D) ${\lim }_{x\to +\infty }f(x)=-\infty$, ${\lim }_{x\to -\infty }f(x)=-\infty$

Worked Solution: First, identify the leading term of $f(x)$: the highest degree term is $-3x^5$, with degree 5 (odd) and leading coefficient $-3$ (negative). From the leading term test, odd degree with negative leading coefficient gives opposite end behavior: right end (as $x\to +\infty$) goes to $-\infty$, and left end (as $x\to -\infty$) goes to $+\infty$. We can verify by sign calculation: $(-3)(+\infty {)}^5=-3(+\infty )=-\infty$, and $(-3)(-\infty {)}^5=-3(-\infty )=+\infty$, which confirms the result. The correct answer is B.

Question 2 (Free Response)

Let $f(x)=(2x-1)(-x+3)(x+4)(x-5)$. (a) Identify the degree of $f(x)$ and the sign of the leading coefficient. (b) Write the end behavior of $f(x)$ using correct limit notation. (c) Describe the overall shape of the graph of $f(x)$ based only on its end behavior.

Worked Solution: (a) There are 4 linear factors, each of degree 1, so total degree is $1+1+1+1=4$ (even). Multiply the leading coefficient of each factor to get the overall leading coefficient: $2\times (-1)\times 1\times 1=-2$, which is negative. Final answer: Degree 4, negative leading coefficient. (b) From the leading term test, even degree with negative leading coefficient has both ends approaching $-\infty$. The end behavior is: $$\underset{x\to +\infty }{\lim }f(x)=-\infty \text{and}\underset{x\to -\infty }{\lim }f(x)=-\infty$$ (c) The graph of $f(x)$ will enter from the bottom left (as $x$ approaches large negative values, $f(x)$ becomes increasingly negative) and exit to the bottom right (as $x$ approaches large positive values, $f(x)$ also becomes increasingly negative), forming an overall downward-opening "hill" shape with up to 3 turning points between the two ends.

Question 3 (Application / Real-World Style)

A city planner models the population of a city over time as $P(t)=0.002t^3-0.1t^2+2.5t+50$, where $P(t)$ is population in thousands of people, and $t$ is time in years after 2000, $t\ge 0$. What does the end behavior of $P(t)$ predict about the city's population far into the future? Justify your answer.

Worked Solution: First, identify the leading term of $P(t)$: $0.002t^3$, which has degree 3 (odd) and leading coefficient $0.002$ (positive). For large positive $t$ (far into the future), ${\lim }_{t\to +\infty }P(t)=+\infty$. This means the model predicts that the city's population will grow without bound as time goes on, increasing at an accelerating rate far into the future.

7. Quick Reference Cheatsheet

8. What's Next

Mastering polynomial end behavior is a critical prerequisite for all upcoming topics in Unit 1: Polynomial and Rational Functions. Immediately next, you will use end behavior as the starting point to sketch full graphs of polynomial functions, identify the number of turning points, and connect end behavior to the number of real roots of a polynomial. Later, when you study rational functions, end behavior of the numerator and denominator polynomials is used to find horizontal and slant asymptotes, and to analyze the long-term behavior of rational models. This topic also forms the foundation for end behavior analysis of other function types later in the course. Without correctly identifying end behavior, you will not be able to correctly sketch graphs or interpret long-term behavior of models on AP exam questions.

Graphing polynomial functions Rational functions and asymptotes Polynomial zeros and multiplicity