Unit Overview

Polynomial and Rational Functions

AP Precalculus· 5 min read 📊 27-32% of overall AP Precalculus exam score

1. Unit at a Glance

This unit builds from foundational function concepts to full, rigorous analysis of polynomial and rational functions, the two most widely used function families in precalculus and calculus. We start with core concepts of how functions change, then move to algebraic manipulation, polynomial analysis, rational function analysis, modeling, and transformations.

The topic sequence connects algebraic manipulation to graphical interpretation and real-world application, helping you link what you see on a graph to what you can compute algebraically — an essential skill for success on the AP exam.

This unit includes the following sub-topics:

AP Precalculus Change in tandem (function behavior)

Introduces how changes in input correspond to changes in output for general functions.

★★⏱ 6 min

AP Precalculus Equivalent representations of polynomial and rational expressions

Covers rewriting expressions to reveal key structural properties of polynomials and rationals.

★★★⏱ 8 min

AP Precalculus Function model construction and application

Walks through building polynomial and rational models from given context constraints.

★★★⏱ 7 min

AP Precalculus Function model selection and assumption articulation

Teaches how to select appropriate models and articulate their underlying assumptions.

★★★⏱ 7 min

AP Precalculus Polynomial functions and complex zeros

Explores the Fundamental Theorem of Algebra and properties of complex roots of polynomials.

★★★★⏱ 8 min

AP Precalculus Polynomial functions and end behavior

Explains how degree and leading coefficient determine end behavior of polynomial graphs.

★★⏱ 5 min

AP Precalculus Polynomial functions and rates of change

Analyzes how rates of change differ across polynomials of different degrees.

★★★⏱ 6 min

AP Precalculus Rates of change (average and over equal intervals)

Foundational topic covering average rates of change over any interval.

★★⏱ 5 min

AP Precalculus Rational functions and end behavior

Covers how degrees of numerator and denominator determine end behavior for rationals.

★★★⏱ 6 min

AP Precalculus Rational functions and holes

Explains how removable discontinuities (holes) arise in rational function graphs.

★★★⏱ 6 min

AP Precalculus Rational functions and vertical asymptotes

Analyzes non-removable discontinuities and behavior around vertical asymptotes.

★★★⏱ 6 min

AP Precalculus Rational functions and zeros

Connects factors of rational functions to their zeros and x-intercepts.

★★★⏱ 6 min

AP Precalculus Transformations of functions

Covers shifts, stretches, and reflections of polynomial and rational function graphs.

★★⏱ 5 min

2. Common Pitfalls

Wrong move:

Confusing holes and vertical asymptotes in rational functions

Why:

Many students treat all discontinuities as identical, regardless of whether they are removable

Correct move:

Check if a shared factor cancels completely to distinguish holes (removable) from vertical asymptotes (non-removable)

Wrong move:

Misidentifying end behavior of rational functions

Why:

Students often mix up the rules for horizontal asymptotes based on degree comparisons

Correct move:

Always compare the degree of the numerator to the degree of the denominator to find end behavior rules

Wrong move:

Forgetting complex roots come in conjugate pairs

Why:

Students often miss that the conjugate root rule only applies to polynomials with real coefficients

Correct move:

Use the Complex Conjugate Root Theorem to find all roots of a real-coefficient polynomial when given one complex root

3. Quick Reference Cheatsheet

Concept / Formula	Key Description
Polynomial end behavior	Determined by leading term $a_{n} x^{n}$ : even $n$ → both ends same direction; odd $n$ → opposite ends
Fundamental Theorem of Algebra	A degree $n$ polynomial has exactly $n$ roots (counting multiplicity) over the complex numbers
Complex Conjugate Root Theorem	If $a + bi$ is a root of a real-coefficient polynomial, then $a - bi$ is also a root
Rational horizontal asymptote rule	deg(num) < deg(den): HA at $y = 0$ ; equal deg: HA at $y = \frac{leading num}{leading den}$ ; deg(num) > deg(den): no HA
Holes in rational functions	Occur at $x = c$ if $(x - c)$ is a common factor of numerator and denominator
Vertical asymptotes in rational functions	Occur at $x = c$ if $(x - c)$ is a factor of the denominator but not the simplified numerator
Average rate of change over $[a, b]$	$\frac{f ( b ) - f ( a )}{b - a}$

What's Next

Begin your study of Unit 1 with the first sub-topic below, which lays the core foundation for all function behavior analysis you will do in this unit. Once you complete all Unit 1 sub-topics, proceed to the first sub-topic of Unit 2, which covers exponential and logarithmic functions.