Polynomial and Rational Functions — AP Precalculus Precalc Study Guide

For: AP Precalculus candidates sitting AP Precalculus.

Covers: Polynomial end behaviour, zeros and multiplicity, vertical and horizontal asymptotes of rational functions, graphing transformations of polynomial and rational functions, and solving associated equations and inequalities.

You should already know: Algebra 1 & 2, basic geometry and trigonometry.

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 papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official College Board mark schemes for grading conventions.

1. What Is Polynomial and Rational Functions?

Polynomial and rational functions are two foundational families of algebraic functions that form 30-40% of the AP Precalculus exam content, tested in both multiple-choice and free-response sections. They are widely used to model real-world phenomena including projectile motion, revenue curves, and population growth. Polynomials have the general form $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_{1}x+a_0$, where $n$ is a non-negative integer called the degree of the polynomial, and $a_n\ne 0$ is the leading coefficient. Rational functions are ratios of two non-zero polynomials, written $f(x)=\frac{P(x)}{Q(x)}$ where $Q(x)$ is not the zero polynomial. These functions are also core prerequisites for AP Calculus AB/BC, where you will analyze their limits, derivatives, and integrals.

2. Polynomial behaviour — end behaviour, zeros, multiplicity

This subtopic is the most frequently tested part of this unit, as examiners often ask you to match polynomial equations to their graphs or sketch graphs from given equations.

Key Definitions

1. End behaviour: Describes the direction the graph of the polynomial extends as $x$ approaches $+\infty$ and $-\infty$, determined entirely by the leading term $a_n{x}^n$:

• If $n$ is even: If $a_n>0$, both ends of the graph point up ($x\to \pm \infty ,f(x)\to +\infty$); if $a_n<0$, both ends point down.
• If $n$ is odd: If $a_n>0$, the left end points down and the right end points up ($x\to -\infty ,f(x)\to -\infty ;x\to +\infty ,f(x)\to +\infty$); if $a_n<0$, the left end points up and the right end points down.

2. Zero: A value $x=c$ where $f(c)=0$, corresponding to an x-intercept at the point $(c,0)$.
3. Multiplicity: If $(x-c{)}^k$ is a factor of $f(x)$ but $(x-c{)}^{k+1}$ is not, the zero $c$ has multiplicity $k$:

• Odd multiplicity: The graph crosses the x-axis at $c$.
• Even multiplicity: The graph bounces (touches the x-axis and turns around) at $c$.

Worked Example

For the polynomial $f(x)=-2(x-3{)}^2(x+1{)}^3$:

• Leading term: $-2x^5$, so degree 5 (odd), leading coefficient negative: end behaviour is $x\to -\infty ,f(x)\to +\infty$ and $x\to +\infty ,f(x)\to -\infty$.
• Zeros: $x=3$ with multiplicity 2 (even, bounces), $x=-1$ with multiplicity 3 (odd, crosses). Exam tip: Always check end behaviour first when matching graphs to equations, as you can eliminate 2-3 wrong options in seconds.

3. Rational functions — vertical and horizontal asymptotes

Asymptotes are lines that the graph of a rational function approaches as $x$ or $y$ approaches $\pm \infty$, and are the most important feature tested for rational functions on the exam.

Key Rules

1. Vertical Asymptotes: Occur at values of $x$ where the simplified denominator of the rational function is zero, and the numerator is non-zero at that value. If both numerator and denominator are zero at a point, that is a hole (removable discontinuity) not an asymptote.
2. Horizontal Asymptotes: Determined by comparing the degree of the numerator ($degP$) and the degree of the denominator ($degQ$):

• If $degP<degQ$: Horizontal asymptote at $y=0$ (the x-axis).
• If $degP=degQ$: Horizontal asymptote at $y=\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$.
• If $degP>degQ$: No horizontal asymptote (you will have a slant/oblique asymptote instead, which is tested occasionally on the exam).

Worked Example

For the rational function $f(x)=\frac{3x^2-12}{x^2-2x-3}$:

• Step 1: Factor fully: Numerator $=3(x-2)(x+2)$, Denominator $=(x-3)(x+1)$. No common factors, so no holes.
• Vertical asymptotes: $x=3$ and $x=-1$, where the denominator is zero.
• Horizontal asymptote: $degP=2$, $degQ=2$, leading coefficients 3 and 1, so $y=3$. Exam note: Always simplify the rational function before finding asymptotes, as failing to cancel common factors will lead you to misidentify holes as vertical asymptotes, a common 1-point deduction on free-response questions.

4. Graphing transformations

Transformations of polynomial and rational parent functions follow a standard set of rules, and are often combined with asymptote or end behaviour questions on the exam. The general form of a transformed function is $g(x)=a\cdot f(b(x-h))+k$, where $f(x)$ is the parent function:

• $a$: Vertical stretch by a factor of $|a|$ if $|a|>1$, vertical compression if $0<|a|<1$; reflect over the x-axis if $a<0$.
• $b$: Horizontal compression by a factor of $\frac{1}{|b|}$ if $|b|>1$, horizontal stretch if $0<|b|<1$; reflect over the y-axis if $b<0$.
• $h$: Horizontal shift right by $h$ units if $h>0$, left by $|h|$ units if $h<0$.
• $k$: Vertical shift up by $k$ units if $k>0$, down by $|k|$ units if $k<0$.

Worked Example

Transform the parent cubic function $f(x)=x^3$ to get $g(x)=-2(x+1{)}^3+4$:

1. $a=-2$: Vertical stretch by factor 2, reflect over the x-axis.
2. $b=1$: No horizontal stretch or reflection.
3. $h=-1$: Shift left by 1 unit.
4. $k=4$: Shift up by 4 units. For rational parent function $f(x)=\frac{1}{x}$, the transformed function $g(x)=\frac{3}{x-2}+1$ shifts the vertical asymptote from $x=0$ to $x=2$ and the horizontal asymptote from $y=0$ to $y=1$.

5. Solving polynomial / rational equations and inequalities

These problems make up the majority of free-response questions for this unit, and require you to show clear step-by-step work to earn full marks.

Polynomial Equations

Solve $P(x)=0$ by factoring, using the rational root theorem, synthetic division, and the quadratic formula for irreducible quadratic factors.

Rational Equations

Solve $\frac{P(x)}{Q(x)}=c$ by multiplying both sides by $Q(x)$ (noting the domain restriction $Q(x)\ne 0$), solving the resulting polynomial equation, and discarding any extraneous solutions that make the original denominator zero.

Inequalities

For both polynomial and rational inequalities:

1. Rearrange the inequality to have all terms on one side, so you are comparing an expression to 0.
2. Find all critical points: zeros of the numerator and denominator (for rational inequalities) or zeros of the polynomial (for polynomial inequalities).
3. Plot critical points on a number line, dividing it into intervals. Use open circles for excluded points (denominator zeros, strict inequalities) and closed circles for included points.
4. Test a value in each interval to see if the expression is positive or negative, and select intervals that satisfy the inequality.

Worked Example

Solve $\frac{x-3}{x+1}\le 0$:

1. Critical points: $x=3$ (numerator zero, included) and $x=-1$ (denominator zero, excluded).
2. Intervals: $x<-1$ (test $x=-2$: $\frac{-5}{-1}=5>0$, invalid), $-1<x\le 3$ (test $x=0$: $\frac{-3}{1}=-3\le 0$, valid), $x>3$ (test $x=4$: $\frac{1}{5}=0.2>0$, invalid).
3. Solution: $(-1,3]$.

6. Common Pitfalls (and how to avoid them)

• Wrong move: Forgetting to simplify rational functions before finding vertical asymptotes, marking holes as asymptotes. Why students do it: Rushing to save time, skipping the factoring step. Correct move: Always factor both numerator and denominator first, cancel common factors, note hole locations, then find asymptotes from the simplified function.
• Wrong move: Confusing horizontal shift direction, interpreting $f(x+2)$ as a shift right 2 units. Why students do it: Associating positive signs with rightward movement on a number line. Correct move: Remember the form $f(x-h)$: $x+2=x-(-2)$, so $h=-2$, shift left 2 units.
• Wrong move: Forgetting to check for extraneous solutions when solving rational equations. Why students do it: Focusing only on solving the resulting polynomial equation, ignoring the domain of the original rational function. Correct move: Plug every solution back into the original denominator to confirm it is non-zero, discard any solutions that make the denominator zero.
• Wrong move: Misidentifying end behaviour of odd-degree polynomials with negative leading coefficients. Why students do it: Memorizing rules without practicing application. Correct move: Test very large positive and negative x values to verify end behaviour, e.g. for $f(x)=-3x^3$, $f(100)=-3,000,000$ (right end down) and $f(-100)=3,000,000$ (left end up).
• Wrong move: Using closed intervals for denominator zeros in rational inequality solutions. Why students do it: Treating all critical points the same regardless of whether the function is defined there. Correct move: Exclude all points where the denominator is zero from your solution set, even for non-strict inequalities ($\le$ or $\ge$).

7. Practice Questions (AP Precalculus Style)

Question 1 (Multiple Choice)

Which of the following functions matches a graph with end behaviour $x\to -\infty ,f(x)\to +\infty$, $x\to +\infty ,f(x)\to -\infty$, an x-intercept at $x=2$ that bounces, and an x-intercept at $x=-1$ that crosses? A) $f(x)=(x-2{)}^2(x+1{)}^3$ B) $f(x)=-3(x-2{)}^2(x+1{)}^3$ C) $f(x)=-2(x-2{)}^3(x+1{)}^2$ D) $f(x)=2(x-2{)}^3(x+1{)}^2$

Solution

Correct answer: B.

• End behaviour: Left end up, right end down indicates odd degree, negative leading coefficient. Eliminate A (positive leading coefficient) and D (positive leading coefficient).
• Bounce at $x=2$ requires even multiplicity, cross at $x=-1$ requires odd multiplicity. Eliminate C (odd multiplicity at 2, even at -1). Only B matches all requirements.

Question 2 (Free Response Part A)

Find all asymptotes of the function $f(x)=\frac{2x^2-2x-12}{x^2-4x+3}$. Show all your work.

Solution

1. Factor numerator and denominator: Numerator: $2x^2-2x-12=2(x^2-x-6)=2(x-3)(x+2)$ Denominator: $x^2-4x+3=(x-3)(x-1)$
2. Simplify: Cancel common factor $(x-3)$, so simplified function is $f(x)=\frac{2(x+2)}{x-1}$, with a hole at $x=3$.
3. Vertical asymptote: Simplified denominator is zero at $x=1$, so vertical asymptote $x=1$.
4. Horizontal asymptote: Degree of original numerator and denominator are both 2, leading coefficients 2 and 1, so horizontal asymptote $y=2$. Final answer: Asymptotes are $x=1$ and $y=2$.

Question 3 (Free Response Part B)

Solve the inequality $\frac{2x+4}{x-5}\ge 1$. Write your solution in interval notation.

Solution

1. Rearrange to isolate zero on the right: $\frac{2x+4}{x-5}-1\ge 0$ → $\frac{2x+4-(x-5)}{x-5}\ge 0$ → $\frac{x+9}{x-5}\ge 0$
2. Critical points: $x=-9$ (numerator zero, included) and $x=5$ (denominator zero, excluded).
3. Test intervals:

• $x<-9$: Test $x=-10$ → $\frac{-1}{-15}=\frac{1}{15}\ge 0$, valid.
• $-9<x<5$: Test $x=0$ → $\frac{9}{-5}=-1.8<0$, invalid.
• $x>5$: Test $x=6$ → $\frac{15}{1}=15\ge 0$, valid. Final solution: $(-\infty ,-9]\cup (5,\infty )$

8. Quick Reference Cheatsheet

9. What's Next

Mastery of polynomial and rational functions is critical for success in the rest of the AP Precalculus syllabus, as the rules for end behaviour, intercepts, and transformations apply directly to exponential, logarithmic, trigonometric, and polar functions you will study later. This content also forms the foundation of AP Calculus AB/BC, where you will use these concepts to compute limits, find derivatives, and evaluate integrals of algebraic and composite functions. If you have any questions about specific subtopics, worked examples, or practice problems, you can ask Ollie, our AI tutor, at any time on the homepage, where you can also access more AP Precalculus study guides, full practice tests, and personalized feedback to target your weak areas before the exam.