Polynomial and Rational Functions — AP Precalculus
1. Core Definitions ★☆☆☆☆ ⏱ 3 min
Polynomial and rational functions are foundational algebraic function families that make up 30-40% of the AP Precalculus exam, tested in both multiple-choice and free-response sections. They model real-world phenomena like projectile motion, revenue curves, and population growth, and are core prerequisites for AP Calculus AB/BC.
2. Polynomial Behaviour: End Behaviour, Zeros & Multiplicity ★★☆☆☆ ⏱ 5 min
This is the most frequently tested subtopic, where examiners commonly ask to match polynomial equations to graphs or sketch graphs from equations. Behaviour is determined entirely by the leading term and factored form of the polynomial.
- If degree $n$ is even: $a_n>0$ → both ends point up; $a_n<0$ → both ends point down
- If degree $n$ is odd: $a_n>0$ → left end down, right end up; $a_n<0$ → left end up, right end down
- Odd multiplicity of a zero: graph crosses the x-axis at the zero
- Even multiplicity of a zero: graph bounces off the x-axis at the zero
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: Asymptotes & Holes ★★☆☆☆ ⏱ 4 min
Asymptotes are lines that a rational function's graph approaches indefinitely, and are the most heavily tested feature of rational functions on the AP exam.
- Vertical Asymptotes: Occur at $x=c$ where the simplified denominator is zero, and numerator is non-zero at $c$
- Holes (removable discontinuities): Occur at $x=c$ where both numerator and denominator are zero before simplification
- Horizontal Asymptotes: $\deg P < \deg Q$ → $y=0$; $\deg P = \deg Q$ → $y = \frac{\text{leading } P}{\text{leading } Q}$; $\deg P > \deg Q$ → no horizontal asymptote
Exam tip: Always simplify before finding asymptotes. Failing to cancel common factors misidentifies holes as asymptotes, a common 1-point FRQ deduction.
4. Graph Transformations ★★☆☆☆ ⏱ 3 min
Transformations of polynomial and rational parent functions follow standard rules, and are often combined with end behaviour or asymptote questions on the exam. The general form of a transformed function is:
g(x) = a \cdot f(b(x - h)) + k
- $a$: Vertical stretch by $|a|$ if $|a|>1$, compression if $0<|a|<1$; reflect over x-axis if $a<0$
- $b$: Horizontal compression by $\frac{1}{|b|}$ if $|b|>1$, stretch if $0<|b|<1$; reflect over y-axis if $b<0$
- $h$: Shift right $h$ units if $h>0$, left $|h|$ units if $h<0$
- $k$: Shift up $k$ units if $k>0$, down $|k|$ units if $k<0$
5. Solving Equations & Inequalities ★★★☆☆ ⏱ 5 min
These problems make up the majority of free-response questions for this unit, requiring clear step-by-step work to earn full marks.
For polynomial equations $P(x)=0$, solve by factoring, rational root theorem, synthetic division, and quadratic formula. For rational equations, multiply through by the denominator to get a polynomial equation, then discard any extraneous solutions that make the original denominator zero.
- Rearrange the inequality to have all terms on one side, comparing to zero
- Find all critical points: zeros of numerator/denominator (rational) or polynomial zeros (polynomial)
- Plot critical points: open circles for excluded points, closed circles for included points
- Test a value in each interval to check sign, select intervals that satisfy the inequality
Common Pitfalls
Why: Rushing to save time, skipping the factoring step
Why: Associating positive signs with rightward movement on a number line
Why: Focusing only on solving the resulting polynomial equation, ignoring the original domain
Why: Memorizing rules without practicing application
Why: Treating all critical points the same regardless of function definition