Precalculus Study Library.

Section I — Multiple Choice (62.5%)

Part A: 28 questions · 80 minutes · no calculator. Tests algebraic fluency and function reasoning.

Part B: 12 questions · 40 minutes · graphing calculator required. Tests applied modeling and numerical reasoning. Four answer choices; no penalty for guessing.

Section II — Free Response (37.5%)

FRQ 1 & 2 (Part A, calc): Function concepts + modeling a non-periodic context (e.g., exponential or rational). 30 minutes, calculator active.

FRQ 3 & 4 (Part B, no-calc): Modeling a periodic context (sinusoidal) + symbolic manipulations (identities, inverses, algebra). 30 minutes, no calculator. Each FRQ worth 6 raw points.

Fixed FRQ Structure Each Year

Score Distributions (2025)

n = 253,596 · Mean: 3.55 · 80.8% scored 3+ · One of the highest pass rates of any AP STEM exam.

What to memorize (no formula sheet)

• Sinusoidal form and all four parameters: $A$, $B$, $C$, $D$
• Period formula: $P = \frac{2\pi}{B}$ for sine/cosine; $P = \frac{\pi}{B}$ for tangent
• All Pythagorean identities: $\sin^2\theta + \cos^2\theta = 1$ and derived forms
• Reciprocal identities: $\csc, \sec, \cot$ in terms of $\sin, \cos, \tan$
• Logarithm rules: product, quotient, power, change of base
• Asymptote rules for rational functions (degree comparison)
• End behavior of polynomials by leading term sign and degree

FRQ scoring priorities

• Show all work — calculator answers without supporting algebra score 0
• Sinusoidal modeling: identify midline, amplitude, period, and horizontal shift from context before writing the equation
• Contextual interpretation: slope of a function model must be stated with units and in the context of the problem
• Inverse functions: verify domain restrictions; state the restricted domain explicitly
• Average rate of change: $\frac{f(b)-f(a)}{b-a}$ — include units in context questions
• Residual plots: explain what a random (non-patterned) residual plot confirms about a model choice

Sinusoidal Functions

$$f(x) = A\sin\!\bigl(B(x-C)\bigr)+D$$ Amplitude $= |A|$, Period $= \dfrac{2\pi}{B}$,
Midline $= D$, Phase shift $= C$

Exponential & Log

$\log_b(xy) = \log_b x + \log_b y$
$\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$
$\log_b(x^r) = r\log_b x$
$\log_b x = \dfrac{\ln x}{\ln b}$

Asymptote Rules (Rational)

Degree num $<$ denom: $y=0$
Equal degrees: $y = \frac{\text{lead}_n}{\text{lead}_d}$
Degree num $=$ denom $+1$: oblique
Degree num $>$ denom $+1$: no HA