AP Precalculus Formula Sheet 2026

Complete formula reference for AP Precalculus · Updated for 2026 exam

Quick Answer: Does AP Precalculus provide a formula sheet? Yes, partially — the exam includes a limited reference sheet with select formulas. However, trig identities, log properties, and transformation rules are not provided and must be memorized.

The AP Precalculus exam provides a limited reference sheet on exam day. This guide covers everything on the official sheet plus the key formulas and identities you must know from memory, organized by unit.

Unit 1 — Polynomial & Rational Functions

Formula / Concept	Notes
$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0$	General polynomial form; degree = $n$
End behavior: even degree, positive lead	Both ends → +∞
End behavior: odd degree, positive lead	Left → −∞, right → +∞
Remainder Theorem	$f(a)$ = remainder when $f(x)$ divided by $(x-a)$
Factor Theorem	$(x-a)$ is a factor iff $f(a) = 0$
Vertical asymptote of $\frac{p(x)}{q(x)}$	Where $q(x) = 0$ and $p(x) \neq 0$
Horizontal asymptote	Compare degrees: if deg $p$ < deg $q$, HA at $y = 0$; if equal, HA at ratio of leading coefficients
Hole in rational function	Common factor in numerator and denominator that cancels

Unit 2 — Exponential & Logarithmic Functions

Formula	Notes
$f(x) = ab^x$	Exponential function; $b > 0$, $b \neq 1$; $a$ = initial value
$f(x) = ae^{kx}$	Natural exponential; $k > 0$ growth, $k < 0$ decay
$\log_b(xy) = \log_b x + \log_b y$	Product rule
$\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$	Quotient rule
$\log_b(x^n) = n \log_b x$	Power rule
$\log_b x = \frac{\ln x}{\ln b}$	Change of base formula
$b^{\log_b x} = x$ and $\log_b(b^x) = x$	Inverse relationship
Half-life: $A = A_0 \left(\frac{1}{2}\right)^{t/h}$	$h$ = half-life period
Doubling time: $A = A_0 \cdot 2^{t/d}$	$d$ = doubling period
Compound interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$	$n$ = compoundings per year
Continuous compounding: $A = Pe^{rt}$	Most common on AP exam

Unit 3 — Trigonometric & Polar Functions

Fundamental Definitions & Identities

Identity	Formula
Pythagorean identity	$\sin^2\theta + \cos^2\theta = 1$
Pythagorean (tan form)	$1 + \tan^2\theta = \sec^2\theta$
Pythagorean (cot form)	$1 + \cot^2\theta = \csc^2\theta$
Reciprocal: $\csc\theta$	$\frac{1}{\sin\theta}$
Reciprocal: $\sec\theta$	$\frac{1}{\cos\theta}$
Reciprocal: $\cot\theta$	$\frac{\cos\theta}{\sin\theta}$
Even/odd: $\cos(-\theta)$	$\cos\theta$ (even)
Even/odd: $\sin(-\theta)$	$-\sin\theta$ (odd)

Key Angle Values (Must Memorize)

θ	sin θ	cos θ	tan θ
0°	0	1	0
30° (π/6)	1/2	√3/2	1/√3
45° (π/4)	√2/2	√2/2	1
60° (π/3)	√3/2	1/2	√3
90° (π/2)	1	0	undefined

Sinusoidal Functions

Property	Formula
General form	$f(x) = A\sin(Bx + C) + D$ or with cos
Amplitude	$\|A\|$
Period	$\frac{2\pi}{\|B\|}$
Phase shift	$-\frac{C}{B}$ (positive = shift right)
Vertical shift (midline)	$D$
Maximum value	$D + \|A\|$
Minimum value	$D - \|A\|$

Polar Coordinates

Conversion	Formula
Polar → Rectangular	$x = r\cos\theta$, $y = r\sin\theta$
Rectangular → Polar	$r = \sqrt{x^2 + y^2}$, $\theta = \arctan(y/x)$
Circle centered at origin	$r = a$
Rose curve	$r = a\cos(n\theta)$ or $r = a\sin(n\theta)$; $n$ petals if odd, $2n$ if even

Unit 4 — Functions Involving Parameters, Vectors & Matrices

Concept	Formula / Rule
Parametric equations	$x = f(t)$, $y = g(t)$; eliminate $t$ to find Cartesian form
Vector magnitude	$\|\vec{v}\| = \sqrt{v_x^2 + v_y^2}$
Vector addition	$(a_1, a_2) + (b_1, b_2) = (a_1+b_1, a_2+b_2)$
Scalar multiplication	$k(a_1, a_2) = (ka_1, ka_2)$
Dot product	$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$
Vectors perpendicular iff	$\vec{a} \cdot \vec{b} = 0$
2×2 matrix multiplication	Row × column; result entry $(i,j)$ = row $i$ dotted with col $j$
2×2 determinant	$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$

Function Transformations (All Units)

Transformation	Effect on $f(x)$
$f(x) + k$	Shift up $k$ units
$f(x) - k$	Shift down $k$ units
$f(x + h)$	Shift left $h$ units
$f(x - h)$	Shift right $h$ units
$af(x)$, $a > 1$	Vertical stretch by factor $a$
$af(x)$, $0 < a < 1$	Vertical compression
$-f(x)$	Reflect over x-axis
$f(-x)$	Reflect over y-axis
$f(bx)$, $b > 1$	Horizontal compression by factor $1/b$

Common AP Precalculus Exam Tasks

Analyze function behavior — identify domain, range, end behavior, intercepts, asymptotes. Every function type (polynomial, rational, exponential, trig) is tested. Know the key features of each.
Apply transformations — given a graph of $f(x)$, describe $f(x-2)+3$ or $-2f(x)$. These appear on both MC and FRQ.
Solve exponential and logarithmic equations — use log properties to isolate variables. Know when to use $\ln$ vs. $\log_{10}$.
Work with sinusoidal models — given a real-world periodic situation (tides, temperature), find amplitude, period, phase shift, and write the equation $A\sin(Bx+C)+D$.
Convert between polar and rectangular — $x=r\cos\theta$, $y=r\sin\theta$, $r=\sqrt{x^2+y^2}$. Practice both directions.

What to Memorize vs. What's on the Reference Sheet

Must Memorize	Likely on Reference Sheet
Trig identities (Pythagorean, reciprocal)	Basic trig definitions
Log properties (product, quotient, power)	Some exponential formulas
Key angle values (30°, 45°, 60°)	Quadratic formula
Transformation rules	Area and volume formulas
Sinusoidal parameter meanings (A, B, C, D)	—

Frequently Asked Questions

Does AP Precalculus give a formula sheet?

Yes, but it's limited. The reference sheet includes some basic formulas but not trig identities, logarithm properties, or transformation rules. Treat it as a safety net, not a substitute for memorization.

Is AP Precalculus hard?

AP Precalculus is relatively new (first offered 2023–24). Early data shows a moderate difficulty — students with strong algebra and trig foundations generally find it manageable. The biggest challenge is the breadth of function types covered in a single course.

Does AP Precalculus count as a math credit in college?

It depends on the university. AP Precalculus is newer than other AP math exams, and fewer schools have established credit policies for it yet. Check directly with your target schools — many accept a 4 or 5 for elective credit but not for fulfilling calculus prerequisites.