AP Precalculus Formula Sheet 2026

Complete formula reference for AP Precalculus · Updated for 2026 exam

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 $1/b$