Pre-Calculus Formula Sheet — Complete Reference A complete pre-calculus formula sheet covering all the topics typically taught in a precalc course: function basics, trigonometry, exponentials and logarithms, sequences and series, conic sections, and an introduction to limits.
Functions & Transformations Function Composition
( f ∘ g ) ( x ) = f ( g ( x ) ) (f \circ g)(x) = f(g(x)) ( f ∘ g ) ( x ) = f ( g ( x )) Inverse Function
f − 1 ( f ( x ) ) = x f^{-1}(f(x)) = x f − 1 ( f ( x )) = x Vertical Shift
y = f ( x ) + k (up by k) y = f(x) + k \text{ (up by k)} y = f ( x ) + k (up by k) Horizontal Shift
y = f ( x − h ) (right by h) y = f(x - h) \text{ (right by h)} y = f ( x − h ) (right by h) Vertical Stretch
y = a ⋅ f ( x ) , ∣ a ∣ > 1 y = a \cdot f(x),\ |a| > 1 y = a ⋅ f ( x ) , ∣ a ∣ > 1 Reflection
y = − f ( x ) (about x-axis) , y = f ( − x ) (about y-axis) y = -f(x) \text{ (about x-axis)},\ y = f(-x) \text{ (about y-axis)} y = − f ( x ) (about x-axis) , y = f ( − x ) (about y-axis)
Polynomial & Rational Functions Quadratic Formula
x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c Vertex of Parabola
x = − b 2 a x = -\frac{b}{2a} x = − 2 a b Discriminant
Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c Rational Root Theorem
Roots = ± p q , p ∣ a 0 , q ∣ a n \text{Roots} = \pm\frac{p}{q},\ p \mid a_0,\ q \mid a_n Roots = ± q p , p ∣ a 0 , q ∣ a n Vertical Asymptote
x = a \text{ where denominator = 0, numerator ≠ 0}
Horizontal Asymptote (deg n = d)
y = a n b d y = \frac{a_n}{b_d} y = b d a n Trigonometry Essentials Pythagorean Identity
sin 2 x + cos 2 x = 1 \sin^2 x + \cos^2 x = 1 sin 2 x + cos 2 x = 1 Reciprocal Identities
csc x = 1 sin x , sec x = 1 cos x , cot x = 1 tan x \csc x = \tfrac{1}{\sin x},\ \sec x = \tfrac{1}{\cos x},\ \cot x = \tfrac{1}{\tan x} csc x = s i n x 1 , sec x = c o s x 1 , cot x = t a n x 1 Double Angle (Sine)
sin 2 x = 2 sin x cos x \sin 2x = 2 \sin x \cos x sin 2 x = 2 sin x cos x Double Angle (Cosine)
cos 2 x = cos 2 x − sin 2 x \cos 2x = \cos^2 x - \sin^2 x cos 2 x = cos 2 x − sin 2 x Sum Formulas
sin ( A + B ) = sin A cos B + cos A sin B \sin(A + B) = \sin A \cos B + \cos A \sin B sin ( A + B ) = sin A cos B + cos A sin B Law of Sines
a sin A = b sin B = c sin C \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} sin A a = sin B b = sin C c Law of Cosines
c 2 = a 2 + b 2 − 2 a b cos C c^2 = a^2 + b^2 - 2ab\cos C c 2 = a 2 + b 2 − 2 ab cos C Exponentials & Logarithms Exponential Form ↔ Log Form
a x = y ⟺ log a y = x a^x = y \iff \log_a y = x a x = y ⟺ log a y = x Product Rule
log a ( M N ) = log a M + log a N \log_a(MN) = \log_a M + \log_a N log a ( M N ) = log a M + log a N Quotient Rule
log a M N = log a M − log a N \log_a\!\tfrac{M}{N} = \log_a M - \log_a N log a N M = log a M − log a N Power Rule
log a M p = p log a M \log_a M^p = p \log_a M log a M p = p log a M Change of Base
log a x = ln x ln a \log_a x = \frac{\ln x}{\ln a} log a x = ln a ln x Exponential Growth/Decay
y = y 0 e k t y = y_0 e^{k t} y = y 0 e k t Compound Interest
A = P ( 1 + r n ) n t A = P\left(1 + \tfrac{r}{n}\right)^{n t} A = P ( 1 + n r ) n t Sequences & Series Arithmetic nth Term
a n = a 1 + ( n − 1 ) d a_n = a_1 + (n-1) d a n = a 1 + ( n − 1 ) d Arithmetic Sum
S n = n 2 ( a 1 + a n ) S_n = \tfrac{n}{2}(a_1 + a_n) S n = 2 n ( a 1 + a n ) Geometric nth Term
a n = a 1 r n − 1 a_n = a_1 r^{n-1} a n = a 1 r n − 1 Geometric Sum
S n = a 1 1 − r n 1 − r S_n = a_1 \tfrac{1 - r^n}{1 - r} S n = a 1 1 − r 1 − r n Infinite Geometric Sum
S = a 1 1 − r , ∣ r ∣ < 1 S = \frac{a_1}{1 - r},\ |r| < 1 S = 1 − r a 1 , ∣ r ∣ < 1 Binomial Theorem
( a + b ) n = ∑ k = 0 n ( n k ) a n − k b k (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k ( a + b ) n = k = 0 ∑ n ( k n ) a n − k b k Conic Sections Circle
( x − h ) 2 + ( y − k ) 2 = r 2 (x - h)^2 + (y - k)^2 = r^2 ( x − h ) 2 + ( y − k ) 2 = r 2 Ellipse
( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1 \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 Parabola (vertex form)
y − k = a ( x − h ) 2 y - k = a(x - h)^2 y − k = a ( x − h ) 2 Hyperbola
( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 Limits & Continuity (Preview) Definition
lim x → a f ( x ) = L \lim_{x \to a} f(x) = L x → a lim f ( x ) = L Continuity
lim x → a f ( x ) = f ( a ) \lim_{x \to a} f(x) = f(a) x → a lim f ( x ) = f ( a ) Special Trig Limit
lim x → 0 sin x x = 1 \lim_{x \to 0} \frac{\sin x}{x} = 1 x → 0 lim x sin x = 1 Definition of e
e = lim n → ∞ ( 1 + 1 n ) n e = \lim_{n \to \infty}\left(1 + \tfrac{1}{n}\right)^n e = n → ∞ lim ( 1 + n 1 ) n
