AP Pre-Calc: All Formulas For AP Exam

Union set

All elements in A or B (or both)

Intersection

All elements in A and B

(x + y)^3

x^3 + 3x^2y + 3xy^2 + y^3

(x - y)^3

x^3 - 3x^2y + 3xy^2 - y^3

Sum of cubes

Difference of cubes

Simple interest formula

I = Prt

I = interest earned

P = principal amount

r = rate of interest

t = time

Work formula

W = rt

W = work

r = rate

t = time

Distance formula

d = rt

d = distance

r = rate

t = time

Midpoint formula

Distance formula

Equation of a circle

Vertical Line Test

Checks if a graph is a function

Average rate of change

Even function

f(-x) = f(x)

Odd function

f(-x) = -f(x)

Composition of functions

One-to-one function

Every x value has a unique y value

Horizontal Line Test

Checks if a graph is one-to-one

Inverse of a function

f^-1(y) = x

Inverse function properties

f^-1(f(x)) = x

f(f^-1(x)) = x

Standard form of parabola

f(x) = a(x - h)^2 + k

If a > 0, parabola opens upwards

If a < 0, parabola opens downwards

Maximum or minimum value of a parabola

x = -b / 2a

Remainder Theorem

If the polynomial P(x) is divided by x - c, then the remainder is the value P(c)

Descartes' Rule Of Signs

Finding asymptotes of rational functions

Exponential function

f(x) = a^x

Compound interest formula

Natural exponential function

Continously compounded interest

Logarithmic function

loga x = y

Natural logarithm

lnx = loge x

Laws of Logarithms

Change of Base formula

Exponential growth (doubling time)

n0 = initial time at t = 0

t = time

a = doubling time

Exponential growth (relative growth rate)

y = population at time t

n0 = initial size of the population

k = relative rate of growth

t = time

Radioactive decay model

pH scale formula

Richter scale formula

Decibel scale formula

I = intensity

I0 = reference intensity

Even-odd properties of trigonometric functions

Pythagorean identities

Shifted sine and cosine curves

y = asink(x - b)

y = acosk(x - b)

Amplitude: |a|

Period: 2π / k

Tangent and cotangent curves

y = atankx

y = acotkx

Period: π / k

Cosecant and secant curves

y = acsckx

y = aseckx

Period: 2π / k

Domain and range of inverse trigonometric functions

Inverse sine:

Domain: [-1, 1]

Range: [-π / 2, π / 2]

Inverse cosine:

Domain: [-1, 1]

Range: [0, π]

Inverse tangent:

Domain: ℝ

Range: (-π / 2, π / 2)

Relationship between degrees and radians

Length of circular arc

s = length

Θ = central angle (radians)

Area of circular sector

Linear and angular speed

Linear: v = s / t = rw

Angular: w = Θ / t

s = rΘ

t = time

1 revolution = 2π

Area of a triangle

A = 1/2absinΘ

Law of Sines

Ambiguous case for Law of Sines

SSA Triangle: can result in 0, 1, or 2 triangles

Law of Cosines

Heron's Formula

Cofunction identities

Addition and subtraction formulas

Double angle formulas

Formulas for lowering powers

Half angle formulas

Relationship between polar and rectangular coordinates

x = rcosΘ

y = rsinΘ

Modulus of complex numbers

Polar form of complex numbers

z = r(sinΘ + isinΘ)

Multiplication and division of complex numbers

De Moivre's Theorem

nth roots of complex numbers

Arithmetic Sequence

Partial sums of an arithmetic sequence

Sn = n/2[2a + (n - 1)d]

Geometric Sequence

Partial sums of a geometric sequence

Sum of an infinite geometric sequence

If |r| < 1, series converges

If |r| >= 1, series diverges

Binomial Coefficient

Term containing a^r in binomial expansion

Finding nth term in binomial expansion