Ap Pre-Calculus Formulas Review

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Complex Numbers Standard Form

a + bi

Complex Number Conjugate (Rationalizing)

a - bi

Point-Slope Form (Given 1 Point & Slope)

y - y₁ = m(x - x₁)

Sum of Cubes (a³ + b³)

(a+b)(a²-ab+b²)

Difference of Cubes (a³ - b³)

(a-b)(a²+ab+b²)

log_b(x) = y

b^y = x

Logarithm Product Rule - log_b​(xy)

log_b​x + log_b​y​

Logarithm Quotient Rule - log_b(x/y)

log_bx - log_by

Logarithm Power Rule - log_b(xⁿ)

n * log_bx

log_b(b^x)

x

b^log_bx

x

ln(e^x)

x

e^{ln x}

x

Vertex / Axis of Symmetry

x = (-b / 2a)

Quadratic Vertex

a(x - h)² + k

Vertical Shift Function Transformation - f(x)

Up k = f(x) + k

Down k = f(x) - k

Horizontal Shift Function Transformation - f(x)

Right h = f(x - h)

Left h = f(x + h)

Vertical Stretch Function Transformation - f(x)

af(x) ; ∣a∣>1

Vertical Compression Function Transformation - f(x)

af(x) ; 0<∣a∣<1

Reflection Across X-Axis - f(x)

af(x) ; a is negative

Reflection Across Y-Axis - f(x)

f(-x)

Change of Base Formula

log_ba = (log a / log b)

Adding Functions - (f + g)(x)

f(x) + g(x)

Subtracting Functions - (f - g)(x)

f(x) - g(x)

Multiplying Functions - (fg)(x)

f(x) * g(x)

Exponential Growth/Decay Formula + Variables

y = a(1 + r)^t

a = initial amount

r = growth/decay rate

t = time

Finding Growth/Decay Factor

b = (y/a)^1/t

Compound Interest Formula + Variables

A = P(1 + (r/n))^nt

A = final amount

P = principal / starting amount

r = yearly interest rate (decimal)

n = times compounded per year

t = time in years

Compounded Continuously Formula

A = Pe^rt

Euler’s Number (e)

2.718

Cylinder Volume

V = πr²h​

Sphere Volume

V=(3/4)​πr³​

45°-45°-90° Triangle

• The two legs are equal

• The hypotenuse is leg√2

30°-60°-90° Triangle

• x = side opposite 30° (short leg)

• x√3 = side opposite 60° (long leg)

• 2x = hypotenuse