ap precalc important stuff

slant asymptote

slant asymptote

numerator > denominator by exactly 1

horizontal asymptote @ y = b

numerator = denominator

horizontal asymptote @ y = 0

numerator < denominator

hole

multiplicity of factor in numerator ≥ multiplicity of factor in denominator

vertical asymptote

multiplicity of factor in numerator < multiplicity of factor in denominator

decreasing @ decreasing rate

graph decreases with flatter slope

decreasing @ increasing rate

graph decreases with steeper slope

increasing @ decreasing rate

graph increases with flatter slope

increasing @ increasing rate

graph increases with steeper slope

exponent product property

b¹b² = b³

exponent quotient property

b⁶/b⁵ = b¹

exponent power property

(b²)⁷ =b¹⁴

log product property

logₙ(3*2) = logₙ3 + logbₙ2

log quotient property

logₙ(8/4) = logₙ8 - logₙ4

log power property

logₙ(3⁶) = 6logₙ3

a

vertical dilation/amplitude

b

horizontal dilation (by factor of 1/b)

c

horizontal translation by -c units (also called the phase shift)

d

vertical translation by d units

sinusoidal equation

f(x) = a sin (b(x + c)) + d

vertical transformations

outside of the function

horizontal transformations

inside of the function

dilations

multiplicative transformations

translations

additive transformations

on a specified domain, a function f is invertible if

each output value (y) is mapped from a unique input value (x)

unit circle

sin = y

cos = x

tan = y/x

[-1, 1] s

sin(sin⁻¹) = x

[-π/2, π/2]

sin⁻¹(sin x) = x

[-1, 1] c

cos(cos⁻¹) = x

[0, π]

cos⁻¹(cos x) = x

all reals

tan(tan⁻¹) = x

(-π/2, π/2)

tan⁻¹(tan x) = x

double angle formula

sin(2u) = 2(sin u)(cos u)

zero with even multiplicity

bounces off the x-axis at that zero

concave up

differences in outputs increase

concave down

differences in outputs decrease

even function

f(-x) = f(x), symmetric over y-axis

odd function

f(-x) = -f(x), symmetric at origin

polar -> rectangular coordinates

x = rcosθ

y = rsinθ

rectangular -> polar coordinates

x² + y² = r²

tanθ = y/x (x ≠ 0)

rectangular -> polar for quadrants I/IV

θ = tan⁻¹(y/x)

rectangular -> polar for quadrants II/III

θ = π + tan⁻¹(y/x)

rectangular -> polar if x or y = 0

sketch graph to find r

absolute value of complex numbers

|a + bi| = √a² + b²

trig identities #1

sin²x + cos²x = 1

trig identities #2

tan²x + 1 = sec²x

trig identities #3

1 + cot²x = csc²x

sin(α + β)

sinα cosβ + sinβ cosα

cos(α + β)

cosα cosβ - sinα sinβ