PreCalc

sin θ

1 / csc θ

cos θ

1 / sec θ

tan θ

1 / sec θ

tan θ

1 / cot θ

tan θ

sin θ / cos θ

cot θ

cos θ / sin θ

sin² θ + cos θ

1

1 + tan² θ

sec² θ

1 + cot² θ

csc² θ

sin(-θ)

- sin θ

cos (-θ)

cos θ

tan(-θ)

- tan θ

sin form

y = a sin(b(x-h))+k

cos form

y = a cos(b(x-h))+k

amplitude

|a|

period (sin cos)

2π / b

phase shift

h

vertical shift

k

midline

y = k

sin parent graph

starts in middle going upwards

cosine parent graph

starts at maximum

tangent form

y = a tan(b(x-h))+k

tangent period

π / b

vertical asymptotes cosine

0

parent asymptotes

x = +- (π / 2)

even degree + positive leading coefficient

both end behaviors ends up

even degree + negative leading coefficient

both end behavior ends down

old degree + positive leading coefficient

left ends down, right ends up

old degree + negative leading coefficient

left ends up, right ends down

old multiplicity (degree)

graph crosses x-axis

even multiplicity (degree)

graph bounces off x-axis

rational functions: vertical asymptotes

set denominator = 0

rational functions: holes

factor and cancel

HA: degree numerator < denominator

y = 0

HA: degrees numerator = denominator

use ratio of leading coefficients

HA: degree numerator > denominator

use long division —> slant asymptote

exponential function general form

y = a(b)^x

growth value rule

b > 1

decay value rule

0<b<1

EF: horizontal asymptotes y usually =

0

EF: y intercept =

a

exponential growth/decay model

y = a(1+r)^t

EF: r means

growth/decay rate in decimal

log_b(x) = y ←-→

←-→ b^y = x

LR: Product Rule

log_b(MN) = log_b M + log_b N

LR: Quotient Rule

log_b (M/N) = log_b M - log_b N

LR: Power Rule

log_b(M^p) = p log_b M

LR: Inverse Relationships

log_b(b^x) =x

b^log_bx = x

LR: Domain

inside of log must be positive

FT: Vertical Changes

Outside function

y = f(x) + k
positive k moves up k

y = f(x) - k
negative k moves down k

FT: Horizontal Changes

Inside function

y = f(x - h)
moves → right h

y = f(x + h)
moves ←- left h

FT: Reflections

y = -f(x)
- outside reflects functions over x axis

y = f(-x)
- inside reflects functions over y axis

Composition (f of g)(x) =

f(g(x))

Inverse Functions Steps (4)

1. Replace f(x) with y

2. Swap x and y

3. Solve for y

4. Rename as f^-1(x)

• A function and its inverse reflects across y = x

AROC

( f(b) - f(a) ) / ( b - a )

• slope between two points

Vector Magnitude Formula

|v→| = √(x² + y²)

Parametric Equations

x = f(t)

y = g(t)

Explicit Arithmetic Sequence

a_n = a₁ + d(n-1)

Recursive Arithmetic Sequence

a_n = a_n-1(r)

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