Precalc final

transforming graph

y=alog(x−h)+k

• h shifts the graph right h units (left if negative).

• k shifts the graph up k units (down if negative).

• a>1 stretches the graph vertically.

• a<0 reflects the graph across the x-axis.

how to compound interest

A=P(1+kr​)^kt

where:

• A = final amount

• P = principal (starting amount)

• r = annual interest rate (decimal)

• k = number of compoundings per year

• t = time in years

A=Pe^rt

combination

C(n,r)=n!/r!(n−r)!

permutation

P(n,r)=n!/(n−r)!

probability

favorable outcomes / all possible outcomes

and = multiply, or = add

sums of series

Sn​=(n/2)​(a1​+an​)

Sn​=a1​(1−rn)​/(1-r)

sum of infinite

s∞=a1/(1-r)

limits of n

same degree = ratio of leading coefficients

numerator greater = infinity

denomenator greater = 0

remainder theorem

remainder of f(x) when x=a is f(a)

factor theorem

a is a zero of f(x) if f(a)=0

all possible rational zeros

factors of constant / factors of leading coefficient

slant asymptotes

when numerator degree is larger, asymptote is the remainder of function

horizontal ellipse

(x-h)²/a² + (y-h)²/b² = 1

vertical ellipse

(x-h)²/b² + (y-h)²/a² = 1

horizontal hyperbola

(x-h)²/a² - (y-k)²/b² = 1

vertical hyperbola

(y-k)²/b² - (x-h)²/a² = 1

hyperbola asymptotes

±b/a (x-h) + k = horizontal

vertical = ±a/b (x-h) + k

determinant of matrices

ad - bc

inverse of matrices

1/determinant [d -b, -c a]