ap precalc gold sheet

concave up

rates of change are increasing

"up like a cup"

concave down

rates of change are decreasing

avg. rate of change from [ a,b ] or x=a to x=b

slope of secant line

change in y/change in x = f(b) - f(a)/b-a

instantaneous rate of change at x = a

slope of tangent line (can estimate by finding AROC for two points)

slope intercept form for linear

y = mx + b

point-slope form for linear

Y-Y₁ = m (X-X₁)

point: (X₁, Y₁)

standard form for linear

ax + by = c

vertex form (quadratic)

y = a (x-h)² + k

V: (h, k)

AOS: x=h

intercept form (quadratic)

y = a (x-p)(x-q)

x-int: (p, 0) (q, 0)

standard form (quadratic)

y = ax² + bx + c

V: (-b/2a, )

fundamental theorem of algebra

a polynomial of degree n has a total of n zeros (real or imaginary)

quadratic formula

x = -b ± √(b² - 4ac)/2a

* finds both real and imaginary zeros

even function

if f(-x) = f(x)

odd function

if f(-x) = -f(x)

end behavior

lim f(x) =

x → ∞/-∞

horizontal asymptotes

compare the degree of the top and bottom

y = #

lim f(x) = #

x → ∞

lim f(x) = #

x → -∞

finding zeros/x-ints of a rational function

factor the numerator and find what values make each factor = to zero

finding vertical asymptotes of a rational function

factor the denominator and find what values make each factor = to zero

X = #

finding holes of a rational function

factor the numerator and denominator, and if there is the same factor on the top and bottom there is a hole (0/0)

slant asymptote

use synthetic or long division to find a linear equation that represents the end behavior

* the degree of the numerator must be one bigger than the degree of the denominator

synthetic division

the divisor must be linear... in the format (x ± #)

arithmetic sequences

aₙ = a₁ + d(n-1)

a₁ is first term

d is common difference

aₙ = a₀ + d(n)

aₙ = aₖ + d(n-k)

k is any term

geometric sequences

aₙ = a₁ (r)ⁿ⁻¹

r is common ratio

aₙ = a₀ (r)ⁿ

aₙ = aₖ (r)ⁿ⁻^K

exponential functions

f(x) = a(b)^x

a is y-int

b is constant proportion/multiplier

f(x) = y₁ (b)^x-x₁

(x₁, y₁) can be any input output pair

interest compounded continuously

A = Pₑ^rt

P is initial amount

r is rate as a decimal

t is times

compound interest

A = P₀ (1+r/n)^(n)(t)

P₀ is initial amount

r is rate as a decimal

n is number of times compounded in a year

half-life

y = y₀ (1/2)^t/n

y₀ is the initial amount

n is the time of half-life