The Ungerecht Family Tree

• Graphing circles, ellipses, parabolas, hyperbolas
• Polynomials, quadratic formula and completing the square

quadratic formula: x = (-b ±√(b²-4ac)) / 2a

2 equations, for 2 unknowns (need n equations to solve for n unknowns)

Example: 5x + 2y = 7, 3x - y = 6
Solve by a) graphing both equations and see where they intersect (can graph multiple equations on desmos)
         b) substitution (using one equation set one variable in terms of the other, the substitute into
               the remaining equation):
                 y = 3x-6           from 2nd equation
                 5x + 2(3x-6) = 7   substitute into first
                 11x - 12 = 7       solve for x, then can find y
         c) addition - put all like terms in columns, 
                       multiply/divide as needed to get one set of terms to cancel
                       add the terms, solve for the remaining term
               5x + 2y = 7
               6x - 2y = 12
               --------------
               11x     = 19, solve for x, then put in to find y 

Define x and y in terms of t (time). Arrows on graph represent increasing values of t. Allows you to create functions, using two graphs, from things are not functions for both x and y together (circles, ellipses, x = y², etc)

Eliminating the parameter: substitute for t and define x in terms of y (or vice versa), aka rectangular equation.

https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/

Basic form to find term n: a(n) = a(1) + d(n-1), where d = step size (difference between terms), a(1) is the first term

Sum of an arithmetic sequence:

S = n/2(a + L)
  S = sum, n = # of terms, a = value of first term, L = value of last term
If don't know last term, just substitute a(n):
  S = n/2 (2a + (n − 1) d)

Combining powers

nᵃnᵇ  = nᵃ⁺ᵇ
nᵃ/nᵇ = nᵃ⁻ᵇ
(nᵃ)ᵇ = nᵃᵇ
n⁻ᵃ   = 1/nᵃ

Factorial:

n! = n * (n-1)!
0! = 1
7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 

inverse of a power

x = bᵉ,   e = logᵦ(x)  (b = base, 10 by default)
eg. 2³ = 8, log₂(8) = 3
log(1) = 0 (for any base), log(x) is undefined for x =< 0
logₐ(x) = logᵣ(x) / logᵣ(a)
        = log(x) / log(a)  for r=10
        = ln(x) / ln(a)    for r=e  (e = Euler's number, 2.718..)
ln(e) = 1, log(10) = 1
ln(x) = logₑx.
eᵏ = c, and k = ln(c) => e^ln(c) = c

a^logₐ(x) = x   (power and log are inverses, cancel each other out)
logₐ(aᵏ) = k   (same reason)
product rule: log(ab) = log(a) + log(b)
quotient rule: log(a/b) = log(a) - log(b)
power rule: log(aᵇ) = b*log(a)

Definitions:

• Function: only has one y value for any x value. Discontinuities are okay (breaks in allowed x values)
• One-to-one function: a function with only one x value for any y value.
• Domain: what values of x are described
• Range: resulting values of y coming from the function

composition of functions: (f’g)(x) = f(g(x)), order of evaluation is important inverse of function - only possible if no two values of x produce the same result graphing an inverse: reflection of the graph about the line y=x

horizontal and vertical asymptotes: y = (quadratic1 of x) / (quadratic2 of x)

• vertical asymptotes (VA) are when denominator goes to zero
• horizontal asymptotes (HA) is when x goes to infinity, look at highest order of x in numerator and denominator:
  • y = axⁿ / bxᵏ
  • if n > k : no HA
  • if n < k : HA = 0
  • if n = k : HA = a/b

Annual rate, continuous rate of growth:

Y = a*bᵗ
a = principle amount, b = annual growth, t = time (years)
y = a*eᵏᵗ = a*(eᵏ)ᵗ, and k = continuous rate of growth  
   k = ln(b)

Half life:

Nt = No(1/2)^(t/t.5)
 Nt = amount at time t, No = initial amount, t.5 = half-life time

This can be rearranged to:

t.5 = t/(log0.5(Nt/No)) = t / (log(Nt/No)/log(1/2))

Also,

Nt = No*e^(-t/tau)
tau = mean lifetime
tau = 1/lambda,  lambda = decay constant
t.5 = ln(2)/lambda
(ln(2) is close to 7 => rule of 70 for doubling returns??)

abs. value of imaginary number

|a + bi| = √(a^2 + b^2)

rational numbers can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats.

irrational includes square roots, pi, etc

• Paul Dawkins notes on Algebra and other topics at https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx
• Math symbols: https://www.rapidtables.com/math/symbols/Set_Symbols.html
• Online graphing calculator: https://www.desmos.com/calculator

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