# The Ungerecht Family Tree

urp:algebra

### Algebra & Pre-Calc

Related pages:

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

PEMDAS/BODMAS - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract

### System of equations

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 ```

### Parametric equations

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.

### Arithmetic/Geometric Sequences

Arithmetic sequence has a constant difference between the terms, such as 1, 3, 5, 7, 9…

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)```

Geometric sequence terms are found by multiplying the previous term by a constant, such as 2, 4, 8, 16 …

a(n) = arⁿ⁻¹

Other sequences exist:

• squares: a(n) = n², cubes, etc.
• triangular numbers: a(n) = n(n+1)/2 (number of dots in a triangle of n rows)
• fibonacci sequence: a(n) = a(n-1) + a(n-2)

Convergence of a power sequence: http://math.bu.edu/people/prakashb/Teaching/32LS10/Lectures/11-2.pdf

### Factorial

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

### Functions

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 a function - only possible if no two values of x produce the same result, ie. must be one-to-one (or limit the domain to make it so)

```Example: To find f⁻¹(x) for f(x) = 5x + 3
y = 5x + 3
y-3 = 5x
x = (y-3)/5
f⁻¹(x) = (x-3)/5   (replace y with x on the last step,
since x is input to the function, and y is output)```

Graphing an inverse: reflection of the graph about the line y=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 : HA = 0
• if n = k : HA = a/b
• if n > k : no HA
• if n = k+1 : oblique (diagonal) asymptote - approaches the line y = mx+b (from polynomial long division)
• if n > k+1 : no asymptote

look halfway down the page here: https://www.mathsisfun.com/algebra/rational-expression.html

### Interest, half life, amortization

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 = 1/lambda,  lambda = decay constant
t.5 = ln(2)/lambda
(ln(2) is close to 7 => rule of 70 for doubling returns??)```

### misc

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