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urp:algebra [2021-11-01]
nerf_herder [Powers & Radicals]
urp:algebra [2022-02-01] (current)
nerf_herder [Arithmetic Sequences]
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 ====Algebra & Pre-Calc==== ====Algebra & Pre-Calc====
 +Related pages:
   * [[Graphing|Graphing circles, ellipses, parabolas, hyperbolas]]   * [[Graphing|Graphing circles, ellipses, parabolas, hyperbolas]]
   * [[poly|Polynomials,​ quadratic formula and completing the square]]   * [[poly|Polynomials,​ quadratic formula and completing the square]]
 +  * [[power|Powers,​ radicals (roots) & logs]]
 +
 +**quadratic formula**: x = (-b ±√(b²-4ac)) / 2a
  
-**quadratic formula**: x = (-b +√(b²-4ac)) / 2a+**PEMDAS/​BODMAS ​** - order of operationsParentheses/​Brackets,​ Exponents/​Order,​ Multiply-Divide, Add-Subtract
  
 ==== System of equations==== ==== System of equations====
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 https://​courses.lumenlearning.com/​suny-osalgebratrig/​chapter/​parametric-equations/​ https://​courses.lumenlearning.com/​suny-osalgebratrig/​chapter/​parametric-equations/​
  
-==== Arithmetic Sequences ====+==== 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 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
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     S = n/2 (2a + (n − 1) d)     S = n/2 (2a + (n − 1) d)
  
-==== Powers & Radicals ====+**Geometric sequence** terms are found by multiplying the previous term by a constant, such as 2, 4, 8, 16 ...
  
-Combining powers +a(n) = arⁿ¹
-  nᵃnᵇ ​ = nᵃ⁺ᵇ +
-  nᵃ/nᵇ = nᵃ⁻ᵇ +
-  ​(n)ᵇ nᵃᵇ +
-  nᵃ   = 1/nᵃ+
  
-Factorial:+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)!   n! = n * (n-1)!
   0! = 1   0! = 1
   7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5    7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 
- 
- 
-==== Logs ==== 
- 
-inverse of a power 
-  x = b^y,   y = logb(x) ​ (b = base, 10 by default) 
-  eg. 2^3 = 8, log2(8) = 3 
-  log(1) = 0 (for any base), log(x) is undefined for x =< 0 
-  logb(x) = logk(x) / logk(b) 
-          = log(x) / log(b) ​ for k=10 
-          = ln(x) / ln(b)    for k=e  (e = Euler'​s number, 2.718..) 
-  ln(e) = 1, log(10) = 1 
-  ln(x)=logex. 
-  e^k = c, and k = ln(c) => e^ln(c) = c 
- 
-  a^loga(x) = x   ​(power and log are inverses, cancel each other out) 
-  loga(a^x) = x   (same reason) 
-  product rule: log(ab) = log(a) + log(b) 
-  quotient rule: log(a/b) = log(a) - log(b) 
-  power rule: log(a^b) = b*log(a) 
  
 ==== Functions ==== ==== 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 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:​ +**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) 
-y = (quadratic1 ​of x) / (quadratic2 ​of x+  ​Example:​ To find f⁻¹(xfor f(x) = 5x + 3 
-vertical asymptotes ​(VA) are when denominator goes to zero +     ​y = 5x + 3 
-horizontal asymptotes ​(HAis when goes to infinity, ​ +     y-3 = 5x 
-  look at highest order of x in numerator and denominator:​ +     (y-3)/5 
-   y = ax^n / bx^m +     f⁻¹(x) ​(x-3)/5   ​(replace y with x on the last step,  
-     if n > m : no HA +                         since x is input to the function, and y is output)
-     if n < m : HA 0 +
-     if n m : HA = a/b+
  
-====misc==== +Graphing an inverse: reflection of the graph about the line y=
-abs. value of imaginary number + 
-  |a + bi| sqrt(a^2 + b^2)+**asymptotes** 
 +(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 : HA 
 +    *  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
  
-rational numbers: can be expressed as a fraction of two integers +look halfway down the page here: https://​www.mathsisfun.com/​algebra/​rational-expression.html
-  ​the decimal expansion either terminates or repeats +
-irrational includes square roots, pi, etc+
  
 +====Interest,​ half life, amortization====
 Annual rate, continuous rate of growth: Annual rate, continuous rate of growth:
-  Y = a*b^t+  Y = a*b
   a = principle amount, b = annual growth, t = time (years)   a = principle amount, b = annual growth, t = time (years)
-  y = a*e^(kt) = a*(e^k)^t, and k = continuous rate of growth  ​+  y = a*eᵏᵗ ​= a*(e), and k = continuous rate of growth  ​
      k = ln(b)      k = ln(b)
  
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   (ln(2) is close to 7 => rule of 70 for doubling returns??)   (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
  
 ==== See also ==== ==== See also ====
urp/algebra.1635734890.txt.gz · Last modified: 2021-11-01 by nerf_herder