User Tools

Site Tools


urp:algebra

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
Next revision Both sides next revision
urp:algebra [2021-11-08]
nerf_herder
urp:algebra [2021-12-19]
nerf_herder [Functions]
Line 1: Line 1:
 ====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
Line 45: Line 46:
     S = n/2 (2a + (n − 1) d)     S = n/2 (2a + (n − 1) d)
  
-==== Powers & Radicals ==== +Convergence of a power sequence: http://math.bu.edu/people/prakashb/Teaching/32LS10/​Lectures/​11-2.pdf 
- +====Factorial====
-Combining powers +
-  nᵃnᵇ ​ = nᵃ⁺ᵇ +
-  nᵃ/nᵇ = nᵃ⁻ᵇ +
-  (nᵃ)ᵇ = nᵃᵇ +
-  n⁻ᵃ ​  = 1/nᵃ +
- +
-(Note: The square root sign, √, refers only to the //positive// rootUse ± to include both roots.) +
- +
-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 ==== 
- 
-Log is the 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) 
  
 ==== Functions ==== ==== Functions ====
Line 99: Line 71:
 Graphing an inverse: reflection of the graph about the line y=x Graphing an inverse: reflection of the graph about the line y=x
  
-**horizontal and vertical ​asymptotes:**+**asymptotes**
 y = (quadratic1 of x) / (quadratic2 of x) y = (quadratic1 of x) / (quadratic2 of x)
   * vertical asymptotes (VA) are when denominator goes to zero   * vertical asymptotes (VA) are when denominator goes to zero
Line 107: Line 79:
     *  if n < k : HA = 0     *  if n < k : HA = 0
     *  if n = k : HA = a/b     *  if n = k : HA = a/b
 +  * oblique (diagonal) - approaches the line y = mx+b
 +    * look halfway down the page here: https://​www.mathsisfun.com/​algebra/​rational-expression.html
 +    * if power of x in numerator > power of x in denominator:​ asymptote = 0
 +    * if power of x in numerator = power of x in denominator:​ asymptote = horizontal, not zero (ratio of largest coefficients)
 +    * if power of x in numerator = 1 less than power of x in denominator:​ asymptote = oblique (definition of line, from polynomial long division)
 +    * if power of x in numerator > 1 less than power of x in denominator:​ asymptote = none
  
 ====Interest,​ half life, amortization==== ====Interest,​ half life, amortization====
urp/algebra.txt · Last modified: 2022-02-01 by nerf_herder