Algebra

• Coordinate plane


  Also called the Cartesian coordinate system, formed by two intersecting lines (x-axis and y-axis), introduced by Rene Descartes
• Ordered pair

  A pair of numbers used to locate a point in the Cartesian plane, the x coordinate is called abscissa, the y coordinate is called ordinate
• Relation


  A law or rule that defines the relationship between the dependent and independent variable, can be many-to-many or one-to-many
• Function


  A special type of relation where one x-input is paired with one y-output, can be one-to-one or many-to-one
• Domain of a function

  Set of x values the function can take
• Range of a function

  Set of y values the function can take
• Solving for x in terms of y

  x left side: y right side
• Interval notation

  Parenthesis = open = excluded, Bracket = close = included
• Operations on functions

  1. Evaluation (substituting an x-value)

  2. Addition (adding the function's equation together)

  3. Subtraction (subtracting the function's equation together)

  4. Multiplication (multiplying the function's equation together)

  5. Division (dividing the function's equation together)
• Distance formula

  d=√((x2-x1)2 + (y2-y1)2)
• Quadratic function


  • Second-degree equation,
• Converting standard form to vertex form

  Add and subtract b^2/2 to the right side or simply complete the square
• Rational function


  • Function written in p(x)/q(x) where p and q are polynomial functions and q is not equal to zero
• Hole of a rational function

  Shared factor of the numerator and denominator which causes cancellation
• Absolute value function


  • Consists of absolute value bars, Range is always non-negative,
• Exponential function


  • Function written in f(x) = b^x where b > 0, and b cannot be equal to one, x cannot be negative -1 < x < 1, the x-intercept is one,
• Exponential growth

  y increases slowly at first then rapidly, the rapid growth is called exponential increase, y = a(1 + r)^x
• Exponential decay

  y decreases rapidly at first then slowly, the rapid decay is called exponential decrease, y = a(1 - r)^x
• Natural exponential function

  Written as f(x) = e^x where e = 2.71828
• Logarithm


  The power or exponent which a base must be raised to yield a certain number, logb a (the logarithm of a with base b) where b is not equal to 1 and a = b^c
• Properties of logarithm

  • logb 1 = 1,
  • logb b=1
  • logb b^x=x
  • logb x= logb y then x = y
  • loga c= logb c + logb a
  • loga b=1/logb a
• Laws of logarithm

  • Law#1: logb (xy) = logb x + logb y,
  • Law#2: logb (x/y) = logb x - logb y,
  • Law#3: logb (x^n) = n logb x
• Antilogarithm


  If log x = y, then x is called the antilogarithm of y, x = antilog y
• Logarithmic function


  • Written as f(x) = logb x, x-intercept is one, no y-intercept, if b > 1, the function is increasing and if b < 1, the function is decreasing, inverse of exponential function,
• Radical function


  • A function that contains a radical with x being the radicand,
• Piecewise function


  A combination of one or more functions whose domain is separated into intervals, the absolute value function is a piecewise function
• Floor function


  A piecewise function with an infinite number of pieces
• LOGARITHMIC FUNCTION

  Standard Form: f(x) = a logb (x - c) + d where

  a = stretching or shrinking,

  d = vertical shift,

  c = horizontal shift,

  Y-intercept = none,

  Horizontal asymptote: none,

  Vertical asymptote: y-axis,

  Domain: (0, ∞), Range: all real numbers
• RADICAL FUNCTION

  Standard Form: f(x) = a√(x - b) + c where

  b = origin of the graph,

  a = orientation and shrinking/stretching,

  c = horizontal shift,

  Y-intercept = none,

  Horizontal asymptote: none,

  Vertical asymptote: y-axis,

  Domain: [0, ∞), Range: all real numbers
• CHANGE OF BASE FORMULA

  logb x =loga x / loga b
• EXPONENTIAL FUNCTION

  Standard Form: f(x) = a * b^(x-c) + d where

  c = horizontal shift,

  d = vertical shift,

  Y-intercept = (0, b^c+d),

  Horizontal asymptote: x-axis,

  Domain: (-∞, ∞), Range: (d, ∞)
• ABSOLUTE VALUE FUNCTION

  Standard Equation: f(x) = a|x - h| + k where

  a = stretch of graph vertically,

  h = horizontal shift,

  k = vertical shift
• Linear function


  • First-degree algebraic equation,
• SET

  collection of well-defined objects or elements.

  • each element in a set is called a subset
• SET NOTATION

  consists of basic symbols that indicate operations across sets

  • all elements except the numbers can be considered notation
• Universal Set - all elements of all the sets

  Null set - a set that does not have any elements or an empty set

  Complement of A set - all elements of the universal set except A

  Subset - represents a set formed from taking subsets of another set.

  Belongs to - used if a particular element belongs to a set 
• Union of a sets - combines all elements into a single set

  Intersection of sets - takes common elements of sets

  • Difference of sets - removes the common elements

  Delta - gives the elements remaining after removing the common elements from the union of two sets
• Types of sets

  • Singleton/unit sets
  • Finite sets
  • Infinite sets
  • Null sets
  • Equal sets
  • Unequal sets
  • Equivalent sets
  • Overlapping sets
  • Disjoint sets
  • Superset
  • Power set
• Singleton/unit sets


  Only has one element
• Finite sets


  Set with countable number of elements