Functions Contents - C
      Functions
      Introduction to Functions
      Function definition, notation and terminology
         Domain, range and codomain
         Evaluating a function
         Composition of functions (a function of a function)
         Inverse function
      The graph of a function
      Functions behavior, properties and characteristic points of the graph
         Domain and range
         Roots or zero function values, x-intercepts, y-intercepts
         Increasing/decreasing intervals
         The instantaneous rate of change or the derivative
         Continuity and discontinuity
         Vertical, horizontal and oblique or slant asymptotes
         Stationary points and/or critical points
         Turning points (extremes, local or relative maximums or minimums)
         Inflection points and intervals of concavity
         Symmetry of a function, parity - odd and even functions
      Transformations of original or source function
         How some changes of a function notation affect the graph of the function
         Translations of the graph of a function
         Reflections of the graph of a function
 
      Types of functions - basic classification
         Algebraic functions and Transcendental functions
      Algebraic functions
         The polynomial function
         Rational functions
         Reciprocal function
      Transcendental functions
         Exponential and logarithmic functions, inverse functions
         Trigonometric or cyclometric functions and inverse trigonometric functions (arc-functions)
      The graphs of the elementary functions
         The graphs of algebraic and transcendental functions
      The graphs of algebraic functions
      The graphs of the polynomial functions
         The source or original polynomial function
         Translating (parallel shifting) of the polynomial function
         Coordinates of translations and their role in the polynomial expression
         The graph of the linear function
         The graph of the quadratic function
         The graphs of the cubic function
         Translated cubic functions
         The graphs of the quartic function
         The basic classification diagram for the quartic function
         The graphs of the quartic function types  4/1 . . . 6
      Coefficients of the source polynomial function are related to its derivative at x0
      Translated power function
         Translated monomial or power function graph example
      The graphs of rational functions
         Basic properties of rational functions
         The graph of the reciprocal function, equilateral or rectangular hyperbola
         Translation of the reciprocal function, linear rational function
      The graphs of transcendental functions
      Exponential and logarithmic functions are mutually inverse functions
         The graph of the exponential function
         The graph of the logarithmic function
      Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
         The graphs of the trigonometric functions and inverse trigonometric functions or arc-functions
         The graph of the sine function
         The graph of the cosine function
         The graph of the arc-sine function and the arc-cosine function
         The graph of the tangent function and the cotangent function
         The graph of the arc-tangent function and the arc-cotangent function
         The graph of the cosecant function
         The graph of the secant function
         The graph of the arc-cosecant and the arc-secant function
         Linear function
      The linear function  f (x) = mx + c
         The graph of the linear function
         Roots or zeros, x- and y-intercepts of a graph
         Properties of the linear function
      Absolute value functions and equations
         The graph of the absolute value function  f (x) = | x |
         The graph of absolute value of a linear function  f (x) = | ax+ b |
         Linear equation with absolute value, graphic solution
      Absolute value inequalities
         Solving linear inequalities with absolute value
      Equations of the straight line
         Definition of the slope of a line
         Slope-intercept form of a line
         The point-slope form of a line
         The two point form of the equation of a line
         Parallel and perpendicular lines
     Polynomial and/or Polynomial Functions and Equations
      Definition of a polynomial or polynomial function
      Division of polynomials
         Division of polynomials examples
      Factoring polynomials and solving polynomial equations by factoring
         Solving quadratic and cubic equations by factoring, examples
      Polynomial functions
         The source or the original polynomial function
         Translating or parallel shifting of the source polynomial function
         Coordinates of translations and their role in the polynomial expression
      Roots or zeros of polynomial function
         Vieta's formulas
      Graphing polynomial functions
         Zero polynomial
         Constant function
      Linear function graph
      Quadratic function and equation
         Transition of the quadratic polynomial from the general to source form and vice versa
         The zeros or the roots of the quadratic function
         Vertex (the turning point, maximum or minimum) - coordinates of translations
         Graphing the quadratic function example
      Cubic function
         Transformation of the cubic polynomial from the general to source form and vice versa
         Coordinates of the point of inflection coincide with the coordinates of translations
         The source cubic functions are odd functions
      There are three types of the cubic functions - the classification criteria diagram
         The graphs of the source cubic functions
      Translated cubic functions
         Translated cubic function, the type 1 - the tangent line at the point of inflection is horizontal
         Translated cubic function, the type 2/1 - no turning points, the tangent at inflection is a slant line
         Translated cubic function, the type 2/2 - with two turning points, the tangent at inflection is a slant line
         Graphing a cubic function the type 2/1 example
         Graphing a cubic function the type 2/2 example
      Quartic function
         Transformation of the quartic polynomial from the general to source form and vice versa
         The coordinates of translations formulas
         The values of the coefficients, a2 and a1 of the source quartic function y = a4x4 + a2x2 + a1x
      The basic classification criteria diagram
         The graphs of quartic functions and their characteristic points
         The quartic  type 1,   y - y0 = a4(x - x0)4a2 = 0 and a1 = 0
         The quartic  type 2,   y - y0 = a4(x - x0)4 + a1(x - x0)a2 = 0
         The quartic  type 3/1,   y - y0 = a4(x - x0)4 + a2(x - x0)2a1 = 0 and  a4a2 > 0
         The quartic  type 3/2,   y - y0 = a4(x - x0)4 + a2(x - x0)2a1 = 0 and  a4a2 < 0
      The graphs and classification criteria diagram for the quartics types 4/1 to 4/6,
                                           y - y0 = a4(x - x0)4 + a2(x - x0)2 + a1(x - x0)
         The zeros and the abscissas of the turning points of the source quartics types 4/1 to 4/6
         Graphing the quartic function example
 
      Transition of the polynomial expression from the general to source form and vice versa
         Deriving the coordinates of translations formulas and the coefficients of the source function
         Quadratic function
         Cubic function
         Quartic function
      Sigma notation of the polynomial
         Coefficients of the source polynomial in the form of a recursive formula
         Coefficients of the source polynomial function are related to its derivative at x0
      The general form or translatable form of the polynomial
         The coordinates of translations and the coefficients of the source function formulas
         Quadratic polynomial formulas
         Cubic polynomial formulas
         Quartic polynomial formulas
         Quintic polynomial formulas
         Sextic polynomial formulas
      Translated monomial (or power) function
         Translated monomial function graphs
         Translated sextic function example
     Rational Functions
      Basic properties of rational functions
         Vertical asymptotes of rational functions
         Horizontal asymptotes of rational functions
         The oblique or slant asymptote of rational functions
      The graph of the reciprocal function  f (x) = c/x, equilateral or rectangular hyperbola
      Translation of the reciprocal function  f (x) = (ax + b)/(cx + d), linear rational function
      Graphing rational functions examples
 
 
 
 
 
 
 
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