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)

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 functions
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 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

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
arc-cosecant and the arc-secant function

f (x) = mx + c
Roots or zeros, x- and y-intercepts of a graph
Absolute value functions and equations
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

Definition of a polynomial or polynomial function
Division of polynomials

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
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
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
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