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