|
|
|
|
|
|
|
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
(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 |
|
The
graphs of the quartic function |
|
Translated
power function |
|
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
(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 |
|
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,
examples |
|
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)4,
a2
= 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)2,
a1
= 0 and
a4a2
> 0 |
|
The quartic
type
3/2, y
-
y0 = a4(x -
x0)4 + a2(x
-
x0)2,
a1
= 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 polynomial, 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 |
|
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 |
|
Vertical asymptotes of
rational functions |
|
Horizontal asymptotes
of rational functions |
|
The oblique or slant
asymptote of rational functions |
|
Graphs of rational
functions, examples |
|
Exponential and Logarithmic
Functions and Equations
|
|
Inverse functions |
|
Exponential functions
|
|
Logarithmic functions |
|
Translated logarithmic
and exponential functions |
|
Rules and properties of logarithms
|
|
Natural
logarithm and
common logarithm
conversions |
|
Changing the base
- different
logarithmic identities
|
|
Exponential and logarithmic equations
|
|
Solving exponential
and logarithmic equations |
|
Exponential equations
|
|
Logarithmic equations
|
|
Trigonometric
Functions |
|
Calculation of values of trigonometric functions
|
|
Trigonometric
reduction formulas - the reference angles |
|
Calculation of
values of trigonometric functions of an arbitrary angle x, examples |
|
Basic relationships
between trigonometric functions of the same angle |
|
Basic relationships
between trigonometric functions of the same angle shown in the
unit circle |
|
Basic relationships
between trigonometric functions of the same angle shown in the
tabular form |
|
Basic relationships
between trigonometric functions of the same angle, examples |
|
The
addition formulas and related identities |
|
The sum and difference formulas
for the trigonometric functions |
|
Deriving the addition formulas
for sine and cosine functions |
|
The
addition formulas for tangent and cotangent functions |
|
Trigonometric
functions of double angles, double angle formulas |
|
Trigonometric functions expressed by the half
angle |
|
Trigonometric
functions of double angles expressed by the tangent function |
|
Trigonometric functions expressed by the
tangent of the half angle |
|
Half angle formulas |
|
Trigonometric functions expressed by the
cosine of the double angle |
|
Sum to product and
product to sum formulas or identities |
|
Sum to product formulas
for the sine and the cosine functions |
|
Sum to product formulas
for the tangent and the cotangent functions |
|
The product to sum
formulas for the sine and cosine functions |
|
Trigonometric identities,
examples |
|
Graphs of
Trigonometric Functions |
|
The Graph of the
Sine Function f
(x) = sin
x |
|
Properties of the sine
function |
|
Domain and range |
|
Zeros of the function |
|
Extremes, maximum and
minimum of the sine function |
|
Parity and periodicity
of the sine function |
|
Behavior of the sine function |
|
The
Graph of the Cosine Function f
(x) = cos
x |
|
Properties of the
cosine function |
|
Domain and range |
|
Zeros of the function |
|
Extremes, maximum and
minimum of the cosine function |
|
Parity and periodicity
of the function |
|
Behavior of the cosine
function |
|
The Graph of the
Tangent Function f
(x) = tan
x |
|
Properties of the tangent function
|
|
Domain and range |
|
Zeros of the tangent
function |
|
Parity and periodicity
of the tangent function |
|
The
tangent function behavior
and monotony |
|
The
Graph of the Cotangent Function f
(x) = cot
x |
|
Properties of the
cotangent function |
|
Domain and range |
|
Zeros of the cotangent
function |
|
Parity and periodicity
of the cotangent function |
|
The
cotangent function behavior
and monotony |
|
The
Graph of the Function y = a
sin
(b
x + c) |
|
The function y = a
sin
x
|
|
The function y = sin
(b
x) |
|
The function y = sin
(x + c)
|
|
The function y = a
sin
(b
x
+ c) |
|
The
Graph of the Function y = a
cos
(b
x + c) |
|
The
graphs of the functions,
y = a
sin
(b
x + c) and
y = a
cos
(b
x + c),
examples |
|
Trigonometric
Functions and Inverse Trigonometric Functions |
|
Trigonometric
functions graphs and relations |
|
The
sine function and the cosine function graphs and relations |
|
The
tangent function and the cotangent function graphs and relations |
|
The
cosecant function
and the
secant function graphs and relations |
|
The
cosecant function y
= csc x |
|
The graph of the cosecant
function |
|
The
secant function
y
= sec x |
|
The graph of the secant
function |
|
Inverse Trigonometric Functions or
Arc-functions and their Graphs |
|
Inverse
functions
|
|
The
arc-sine function and the arc-cosine function |
|
The
arc-sine function y = arcsin
x or y = sin-1x |
|
The
arc-cosine function y = arccos
x or y = cos-1x
|
|
The
graph of the
arc-sine
function and the arc-cosine
function |
|
The
arc-tangent function and the
arc-cotangent function |
|
The
arc-tangent function y = arctan
x or y = tan-1x |
|
The
arc-cotangent function
y = arccot
x or y = cot-1x |
|
The
graph of the
arc-tangent function and the
arc-cotangent function |
|
The
arc-cosecant function and the arc-secant function |
|
The
graph of the
arc-cosecant and the
arc-secant function |
|
Trigonometric
Equations |
|
Basic Trigonometric
Equations |
|
The equation sin
x = a |
|
The equation
cos
x = a |
|
The equation
tan
x = a |
|
The equation cot
x = a |
|
The
Equations,
sin
(bx + c) = m, -1
<
m <
1,
cos
(bx + c) = m,
-1
<
m <
1,
|
|
tan
(bx
+ c) = m and cot
(bx
+ c) = m,
where
b,
c and
m are real
numbers. |
|
The
Equation
sin
(bx + c) = m, -1
<
m <
1, example |
|
The
Equation
cos
(bx + c) = m,
-1
<
m <
1, example |
|
The
Equation
tan
(bx
+ c) = m, example |
|
The
Equation
cot
(bx
+ c) = m, example |
|
Equations that can be
written as f
· g = 0 |
|
Trigonometric
equations of quadratic form |
|
Equations of the Type
a
· cos
x +
b
· sin
x = c |
|
Introducing an auxiliary angle
method |
|
Introducing an auxiliary angle
method example |
|
Introducing new unknown
t
= tan x/2 |
|
Introducing new unknown
t
= tan x/2
example |
|
Homogeneous Equations
in sin
x and cos
x |
|
Homogeneous equations
of first degree a×sin x
+ b×cos
x = 0 |
|
Homogeneous equations
of second degree a
sin2
x
+ b sin x
· cos
x + c
cos2
x = 0 |
|
The
Basic Strategy for Solving Trigonometric Equations |
|
Trigonometric equations
examples |
|
|
|
|
|
| |
| <=
Back to Function contents - A
Functions
contents continue => |
| |
 |
| |
| |
| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |