|
|
|
|
|
|
|
|
|
|
Real
Numbers |
The set of real
numbers |
|
Rational numbers |
|
Decimal representation of rational
numbers or fractions
|
|
Expanded form of decimal number,
decimal fractions
|
|
Terminating decimals
|
|
Recurring decimals
(Infinite decimals, period) |
|
Purely recurring decimals |
|
Mixed recurring decimals
|
|
Converting decimal number to a
fraction |
|
Converting
terminating decimal to a fraction |
|
Converting the purely recurring decimal
to a fraction |
|
Converting the mixed recurring decimal
to a fraction |
|
Exponential or
scientific notation of decimal numbers
|
|
Converting
from a number to scientific notation |
|
Converting
from scientific notation to a decimal number |
|
Irrational numbers |
|
Rationalizing
a denominator |
|
The
Real Number Line, and Relations |
|
Inequalities of real
numbers |
|
Relations,
less than and greater than |
|
Interval
definition and notation |
|
Closed and open intervals
(unbounded
intervals) |
|
Distance
and absolute value |
|
Properties of absolute
value |
|
Properties of the
Real Numbers |
|
Properties of
exponents |
|
The
rules for powers (or exponents) |
|
Simplifying an exponential expression |
|
Order
of Operations |
|
Absolute
Value Equations |
|
Solving absolute value
equations |
|
Linear Inequalities
|
|
Solving
inequalities |
|
Properties of
inequalities |
|
Examples
of solving single linear inequalities |
|
Solving
compound (double) inequalities |
|
Absolute
Value Inequalities |
|
Solving
linear inequalities with absolute value |
|
Examples
of solving
linear inequalities with absolute value |
|
Quadratic
Equation x2
= a,
a
> 0 |
|
Radicals
(roots) |
|
Square
root |
|
Properties
of square roots |
|
Adding,
subtracting, multiplying and dividing square roots |
|
Rationalizing
a denominator |
|
The
Graph of the Quadratic Function
f
(x)
= x2 |
|
Translation
of the source quadratic function in the direction of the y-axis, |
|
quadratic
function of the form f
(x)
= x2 + y0 |
|
Quadratic
equation
x2
= a,
a
>
0 |
|
The
Principal Square Root Function
- the inverse of the square of
x
(or quadratic) function |
|
Definition
of the inverse function |
|
The
graph of the principal square root function |
|
Translation
of the principal square root function in the direction of the
x-axis |
|
Radicals
(roots) and/or Exponentiation
with Fractional (rational) Exponent
|
|
Rules and properties of radicals
|
|
Rules and properties of radicals
and/or fractional exponents
|
|
Simplifying radical expressions
|
|
Operations
on radical expressions |
|
Adding, subtracting and multiplying
radicals
|
|
Rationalizing a denominator
|
|
Radicals
and/or fractional
(rational) exponents |
|
Rational Expressions
|
|
Simplification of rational expressions, reducing to lowest
terms
|
|
Addition and subtraction of rational
expressions
|
|
Multiplication and division of rational
expressions
|
|
Solving complex rational expressions
|
|
Imaginary
and Complex Numbers |
|
Imaginary numbers basic definitions |
|
Imaginary unit |
|
Complex Numbers |
|
Real and imaginary
parts |
|
The complex plane |
|
The set of all complex
numbers C |
|
Absolute value,
modulus of a complex number |
|
Complex conjugates |
|
Addition and
subtraction of complex numbers |
|
Multiplication and
division of
complex numbers |
|
Polar or
Trigonometric Notation of Complex Numbers |
|
Multiplication
and division of complex numbers in the polar form |
|
Exponentiation
and root extraction of complex numbers in the polar form |
|
Powers and roots of
complex numbers, use of de Moivre’s formulas |
|
Euler’s
formula,
relationship between trigonometric functions and the complex exponential
function |
| |
| Equations
and Inequalities, Functions |
|
|
|
Linear
equation
in one variable
|
|
Linear
equation |
|
Solving linear equations |
|
Linear
equation
in one variable, examples
|
|
Linear Inequalities
|
|
Solving inequalities |
|
Properties of
inequalities |
|
Examples
of solving single linear inequalities |
|
Solving
compound (double) inequalities |
|
Equations
with Rational Expressions |
|
Solving rational
equations |
|
Rational
equations - Linear equations |
|
Rational
Inequalities |
|
Method of solving
rational inequalities
|
|
The
graph of the translated equilateral (or rectangular) hyperbola
|
|
Rectangular
(Two-dimensional, Cartesian) Coordinate System |
|
Coordinate axes, x-axis
and y-axis,
origin, quadrants
|
|
Points in the
Coordinate plane |
|
Midpoint of a line segment
|
|
The distance formula
|
|
Dividing
a line segment in a given ratio |
|
The
area
of a triangle |
|
The
coordinates of the centroid of a triangle |
|
Lines
parallel to the axes, horizontal and vertical lines
|
|
Polar Coordinate System
|
|
Polar
and Cartesian coordinates relations |
|
Conversion from polar
to rectangular coordinates |
|
Conversion from
rectangular to polar coordinates |
|
Polar coordinates of a point |
|
Equation
of a line in polar form |
|
Lines
parallel to the axes, horizontal and vertical lines |
|
Lines
running through the origin or pole (radial lines) |
|
Polar
equation
of a line |
|
Equation
of a circle in polar form |
|
General
equation of a circle in polar coordinates |
|
Polar
equation of a circle with a center on the polar axis running
through the pole |
|
Polar
equation of a circle with a center at the pole |
|
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 |
|
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 |
|
Transcendental
Functions - 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 |
| 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 Straight Line
|
|
Slope of a line |
|
Slope-intercept form
of a line |
|
The point-slope form
of a line |
|
The two points form of
the equation of a line |
|
Parallel and
perpendicular lines |
|
Simultaneous
Linear Equations
|
|
System of two linear equations in
two unknowns (variables)
|
|
The
system has, a single solution, no solution or has infinitely many
solutions
|
|
Solving
systems of equations
|
|
Substitution and comparison
method
|
|
Addition or elimination method
|
|
System
of linear equations word problems
|
|
Solving
systems of equations graphically
|
|
Independent
equations, inconsistent equations and dependent equations
|
|
Cramer’s
rule (using determinants) to solve systems of linear equations
|
|
Solving system
of two equations in two unknowns using Cramer's rule
|
|
Solving system of three equations
in three unknowns using Cramer's rule
|
|
Method
of expanding a determinant of a rank n to cofactors |
|
Systems
of Linear Inequalities
|
|
Solving and graphing systems of
linear inequalities
|
|
Solving and graphing systems of
linear inequalities in two variables
|
|
Rational
Inequalities |
|
Method of solving
rational inequalities
|
|
The
graph of the translated equilateral (or rectangular) hyperbola
|
| Polynomial and/or Polynomial
Functions and Equations |
|
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 |
|
Source or original polynomial function |
|
Translating (parallel shifting) of the polynomial function |
|
Coordinates of translations and their role in the polynomial
expression |
|
Transformations of the polynomial function applied to the
quadratic and cubic functions |
|
Roots or zeros of
polynomial function |
|
Graphing
polynomial functions |
|
Zero polynomial |
|
Constant function |
|
Linear function |
|
Quadratic
Function
and Equation |
|
The
graph of quadratic function |
|
Translated form of
quadratic function |
|
The sum and product of
the roots |
|
Vertex
(the turning point, maximum or minimum) - coordinates of
translations |
|
Transformations
of the graph of the quadratic function |
|
Quadratic equation
word problems |
|
Cubic
Functions |
|
Graphs
of cubic functions |
|
Graphing a cubic function,
example |
|
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 |
|
Binomial
equations |
|
Solving binomial equations |
|
Equations
reducible to quadratic, bi-quadratic equations |
|
Solving equations reducible to quadratic |
|
Quadratic
inequalities
|
|
Solving
quadratic inequalities |
|
Graphic solution of
quadratic inequalities |
|
Absolute value
equations, quadratic
equations and functions |
|
Solving quadratic
equations with absolute value |
|
Graphing absolute value
functions or equations, examples |
|
Radical
or irrational equations |
|
Solving
irrational or radical equations |
|
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 |
|
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, common logarithm
|
|
Changing the base – different
logarithmic identities
|
|
Exponential and logarithmic equations
|
|
Solving exponential
and logarithmic equations |
|
Exponential equations
|
|
Logarithmic equations
|
|
|
|
|
|
Circle |
|
General equation of a circle with the center
S(p, q) - translated circle |
|
The
equation of the circle, example |
|
Equation of the circle with the
center at the origin O(0, 0)
|
|
Circle through three points
|
|
The
equation of the circle
through three points,
example |
|
Circle
and Line
|
|
Line circle intersection
|
|
Circle
and line, examples
|
|
Equation of a tangent at a point
of a circle with the center at the origin
|
|
Equation of a tangent at a point
of a translated circle
|
|
Equation of a tangent at a point
of a circle, examples
|
|
Condition of tangency - Condition for a line to be the tangent to a circle
|
|
Condition for a line to be the tangent to the circle with
the center at the
origin
O(0, 0)
|
|
Condition for a line to be the tangent to the translated circle
|
|
Tangents to a circle from a point outside the circle - use of the
tangency condition
|
|
Angle between a line and a circle
|
|
Mutual position of two circles
|
|
The radical line or the radical
axis
|
|
The pole and the polar
|
|
Angle between two circles
|
|
Ellipse
|
|
Definition and construction |
|
Eccentricity and linear
eccentricity
|
|
Constructions of an ellipse |
|
Equation of the ellipse, standard
equation of the ellipse
|
|
Major axis, minor axis, and
vertices
|
|
The focal parameter, latus rectum
|
|
The parametric equations of the ellipse |
|
Equation of a translated ellipse
|
|
Ellipse
and Line
|
|
Intersection of ellipse and line - tangency condition
|
|
Equation of the tangent at a point on the ellipse
|
|
Construction of the tangent at a point on the ellipse
|
|
Angle between the focal radii at a point of the ellipse
|
|
Tangents to an ellipse from a
point outside the ellipse - use of the tangency condition
|
|
Construction of tangents from a point outside the ellipse
|
|
Polar and pole of the ellipse
|
|
Equation of the polar of the given
point
|
|
Ellipse and line examples
|
|
Hyperbola
|
|
Definition and construction
of the hyperbola
|
|
Construction of the hyperbola
|
|
Equation of the hyperbola
|
|
Properties of the hyperbola
|
|
Examining equation of the hyperbola
|
|
Equilateral or rectangular
hyperbola
|
|
Translated hyperbola
|
|
Equation of the hyperbola in vertex form
|
|
Parametric equation of the hyperbola
|
|
Examples of hyperbola |
|
Equilateral or rectangular hyperbola with the coordinate axes as its
asymptote
|
|
Translation
of equilateral or rectangular hyperbola with the coordinate axes as its
asymptote |
|
Hyperbola
and Line
|
|
Hyperbola and line relationships
|
|
Condition for a line to be the tangent to the hyperbola -
tangency condition |
|
The equation of the tangent at the point on the hyperbola
|
|
Polar and pole of the hyperbola
|
|
Construction of the tangent at the point on the hyperbola
|
|
Construction of tangents from a point outside the hyperbola
|
|
Properties of the hyperbola
|
|
The area of a triangle which the tangent at a point on the hyperbola forms with
asymptotes |
|
The tangency point bisects the line
segment of the tangent between
asymptotes |
|
The parallels to the asymptotes through the tangency point intersect asymptotes |
|
The equation of the equilateral or rectangular hyperbola with the
coordinate axes as its
asymptotes |
|
Hyperbola and line examples
|
|
Parabola
|
|
Definition and construction
of the parabola
|
|
Construction of the parabola
|
|
Vertex form of the equation of a parabola
|
|
Transformation of the equation of a parabola
|
|
Equation of a translated parabola - the standard form
|
| The parabola whose axis
of symmetry is parallel to the y-axis |
| Equation of the parabola
written in the general form |
|
Parametric equations of the parabola
|
|
Examples
|
|
Parabola
and Line
|
|
Common points of a line and a parabola |
|
Condition for a line to be the tangent to the parabola
- tangency condition
|
|
The equation of the tangent and the normal at the point on the parabola
|
|
Properties of the parabola
|
|
Polar of the parabola
|
|
Construction of the tangent at the point on the parabola
|
|
Construction of the tangents from a point exterior to the parabola
|
|
Parabola and line, examples |
|
Conics,
a Family of Similarly Shaped Curves - Properties of Conics |
|
Conics,
a Family of Similarly Shaped Curves – Properties of Conics |
|
Dandelin's
Spheres - proof of conic sections focal properties
|
|
Proof that conic section curve is
the ellipse
|
|
Proof that conic section curve is the hyperbola
|
|
Proof that conic section curve is the parabola
|
|
Conics - a family of similarly shaped curves |
|
| Vectors in a Plane and Space |
|
|
|
Vectors
in a Plane |
|
Vectors - introduction |
|
Length, magnitude or
norm of the vector |
|
Collinear, opposite
and coplanar vectors |
|
Addition of vectors |
|
Triangle rule (law)
and parallelogram rule |
|
Zero or null vector |
|
Subtraction of vectors |
|
Scalar multiplication
or multiplication of a vector by scalar |
|
Unit vector |
|
Addition, subtraction
and scalar multiplication of vectors, examples |
|
Linear combination of
vectors |
|
Linear dependence of
vectors |
|
Vectors
and a coordinate system, Cartesian vectors |
|
Vectors in
a coordinate plane (a two-dimensional system of coordinates), Cartesian
vectors |
|
Radius vector or position vector |
|
Vector components |
|
Vectors in
a two-dimensional system, examples |
|
Vectors in
three-dimensional space in terms of Cartesian coordinates |
|
Angles of
vectors in relation to coordinate axes, directional cosines - scalar
components of a vector |
|
The unit vector of a vector |
|
Vectors in
a three-dimensional coordinate system, examples |
|
Scalar product or dot
product or inner product |
|
Orthogonality or
perpendicularity of two vectors |
|
Different positions of two vectors and the corresponding values of
the scalar product |
|
Square of magnitude of
a vector |
|
Scalar product of unit
vectors |
|
Scalar or dot product
properties |
|
Scalar product in the
coordinate system |
|
Angle between vectors
in a coordinate plane |
|
Projection of a vector in the direction of another vector,
the scalar and vector components |
|
Vector
product or cross product |
|
Vector
product |
|
Right-handed system |
|
An example for the vector product in physics |
|
The condition for two vectors to be parallel |
|
The vector products of the standard unit vectors |
|
The vector product properties |
|
The vector product in
the component form |
|
The mixed product
or the scalar triple product |
|
The mixed product
or scalar triple product definition |
|
The
mixed product properties |
|
The condition for
three vectors to be coplanar |
|
The mixed product
or scalar triple product expressed in terms of components |
|
The vector product
and the mixed product use, examples |
| |
|
|
|
| Sequences
|
|
Arithmetic sequence/progression
|
|
General term of an arithmetic
sequence
|
|
The sum of the first n terms of an arithmetic sequence |
|
The sum of the first n natural numbers |
|
Arithmetic sequences, examples
|
|
Geometric sequence/progression |
|
General term of a geometric sequence
|
|
The sum of the first n terms of a finite geometric
sequence, geometric series |
|
Geometric sequences, examples
|
|
Recursive
definition and the recursion
formula |
| Series
|
|
Finite and infinite, convergent and
divergent series |
|
Geometric series
|
|
The sum of an infinite geometric
sequence, infinite geometric series |
|
The sum of an infinite converging
geometric series, examples
|
|
Converting recurring decimals
(infinite decimals) to fraction
|
| |
|
Mathematical Induction and Binomial Theorem |
|
|
|
Mathematical
Induction
|
|
Mathematical
induction examples
|
|
The Binomial Theorem
|
|
Factorial
|
|
Binomial coefficients and
Pascal's triangle |
|
The
binomial theorem, sigma notation
and binomial expansion
algorithm |
|
The
binomial theorem and binomial expansion algorithm examples |
| |
|
Combinatorics and Probability |
|
|
|
Permutations,
Combinations and Variations |
|
Permutations
|
|
Permutations
of n objects some of which are the
same |
|
Combinations
|
|
Combinations
with repetition
|
|
Variations or permuted
combinations (permutations without repetition) |
|
Variations with repetition (or permuted
combinations with repetition) |
|
Probability
|
|
Probability
definition, terms
and notation |
|
Sample space, event |
|
Probability formula |
|
The probability of
mutually exclusive events |
|
The probability of not
mutually exclusive events |
|
The probability of two
independent events |
|
The probability of two
dependent events |
|
Calculating
probabilities, examples |
|
|
|
|
| |
| |
|
|
| |
| |
| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
