|
|
|
|
|
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The set of real
numbers |
|
Natural numbers |
|
Whole numbers |
|
Integers |
|
Rational numbers |
|
Irrational numbers |
|
Real numbers |
|
Operations on real
numbers |
|
Properties of real
numbers |
|
Decimal Numbers |
|
Decimal representation
of rational numbers or fractions |
|
Expanded form of decimal number,
decimal fractions
|
|
Terminating decimals |
|
Recurring decimals
(Infinite decimals)
|
|
Purely recurring decimals
|
|
Mixed recurring
decimals
|
|
Converting decimal
number to a fraction
|
|
Terminating decimals
to fraction |
|
Purely
recurring decimals to fraction |
|
Mixed recurring
decimals to fraction
|
|
Exponential or
scientific notation of decimal numbers |
|
Order
of operations |
|
Properties of
exponents |
|
The
rules for powers (or exponents) |
|
Simplifying an exponential expression |
|
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 |
|
Algebraic Expressions
|
|
Algebraic expressions - preliminaries |
|
Evaluating algebraic
expressions |
|
Simplifying algebraic
expressions |
|
Expanding
algebraic expression by removing parentheses (i.e. brackets) |
|
The square of a
binomial - the perfect square trinomial |
|
The square of a trinomial |
|
The cube of binomial |
|
The binomial expansion
algorithm |
|
The difference of two
squares, multiplying |
|
The difference of two
squares, factoring |
|
The difference of two
cubes |
|
The sum of two cubes |
|
Factoring and expanding algebraic expressions, rules for transforming algebraic expressions |
|
Factoring
algebraic
expressions - methods, the
greatest common factor |
|
Factoring by grouping |
|
Perfect
square trinomials - the
square of a binomial |
|
The difference of
two
squares |
|
Factoring
quadratic trinomials |
|
The sum and difference of cubes |
|
Factoring polynomials |
|
Using a variety of methods including combinations of the above to factorize
algebraic expressions |
|
Linear Equations
in One Variable
|
|
Solving linear
equations |
|
Word
problems that lead to simple linear equations |
|
Number problems |
|
Age
problems
|
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Mixture
problems |
|
Work problems |
|
Time and travel (distance) problems |
|
Geometry word problems |
|
Miscellaneous word
problems |
|
Variable expressions and formula problems |
|
Solving
a formula for a specified variable, transposition of a
formula (changing the subject of
a formula) |
|
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 |
|
Polynomial
and/or Polynomial function
|
|
Definition
of a polynomial or polynomial
function |
|
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 |
|
Zeros of
a polynomial function |
|
Addition and subtraction of
polynomials
|
|
Multiplication
of polynomials |
|
Division
of polynomials |
|
Factoring
polynomials and solving
polynomial equations by factoring |
|
Graphing
polynomial functions |
|
Zero polynomial |
|
Constant function |
|
Linear function |
|
Quadratic function |
|
Transformations
of the graph of the quadratic function |
|
Cubic functions |
|
Graphs
of cubic functions |
|
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
|
|
Graphing
Linear Equation, Linear Function (First Degree Polynomial)
|
|
Linear
function f(x) = mx + c |
| Slope-intercept form of a line
|
| Slope
or gradient, y-intercept and x-intercept or zero of a function |
| The graph of
the linear function |
|
Properties
of the linear function
|
|
Lines
parallel to the axes, horizontal and vertical lines
|
|
The
point-slope form of a line
|
|
Parallel
and perpendicular lines
|
|
General
form of the equation of a line
|
|
The two point form of the equation of a line
|
|
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 the determinant) 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 |
|
Equations
with Rational Expressions |
|
Solving rational
equations |
|
Rational
equations - Linear equations |
|
Rational
equations - Quadratic equations |
|
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 |
|
Interest
Calculations |
|
Simple interest |
|
Amount of interest,
principal (initial amount), interest rate and amount after n
years |
|
Compound interest |
|
Periodic compounding |
|
Continuous compounding |
|
Exponential
growth and decay, application of the natural exponential
function |
|
Quadratic Equations and Quadratic
Function |
|
Quadratic equation |
|
Solving quadratic equations by completing the
square, the quadratic formula |
|
Solving quadratic equations by factoring, Vieta’s formula |
|
Quadratic function or
the second-degree polynomial |
|
Translated form of
quadratic function |
|
Vertex (maximum/minimum) - coordinates of translation
|
|
Roots or zeros of the
function, axis of symmetry and y-intercept |
|
Graphing a quadratic
function |
|
Transformations
of the graph of the quadratic function |
|
Quadratic Inequalities
|
|
Solving quadratic inequality graphically
|
|
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
|
|
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
graph of a function |
|
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 |
|
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 the algebraic and transcendental functions |
|
Transcendental
functions - The graphs of transcendental 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 |
|
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
|
|
Exponential equations
|
|
Logarithmic equations
|
|
|
|
|
|
Circle |
|
General
equation of a circle with the center S(p, q) - translated circle |
|
Equation of the circle with the
center at the origin O(0, 0)
|
|
Circle through three points
|
|
Circle
and Line
|
|
Line circle intersection
|
|
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
|
|
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 equations 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 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 |
| Equations 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 |
| |
|
|
|
| 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
|
|
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 -
Combinatorial
Analysis |
|
|
|
Permutations,
Combinations and Variations |
|
Permutations
|
|
Permutations
of n objects some of which are the
same |
|
Combinations
|
|
Combinations
|
|
Combinations
with repetition
|
|
Variations |
|
Variations or permuted
combinations (permutations without repetition) |
|
Variations with repetition (or permuted
combinations with repetition) |
|
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
