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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
|
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
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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
|
Graphs of 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 |
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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 |
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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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