Algebra and Functions

Indices

• Introduction
• Multiplication rule 1
• Multiplication rule 2
• Division
• Negative indices
• Fractions to negative indices
• Fractional type
• Simplifying (negative powers)
• Expressing terms in the form axⁿ
• Exam questions

Surds

• multiplication rule
• combining and simplifying
• division rule
• rationalising
• surds on a calculator
• Exam questions

Functions

• Functions - notation

Factorising

• Introduction
• HCF types
• Grouping types
  • Quadratic expressions
    • Introduction
    • HCF (Highest Common Factor) types
    • Difference of two squares type
    • Trinomials : (grouping method)
    • Trinomials : (inspection method) 1 | 2 | 3 | 4 | 5 | 6 | 7
• Miscellaneous Exercise
  • Exercise
• Exam questions

Completing the Square

• How to do it
  • Various other types : 1 | 2 | 3
  • Applications
    • Sketching graphs
• Exam questions

Quadratic Equations

• Equations
  • Solve by factorising
  • Solve by completing the square
  • Solve by formula
  • Proof of quadratic formula
  • Extension - Solving in some function of x
  • Roots and the Discriminant
    • Roots and Discriminant
  • Exam questions

Quadratic Graphs

• Sketching quadratic graphs

Polynomials

• Polynomials

Graph Transformations

• Learn these base graphs
  • Basic graphs used in transformations
• Translations
  • y = f(x ± a ) and y= f(x) ± a
    • Why does the transformation do this?
• Reflections
  • y = - f(x) and y = f(-x)
    • Why does the transformation do this?
• Stretches
  • y = a f(x)
    • Why does the transformation do this?
  • y = f (ax)
    • Why does the transformation do this?
• Exam Questions on transformations
  • Exam questions

Simultaneous Equations

• Elimination method for linear types
  • Examples :
    • 1 | 2 | 3 | 4 | 5 (method 1) | 5 (method 2)
• Substitution Method for linear and quadratic types
  • Examples :
    • 1 | 2
• Exam Questions

Inequalities

• Linear inequalities
  • introduction
  • rules for reversing the inequality sign
  • solving a linear type
  • solving a double inequality
  • Exam Questions
• Quadratic inequalities
  • Examples :
    • 1 | 2 | 3 | 4 | 5 | 6
    • Exam Questions

Sequences and series

Recurrence relationships

• Recurrence relationships
• Exam questions

Sigma notation

• Sigma notation
• Exam questions

Arithmetic sequences and series

• Arithmetic sequences and series
• Sum of the first n terms
• Exam questions

Coordinate geometry

Gradient

• Line segment
• Parallel lines
• Perpendicular lines

Lines

• Equation of a line
• Distance between two points
• Mid-point of a line segment
• Equation of a parallel line
• Equation of a perpendicular bisector

Intersection of graphs

• Two straight lines
• Parabola and a straight line
  • Nature of intersection
  • Tangent to a curve (one root)
• Hyperbola and straight line

Exam questions

Differentiation

Introduction

• The gradient function dy/dx
  • Differentiation from 1st principles
• terms of the form axⁿ
  • Extending to root types
  • Extending to fractional types
• Second Derivatives
  • The second derivative
• Exam Questions
  • Exam questions

Tangents and Normals

• Tangents and normals
• Exam Questions
  • Exam questions

Integration

Introduction

• terms of the form axⁿ
  • fractional and root type terms
• Exam questions

Equations of curves

• Finding the equation of a curve given dy/dx
  • Using f'(x) notation
• Exam questions