Units 1 & 2 (Quadratic Functions & Equations)
Focus on the structure, behavior, and algebra of quadratic functions. Explore factorization, solving techniques (including completing the square and the quadratic formula), and graphical interpretation (axis of symmetry, vertex, and intercepts). Emphasis is placed on the application of quadratics in modelling and problem solving.

Unit 2 (Algebraic Manipulation & Graphical Connections)
Develop fluency in manipulating quadratic expressions, solving equations analytically and graphically, and interpreting transformations of the basic parabola y=x2y = x^2y=x2. Reinforce connections between algebraic and graphical representations to support modelling contexts and real-world applications.

Unit 1: Foundations of Quadratic Algebra and Graphs

1.1 Structure of Quadratic Expressions

• Standard Form: ax2+bx+cax^2 + bx + cax2+bx+c; vertex form and factored form

• Terminology: leading coefficient, axis of symmetry, discriminant

Quadratic Arithmetic

• Expanding binomials and perfect squares

• Factorization techniques:

  • Common factor extraction

  • Simple trinomials (e.g. x2+bx+cx^2 + bx + cx2+bx+c)

  • Hard trinomials (e.g. ax2+bx+cax^2 + bx + cax2+bx+c, where a≠1a \neq 1a=1)

  • Completing the square

1.2 Solving Quadratic Equations

• By Factorisation

• By Completing the Square

• By Quadratic Formula: derivation from completing the square

• Discriminant Analysis: nature of roots (real, repeated, complex)

1.3 Graphing Quadratic Functions

• Key Features: vertex, y-intercept, x-intercepts, axis of symmetry

• Graph Shape: concavity based on sign of leading coefficient

• Graphing from Different Forms: standard, vertex, and factored

• Use of Technology: plotting with TI-Nspire/Casio, confirming algebraic work

1.4 Transformations of Quadratic Graphs

• Vertical & Horizontal Translations

• Reflections in x-axis and y-axis

• Vertical Stretches/Compressions

• Application in Modelling: e.g. projectile motion, revenue maximization

Unit 2: Further Techniques and Applications

2.1 Applications of Quadratics in Context

• Worded Problems: motion under gravity, area and optimisation scenarios

• Modelling with Parabolas: fitting equations to real data

2.2 Graphical and Numerical Solutions

• Using Graphs to Estimate Roots

• Understanding Parabolas that Don’t Intersect the x-axis

• Vertex Form for Maximum/Minimum Values in Applications

2.3 Systems Involving Quadratics

• Simultaneous Equations: linear-quadratic systems

• Graphical Interpretation: points of intersection

• Algebraic Solutions: substitution and elimination

2.4 Exploring Rates of Change (Prelude to Calculus)

• Average Rate over an Interval on a Quadratic Curve

• Concept of Gradient of a Chord

• Estimating Instantaneous Rate: leading into the idea of derivatives