Units 1 & 2 (Methods, Transformations)
Focus on understanding algebraic and geometric transformations of functions, including linear, quadratic, and other polynomial forms. Emphasis is placed on domain/range, graphical representations, and algebraic rules governing transformations. These units also cover foundational skills for analyzing functions and modelling real-world scenarios.

Unit 1: Algebraic and Graphical Foundations

1.1 Introduction to Functions and Notation

• Definition of a function

• Function notation and evaluation

• Domain and range (set notation and inequalities)

1.2 Linear and Quadratic Functions

• Forms of linear functions: gradient-intercept, point-slope

• Quadratics in standard, factored, and vertex form

• Techniques for graphing from algebraic form

• Axis of symmetry, vertex, and intercepts

1.3 Transformations of the Plane

• Translations: horizontal and vertical shifts

• Reflections: across the x-axis and y-axis

• Dilations: vertical and horizontal stretches/compressions

• Combined (composite) transformations

• Impact on graphs and equations

• Identifying transformations from graphs and equations

1.4 Modelling with Functions

• Using linear and quadratic models in applied contexts

• Interpretation of parameters (slope, intercept, vertex)

• Fitting models to data (by hand and with technology)

Unit 2: Further Transformations and Graphical Reasoning

2.1 General Function Transformations

• Algebraic rules for y = af(b(x + c)) + d

• Interpretation of parameters a, b, c, and d

• Application to polynomial and other non-linear functions

• Transformation of piecewise and absolute value functions

2.2 Inverse and Composite Functions

• Definition of inverse functions and notation f⁻¹(x)

• Finding inverses graphically and algebraically (when they exist)

• Domain and range considerations

• Composite functions (f ∘ g)(x), evaluation and domain constraints

2.3 Graph Sketching and Analysis

• Determining function behavior under transformations

• Symmetry, intercepts, and turning points

• Sketching without technology

• Use of CAS/graphing calculators for verification and modelling

2.4 Real-World Modelling and Problem Solving

• Applying transformations to contextual problems

• Rewriting equations to interpret features (e.g. maximum profit, minimum cost)

• Exploring multiple transformations in modelling scenarios

• Introduction to piecewise models for hybrid systems