Units 1 & 2 (Methods – Gallery of Graphs)
Focus on understanding a wide variety of function families, their key features and transformations, techniques for accurate sketching (including by hand and with technology), composition and inversion of functions, and the earliest connections between graph behaviour and rate of change.

Unit 1: Graphical Foundations and Function Families
1.1 Function Types and Their Features

• Polynomial (quadratic, cubic), power, exponential, logarithmic, reciprocal, and simple trigonometric functions

• Domain, range, intercepts, asymptotes (where applicable), periodicity (for trig) and end-behaviour

1.2 Transformations of Graphs

• Translations (horizontal, vertical)

• Reflections in the x- and y-axes

• Stretches and compressions (vertical and horizontal scaling)

• Composite transformations and tracking key points

1.3 Combining, Composing and Inverting Functions

• Sum, difference, product and quotient of functions and their graphical impact

• Composition (f∘g)(x)(f\circ g)(x)(f∘g)(x) and domain restrictions

• Finding and sketching inverses f−1(x)f^{-1}(x)f−1(x), including horizontal–vertical swap and reflection

1.4 Graphing Techniques and Technology Integration

• Plotting by hand using key features and “shape templates” for each family

• Using CAS or graphing calculators to verify and explore more complex cases

Unit 2: Advanced Graphical Analysis & Prelude to Calculus
2.1 Rational and Transcendental Function Sketching

• Sketching rational functions: identifying holes, vertical/horizontal/oblique asymptotes

• Exponential and logarithmic curves: long-term growth/decay, translations, reflections

• Higher-frequency and phase-shifted trigonometric graphs (sine, cosine, tangent)

2.2 Rates of Change from Graphs

• Average rate of change as gradient of a secant line between two points

• Instantaneous rate of change as gradient of a tangent line (graphical approximation)

• Interpreting rate of change in context (e.g. population growth, decay processes)

2.3 Introduction to the Derivative Concept

• Intuitive “limit of secants” definition on a graph

• Sketching a graph of derivative f′(x)f′(x)f′(x) alongside f(x)f(x)f(x) based on increasing/decreasing and turning points

2.4 Area under the Curve (Introductory)

• Riemann sums and estimating area from a graph

• Connection between accumulation (area) and rate of change (prelude to the Fundamental Theorem)