H2 Maths Functions and Graphs | Free Notes

Study guide07 Oct 2025, 00:00 Z

H2 Maths notes on functions and graphs: key formulas, worked examples, and exam techniques for sketching, transformations, and inequalities.

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Q: What does H2 Maths Notes (JC 1-2): 1) Functions and Graphs cover?
A: Detailed guide to MOE Topic 1 -- functions, transformations, equations, and inequalities -- with worked examples and calculator workflows.

Before you begin\ Make sure your Additional Maths foundations are fresh (logarithms, surds, quadratic inequalities). Keep the graphing calculator (GC) in Exam Mode while you revise so keystrokes become automatic.

Functions define inputs; graphs show behaviour: State domain and key features.
Most marks come from restrictions, transformations, and intervals: Track what changes and what stays fixed.
A complete solution combines algebra, sketch, and GC evidence where allowed: Write the method before the final answer.

Concrete example: Before finding an inverse for a quadratic, restrict its domain so it is one-to-one. Without that restriction, the inverse is not a function.

For the full topic map, paper weightings, and official PDF link, see our H2 Maths Syllabus 2026-27 overview.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 1 coverage is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

1.1 | Functions

Core ideas

A function maps each input in its domain to exactly one output in its range.
Injective (one-to-one) functions pass the horizontal-line test and admit inverses on their natural domain.
Composition $(f \circ g)(x) = f\bigl(g(x)\bigr)$

Reviewed by

Marcus Pang·Managing Director (Maths)

Step	What to check	Example signal	Common trap
1. Start with $g$	Values of $x$ allowed in $g(x)$ .	Denominators, square roots, and logarithms inside $g$ .	Keeping an $x$ -value that makes $g$ undefined.
2. Feed into $f$	Values of $g(x)$ allowed as inputs to $f$ .	If $f(u)=\sqrt{u}$
3. State the final domain	Intersection of all restrictions from steps 1 and 2.	Solve the inequalities together.	Writing a formula with no domain statement.

Transformation	Effect on $y = f(x)$	Notes
$y = a f(x)$	Vertical stretch by $\lvert a \rvert$ (reflect if $a \lt 0$ )	Multiply y-values.
$y = f(x) + a$	Shift up by $a$	Add to the output.
$y = f(x + a)$	Shift left by $a$	Replace $x$ with $x + a$
$y = f(ax)$	Horizontal compression by $\lvert a \rvert$ (reflect if $a \lt 0$ )	Divide x-values by $a$
$y = \lvert f(x) \rvert$	Reflect negative parts above the x-axis	Ensure the range is non-negative.
$y = f(\lvert x \rvert)$	Mirror the right-half graph across the y-axis	Only the right half is independent.
$y = \frac{1}{f(x)}$	Reciprocal graph	Watch for new vertical asymptotes where $f(x) = 0$

H2 Maths Functions and Graphs | Free Notes

1.1 | Functions

Core ideas

Domain/range checks

Composition pitfalls

Composition domain checkpoint

1.2 | Graphs and Transformations

Essential graph features

Transformation cheat sheet

Parametric sketches

1.3 | Equations and Inequalities

General equation strategy

Systems of linear equations

Inequalities

Graphical method with GC

Practice Quiz

1.4 | Quick Revision Checklist

Sources

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