H2 Maths Graphs & Transformations Formula Sheet

Study guide07 Oct 2025, 00:00 ZUpdated 17 Jul 2026

H2 Maths graphs and transformations formula sheet: translations, stretches, reflections, the modulus and reciprocal transforms, and asymptote rules - every key result on one pag...

Q: What does H2 Maths Notes (JC 1-2): 1.2) Graphs and Transformations cover?
A: Sketching workflows, asymptotes, parametric curves, and transformation chains for H2 Maths Topic 1.2.

Download: Get the H2 Maths Graphs and Transformations formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Keep a doodle pad next to your graphing calculator (GC). Sketch every curve by hand even if technology can plot it-the mark scheme rewards annotations and reasoning, not just the final picture.

A sketch is a labelled argument: Mark intercepts and asymptotes.
Transformations move points, intercepts, and asymptotes: Track one feature at a time.
Good sketches show behaviour, not decoration: Add turning points, end behaviour, and key restrictions.

Concrete example: For $y=\frac{2x-3}{x+1}$ , the vertical asymptote is $x=-1$ and the horizontal asymptote is $y=2$

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 1.2 scope is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

Every transformation rule the 9758 syllabus expects you to apply, on one screen. Track how each rule moves intercepts, asymptotes, and turning points - not just the curve shape. Worked examples for each appear in the sections below.

Transformation rules

Transformation	Effect on the graph
$y = f(x) + a$	Shift up by $a$ (shift down if $a < 0$ ).

Asymptote quick reference

Asymptote type	How to find it
Vertical	Set the denominator equal to zero; check for cancellable factors first.
Horizontal	Examine behaviour as $x \to \pm\infty$ ; compare degrees of numerator and denominator.
Oblique	Perform polynomial long division when the numerator degree exceeds the denominator by exactly 1.

Symmetry quick reference

Test	Conclusion
$f(-x) = f(x)$	Even function; graph symmetric about the y-axis.
$f(-x) = -f(x)$

Core Sketching Strategy

Intercepts: Solve $f(x) = 0$ for x-intercepts; evaluate $f(0)$ for the y-intercept.
Asymptotes: Inspect denominators for vertical asymptotes and examine end behaviour (long division) for horizontal or oblique asymptotes.
Stationary Points: Solve $f'(x) = 0$

Common Function Families

Polynomials: End behaviour determined by degree and leading coefficient; turning points up to $n - 1$ for degree $n$ .
Rational functions: Simplify to remove removable discontinuities; locate vertical, horizontal, and oblique asymptotes.
Exponentials and logarithms: $y = a^x$ always positive; $y = \ln x$

Example -- Rational sketch

Sketch $y = \dfrac{2x - 3}{x + 1}$ .

Vertical asymptote at $x = -1$ ; horizontal asymptote from division: $y = 2 - \dfrac{5}{x + 1}$ so $y = 2$

Transformation Toolkit

Apply transformations in reverse order of operations - inside-out for $x$ changes, outside-in for $y$ changes.

A graph transformation moves or reshapes the whole curve. Here y = f(x) is translated upward by b to y = f(x) + b; the same family also covers horizontal translations f(x − a), stretches, and reflections about the axes.

Expression	Effect on base graph $y = f(x)$
$y = a f(x)$	Vertical stretch by $\lvert a \rvert$

Example -- Transformation chain

Given $y = f(x)$ , sketch $y = -3 f\bigl(2(x - 1)\bigr) + 2$ .

Compress horizontally by factor $\tfrac{1}{2}$ (replace $x$ with $2x$ ).
Shift right by 1 (replace $x$ with $x - 1$

Document each step and track how intercepts/asymptotes move.

Feature-tracking checkpoint

For transformation questions, track features instead of redrawing the whole graph from scratch at every step.

Feature to track	What to write before transforming	How it changes	Trap to avoid
x-intercept	Original coordinate such as $(a, 0)$	Apply horizontal changes first, then vertical changes to the new point	Moving only the curve shape but leaving intercept labels unchanged
y-intercept	Original coordinate where $x = 0$	Recheck after horizontal transformations, because the point crossing the y-axis may change

Misconception check: a transformation moves every feature attached to the graph. If the curve shifts right by 2 but the vertical asymptote is still labelled with the old x-value, the sketch is internally inconsistent.

Reciprocal sketch checkpoint

For $y=\frac{1}{f(x)}$ , do not try to "flip the graph" visually. Convert the important features of $y=f(x)$ one by one.

Feature on

y=f(x)

What happens on

y=\frac{1}{f(x)}

Worked check: if $f(x)=x-2$ , then $y=\frac{1}{f(x)}=\frac{1}{x-2}$

Misconception check: reciprocal graphs preserve the x-coordinate of each usable point. Only the y-value is reciprocated, while zeros of the original become places where the reciprocal graph is undefined.

Parametric and Polar Sketches

When $x = g(t)$ and $y = h(t)$ , produce a parameter table and arrow the direction of increasing $t$ .
To convert to Cartesian form, eliminate $t$ algebraically or use trigonometric identities.

Example -- Parametric curve

For $x = 2 \cos t$ , $y = 3 \sin t$ , eliminating $t$ gives $\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1$

Graphs of Inverses and Moduli

Plot the line $y = x$ to reflect the original graph when sketching $y = f^{-1}(x)$ .
For $y = \lvert f(x) \rvert$

Example -- Modulus manipulation

Sketch $y = \lvert 2x - 5 \rvert - 3$ .

Sketch $y = 2x - 5$ first.
Reflect the negative portion above the axis to get $y = \lvert 2x - 5 \rvert$ .
Translate down by 3.
Locate vertices and intercepts precisely.

Calculator Support

GC TRACE verifies intercepts and intersections quickly-record the coordinates used.
dy/dx feature confirms the location of stationary points.
For parametric plots, use GC parametric mode and state the viewing window chosen.

Exam Watch Points

Always annotate asymptotes with equations and show arrows approaching them.
Use exact values for key points (fractions, radicals) rather than decimals when possible.
For modulus and reciprocal graphs, indicate excluded points clearly.
Double-check final sketches against derivative sign charts to avoid contradicting monotonicity.

Practice Quiz

Check that you can run the full sketch workflow, including modulus and transformation sequences, without referring to notes.

Quick Revision Checklist

Execute full sketching workflow: intercepts, asymptotes, stationary points, symmetry.
Apply transformation chains accurately with correct order and orientation.
Translate between parametric/polar and Cartesian forms fluently.
Explain graph features (turning points, asymptotes) in written solutions, not just diagrams.

Next steps: Stay on the H2 Maths notes hub and continue with Topic 1.3 - Equations & Inequalities to consolidate solution techniques.

Want weekly guided practice on Graphs and Transformations? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

Horizontal translation direction: $y = f(x - a)$ shifts the graph RIGHT by $a$ , not left. Students who confuse the sign lose the transformation mark and misplace every annotated intercept and asymptote.
Wrong order for combined transformations: When the expression is $y = af(bx + c) + d$

Frequently asked questions

Is there a formula sheet for H2 Maths graphs and transformations?
Yes - the "Formulas at a glance" section near the top of this page collects every transformation rule you need: translations, stretches, reflections, the modulus and reciprocal transforms, and asymptote identification rules. Note that MF27 does not list these transformation rules, so you must recall them in the exam.

Which paper does Graphs and Transformations appear in?
Topic 1.2 is Pure Mathematics content and is assessed in both Paper 1 (100 marks, pure only) and Paper 2 Section A (40 marks, pure). [1] Curve sketching and transformation questions appear in either paper, so you cannot skip revision on the basis that it is "only Paper 1 content".

Are parametric curves part of Topic 1.2?
No. Parametric equations - differentiating $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

Do I need to memorise the standard curve shapes, or are they given in the exam?
Standard curve shapes are not provided in the formula list (MF27). [1] You are expected to recall the general forms of rational graphs (including hyperbola branches and oblique asymptotes), modulus graphs, exponentials, and logarithms. The GC is permitted and can verify your sketch, but you must produce the annotated hand-drawn sketch yourself for full marks.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

Functions & graphs: Functions · Graphs & Transformations (this page) · Equations & Inequalities
Sequences & series: Sequences & Series
Vectors: Vectors
Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations
Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 1.2 Graphs and transformations (sketching, asymptotes, parametric curves, transformations, modulus/reciprocal graphs) in Section A Pure Mathematics: https://isomer-user-content.by.gov.sg/334/f27e37f7-f0ec-4a35-b1e8-3a5e88ae2f81/9758_y26_sy.pdf