H2 Maths graphs and transformations formula sheet: translations, stretches, reflections, the modulus and reciprocal transforms, and asymptote rules - every key result on one pag...
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.
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).
y=f(x+a)
Shift left by a (shift right if a<0).
y=af(x)
Vertical stretch by factor ∣a∣; reflection in the x-axis if a<0.
y=f(ax)
Horizontal compression by factor ∣a∣; reflection in the y-axis if a<0.
y=−f(x)
Reflection in the x-axis.
y=f(−x)
Reflection in the y-axis.
y=∣f(x)∣
Reflect sections below the x-axis upward; keep sections above unchanged.
y=f(∣x∣)
Discard the left half of the original graph; mirror the right half in the y-axis.
y=f(x)1
Zeros of f become vertical asymptotes; horizontal asymptote y=k
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→±∞; 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)
Odd function; graph has rotational symmetry about the origin.
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, determine nature via f′′(x) or sign chart.
Symmetry: Test f(−x)=f(x) (even) or f(−x)=−f(x) (odd). For rational graphs, look for rotational symmetry.
Sign Diagram: Tabulate sign changes across critical points to understand where the graph lies above or below the x-axis.
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=ax always positive; y=lnx defined for x>0.
Trigonometric graphs: Document period, amplitude, phase shift, and vertical displacement before sketching.
Example -- Rational sketch
Sketch y=x+12x−3.
Vertical asymptote at x=−1; horizontal asymptote from division: y=2−x+15 so y=2.
Intercepts: x-intercept at x=23, y-intercept y=−3.
Sketch the hyperbola considering symmetry about the intersection of asymptotes.
Transformation Toolkit
Apply transformations in reverse order of operations - inside-out for x changes, outside-in for y changes.
Expression
Effect on base graph y=f(x)
y=af(x)
Vertical stretch by ∣a∣ and reflection if a<0.
y=f(x)+b
Shift up by b.
y=f(x−h)
Shift right by h.
y=f(ax)
Horizontal compression by ∣a∣.
y=f(∣x∣)
Mirror the right-half across the y-axis.
y=∣f(x)∣
Reflect sections below the x-axis upward.
y=f(x)1
Reciprocal curve with vertical asymptotes where f(x)=0.
Example -- Transformation chain
Given y=f(x), sketch y=−3f(2(x−1))+2.
Compress horizontally by factor 21 (replace x with 2x).
Shift right by 1 (replace x with x−1 inside the bracket).
Stretch vertically by 3 and reflect in x-axis (multiply by −3).
Shift up by 2.
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
Assuming every old y-intercept remains on the y-axis
Vertical asymptote
Original equation such as x=k
Apply the same horizontal transformation as the x-coordinate
Transforming it like a y-value
Horizontal asymptote
Original equation such as y=m
Apply the same vertical transformation as the y-coordinate
Transforming it like an x-value
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=f(x)1, 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=f(x)1
What to mark first
Common trap
Zero at x=a
Vertical asymptote x=a
Draw the new vertical asymptote before shaping the branches
Plotting (a,0) again on the reciprocal graph
Horizontal asymptote y=k, where k=0
Horizontal asymptote y=k1
Point (p,q), where q=0
Point (p,q1)
Sign of f(x)
Same sign for the reciprocal
Use the old sign diagram to decide whether each branch is above or below the x-axis
Drawing a negative branch above the axis after inversion
Worked check: if f(x)=x−2, then y=f(x)1=x−21 has vertical asymptote x=2, not an x-intercept at x=2. Since f(3)=1, the reciprocal graph still passes through (3,1). Since f(1)=−1, it also passes through (1,−1).
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.
For polar coordinates (r,θ), remember x=rcosθ, y=rsinθ. Sketch over one full period.
Example -- Parametric curve
For x=2cost, y=3sint, eliminating t gives 4x2+9y2=1, an ellipse centred at the origin.
Graphs of Inverses and Moduli
Plot the line y=x to reflect the original graph when sketching y=f−1(x).
For y=∣f(x)∣, copy the portions above the axis and reflect negative sections.
For y=f(∣x∣), ignore the left-half of the original and mirror the right-half.
Example -- Modulus manipulation
Sketch y=∣2x−5∣−3.
Sketch y=2x−5 first.
Reflect the negative portion above the axis to get y=∣2x−5∣.
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.
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, the horizontal shift by −c/b must be applied AFTER the horizontal stretch by 1/b, not before. Reversing the order shifts intercepts to wrong positions.
Forgetting to transform asymptotes: After applying a transformation such as y=f(x−2)+3, the vertical asymptote moves from x=k to x=k+2
Unlabelled key features: The mark scheme for graph sketching questions explicitly awards marks for labelling intercepts with coordinates, stating asymptote equations, and marking stationary points. A correct shape without annotations scores zero for those marks.
Confusing y=f(∣x∣) with y=∣f(x)∣: For y=f(∣x∣)
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 dxdy=dx/dtdy/dt and finding tangents/normals - fall under Topic 5.1 Differentiation. [1] Topic 1.2 may ask you to sketch a parametric curve by producing a parameter table and converting to Cartesian form, but the calculus of parametric curves is examined separately under Topic 5.
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:
