Top 200 Secondary 4 Additional Mathematics Vocabulary

Secondary 4 Additional Mathematics vocabulary makes algebra and calculus easier to reason about. This guide explains 200 terms in ten connected chapters, from quadratic equations and surds to functions, logarithms, trigonometry, differentiation and integration. The aim is not to memorise unusual words; it is to understand what each mathematical instruction requires and why each step works.

For the 2026 GCE O-Level, Additional Mathematics is subject code 4049. The 2027 Singapore-Cambridge SEC G3 subject code is K341. It is distinct from G3 Mathematics K310 and from other subject-level routes. SEAB’s official 2027 syllabus organises G3 Additional Mathematics into algebra, geometry and trigonometry, and calculus. These 200 words form an independent study glossary, not a compulsory SEAB word list; some entries clarify related mathematical language beyond the core wording.

Quick guide:

Get meanings for 200 mathematical words and expressions at the top of the article.
Learn why the discriminant, domain, composite order and derivative signs change the answer.
Follow worked examples for quadratic functions, logarithms, trig equations and calculus.
Use a four-week vocabulary-plus-working revision plan and official syllabus links.

Top 200 Secondary 4 Additional Mathematics Vocabulary Terms

Use these in numerical and algebraic context. A student who can explain a term, draw the corresponding graph and solve a new problem owns the vocabulary more securely than a student who can recite a definition alone.

1. Quadratic functions, equations and inequalities — terms 1–20

Quadratic terminology tells you whether a graph crosses, touches or misses an axis. Interpret the discriminant before assuming a pair of real roots.

quadratic function — A function whose highest power of the variable is two.
quadratic equation — An equation reducible to a polynomial of degree two.
leading coefficient — The multiplier of the highest-degree term.
constant term — A term containing no variable.
discriminant — The quantity b² − 4ac for ax² + bx + c.
real roots — Solutions expressible as real numbers.
repeated root — A solution occurring more than once algebraically.
distinct roots — Different solution values of an equation.
completing the square — Rewriting a quadratic in vertex form.
vertex form — An expression of the form a(x − h)² + k.
turning point — A point where a curve changes from rising to falling or conversely.
axis of symmetry — A line dividing a quadratic graph into mirror halves.
maximum value — The greatest possible output on the specified domain.
minimum value — The least possible output on the specified domain.
quadratic inequality — A second-degree expression compared using an inequality.
solution interval — A continuous range of permitted solution values.
factor form — A product of factors equivalent to an expression.
simultaneous equations — Equations solved together for shared values.
substitution method — Solving by replacing a variable using another equation.
graph intersection — A point belonging to two graphs at once.

2. Surds, polynomials and factor theorems — terms 21–40

Exact surds avoid rounding away important information. Remainder and factor theorems give short tests for polynomial behaviour without a full graph.

surd — An irrational root expression retained in exact form.
radical — A root symbol and the expression beneath it.
radicand — The expression inside a root.
rationalise — Remove irrational expressions from a denominator.
conjugate — A paired expression whose sign difference simplifies a product.
exact form — An answer retained without decimal approximation.
polynomial — A sum of terms with nonnegative integer powers.
polynomial degree — The highest variable power with a nonzero coefficient in a polynomial.
cubic polynomial — A polynomial of degree three.
polynomial division — Division of one polynomial by another.
quotient — The result of a division before any remainder.
remainder — The leftover part after division.
factor theorem — A criterion linking f(a) = 0 to the factor (x − a).
remainder theorem — A rule stating that division by (x − a) leaves remainder f(a).
linear factor — A polynomial factor of degree one.
factorisation — Writing a polynomial as a product of factors.
root of polynomial — A value giving polynomial output zero.
algebraic identity — An equality true for all permitted variables.
difference of cubes — The expression a³ − b³ and its factorisation pattern.
sum of cubes — The expression a³ + b³ and its factorisation pattern.

3. Binomial expansions and algebraic reasoning — terms 41–60

Binomial algebra requires choosing the appropriate term, not just copying Pascal’s triangle. Pay attention to coefficients and powers.

binomial — An algebraic expression with two terms.
binomial theorem — A rule for expanding powers of a binomial.
binomial coefficient — A numerical multiplier of a term in binomial expansion.
factorial — The product of positive integers from one through n.
combination — A selection where order does not matter.
general term — A formula describing an arbitrary term in a sequence or expansion.
power — An exponent indicating repeated multiplication or its extension.
index law — A valid rule for manipulating powers.
coefficient of x — The multiplier of the term containing x to the specified power.
constant coefficient — The coefficient of the term with no variable.
term position — The place of a term in a specified expansion.
expansion — Writing a product as a sum of terms.
simplification — Rewriting a result into an equivalent clearer form.
rearrangement — Changing an equation’s form while preserving validity.
common factor — A factor shared among several terms.
equivalent expression — A rewritten form equal for every allowed input.
parameter — A symbol treated as fixed within a stated model.
condition — A requirement that must be met for a result.
restriction — A limit on the permitted values of variables.
counterexample — One valid case showing a general claim is false.

4. Functions, composite functions and inverse functions — terms 61–80

Function order matters: applying g after f is usually different from applying f after g. Inverses require one-to-one behaviour on the chosen domain.

function — An assignment giving each input exactly one output.
domain — The allowed set of input values.
range — The set of values actually produced.
codomain — The set into which a function maps its outputs.
mapping — The rule associating permitted inputs with outputs.
function notation — Writing a function’s output as f(x).
image — The output associated with a given input.
preimage — An input producing a specified output.
one-to-one function — A function where distinct inputs give distinct outputs.
many-to-one function — A function where several inputs may share one output.
composite function — A function formed by applying one function after another.
composition order — The sequence in which nested functions are applied.
inverse function — A function reversing a one-to-one mapping on its chosen domain.
domain restriction — Limiting inputs to make a function suitable for an operation.
function graph — The set of coordinate points representing a mapping.
horizontal line test — A graphical test for whether a function is one-to-one.
vertical line test — A graphical test that a relation represents a function.
transformation — A systematic change of a graph’s position or shape.
translation of graph — A shift applied to all graph points.
reflection of graph — A mirror transformation of a graph across an axis or line.

5. Indices, exponentials and logarithms — terms 81–100

Logs are the inverse language of powers. A real logarithm cannot have a nonpositive argument, so domain restrictions matter before solving.

exponential function — A function with the variable appearing in an exponent.
logarithm — The power to which a positive base must be raised to obtain a number.
logarithmic function — The inverse of an exponential function on suitable domains.
base of logarithm — The positive number other than one being raised to a power.
common logarithm — A logarithm with base ten.
natural logarithm — A logarithm with base e.
Euler’s number — The constant e, approximately 2.71828.
exponential growth — Increase proportional to current value in a continuous idealised model.
exponential decay — Decrease proportional to current value in a continuous idealised model.
logarithm law — A valid identity used to combine or split logarithms.
change of base — Converting a logarithm into an equivalent logarithm of another base.
inverse operation — An operation that undoes a preceding operation.
logarithmic domain — Positive permitted arguments for real-valued logarithms.
exponential asymptote — A line approached by an exponential graph under relevant conditions.
logarithmic asymptote — A line approached by a logarithmic graph under relevant conditions.
growth constant — A parameter controlling exponential increase.
decay constant — A parameter controlling exponential decrease.
initial value — The value of a model at its stated starting point.
doubling time — Time for an exponentially growing quantity to double.
half-life — Time for an exponentially decaying quantity to halve.

6. Coordinate geometry, straight lines and curves — terms 101–120

Coordinates connect algebra to geometry: derive a line from enough information, check intersection points and avoid confusing a tangent with a normal.

Cartesian coordinates — Ordered numbers locating a point on a plane.
coordinate plane — A plane with perpendicular numerical axes.
origin — The intersection of the coordinate axes.
gradient — Change in vertical coordinate per unit horizontal change.
slope-intercept form — The line equation y = mx + c.
point-slope form — A line equation based on one point and its gradient.
two-point formula — A method finding a line’s gradient from two points.
parallel lines — Lines in one plane that do not meet.
perpendicular lines — Lines meeting at ninety degrees.
normal — A line perpendicular to a curve’s tangent at a point.
tangent — A line sharing a curve’s instantaneous direction at a point.
midpoint — The point halfway between the ends of a segment.
distance formula — An expression calculating separation between coordinate points.
collinear — Lying on the same straight line.
locus — A set of points satisfying a geometric condition.
circle equation — An equation expressing the points at a fixed distance from a centre.
radius — The distance from a circle’s centre to a point on it.
centre — The point equidistant from all points on a circle.
intersection point — A coordinate satisfying two geometric relations.
gradient of tangent — The instantaneous slope of a curve at a point.

7. Trigonometric ratios and identities — terms 121–140

Trigonometric equations may have multiple solutions within a stated interval. Unit-circle reasoning is safer than recalling one calculator output.

sine — A trigonometric ratio or circular function written sin.
cosine — A trigonometric ratio or circular function written cos.
tangent ratio — The ratio sin θ divided by cos θ where defined.
sine rule — A relationship connecting triangle sides and opposite-angle sines.
cosine rule — A relationship between triangle sides and an included angle.
trigonometric identity — A relationship true for all angles in its permitted domain.
trigonometric equation — An equation involving trigonometric functions.
reference angle — An acute angle used to identify a trigonometric value’s magnitude.
quadrant — One of four regions of the coordinate plane.
unit circle — A circle of radius one centred at the origin.
radian — An angle unit based on arc length divided by radius.
degree — An angle unit dividing a full turn into 360 parts.
periodic function — A function repeating its values at regular input intervals.
amplitude — Maximum distance from a periodic midline.
trigonometric period — The smallest positive repeat interval, when it exists.
principal value — The selected standard output of an inverse trig function.
inverse sine — A function returning an allowed angle whose sine is specified.
inverse cosine — A function returning an allowed angle whose cosine is specified.
inverse tangent — A function returning an allowed angle whose tangent is specified.
all-solutions form — An expression describing every permissible solution of a trig equation.

8. Angular measure, curves and applications — terms 141–160

Radian measure connects arcs and angles: s = rθ and sector area = ½r²θ use θ measured in radians. Degrees must be converted first.

arc length — Distance measured along a circle’s curved edge.
sector — The area bounded by two radii and the included circular arc.
sector area — Area enclosed by a circular sector.
central angle — An angle with its vertex at a circle centre.
angular displacement — The angle through which a rotating object turns.
angular measure — The size of an angle expressed in a chosen unit.
circular measure — Measuring angles using radians.
positive rotation — Anticlockwise turning under the standard convention.
negative rotation — Clockwise turning under the standard convention.
restricted interval — A specified range limiting which angles count as solutions.
general solution — A formula accounting for a full repeating family of solutions.
periodicity — Repeated behaviour after a fixed interval.
trig graph — A graph of a trigonometric function.
phase shift — Horizontal displacement of a periodic graph.
vertical shift — An upward or downward shift of a graph.
horizontal stretch — Changing distances between graph points along the x-axis.
vertical scale factor — The multiplier changing vertical distances from a reference line.
trig modelling — Using periodic functions to represent repeating patterns.
angle conversion — Changing angle values between degrees and radians.
arc-radius relationship — The formula s = rθ when θ is in radians.

9. Differentiation and rates of change — terms 161–180

A derivative describes local change. A zero gradient may identify a stationary point, but further reasoning is needed to classify it.

limit — The value approached by a function as input approaches a point.
derivative — The instantaneous rate of change of a function.
differentiation — The process of finding derivatives.
first derivative — The rate at which the function value changes with input.
second derivative — The rate of change of the first derivative.
chain rule — A rule for differentiating a function nested inside another.
product rule — A rule for differentiating a product of functions.
quotient rule — A rule for differentiating a ratio of functions.
implicit differentiation — Differentiating a relation without isolating one variable first.
rate of change — Change in a quantity per change in another.
instantaneous gradient — The slope of a curve at one point.
stationary point — A point on a differentiable curve where its first derivative is zero.
local maximum — A nearby highest value of a function.
local minimum — A nearby lowest value of a function.
point of inflection — A point where the concavity of a curve changes.
concavity — The bending direction of a curve.
increasing function — A function whose values rise as inputs rise on an interval.
decreasing function — A function whose values fall as inputs rise on an interval.
derivative test — Using derivative signs to classify behaviour or extrema.
related rates — Connecting rates of change among related quantities.

10. Integration, area and mathematical communication — terms 181–200

Integration and differentiation are linked. Check the integrand, limits, sign and constants; not every definite integral represents a positive geometric area.

integration — A mathematical process that can recover a function from its derivative.
antiderivative — A function whose derivative is the given function.
indefinite integral — A family of antiderivatives including a constant.
constant of integration — An arbitrary constant representing the integration family.
definite integral — The accumulated signed change over a specified interval.
limits of integration — The endpoints specifying the interval of a definite integral.
integrand — The function being integrated.
integration variable — The variable with respect to which integration occurs.
power rule of integration — A rule for integrating suitable powers of a variable.
reverse chain rule — Recognition of an antiderivative via the chain rule in reverse.
area under a curve — Geometric area between a curve and a reference axis on an interval.
signed area — An integral contribution positive above and negative below an axis.
area between curves — The region bounded between two function graphs.
fundamental theorem of calculus — A result connecting integration and differentiation.
displacement from velocity — The signed change of position obtained by integrating velocity.
total distance from velocity — The accumulated magnitude of motion found by integrating speed.
initial condition — A specified value used to determine a constant.
boundary condition — A required value or behaviour at the edge of a domain.
verification by differentiation — Checking an antiderivative by differentiating it.
reasoned working — A clear chain of valid steps that justifies a result.

Seven Worked Examples: How the Words Determine the Maths

1. Completing the square reveals a turning point

Suppose f(x) = x² − 6x + 5. Complete the square: f(x) = (x − 3)² − 4. Because a square is never negative, the function has a minimum value −4 when x = 3. The turning point is (3, −4), and its axis of symmetry is x = 3. When a question asks for a maximum or minimum, look for the graph’s behaviour and domain rather than guessing from a sketch.

2. The discriminant checks the number of real roots

For x² − 4x + 3 = 0, we have a = 1, b = −4 and c = 3. The discriminant is b² − 4ac = 16 − 12 = 4, which is positive. Therefore the equation has two distinct real roots, 1 and 3. A discriminant of zero indicates a repeated real root; a negative discriminant means no real roots for a quadratic with real coefficients.

3. Composite functions depend on order

Let f(x) = 2x + 3 and g(x) = x². Then g(f(x)) = (2x + 3)², whereas f(g(x)) = 2x² + 3. Even though the two operations use the same functions, the composite results differ. Write the inside function first, then apply the outer one.

4. A logarithm asks for an exponent

Since 2³ = 8, log₂8 = 3. Similarly, ln(e²) = 2. A basic domain check is essential: log₂(x − 1) is defined for real values only when x > 1. Solving an equation using algebraic manipulation does not make an invalid logarithm argument acceptable.

5. One sine equation may have more than one angle

On the interval 0° ≤ θ < 360°, solve 2sin θ = 1. This gives sin θ = 1/2 and therefore θ = 30° or 150°. A calculator returning 30° may show one principal value, but the diagram and interval reveal a second valid answer. Do not add 360° when the endpoint is excluded.

6. Differentiation classifies stationary points

For y = x³ − 3x² + 2, dy/dx = 3x² − 6x = 3x(x − 2). The stationary points occur at x = 0 and x = 2. The second derivative is 6x − 6; it is negative at x = 0, indicating a local maximum (0, 2), and positive at x = 2, indicating a local minimum (2, −2). A zero first derivative alone does not identify the type of point.

7. Integration turns a rate into accumulated change

A hypothetical object’s velocity along a straight line is v(t) = 4t m/s for 0 ≤ t ≤ 3 s. Its displacement is ∫₀³ 4t dt = [2t²]₀³ = 18 m. Because velocity is nonnegative throughout this interval, total distance is also 18 m here. If velocity becomes negative, integrating velocity still gives displacement, while total distance requires accounting for the magnitude of motion.

Why Students Lose Marks Even When They Know the Formulas

Missing domain restrictions: Squaring both sides or taking logarithms can produce invalid candidates. Check each proposed solution against the original equation, especially for surds, rational expressions and inverse functions.

Using the wrong graph transformation: f(x − 3) shifts the graph of y = f(x) three units to the right, while f(x) − 3 shifts it down. The symbols are similar but the effects differ.

Forgetting the angle mode: A calculator in radians gives different numerical results from one in degrees. The problem must determine the mode; when using s = rθ, θ is in radians.

Confusing the gradient with the gradient function: The derivative f′(x) is an expression giving gradients at different x-values. A tangent gradient at a specified point is obtained by substituting that x-value into f′(x).

Omitting an integration constant: The indefinite integral of 2x is x² + C. The arbitrary C is necessary because all functions x² + C have derivative 2x. In a definite integral, evaluate using the specified endpoints rather than appending an arbitrary constant to the result.

Using a formula without justifying the conditions: A tangent intersects a curve with the same instantaneous slope at the contact point. To prove tangency using algebra, you may need to show a repeated intersection root or a matching gradient, depending on the question.

What Additional Mathematics Is Really Building

G3 Mathematics gives a broad foundation in numerical methods, geometry, statistics and problem-solving. Additional Mathematics asks students to work more abstractly with functions and changing quantities. A quadratic may describe a physical path in a simplified model; a derivative gives the rate of change; a definite integral accumulates that rate over an interval.

A student should be able to move from word to symbol to graph to real-world meaning. For a function describing cost or distance, name the input, units and valid domain. Not every mathematical value makes sense in the chosen situation: a model predicting a negative number of whole objects needs interpretation or domain restriction.

In scientific applications, logarithms and exponential functions can represent growth and decay under certain assumptions. Trigonometric functions describe repeating processes. These links help students appreciate the mathematics but do not imply that every real-world phenomenon exactly follows a simple formula.

Four-Week Secondary 4 Additional Mathematics Study Plan

Week 1 — Algebra control: Learn terms 1–50. Complete the square in four expressions, solve quadratic inequalities and test a factor theorem result.
Week 2 — Functions and logarithms: Learn 51–100. Draw a composite-function arrow diagram, find a domain restriction and verify a logarithmic solution against its original equation.
Week 3 — Geometry and trig: Learn 101–150. Sketch transformed graphs, convert angles and solve trig equations over specific intervals.
Week 4 — Calculus: Learn 151–200. Interpret derivatives in words, classify stationary points and verify antiderivatives by differentiation.

Keep a notebook with the columns term misread, correct mathematical definition, original question and fresh successful solution. Begin each session by revisiting three previously missed terms. The goal is fewer repeated misunderstandings, not simply more completed worksheets.

Frequently Asked Questions

Are these 200 words an official Additional Mathematics syllabus list?

No. They are a curated independent glossary. Use the official syllabus to determine precisely what is assessed; some entries clarify wider mathematical language.

Is G3 Mathematics the same as G3 Additional Mathematics?

No. The 2027 SEC G3 codes are Mathematics K310 and Additional Mathematics K341. Additional Mathematics assumes knowledge of G3 Mathematics but has distinct subject content and assessments.

What is the difference between O-Level 4049 and SEC K341?

4049 identifies the 2026 O-Level Additional Mathematics examination; K341 identifies the 2027 SEC G3 Additional Mathematics examination. Do not combine details from different years without checking the specific official syllabus.

Why are composite functions and domains difficult?

The same symbol x can enter different rules in sequence, and not every input is valid. Draw a mapping diagram or state the domain before simplifying the expression.

Do I need a tutor to memorise mathematical vocabulary?

Not necessarily. First identify whether the difficulty is terminology, foundational algebra, conceptual reasoning or checking steps. Choose focused help after that starting position is clear, rather than adding extra work without diagnosis.

How much calculus should a student learn from this list?

Enough to use the definitions within their current school topics. Connect differentiating to a rate and integrating to an accumulated quantity. Always follow the current year and subject syllabus for assessed techniques.

Official References and Related eduKate Guides

Updated 7 November 2026. This is an independent educational reference, not an official examination vocabulary specification.

Top 200 Vocabulary for Secondary 4 Additional Mathematics | O-Level and SEC 2027 Algebra, Trigonometry and Calculus