IGCSE Additional Mathematics (0606)
Syllabus Notes (2025-2027)

1. Functions

Understanding rigorous mapping notation, domain constraints, and functional operations is critical for advanced algebraic calculus.

• Domain: The complete set of all possible independent input values (\(x\)) for which the function is mathematically defined. Always check for denominators \(\neq 0\) and expressions under even roots \(\ge 0\).
• Range (Image set): The complete set of all dependent output values (\(y\) or \(f(x)\)) produced by the function over its specified domain. It is often determined by analyzing turning points or boundary values.
• Function conditions: A mapping is defined as a function if and only if each distinct input element maps to exactly one unique output element (valid for one-to-one or many-to-one mappings). Many-to-many or one-to-many mappings are relations, not functions.
• Inverse Function (\(f^{-1}\)): Exists if and only if the original function is strictly one-to-one. Geometrically, the graph of \(f^{-1}(x)\) is the exact reflection of \(f(x)\) across the line \(y = x\). Crucially:
  \(\text{Domain of } f^{-1} = \text{Range of } f\) and \(\text{Range of } f^{-1} = \text{Domain of } f\).
• Composite Functions (\(gf(x)\) or \(g[f(x)]\)): The compound operation is valid if and only if the Range of the inner function \(f\) is a subset of the Domain of the outer function \(g\) (\(\text{Range of } f \subseteq \text{Domain of } g\)). Note that order matters explicitly: \(fg(x) \neq gf(x)\).

2. Quadratic Functions

Quadratic expressions follow the structural standard form \(ax^2 + bx + c\). Mastering their representations yields vital information about their roots and geometric nature.

Completing the Square (Vertex Form)

Transform standard general quadratics into vertex form: \(f(x) = a(x-h)^2 + k\).

• The turning point (vertex) of the parabola is located directly at the coordinate pair \((h, k)\).
• If \(a > 0\), the curve opens upward, yielding a minimum value of \(k\) at \(x = h\).
• If \(a < 0\), the curve opens downward, yielding a maximum value of \(k\) at \(x = h\).

The Discriminant (\(\Delta = b^2 - 4ac\))

Used to determine the nature of the roots or the intersection status between a straight line and a quadratic curve.

Syllabus Tip: If a question states that a line simply meets a curve, you must apply the combined condition \(b^2 - 4ac \ge 0\).

3. Factors of Polynomials

Applies systematic algebraic analysis to higher-degree polynomials, specifically focusing on cubic functions formatted as \(f(x) = ax^3 + bx^2 + cx + d\).

Step-by-Step Factoring of Cubics

1. Use trial and error alongside the Factor Theorem to locate a scalar value where \(f(k) = 0\), establishing your first linear factor \((x - k)\).
2. Employ algebraic long division or synthetic matching to divide \(f(x)\) by \((x - k)\), reducing the expression to a remaining quadratic quotient: \((x - k)(px^2 + qx + r)\).
3. Factorize the remaining quadratic component completely using standard splitting methods or the quadratic formula to obtain all three roots.

4. Equations, Inequalities and Graphs

Covers absolute value absolute modulus mathematics alongside mechanical methods for mapping linear, quadratic, and cubic inequalities.

Modulus Functions

The modulus operator \(|f(x)|\) isolates absolute numerical magnitude, transforming all negative outputs to non-negative expressions: \(|x| \ge 0\).

• Solving Equations: To solve a system like \(|ax + b| = |cx + d|\), handle it analytically by squaring both operational sides or breaking it down into split linear equations: \(ax + b = \pm(cx + d)\). Always verify solutions to filter out extraneous values.
• Graphing Modulus Curves: First, sketch the unmodulated base function \(y = f(x)\) using standard dashed parameters. Then, reflect any portions that drop below the x-axis (negative y-zone) vertically upward into the positive quadrant, creating distinct sharp local turning points called cusps.

Inequalities Guidance

For quadratic inequalities (\(ax^2 + bx + c > 0\)), determine the critical values by solving the equation variant. Then, sketch the path of the curve to determine whether the target regions lie outside the roots (\(x < x_1\) or \(x > x_2\)) or between them (\(x_1 < x < x_2\)).

5. Simultaneous Equations

Focuses on determining the exact intersections of mathematical lines and curves by solving a system containing one linear equation and one non-linear equation simultaneously.

Universal Operational Protocol

1. Isolate one easy independent variable within the linear expression (e.g., transforming the linear statement above into \(y = x - 3\)).
2. Substitute this substitution expression directly into every instance of that variable within the non-linear equation. This safely eliminates one variable.
3. Expand all terms carefully and simplify to construct a standard single-variable quadratic format: \(Ax^2 + Bx + C = 0\).
4. Solve the quadratic equation via factorization or formula to discover the primary values of \(x\), then substitute them back into the linear expression to secure the matching \(y\) coordinates.

6. Logarithmic and Exponential Functions

Logarithmic bases and exponential models operate as direct mathematical inverses. The standard operational equivalency statement forms the core of all manipulation:

Core Laws of Logarithms

• Product Law: \(\log_a (xy) = \log_a x + \log_a y\)
• Quotient Law: \(\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\)
• Power Law: \(\log_a (x^n) = n \log_a x\)
• Special Evaluators: \(\log_a a = 1\), \(\log_a 1 = 0\), and \(a^{\log_a x} = x\).
• Change of Base Theorem: Necessary for calculating non-standard formats manually: \[ \log_a b = \frac{\log_c b}{\log_c a} \implies \log_a b = \frac{1}{\log_b a} \]

7. Straight-line Graphs

Analytical geometric properties governing linear coordinate sets labeled cleanly as \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\).

• Gradient Formulation: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
• Parallel Orientation Alignment: \(m_1 = m_2\) (Equal steepness).
• Perpendicular Intersection Alignment: \(m_1 \times m_2 = -1 \implies m_2 = -\frac{1}{m_1}\) (Negative reciprocal).
• Midpoint Vector Coordinate: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
• Straight-line Length (Distance Formula): \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Linearisation of Non-Linear Relations

Transforming complex non-linear curves into standard straight-line formats (\(Y = mX + C\)) by applying strategic logarithmic functions to axis plots.

8. Coordinate Geometry of the Circle

Key Geometric Problem Types

• Completing the Square for Circles: Convert general expanded equations back into standard form to easily read out the center coordinates and radius.
• Tangent Properties: The radius of a circle is always perpendicular to the tangent line at the point of contact. Therefore, \(m_{\text{radius}} \times m_{\text{tangent}} = -1\).
• Chords and Perpendicular Bisectors: The perpendicular bisector of any chord on the circle will pass directly through the circle's center point.

9. Circular Measure

All formulas within this core operational framework strictly demand that rotational angles are calculated and expressed in radians rather than degrees.

For a sector within a circle of radius \(r\) subtending an angle \(\theta\) (strictly in radians):

• Arc Length Formula: \[ s = r\theta \]
• Sector Area Formula: \[ A = \frac{1}{2}r^2\theta \]
• Perimeter of a Sector: \[ P = 2r + s = 2r + r\theta \]
• Area of a Segment: Calculated by subtracting the area of the central triangle from the overall sector area: \[ A_{\text{segment}} = \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta = \frac{1}{2}r^2(\theta - \sin\theta) \]

10. Trigonometry

Extends basic foundational acute definitions to encompass all real circular coordinate frameworks across four distinct algebraic quadrants.

Reciprocal Trigonometric Functions

Pythagorean Fundamental Identities

Essential for proving trigonometric identities and solving complex equations:

• \(\sin^2 A + \cos^2 A = 1\)
• \(\sec^2 A = 1 + \tan^2 A\)
• \(\csc^2 A = 1 + \cot^2 A\)

Non-Calculator Reference Table (Crucial for Paper 1)

11. Permutations and Combinations

Differentiates systematically between strict ordered tracking lists and unstructured group cluster selections.

• Permutations (\(^nP_r\)): Applied when the internal **arrangement order matters** explicitly (e.g., creating specific numerical codes, assigning unique award placements, or scheduling seating lines). \[ ^nP_r = \frac{n!}{(n-r)!} \]
• Combinations (\(^nC_r\)): Applied when selecting items where the **order does not matter** (e.g., picking a committee, selecting a subset of cards, or choosing team combinations). \[ ^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Strategy for Constraint Problems

• Items Kept Together: Tie the restricted items together and treat them as a single macro-object. Permute the outer group, then multiply by the internal permutations of the tied items.
• Items Separated: Arrange the unrestricted items first, then place the restricted items into the empty gaps created between them.

12. Series

Deals with algebraic expansions of binomial expressions alongside structural modeling of progressive mathematical sequences.

Binomial Expansion Theorem

For any positive integers \(n\), the systematic expansion expression is structured as:

The general \((r+1)\)-th term is defined as: \(T_{r+1} = \binom{n}{r}a^{n-r}b^r\).

Progressions Matrix

Convergence and Infinite Geometric Series

A geometric progression will converge toward a constant value if and only if its common ratio satisfies **\(|r| < 1\)**. Under this condition, the sum to infinity is calculated using:

13. Vectors in Two Dimensions

Vectors represent spatial shifts defined by both a clear physical direction and a precise numerical scalar length.

Notation Options

Vectors can be written as a column matrix or using standard Cartesian unit vectors \(\mathbf{i}\) and \(\mathbf{j}\):

Key Mathematical Vector Operators

• Magnitude (Length): Calculated using the standard Pythagorean distance formula: \[ |\mathbf{r}| = \sqrt{x^2 + y^2} \]
• Unit Vector (\(\hat{\mathbf{r}}\)): A vector pointing in the exact same direction as \(\mathbf{r}\) but with a magnitude of exactly 1 unit: \[ \hat{\mathbf{r}} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{1}{\sqrt{x^2 + y^2}}\begin{pmatrix} x \\ y \end{pmatrix} \]
• Collinear Vectors: Two vectors are parallel and share a common line of action if one is a scalar multiple of the other: \(\mathbf{a} = k\mathbf{b}\).

14. Calculus

Crucial Examination Alert: No calculus formula sheets are provided in your examination booklet! You must fully commit all derivative and integral rules to memory.

Differentiation Master Rules Reference

Advanced Tools for Differentiation

• Chain Rule (Composite Functions): \[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \]
• Product Rule (Multiplied Functions): For \(y = uv\): \[ \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \]
• Quotient Rule (Divided Functions): For \(y = \frac{u}{v}\): \[ \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \]

Integration (Inverse Differentiation Process)

Do not forget to always include the arbitrary constant of integration (\(+ c\)) for all indefinite integrals.

• Power Rule for Integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + c \quad (\text{Valid for all } n \neq -1) \]
• The Special Exponent Case (\(n = -1\)): \[ \int \frac{1}{x} \, dx = \ln|x| + c \qquad \implies \qquad \int \frac{1}{ax+b} \, dx = \frac{1}{a}\ln|ax+b| + c \]
• Trigonometric Integrals: \[ \int \cos(ax+b) \, dx = \frac{1}{a}\sin(ax+b) + c \] \[ \int \sin(ax+b) \, dx = -\frac{1}{a}\cos(ax+b) + c \]

Applications of Calculus

• Stationary Points: Set \(\frac{dy}{dx} = 0\) to solve for critical coordinates. Check the second derivative (\(\frac{d^2y}{dx^2}\)) to determine their nature:
  • If \(\frac{d^2y}{dx^2} > 0\), the point is a Minimum.
  • If \(\frac{d^2y}{dx^2} < 0\), the point is a Maximum.
• Definite Integrals and Area: The total physical geometric area bound between a curve \(y=f(x)\), the x-axis, and vertical lines \(x=a\) and \(x=b\) is calculated via: \[ \text{Area} = \int_{a}^{b} f(x) \, dx \]
• Kinematics: Calculus pathways connecting Displacement (\(s\)), Velocity (\(v\)), and Acceleration (\(a\)) over time variable \(t\): \[ s \xrightarrow{\text{Differentiate } \frac{ds}{dt}} v \xrightarrow{\text{Differentiate } \frac{dv}{dt}} a \] \[ a \xrightarrow{\text{Integrate } \int a \, dt} v \xrightarrow{\text{Integrate } \int v \, dt} s \]