➕ Year 9 Mathematics

Functions

Function Notation

• A function maps an input to exactly one output: f(x) = 2x + 3 means "multiply x by 2, then add 3"
• f(4) means substitute x = 4: f(4) = 2(4) + 3 = 11
• The domain is the set of allowed inputs; the range is the set of possible outputs

Composite Functions

• fg(x) means "apply g first, then f to the result": fg(x) = f(g(x))
• Order matters: fg(x) ≠ gf(x) in general

Inverse Functions

• The inverse function f⁻¹(x) undoes f(x). If f maps x → y, then f⁻¹ maps y → x
• To find f⁻¹: write y = f(x), rearrange to make x the subject, then replace x with f⁻¹(x) and y with x
• The graph of f⁻¹ is the reflection of f in the line y = x

Quadratics & Completing the Square

Solving Quadratics — Methods

• Factorising: find two brackets that multiply to give the quadratic. Works when the equation factorises neatly.
• Quadratic formula: always works. x = (−b ± √(b²−4ac)) / 2a for ax² + bx + c = 0
• Completing the square: rewrites ax² + bx + c in the form a(x + p)² + q

Completing the Square

• For x² + bx + c: x² + bx + c = (x + b/2)² − (b/2)² + c
• The vertex of the parabola y = (x + p)² + q is at (−p, q)
• If the coefficient of x² ≠ 1, factor it out first

The Discriminant

• The discriminant Δ = b² − 4ac tells you how many real roots the quadratic has:
• Δ > 0 → two distinct real roots (the parabola crosses the x-axis twice)
• Δ = 0 → one repeated real root (the parabola touches the x-axis at its vertex)
• Δ < 0 → no real roots (the parabola does not cross the x-axis)

Inequalities & Regions

Quadratic Inequalities

• To solve x² − 5x + 6 > 0: first solve x² − 5x + 6 = 0 → (x−2)(x−3) = 0 → x = 2 or x = 3
• Sketch the parabola (opens upward since coefficient of x² is positive)
• The parabola is above zero outside the roots: x < 2 or x > 3
• For x² − 5x + 6 < 0: the parabola is below zero between the roots: 2 < x < 3

Graphical Inequalities and Regions

• To show a region satisfying multiple inequalities: draw each boundary line, shade the required region, and use a test point to confirm
• Dashed line: strict inequality (not equal, < or >)
• Solid line: non-strict inequality (≤ or ≥)
• Linear programming: find the optimal value of an expression within a feasible region (relevant for GCSE and beyond)

Circle Theorems

• Angle at the centre = 2 × angle at the circumference (both subtended by the same arc)
• Angles in the same segment are equal (both subtended by the same chord, on the same side)
• Angle in a semicircle = 90° (the diameter subtends a right angle at the circumference)
• Opposite angles in a cyclic quadrilateral add up to 180° (a cyclic quadrilateral has all four vertices on the circle)
• Tangent–radius angle = 90° (a tangent to a circle is perpendicular to the radius at the point of contact)
• Tangents from an external point are equal in length
• Alternate segment theorem: the angle between a tangent and a chord equals the inscribed angle in the alternate segment

Vectors

What Is a Vector?

• A vector has both magnitude (size) and direction. Shown as a column vector: (x over y) or in bold: a
• A scalar has magnitude only (e.g. speed, temperature)
• On a grid: the vector (3 over 2) means 3 right, 2 up

Vector Operations

• Addition: a + b = (a₁+b₁ over a₂+b₂). Geometrically: place vectors end to end.
• Subtraction: a − b = a + (−b)
• Scalar multiplication: ka scales the vector by k. If k is negative, the direction reverses.
• The negative vector: −a has the same magnitude as a but opposite direction

Magnitude

Using Vectors for Geometry

• If OA = a and OB = b, then AB = b − a (go back from A to O, then forward to B)
• Midpoint M of AB: OM = ½(a + b)
• If two vectors are parallel: one is a scalar multiple of the other
• If vectors share a point and are parallel, the points must be collinear (on the same straight line)

Transformations

The Four Transformations

• Translation: described by a column vector. Every point moves the same amount.
• Reflection: described by a mirror line (e.g. y = x, y = −x, x = 2). Each point maps to its mirror image.
• Rotation: described by centre, angle, and direction (clockwise/anticlockwise)
• Enlargement: described by centre and scale factor. Scale factor > 1 enlarges; 0 < SF < 1 reduces; negative SF enlarges and rotates 180°.

Combined Transformations

• Applying two transformations in sequence: the order matters
• Describing the single transformation equivalent to two combined ones is a common exam question
• Area scale factor = (length scale factor)² for enlargements

Advanced Statistics

Histograms

• Histograms display continuous data grouped into class intervals (which may be unequal in width)
• The y-axis shows frequency density, NOT frequency: Frequency density = Frequency ÷ Class width
• Area of each bar = frequency. This is what makes unequal class widths work correctly.

Cumulative Frequency and Box Plots

• Cumulative frequency: running total of frequencies. Plot against the upper class boundary.
• From cumulative frequency curve (ogive): read off median (50th percentile), lower quartile (25th), upper quartile (75th)
• Interquartile range (IQR) = UQ − LQ. Measures the spread of the middle 50% of data. Less affected by outliers than the range.
• Box plot (box-and-whisker): shows min, LQ, median, UQ, max. Any value more than 1.5 × IQR beyond the quartiles is an outlier (shown as a cross).

Comparing Distributions

• Always compare: measure of average (mean, median) AND measure of spread (range, IQR, standard deviation)
• State which distribution has the higher/lower average and which is more/less spread out — in context of the data

Combined Probability

Mutually Exclusive Events

• Two events are mutually exclusive if they cannot both happen at the same time (e.g. rolling a 3 and a 5 on one die)
• P(A or B) = P(A) + P(B) for mutually exclusive events
• All outcomes in a sample space are mutually exclusive and exhaustive: their probabilities sum to 1

Independent Events

• Two events are independent if the outcome of one does not affect the outcome of the other
• P(A and B) = P(A) × P(B) for independent events

Conditional Probability

• P(B|A) = the probability of B given that A has already happened
• P(A and B) = P(A) × P(B|A)
• This is the foundation of tree diagrams for dependent events

Venn Diagrams for Probability

• Two overlapping circles in a rectangle (the universal set ξ)
• A ∩ B (intersection): elements in both A and B
• A ∪ B (union): elements in A or B or both
• A' (complement): elements NOT in A
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (the addition formula for non-mutually exclusive events)