3.2 · Algebra

OxfordAQA 9260 unit 3.2 (specification for first assessment 2018) — the largest unit at 35% of each Extension paper. Core rows A1–A26 cover notation through graphs, equations, inequalities and sequences; the Extension additions add subject-twice rearrangement, triple expansions, all quadratic factorisation, fractional powers, quadratic algebraic fractions, formal proof, the full functions toolkit (domain, range, inverse 反函数, composite 复合函数), line equations, exponential and trigonometric graphs, differentiation 微分 with tangents and stationary points 驻点, completing the square, gradients and areas under graphs, variation, every simultaneity pair, quadratic inequalities and quadratic sequences. The 2E specimen samples this unit heavily (algebra is its spine); the unsampled lines are labelled specification originals.

Must-know for unit 3.2 (NO formulae sheet):

Letters express generalised number; substitute numbers into formulae and expressions (positive, negative, fractional). The Extension addition rearranges complex formulae including when the subject appears twice: from $ax + 3 = 5(xt) - 2a$, collect the subject's terms first — $ax + 2a = 5xt - 3$, so $a(x + 2) = 5xt - 3$ and $a = \dfrac{5xt - 3}{x + 2}$. Check every rearrangement numerically.

Collect like terms; expand single brackets and (Extension) products of two or three binomials 二项式: $(x + 2)(x - 3)(x + 1) = (x^2 - x - 6)(x + 1) = x^3 - 7x - 6$. Factorise by common factors and (Extension) every quadratic $ax^2 + bx + c$, including the difference of two squares: $2x^2 - 7x - 15 = (2x + 3)(x - 5)$ and $9x^2 - 25 = (3x - 5)(3x + 5)$. Index laws run to fractional powers: $4c^5 \div 3d^3$ stays as $\tfrac{4c^5}{3d^3}$ in index-form answers (the 1E specimen's answer shape).

Rational expressions 有理式 (Extension) add, subtract, multiply and divide linear or quadratic algebraic fractions — factor first, cancel whole factors, common denominators. The 2E specimen's $\dfrac{11m - 14}{m^2 - 3m - 4}$ and $\dfrac{2x + 3}{x + 4}$ answers come from this machinery.

Worked example: simplify, then subtract. (a) Simplify $\dfrac{x^2 - 9}{x^2 + x - 6}$. (b) Write $\dfrac{3}{x + 2} - \dfrac{1}{x - 1}$ as a single fraction.

(a) Factorise top and bottom, then cancel the whole common factor $(x + 3)$:

(b) The common denominator is $(x + 2)(x - 1)$. Put the second numerator in a bracket so the minus sign reaches both terms:

Cancelling single terms (such as the $x^2$ in part (a)) is never allowed: only whole factors cancel.

Argue mathematically that identities 恒等式 hold for every value — with the Extension demand for formal proofs 证明. Structure: state the algebra, simplify, compare with the claim. Classic (2E specimen family): prove that $n^2 + n + 9$ is odd for every integer $n$. Write $n^2 + n = n(n + 1)$: a product of two consecutive integers 连续整数, one of which is even, so $n(n + 1)$ is even, and even $+ 9$ is odd. Another parity proof: $(2n + 1)^2 - (2n - 1)^2 = (4n^2 + 4n + 1) - (4n^2 - 4n + 1) = 8n$ — a multiple of 8 for every integer $n$. Numeric examples are never a proof.

A function 函数 maps each input to exactly one output (Core: simple input–output machines; Extension: the formal definition, $\mathrm{f}(x) = \ldots$ notation, domain 定义域 and range 值域). Composite: $\mathrm{fg}(x) = \mathrm{f}(g(x))$, apply $g$ first — the 2E specimen builds $\mathrm{fg}(x) = (x - 5)^2 + 1$ and solves $\mathrm{fg}(x) = 5$: $(x - 5)^2 = 4$ gives $x = 3$ (or $x = 7$; check: $(3 - 5)^2 + 1 = 5$). Inverse: swap and rearrange — the 1E specimen's $\mathrm{f}^{-1}(x) = \tfrac x3$ comes from $\mathrm{f}(x) = 3x$.

Coordinates in all four quadrants 象限; plot $y = mx + c$; the Extension additions find the equation through two points or through one point with a given gradient, and use $y - y_1 = m(x - x_1)$: through $(1, 7)$ and $(2, 9)$, $m = 2$ and $y = 2x + 5$. Parallel lines 平行线 share $m$; perpendicular 垂直 gradients multiply to $-1$ (specification line, not sampled).

Sketch and interpret linear, quadratic, cubic 三次 and reciprocal graphs; the Extension additions add exponential functions $y = k^x$ for positive $k$ and the trigonometric graphs (degrees) — both specification lines not sampled by the specimens. Identify roots, intercepts 截距 and turning points 转折点; the Extension addition DEDUCES turning points by completing the square 配方: $y = x^2 - 6x + 5 = (x - 3)^2 - 4$ has its minimum at $(3, -4)$ (specification line, not sampled). Gradients and areas under graphs (including quadratic and non-linear graphs) are estimated by tangents and trapezia 梯形, and — at Extension — computed exactly by differentiation (next section).

The Extension calculus block: understand and use the gradient function 导函数 $\dfrac{dy}{dx}$, differentiate $kx^n$ for positive integer $n$ (and sums), and know that the gradient of a function at a point IS the gradient of the tangent 切线 there — then work out the tangent's equation.

The 2E specimen's values (family citation): a curve with $\dfrac{dy}{dx} = 15$ at the point carries tangent $y = 15x - 36$. Stationary points solve $\dfrac{dy}{dx} = 0$, classified as maxima, minima 极值 or points of inflection 拐点, with curves sketched from the derivatives 导数:

Worked example: stationary points and a tangent. $y = x^3 - 12x + 5$: $\dfrac{dy}{dx} = 3x^2 - 12 = 0$ gives $x = \pm 2$. At $x = -2$, $y = 21$ (gradient changes $+$ to $-$: maximum); at $x = 2$, $y = -11$ (changes $-$ to $+$: minimum). At $x = 3$ the gradient is $3(9) - 12 = 15$ and the tangent is $y + 4 = 15(x - 3)$, i.e. $y = 15x - 49$.

Solve linear equations (unknown both sides), quadratics by factorising and (Extension) by completing the square and the quadratic formula — the 1E specimen's $x = -3 \pm \sqrt7$ comes from $x^2 + 6x + 2 = 0$: $x = \tfrac{-6 \pm \sqrt{36 - 8}}{2}$. Simultaneous pairs: two linear equations, and (Extension) one linear with one quadratic 二次方程. Variation (Extension): direct $y = kx^n$ and inverse $y = \tfrac{k}{x^n}$ in algebraic terms (specification line, not sampled). Worked example: a linear and a quadratic. Solve $y = x + 1$ and $x^2 + y^2 = 13$ together. Substitute the linear equation into the quadratic:

So $x = 2$, $y = 3$ or $x = -3$, $y = -2$. Give the answers as PAIRS, and find each $y$ from the linear equation: squaring $y^2 = 9$ would wrongly allow $y = -3$ with $x = 2$.

Inequalities: linear in one variable, and (Extension) in one or two variables plus quadratic inequalities 二次不等式 — solve $x^2 - x - 6 > 0$ by critical values $x = -2, 3$: positive quadratic, so $x < -2$ or $x > 3$ (specification line, not sampled).

Generate terms from term-to-term and position-to-term rules; use triangular, square and cube number sequences (Core) with the Extension additions covering quadratic sequences 二次数列: second differences 二阶差 are constant, and the nth term 第n项 is $an^2 + bn + c$ with $a = \tfrac12(\text{second difference})$. For $5, 9, 17, 29, 45, \ldots$: first differences $4, 8, 12, 16$, second difference $4$, so $a = 2$. Take $2n^2$ (which gives $2, 8, 18, 32, 50$) and subtract: the residuals $3, 1, -1, -3, -5$ are linear, $-2n + 5$. So the nth term is $2n^2 - 2n + 5$ — check $n = 4$: $32 - 8 + 5 = 29$. ✓

• Generalised notation; substitution; rearrangement with the subject twice (A1–A3, Extension).
• Expanding two or three binomials; factorising all quadratics and the difference of two squares; fractional index answers; quadratic algebraic fractions (A4–A7, Extension; 1E/2E specimens).
• Formal algebraic proof (A8, Extension; 2E n² + n + 9 family).
• Functions: definition, domain, range, composite fg, inverse (A9, Extension; 2E fg(x) = (x−5)² + 1, 1E f⁻¹(x) = x/3).
• Coordinates; line equations through two points or point + gradient; perpendicular gradients (A10–A11; the latter not sampled).
• Linear, quadratic, cubic, reciprocal, exponential and trigonometric graphs; turning points by completing the square; gradient/area estimates (A12, A16–A17; several lines not sampled).
• Differentiation of kxⁿ; gradient = tangent gradient; tangent equations; stationary points classified with sketches (A13–A15, Extension; 2E tangent family).
• Linear equations; quadratics by all three methods; linear–linear and linear–quadratic simultaneity (A19–A21, Extension).
• Context into algebra (A22, Extension); linear (one/two variable) and quadratic inequalities with solution sets (A23, Extension; not sampled).
• Sequence types; nth terms linear and quadratic (A24–A26, Extension).

Boundary notes: NO formulae sheet in 9260. Lines marked "not sampled" (perpendicular gradients, exponential/trig sketches, completing-the-square turning points, variation, quadratic and two-variable inequalities) are specification originals; the specimens are Extension-tier only, so Core-tier demand has no local evidence. Worked examples are adapted from the 2016 specimen pair or original.