Algebra and Graphs

• If you remember only one sentence: every algebraic move is a claim that two expressions are equal for all permitted values — so check both halves of that claim, the equality and the permitted values.
• Nine and thirteen. Nine active Core rows: 2.1, 2.2, 2.4, 2.5, 2.6, 2.7, 2.9, 2.10 and 2.11. Thirteen Extended rows: 2.1 through 2.13 without a gap. Every one of them has its own teaching section in this chapter, and every one is tagged where it is taught.
• All four describe the same quadratic. Choosing which to produce is the real question being asked: “solve” wants factorised, “find the minimum” wants completed square, “where does it cross the \(y\)-axis” wants standard form.
• Under time pressure, do these two things. First, name the object before choosing a method — expression, equation, inequality, formula, sequence, function. Second, check by substitution: put one convenient number into the original and into your answer. Those two habits together catch more errors than any amount of re-reading, and each costs about fifteen seconds.
• If you are down to one week. Do Day 1 and Day 3 back to back, then the mixed challenge. Skip the re-reading entirely — with limited time, testing yourself and correcting errors is worth several times as much as reading the sections again, however uncomfortable that feels.

• C2.1 — Use a letter as an unknown, a variable or a generalised number, and say which one a given problem needs.
• C2.1 — Substitute a value — including a negative value, in brackets — into an expression or a formula.
• C2.2 — Collect like terms, recognising that \(a^2\) and \(a\) are not like terms.
• C2.2 — Expand a single bracket, and expand a product of two brackets in one variable such as \((x+1)(x-3)\).
• C2.2 — Factorise by extracting common factors, taking out the highest common factor so that the answer is factorised fully.
• C2.4 — Use positive, zero and negative indices, and apply every index law to expressions containing letters.
• C2.4 — Solve an index equation by rewriting both sides with a common base, without using logarithms.
• C2.5 — Construct simple expressions, equations and formulas from a verbal description, including consecutive, even and odd integers.
• C2.5 — Solve a linear equation in one unknown, including one with brackets and one with the unknown on both sides.
• C2.5 — Solve simultaneous linear equations in two unknowns by elimination and by substitution, and construct such a pair from a context.
• C2.5 — Change the subject of a simple formula, where the subject appears once and is not under a power or a root.
• C2.6 — Represent an inequality on a number line and interpret one drawn for you, with open circles for \(<\) and \(>\) and closed circles for \(\le\) and \(\ge\).
• C2.7 — Continue a number sequence or pattern and state its term-to-term rule.
• C2.7 — Find and use the \(n\)th term of a linear sequence, and of a simple quadratic or simple cubic sequence such as \(n^2+1\) or \(2n^3\).
• C2.9 — Draw a graph from given data with sensible scales and fully labelled axes.
• C2.9 — Read and interpret travel graphs and conversion graphs, and interpret the gradient of a straight-line graph as a rate of change.
• C2.10 — Construct a table of values and draw the graph of \(ax+b\), of \(x^2+ax+b\), and of \(\dfrac{a}{x}\) for \(x\ne0\), with \(a\) and \(b\) integers.
• C2.10 — Solve associated equations graphically: read roots where the curve meets the \(x\)-axis, and find the intersection of a line and a curve.
• C2.11 — Recognise, sketch and interpret linear and quadratic graphs, showing roots, the \(y\)-intercept and the axis of symmetry.
• E2.2 — Expand a product of more than two brackets, and products in two variables such as \((x+y)(x-y)\).
• E2.2 — Factorise \(ax+bx+kay+kby\) by grouping, \(a^2x^2-b^2y^2\) as a difference of two squares, \(a^2+2ab+b^2\) as a perfect square, \(ax^2+bx+c\), and \(ax^3+bx^2+cx\).
• E2.2 — Complete the square for \(ax^2+bx+c\), including when \(a\ne1\), and read the turning point from the result.
• E2.3 — Add, subtract, multiply and divide algebraic fractions with numerical and algebraic denominators.
• E2.3 — Factorise and simplify a rational expression, cancelling only common factors, and state every restriction the original expression carries.
• E2.4 — Use fractional indices as well as positive, zero and negative ones.
• E2.5 — Solve a fractional equation with numerical and linear algebraic denominators, stating excluded values first and testing solutions against them.
• E2.5 — Solve simultaneous equations where one is linear and one is non-linear, with powers no higher than two.
• E2.5 — Solve a quadratic by factorisation, by completing the square and by the quadratic formula, giving exact answers in surd form when asked.
• E2.5 — Change the subject of a formula when the subject appears twice, and when it sits under a power or a root.
• E2.6 — Construct, solve and interpret linear inequalities, reversing the sign when multiplying or dividing by a negative.
• E2.6 — Represent a linear inequality in two variables graphically, with the correct broken or solid boundary, shading the unwanted region.
• E2.6 — List the inequalities that define a given region.
• E2.7 — Find the \(n\)th term of a general quadratic or cubic sequence from its differences, and of an exponential sequence from its ratio, using \(T_n\) subscript notation.
• E2.7 — Recognise a simple combination of two families, such as \(n^2+2^n\).
• E2.8 — Write a proportion statement with \(\propto\), introduce a constant \(k\), find \(k\) from given data, and handle direct and inverse proportion involving \(x\), \(x^2\), \(\sqrt{x}\), \(x^3\) and \(\sqrt[3]{x}\).
• E2.9 — Apply rate of change to simple kinematics on distance–time and speed–time graphs, and describe acceleration and deceleration physically.
• E2.9 — Calculate distance travelled as the area under a speed–time graph built from linear sections.
• E2.9 — Estimate and interpret the gradient of a tangent drawn to a curve at a point.
• E2.10 — Draw and interpret graphs of sums of up to three terms of the form \(ax^n\) with \(n\in\{-2,-1,-\tfrac12,0,\tfrac12,1,2,3\}\), and of \(ab^x+c\).
• E2.10 — Draw and interpret graphs representing exponential growth and decay problems.
• E2.11 — Sketch cubic, reciprocal and exponential curves as well as linear and quadratic ones, showing turning points, roots, symmetry and both vertical and horizontal asymptotes.
• E2.11 — Find the turning point of a quadratic by completing the square.
• E2.12 — Differentiate a sum of up to three terms of the form \(ax^n\), where \(a\) is rational and \(n\) is a positive integer or zero, using \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) notation.
• E2.12 — Use the derivative to find the gradient of a curve at a point, and to locate stationary points by solving \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=0\).
• E2.12 — Discriminate between a maximum and a minimum by an accurate sketch, by the second differential, or by inspecting the gradient either side.
• E2.13 — Use function notation, and state the domain and range of a simple function.
• E2.13 — Draw and read a mapping diagram.
• E2.13 — Find \(f^{-1}(x)\), and explain why it is not \(\dfrac{1}{f(x)}\).
• E2.13 — Form the composite \(gf(x)=g(f(x))\), applying the function nearest \(x\) first, and show that order matters.

Why it matters: generalisation is the entire economic value of algebra. “The perimeter of this rectangle is 34 cm” is one fact. \(P=2(l+w)\) is every such fact, for every rectangle that will ever be measured, written once. Every formula you will meet in Topics 3 to 9 is a letter doing this third job.

Indices II

The extension of index notation and the index laws from numbers to algebraic expressions and equations. A positive index counts repeated multiplication, a zero index gives one, a negative index denotes a reciprocal and a fractional index denotes a root, with the denominator of the fraction naming the root and the numerator the power. An index equation is solved by rewriting both sides as powers of the same base and then equating the indices; logarithms are not required at this level.

Introduction to Algebra

The use of letters to stand for numbers, so that a relationship can be written down before the numbers are known. A letter may act as a specific unknown to be found, as a variable that ranges over many values, or as a generalised number in a rule true for all values. Substituting a number for a letter replaces the letter everywhere it appears, and a negative value must be enclosed in brackets so that powers and signs act on the whole value.

Algebraic Fractions

A fraction whose numerator or denominator contains a variable, handled by exactly the rules that govern numerical fractions: a common denominator for addition and subtraction, multiplication of numerators and denominators for products, and multiplication by the reciprocal for division. Because a denominator may not be zero, every algebraic fraction carries restrictions on its variable, and those restrictions come from the original expression and survive any later cancellation.

Proportion

A relationship in which one quantity is a fixed multiple of a power or root of another. In direct proportion the two quantities rise and fall together and y equals k times that power; in inverse proportion one rises as the other falls and y equals k divided by that power. The symbol for is proportional to must be replaced by an equals sign and a constant of proportionality k before any calculation, and k is found by substituting one given pair of values.

Graphs of Functions

The construction and interpretation of the graph of a function from a table of values, for functions built from at most three terms of the form a times x to the power n, where n is minus two, minus one, minus a half, zero, a half, one, two or three, and for exponential functions of the form a times b to the power x plus c. The roots of an equation appear as the points where the curve crosses the x-axis, the solutions of one function equalling another appear as the x-coordinates of their intersections, and a line may be added to a drawn curve so that a transformed equation becomes an intersection problem.

Sequences

An ordered list of numbers generated by a rule, whose nth term is written using subscript notation such as T sub n. A term-to-term rule says how to get from one term to the next; a position-to-term rule gives any term directly from its position. The family is identified from differences: a constant first difference means linear, a constant second difference means quadratic, and a constant third difference means cubic, while a constant ratio between consecutive terms means exponential.

Equations

A statement that two expressions are equal, true only for particular values of the unknown, which solving is the process of finding. Linear equations are solved by inverse operations, fractional equations by clearing denominators after recording their excluded values, simultaneous linear equations by elimination or substitution, and quadratic equations by factorisation, by completing the square or by the quadratic formula. Changing the subject rearranges a formula so that a different letter stands alone, and any solution that makes an original denominator zero must be rejected. Linear equations, simultaneous linear equations and simple subject changes are required on both routes; fractional equations, mixed linear and non-linear pairs, quadratic equations and harder subject changes are Extended requirements.

Algebraic Manipulation

The rewriting of an algebraic expression into a different but equivalent form, so that the same quantity is expressed in a shape better suited to the question. Expanding removes brackets by multiplying out and factorising restores brackets by extracting common structure; on the Extended route the same idea extends to a full set of factorising patterns and to completing the square, which rewrites a quadratic as a squared bracket plus a constant. Every such rewrite must hold for every value of the letter, and in Cambridge usage factorise always means factorise fully.

Inequalities

A statement that one quantity is less than, greater than, or not more or less than another, whose solution is a range of values rather than a single value. Representing and interpreting an inequality is required on both routes; solving one, and working in two variables, are Extended requirements. Linear inequalities are solved by the same operations as equations, with one exception: multiplying or dividing both sides by a negative number reverses the direction of the inequality. In one variable the solution is an interval shown on a number line with open circles for strict inequalities and closed circles for inclusive ones; in two variables it is a region of the plane bounded by broken lines for strict inequalities and solid lines for inclusive ones.

Graphs in Practical Situations

The use of graphs to represent real quantities, where the gradient and the area beneath the line both carry physical meaning fixed by the units of the axes. On a distance-time graph the gradient is speed and a horizontal section means stationary; on a speed-time graph the gradient is acceleration and the area beneath the graph is the distance travelled. Where the graph curves, an instantaneous rate is estimated by drawing a tangent at the point and finding the gradient of that tangent from two widely separated points on it. A conversion graph instead turns one unit into another by reading from one axis to the line and then to the other axis, and it need not pass through the origin.

Functions

A rule that assigns to each input in its domain exactly one output, the set of outputs actually produced being its range. Function notation writes the output of f at x as f of x. The inverse function reverses a one-to-one function, so that if f maps four to seven then its inverse maps seven to four, and it is found by swapping x and y and making y the subject. A composite function applies two functions in succession, and Cambridge writes gf of x to mean g of f of x, so the function written nearest to x acts first.

Differentiation

The process that turns the equation of a curve into a gradient function, written dy by dx, giving the exact gradient of the curve at every value of x rather than the single average gradient between two points. For a term a times x to the power n the derivative is a times n times x to the power n minus one, so a constant term differentiates to zero and the derivatives of a sum are added term by term. A stationary point is a point where dy by dx equals zero, and it is classified as a maximum or a minimum by an accurate sketch, by the sign of the second differential, or by inspecting the sign of the gradient on either side of it.

Sketching Curves

The rapid drawing of a curve from its equation alone, showing the features that identify it rather than accurately plotted points. The five required families are linear, quadratic, cubic, reciprocal and exponential. A sketch must show the applicable intercepts, roots, turning points and axis of symmetry, any vertical or horizontal asymptotes, and the correct behaviour at each end. A reciprocal curve approaches its asymptotes without ever meeting or crossing them.

• Core C2.1 “\(-2^2=4\), so \(-2\) squared is 4.” Correct \(-2^2\) means \(-(2^2)=-4\); \((-2)^2=4\). They are different expressions. When you substitute a negative value, write the brackets — every time, without deciding whether they matter.
• Extended E2.3 “\(\dfrac{x+3}{x}=3\), because the \(x\)s cancel.” Correct Cancellation removes a common factor of the whole numerator and the whole denominator. Here \(x\) is a factor of the denominator but only a term of the numerator. Test it: at \(x=1\) the expression is 4, not 3.
• Extended E2.3 “I cancelled the \((x-2)\), so \(x=2\) is fine now.” Correct Restrictions are inherited from the original expression and survive every simplification. \(\frac{(x-2)(x-3)}{(x-2)(x+2)}\) simplifies to \(\frac{x-3}{x+2}\), but \(x=2\) remains barred, because the expression you started with was undefined there.
• Extended E2.6 “\(-2x>6\) gives \(x>-3\).” Correct Multiplying or dividing an inequality by a negative number reverses it: \(x<-3\). Adding and subtracting never do. If you are unsure, test one value from each side.
• Core C2.7 “The first differences are 5, 7, 9, so the sequence is linear with a changing gradient.” Correct A sequence is linear only when the first difference is constant. Differences of 5, 7, 9 are not constant, so this sequence is not linear; a constant second difference identifies a quadratic. Extended a constant third difference identifies a cubic, and a constant ratio identifies an exponential sequence.
• Extended E2.13 “\(f^{-1}(x)=\dfrac{1}{f(x)}\).” Correct The \(-1\) in \(f^{-1}\) is not an index. \(f^{-1}\) is the function that reverses \(f\): if \(f(4)=7\) then \(f^{-1}(7)=4\). For \(f(x)=3x-5\), \(f^{-1}(x)=\frac{x+5}{3}\), while \(\frac{1}{f(x)}=\frac{1}{3x-5}\) — nothing like it.
• Extended E2.9 “The area under the speed–time graph gives the acceleration.” Correct On a speed–time graph the gradient is acceleration and the area is distance travelled. Check with units: \(\text{m/s}\div\text{s}=\text{m/s}^2\) for the gradient, and \(\text{m/s}\times\text{s}=\text{m}\) for the area. The units settle the argument every time.
• Extended E2.11 “The reciprocal curve eventually meets its asymptote.” Correct For \(y=\frac{a}{x}+b\) with \(a\ne0\), the value \(y=b\) would need \(\frac{a}{x}=0\), which no real \(x\) achieves. The branches approach \(y=b\) and never reach it, and they never cross \(x=0\) either. Draw them approaching, not touching.
• Core C2.2 “\(6x^2+9x=3(2x^2+3x)\), so that is factorised.” Correct Every term also contains \(x\), so the highest common factor is \(3x\), not 3. The full factorisation is \(3x(2x+3)\). Cambridge defines “factorise” as factorise fully, so take out the highest common numerical factor and the lowest power of every letter present in all terms, in one move.
• Core C2.2 “\((x+3)^2=x^2+9\).” Correct A squared bracket is a product of two brackets, so it has four parts: \((x+3)(x+3)=x^2+3x+3x+9=x^2+6x+9\). Squaring is not distributive over addition. Test it at \(x=1\): the left side is \(16\), and \(1+9=10\). The middle term is the one that disappears, and it is the one that carries the marks.

• If you are on the Core route, you can trust the boundary. No Core explanation, worked example, formula, diagram or answer in this chapter relies on anything inside an Extended depth panel or an Extended-only section. Skipping every Extended block leaves a complete, self-contained course.
• Use the reveals honestly. Every answer in this chapter is hidden behind a button for one reason: retrieval practice only works if you actually retrieve. Write something down — even a wrong something — before you press “Show answer”. If you are printing this chapter, all answers print in full.
• Paper numbers are not interchangeable between syllabuses. If a book, a worksheet or a website talks about “Paper 1 and Paper 2” as the whole qualification, it is describing a different syllabus. In 0580, Paper 1 is a Core component and Paper 2 is an Extended component, and they are not the same examination.
• The List of formulas. A List of formulas is printed in the question paper. For this topic the one that matters is the quadratic formula: for \(ax^2+bx+c=0\) with \(a\ne0\), \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] That is an Extended requirement (E2.5) — quadratic equations are not part of the Core course — so it matters on Papers 2 and 4 only. You do not need to memorise it, but you do need to be fluent enough with it that reading it off the sheet costs seconds, not minutes. Nothing else in Topic 2 is given: the difference of two squares, the completed-square procedure, the index laws, the \(n\)th-term methods and the derivative of \(ax^n\) are all recall.
• One check is worth ten re-readings. The E step takes about fifteen seconds and catches the majority of sign slips, dropped factors and arithmetic errors. Substituting a single convenient number — usually \(x=0\), \(x=1\) or \(x=2\) — into both the original and your answer is the fastest verification in algebra. Choose a value that is allowed; on the Extended route, also make sure it is not a root of anything you cancelled.
• “Factorise” means “factorise fully”. The Cambridge syllabus states this explicitly for both routes, so it is never a partial-credit judgement call. On the Core route that means the common factor must be the highest one; on the Extended route it also means asking whether the bracket you have just written factorises again.
• Logarithms are not required. The syllabus states this explicitly, and it is a useful piece of information rather than merely a restriction: it tells you that a common base must exist for any index equation you are set. If you cannot find one, you have misread a number — check whether that 32 is really 32 (\(=2^5\)) and not 33.
• Shade the unwanted region. This surprises students who have learned the opposite convention elsewhere, and it is stated explicitly in the syllabus. The practical reason is good: when three or four inequalities overlap, shading what you don't want leaves the answer as the one clear, unshaded patch, instead of an unreadable pile of overlapping shading. Always label that clear patch \(R\), and always read the question in case it directs otherwise.
• The shortcut, and why it works. For a linear sequence, \(T_n=dn+(T_1-d)\) where \(d\) is the common difference. The constant is “the term before the first” — the zeroth term. For \(5,9,13,\ldots\) with \(d=4\), the zeroth term is \(5-4=1\), giving \(T_n=4n+1\) immediately.
• Describe deceleration physically. The gradient of the final phase is \(\frac{0-16}{4}=-4\ \mathrm{m/s^2}\). The correct description is “the car decelerates at \(4\ \mathrm{m/s^2}\)” or “the acceleration is \(-4\ \mathrm{m/s^2}\)”. What you must not write is “the speed is \(-4\)” — the speed is falling, but it is never negative on this graph. Say what is physically happening, and let the sign describe the change.
• Exponential curves never repeat a \(y\)-value. Unlike a quadratic, an exponential graph is always increasing or always decreasing, so it crosses any horizontal line at most once. If your plotted exponential turns around, a table value is wrong — the usual cause is evaluating \(2^{-3}\) as \(-8\) instead of \(\frac18\).
• A clarification worth knowing about. Version 2 of this syllabus, published in February 2024, clarified the E2.11 graph expectations; the change is carried in the update notice that accompanies the current version 3. In practice the families that gained the most from that clarification are the reciprocal and the exponential: both now clearly require asymptote behaviour to be shown correctly. If you learned this topic from older material that treated those two lightly, they are the two to check.
• The cross-check that costs one line. The turning point of a parabola always sits on its axis of symmetry, and the axis of symmetry sits halfway between the roots. Here \(\frac{-1+5}{2}=2\) ✓, which confirms the completed square independently. If the two disagree, the completing-the-square arithmetic is where to look, and it is almost always the constant term. The full method for completing the square is in section 2.2.
• Two boundaries this row draws explicitly. First, the powers you will be asked to differentiate are positive integers or zero — there is no \(x^{-1}\) or \(x^{1/2}\) to differentiate at 0580, even though such powers do appear in the graphs of E2.10. Second, the syllabus states that candidates are not expected to identify points of inflection. If a stationary point is neither a maximum nor a minimum, no 0580 question will require you to name it.
• Range is about outputs, not inputs. The two are easy to swap under pressure. Domain = what goes in; range = what comes out. For \(g(x)=x^2\) with domain \(\{-2,-1,0,1,2\}\), the range is \(\{0,1,4\}\) — three values, not five, because \(-2\) and \(2\) both produce 4 and a set lists each value once.
• Use these tables actively. Cover the right-hand column and reconstruct it from the left. Recognition is not the same as recall, and only recall is available to you in an exam hall.
• Work it as a test. No notes, no calculator: allow about fifteen minutes on the Core route and thirty-five on the Extended route. Write full working. Then mark yourself against the answers and, for every mark lost, write down which subtopic it belonged to.
• Marking guide — read the line for your route. Core route, out of 5. Five correct: this chapter is secure — move to the mixed challenge. Three or four: you know the methods but are losing marks to signs, brackets or “fully”, so work the mistake clinic before re-testing. Two or fewer: the gaps are in the methods themselves — note which subtopic each lost question belonged to and re-read those sections rather than doing more questions. Extended route, out of 14. Twelve or more: secure — move to the mixed challenge. Eight to eleven: method is there, detail is leaking; work the mistake clinic. Seven or fewer: re-read the sections the lost questions came from, in order, rather than attempting more questions.
• Read the marks before you start writing. A 1-mark question wants a number; a 4-mark question wants several separate, visible steps of working. If you find yourself writing half a page for one mark, you have misread the question.
• Scoring the mixed challenge — read the line for your route. Core route, out of 13. 12–13: this chapter is done; keep it warm with the spaced-review plan. 9–11: solid method, marks leaking at the detail level — signs, brackets and “fully”. 6–8: the methods are there but not yet automatic; redo the Core items in the section labs. Below 6: work through the nine Core teaching sections in order rather than doing more questions. Extended route, out of 47. 41–47: done; move to the spaced-review plan. 32–40: solid method, marks leaking at the detail level — restrictions, signs, and “fully”. 22–31: the methods are there but not yet automatic; redo the section labs. Below 22: work through the thirteen teaching sections in order, because more questions will only re-confirm the same gaps.
• Interleave, do not block. Once Chapter 2 is secure, stop practising it on its own. Mix its questions in with Coordinate Geometry and Mensuration, because the difficulty in a real paper is recognising which topic a question belongs to. Practising twenty algebra questions in a row removes that difficulty artificially and flatters your sense of how ready you are.