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.
• Open or closed, solid or broken — one idea. The question is always “is the boundary value itself included?” If the symbol has the little bar underneath (\(\le\), \(\ge\)) the boundary is included, so you draw something filled in: a closed circle, or a solid line. No bar means not included, so you draw something hollow: an open circle, or a broken line.
• 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.

• Use a letter as an unknown, a variable or a generalised number, and say which one a given problem needs.
• Substitute a value — including a negative value, in brackets — into an expression or a formula.
• Construct an expression from words, including consecutive, consecutive even and consecutive odd integers.
• Collect like terms, recognising that \(a^2\) and \(a\) are not like terms.
• Expand a single bracket, a product of two brackets, and a product of three brackets.
• Factorise fully by extracting a common factor, by grouping, and using the difference of two squares.
• Recognise and factorise a perfect square, a quadratic \(ax^2+bx+c\) and a cubic \(ax^3+bx^2+cx\).
• Complete the square for \(ax^2+bx+c\), including when \(a\ne1\), and read the turning point from the result.
• Add, subtract, multiply and divide algebraic fractions with numerical and algebraic denominators.
• Factorise before cancelling, and cancel only common factors — never terms in a sum.
• State every restriction, including the one that a divisor may not be zero, and keep it after cancelling.
• Use positive, zero, negative and fractional indices and every index law on variable expressions.
• Solve an index equation by rewriting both sides with a common base.
• Distinguish an expression, an identity, an equation and a formula, and say what each is for.
• Solve a linear equation, including one with brackets and one with the unknown on both sides.
• Solve a fractional equation, stating excluded values first and testing solutions against them at the end.
• Solve simultaneous linear equations by elimination and by substitution, and construct them from a context.
• Solve a quadratic by factorisation, by completing the square, and by the quadratic formula.
• Give exact solutions in surd form when an exact answer is required.
• Change the subject of a formula when the subject appears twice, and when it sits under a power or a root.
• Solve simple and compound linear inequalities, reversing the sign when multiplying or dividing by a negative.
• Represent a solution on a number line with correct open and closed circles.
• Draw a two-variable inequality with the correct broken or solid boundary, shading the unwanted region.
• List the inequalities that define a given region.
• Continue a sequence, give its term-to-term rule, and find its \(n\)th term using \(T_n\) notation.
• Identify a linear, quadratic, cubic or exponential sequence from its differences or its ratio.
• Find the \(n\)th term of a quadratic sequence from the second difference, and verify it against several terms.
• Write a proportion statement with \(\propto\), introduce a constant \(k\), and find \(k\) from given data.
• Handle direct and inverse proportion involving \(x\), \(x^2\), \(\sqrt{x}\), \(x^3\) and \(\sqrt[3]{x}\).
• Read and draw distance–time, speed–time and conversion graphs, with correct units.
• Interpret gradient as a rate of change, and describe acceleration and deceleration physically.
• Find distance travelled as the area under a speed–time graph built from linear sections.
• Estimate the gradient of a curve by drawing a tangent and using two well-separated points on the tangent.
• Construct a table of values and draw an accurate graph of a function built from up to three \(ax^n\) terms, or \(ab^x+c\).
• Find roots graphically, and add a line to a drawn curve to solve a transformed equation.
• Sketch linear, quadratic, cubic, reciprocal and exponential curves showing intercepts, roots, turning points, symmetry, asymptotes and end behaviour.
• Use function notation, and state the domain and range of a simple function.
• Draw and read a mapping diagram.
• Find \(f^{-1}(x)\), and explain why it is not \(\dfrac{1}{f(x)}\).
• 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.

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 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; factorising restores brackets by extracting common structure; completing the square 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.

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.

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.

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.

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.

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. 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 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.

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.

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.

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.

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.

• “\(-2^2=4\), 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.
• “\(\dfrac{x+3}{x}=3\), 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.
• “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.
• “\(-2x>6\) 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.
• “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. A constant second difference identifies a quadratic; a constant third difference identifies a cubic; a constant ratio identifies an exponential sequence.
• “\(f^{-1}(x)=\dfrac{1}{f(x)}\).” 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.
• “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.
• “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.

• 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.
• The formula list. A List of formulas is printed on page 2 of both papers. For this chapter, the one that matters is the quadratic formula: for \(ax^2+bx+c=0\) with \(a\ne0\), \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] You do not need to memorise it — but you do need to be fluent enough with it that reading it off the sheet costs you seconds, not minutes. Nothing else in Topic 2 is given: the difference of two squares, the completed-square procedure, the index laws and the nth-term methods 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, and one that is not a root of anything you cancelled.
• “Factorise” means “factorise fully”. The Cambridge syllabus states this explicitly, so it is never a partial-credit judgement call. Stopping at \(2x(x^2-4x+3)\) leaves a quadratic that plainly factorises; the complete answer is \(2x(x-1)(x-3)\). Step 4 of the method above exists for exactly this.
• 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 \(\frac18\).
• The version 2 clarification. The February 2024 update to this syllabus specifically clarified the expectations for reciprocal and exponential graphs in 2.11. Both now clearly require asymptote behaviour to be shown correctly. If you learned this topic from older material that treated those two families lightly, they are the two to check.
• 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. Twenty-five minutes, no notes, no calculator. Write full working. Then mark yourself against the answers and, for every mark lost, write down which of the twelve subtopics it belonged to.
• 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.
• 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.