Algebra and Graphs — CIE IGCSE Mathematics Revision Notes
What you'll learn
- How to manipulate algebraic expressions confidently: expanding, factorising and simplifying fractions
- The laws of indices and how they apply to algebraic terms
- Three methods for solving quadratic equations and when each is quickest
- How to solve simultaneous equations, including one linear and one non-linear
- Sequences: linear, quadratic and geometric, and how to find the nth term
- Function notation, composite functions and inverse functions
- The shapes of the standard graph types, and how to read solutions and gradients from a curve
Key terms and definitions
Expression — a collection of terms with no equals sign, such as 3x + 5.
Equation — a statement that two expressions are equal, which can be solved.
Identity — a statement true for every value of the variable, written with ≡.
Coefficient — the number multiplying a variable. In 7x² the coefficient is 7.
Factorise — write an expression as a product of factors. The reverse of expanding.
Discriminant — the quantity b² − 4ac inside the quadratic formula. It tells you how many real roots there are.
Subject of a formula — the variable on its own, usually on the left.
nth term — a rule giving any term of a sequence directly from its position n.
Function — a rule assigning exactly one output to each input, written f(x).
Composite function — one function applied to the output of another, written fg(x), meaning do g first.
Inverse function — the function that undoes f, written f⁻¹(x).
Asymptote — a line a curve approaches but never reaches.
Core concepts
Algebraic manipulation
Expanding brackets. Multiply every term inside by the term outside. For two brackets, multiply each term in the first by each in the second: (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15.
Factorising comes in four common forms and recognising which one you are looking at saves most of the work:
- Common factor: 6x² + 9x = 3x(2x + 3)
- Difference of two squares: x² − 49 = (x + 7)(x − 7). This only works for a subtraction of two squares — x² + 49 does not factorise.
- Quadratic trinomial: find two numbers that multiply to give ac and add to give b
- Grouping for four terms: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
Algebraic fractions follow exactly the same rules as numerical ones. Factorise the top and bottom first, then cancel common factors — never individual terms. In (x² − 9)/(x + 3) you factorise to (x + 3)(x − 3)/(x + 3) and cancel to x − 3.
Rearranging formulae. Do the same operation to both sides, working the reverse order of operations, and collect every term containing the new subject on one side before factorising it out.
Laws of indices
These apply to numbers and to algebraic terms identically:
- xᵃ × xᵇ = xᵃ⁺ᵇ
- xᵃ ÷ xᵇ = xᵃ⁻ᵇ
- (xᵃ)ᵇ = xᵃᵇ
- x⁰ = 1 for any non-zero x
- x⁻ᵃ = 1 ÷ xᵃ
- x^(1/2) = √x, and more generally x^(a/b) = the b-th root of xᵃ
A fractional index therefore means a root and a negative index means a reciprocal. Both appear regularly on the extended paper.
Solving equations
Linear equations. Expand, collect like terms, and isolate the variable.
Quadratic equations have three methods, and choosing well saves time:
Factorising is fastest when it works. Set the equation to zero, factorise, and set each factor to zero.
The quadratic formula always works: x = (−b ± √(b² − 4ac)) ÷ 2a. Use it when factorising is not obvious or the roots are not whole numbers.
Completing the square writes the equation as (x + p)² + q = 0. It is the method to use when a question asks for the turning point, or for the minimum or maximum value.
The discriminant b² − 4ac tells you the number of roots before you solve: positive gives two distinct roots, zero gives one repeated root, and negative gives none.
Simultaneous equations. For two linear equations, use elimination — scale one or both until a variable's coefficients match, then add or subtract. For one linear and one quadratic, substitution is the only route: rearrange the linear equation for one variable, substitute into the quadratic, and solve. Remember to find both coordinates for each solution.
Inequalities are solved like equations with one crucial exception: multiplying or dividing by a negative number reverses the inequality sign.
Sequences
Linear sequences have a constant first difference d. The nth term is dn + (the term that would sit at position zero). For 5, 8, 11, 14 the difference is 3, so the rule is 3n + 2.
Quadratic sequences have a constant second difference. The coefficient of n² is half that second difference; subtract that part and the remainder is a linear sequence you already know how to handle.
Geometric sequences multiply by a constant ratio r. The nth term is ar^(n−1), where a is the first term.
Always verify your rule on two terms before using it — testing only the first term catches almost nothing.
Direct and inverse proportion
Proportion questions are algebra in disguise and follow one reliable procedure.
Direct proportion means one quantity is a constant multiple of another, written y ∝ x and turned into the equation y = kx. As one doubles, so does the other. The variation may involve a power — y ∝ x² gives y = kx².
Inverse proportion means one rises as the other falls, written y ∝ 1 ÷ x and turned into y = k ÷ x. Their product is constant. Again a power is possible: y ∝ 1 ÷ x² gives y = k ÷ x².
The method never changes:
- Write the proportion statement as an equation with the constant k.
- Substitute the pair of values you are given and solve for k.
- Rewrite the equation with k now known.
- Use it to answer the question.
The most common error is skipping step 2 and trying to scale the numbers directly, which only works for simple direct proportion and fails as soon as a square or an inverse is involved.
Functions
Notation. f(x) = 3x − 1 means the rule triples the input and subtracts one. To find f(4), substitute: 3(4) − 1 = 11.
Composite functions. fg(x) means apply g first, then f. The order matters and reversing it is the single most common error here.
Inverse functions. To find f⁻¹(x): write y = f(x), swap x and y, then rearrange for y. The inverse undoes the function, so f(f⁻¹(x)) = x.
Graphs and their shapes
Recognising a graph's family from its equation is worth marks on its own:
| Equation | Shape |
|---|---|
| y = mx + c | straight line, gradient m, y-intercept c |
| y = ax² + bx + c | parabola — opens upward if a > 0, downward if a < 0 |
| y = ax³ + … | cubic — one or two turning points |
| y = a ÷ x | reciprocal — two branches, asymptotes on both axes |
| y = aˣ | exponential — rapid growth, never touches the x-axis |
| y = sin x, cos x, tan x | periodic waves |
Straight lines. Gradient m = rise ÷ run. Parallel lines share a gradient; perpendicular gradients multiply to −1.
Reading solutions from a graph. Where a curve crosses the x-axis, y = 0, so those are the roots. To solve a different equation from the same curve, rearrange it so one side matches the plotted function and draw the other side as a line — the intersections give the solutions.
Gradient of a curve. The gradient changes continuously, so you find it at a single point by drawing a tangent there and calculating the gradient of that tangent. Draw the tangent long enough to read two clear points, since a short tangent produces a badly inaccurate answer.
Worked examples
Example 1: Solving a quadratic with the formula
Solve 2x² + 5x − 3 = 0.
Here a = 2, b = 5, c = −3.
The discriminant is b² − 4ac = 5² − 4(2)(−3) = 25 + 24 = 49. It is positive, so expect two distinct roots — and it is a perfect square, which tells you the equation would also have factorised.
x = (−5 ± √49) ÷ 4 = (−5 ± 7) ÷ 4
x = 2 ÷ 4 = 0.5 or x = −12 ÷ 4 = −3
Check by substituting x = 0.5: 2(0.25) + 2.5 − 3 = 0.5 + 2.5 − 3 = 0 ✓
Example 2: The nth term of a linear sequence
Find the nth term of 5, 8, 11, 14, … and the 10th term.
The first differences are 3, 3, 3 — constant, so the sequence is linear with d = 3.
The rule begins 3n. At n = 1 that gives 3, but the first term is 5, so add 2.
nth term = 3n + 2
The 10th term is 3(10) + 2 = 32.
Test it on a second term before trusting it: n = 3 gives 3(3) + 2 = 11 ✓
Example 3: Simultaneous equations, one linear and one quadratic
Solve y = x + 1 and x² + y² = 25.
Substitute the linear equation into the quadratic:
x² + (x + 1)² = 25 x² + x² + 2x + 1 = 25 2x² + 2x − 24 = 0 x² + x − 12 = 0 (x + 4)(x − 3) = 0
So x = −4 or x = 3. Now find the matching y values from y = x + 1:
When x = −4, y = −3. When x = 3, y = 4.
The solutions are (−4, −3) and (3, 4). Check the second in the original: 9 + 16 = 25 ✓
Leaving the y values out is the commonest way to lose marks on this type.
Example 4: Inverse proportion
y is inversely proportional to the square of x. When x = 2, y = 9. Find y when x = 3.
Write the relationship: y = k ÷ x²
Substitute the known pair: 9 = k ÷ 2² = k ÷ 4, so k = 36
The equation is y = 36 ÷ x².
When x = 3: y = 36 ÷ 9 = 4
A sense check: x increased, and because the proportion is inverse, y fell — which it did, from 9 to 4.
Common mistakes and how to avoid them
Cancelling terms instead of factors. In (x + 6)/2 you cannot cancel the 6 with the 2. Factorise first; only a whole bracket can cancel with a whole bracket.
Losing a sign in the quadratic formula. When c is negative, −4ac becomes an addition. Write the substitution out in full before simplifying.
Forgetting to reverse an inequality. Dividing by a negative flips the sign every time.
Getting fg(x) the wrong way round. The function nearest the x acts first.
Giving only one coordinate. A non-linear simultaneous pair has two full solutions, each with an x and a y.
A tangent drawn too short. Extend it right across the grid so the two points you read are far apart, or the gradient will be well out.
Assuming every quadratic factorises. Check the discriminant first; if it is not a perfect square, go straight to the formula.
Exam technique for Algebra and Graphs
This topic appears across both papers and is one of the heaviest-weighted on the syllabus, so the time spent here pays back widely.
Show every line of working. Method marks are available even when the final answer is wrong, and an unsupported answer scores nothing if it is incorrect.
Read what the question asks for. "Solve" wants values; "factorise" wants a product; "simplify" wants a tidier expression, not an answer.
Watch the accuracy instruction. Give three significant figures unless told otherwise, and keep full accuracy in your calculator until the final step rather than rounding partway.
Use the graph you are given. If a grid is printed, the question expects a graphical method, and reading values off accurately is the intended route.
Check by substitution. Putting your solution back into the original equation takes seconds and catches sign errors reliably.
Label your sketches. Mark intercepts and turning points; a sketch with no labelled features rarely earns full marks.
Quick revision summary
- Four factorising patterns: common factor, difference of two squares, trinomial, grouping
- Indices: add to multiply, subtract to divide, multiply to raise a power; negative means reciprocal, fractional means root
- Quadratics: factorise if you can, formula if you cannot, complete the square for turning points
- Discriminant b² − 4ac: positive gives two roots, zero gives one, negative gives none
- Simultaneous equations — elimination for two linear, substitution when one is quadratic, and give both coordinates
- Linear sequence dn + c from the first difference; quadratic from the second difference; geometric ar^(n−1)
- fg(x) means g first; find f⁻¹ by swapping x and y then rearranging
- Know the six standard graph shapes on sight
- Curve gradient = gradient of a tangent drawn at that point, extended well across the grid
- Proportion: write the equation with k, substitute to find k, then use it — never scale numbers directly
- Reversing an inequality when dividing by a negative is the most-missed single rule in this topic