Revision Notes

Algebra and Graphs

These comprehensive revision notes cover everything you need to know about algebra for the Cambridge IGCSE Mathematics 0580 examination. Written by Teacher Rig, each section includes key concepts, essential formulas, and practical exam tips to help you achieve your best grade.

Expanding and Factorising

Expanding means removing brackets by multiplying. Single bracket: multiply each term inside by the term outside. Double brackets: use FOIL (First, Outer, Inner, Last). Factorising is the reverse - putting an expression into brackets. Always look for a common factor first, then try the difference of two squares or trinomial factorisation.

Key Formulas

a(b + c) = ab + ac
(x + a)(x + b) = x squared + (a+b)x + ab
a squared - b squared = (a+b)(a-b)

Exam Tips

Always check your expansion by substituting a value like x = 1
For factorising trinomials, find two numbers that multiply to give c and add to give b
Difference of two squares is very common - look for it whenever you see a subtraction of perfect squares

Solving Linear Equations

The goal is to isolate x by performing the same operation to both sides. Work systematically: expand brackets first, collect x terms on one side, collect number terms on the other, then divide. With fractions, multiply through by the LCM of the denominators first.

Exam Tips

Show every step of working - method marks are awarded even if the final answer is wrong
With fractions, multiply EVERY term by the common denominator
Always check your answer by substituting back into the original equation

Solving Quadratic Equations

There are three methods: (1) Factorising - when the expression can be written as two brackets. (2) Quadratic formula - works for any quadratic but may give irrational answers. (3) Completing the square - useful for finding the turning point. Always try factorising first as it is quickest.

Key Formulas

x = (-b +/- sqrt(b squared - 4ac)) / (2a)
ax squared + bx + c = a(x + b/(2a)) squared + c - b squared/(4a)

Exam Tips

If the question says 'give answers to 2 decimal places', you MUST use the quadratic formula (it will not factorise)
The discriminant b squared - 4ac tells you the number of solutions: positive = 2, zero = 1, negative = 0
When using the formula, write it down before substituting values

Simultaneous Equations

Two equations with two unknowns can be solved by elimination or substitution. Elimination: make the coefficients of one variable the same, then add or subtract. Substitution: rearrange one equation to express one variable in terms of the other, then substitute. For one linear and one quadratic, you must use substitution.

Exam Tips

For elimination, multiply one or both equations to match coefficients before adding/subtracting
For linear-quadratic pairs, substitute the linear into the quadratic, not the other way around
Always check your answer in BOTH original equations

Rearranging Formulas

Treat formula rearrangement exactly like solving an equation, but instead of getting a number, you get an expression. The subject is the variable you are solving for. Perform inverse operations in the reverse order to how they would be applied.

Exam Tips

If the new subject appears more than once, factorise to collect all instances together
Squaring or square-rooting may be needed - remember to consider both positive and negative roots where appropriate
Write down each step clearly showing what you did to both sides

Function Notation

f(x) represents a function of x. f(3) means substitute x = 3 into the function. Functions can be thought of as machines: input goes in, the rule is applied, output comes out. The domain is the set of allowed inputs, and the range is the set of possible outputs.

Key Formulas

f(a) means replace x with a in the expression for f(x)

Exam Tips

f(x) is NOT f multiplied by x
When finding f(3), replace EVERY x in the expression with 3
The domain tells you what x values are allowed; the range tells you what y values are possible

Composite Functions

A composite function applies one function then another. fg(x) means apply g first, then apply f to the result. This is read as 'f of g of x'. The order matters: fg(x) is usually different from gf(x).

Key Formulas

fg(x) = f(g(x)): apply g first, then f
gf(x) = g(f(x)): apply f first, then g

Exam Tips

Read fg(x) from RIGHT to LEFT: g first, then f
To find fg(x) as an expression, replace every x in f with the whole expression for g(x)
fg(x) and gf(x) are usually different - order matters

Inverse Functions

The inverse function f inverse 'undoes' what f does. If f(3) = 7, then f inverse(7) = 3. To find the inverse algebraically: write y = f(x), swap x and y, then solve for y. The graph of the inverse is a reflection of the original in the line y = x.

Key Formulas

f(f inverse(x)) = x
f inverse(f(x)) = x

Exam Tips

The three steps: write y = f(x), swap x and y, solve for y
Check by verifying that f(f inverse(x)) = x
Not all functions have inverses - a function must be one-to-one to have an inverse

Linear (Arithmetic) Sequences

A linear sequence has a constant difference between consecutive terms. The nth term is an + b where a is the common difference and b = first term - a. The general term can also be written as a + (n-1)d where a is the first term and d is the common difference.

Key Formulas

nth term = dn + (first term - d)
nth term = a + (n-1)d
Common difference d = second term - first term

Exam Tips

Always verify your formula works for n = 1, 2, and 3
The common difference d is the coefficient of n
If the sequence decreases, d is negative

Quadratic Sequences

A quadratic sequence has a constant second difference. The nth term is an squared + bn + c. To find a: a = second difference / 2. Then subtract an squared from each term to get a linear sequence, which gives bn + c.

Key Formulas

Second difference = 2a
nth term = an squared + bn + c

Exam Tips

Always check first differences first - if they are constant, it is linear, not quadratic
After finding the an squared part, the remaining terms should form a linear sequence
Some quadratic sequences can be spotted as perfect squares or other patterns

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms. Each term is found by multiplying the previous term by the common ratio r. The nth term is ar^(n-1) where a is the first term.

Key Formulas

nth term = ar^(n-1)
Common ratio r = second term / first term
Sum of n terms = a(1 - r^n) / (1 - r)

Exam Tips

The common ratio can be negative (terms alternate in sign)
If |r| < 1, the terms get smaller
Check: multiply any term by r to get the next term

Sum of Arithmetic Series

The sum of the first n terms of an arithmetic sequence can be found using a formula. This is useful for finding the total of many terms without adding them individually.

Key Formulas

S(n) = n/2 (2a + (n-1)d)
S(n) = n/2 (a + l) where l is the last term

Exam Tips

The second formula is easier if you know the last term
Both formulas give the same result
Remember Gauss's trick: pair first and last terms

Linear Inequalities

Solve linear inequalities using the same methods as equations, with one crucial difference: if you multiply or divide by a negative number, you must reverse the inequality sign. The solution is a range of values, not a single value.

Key Formulas

Solving is like equations, but flip sign when multiplying/dividing by negative

Exam Tips

FLIP the inequality sign when multiplying or dividing by a negative
Keep the variable on the left for clarity
Represent solutions on a number line: open circle for < or >, closed circle for <= or >=

Double Inequalities

A double inequality like a < x + 2 < b can be solved by performing the same operation on all three parts simultaneously. List integer values carefully, noting which endpoints are included.

Exam Tips

Apply operations to ALL THREE parts simultaneously
< means the endpoint is NOT included, <= means it IS included
List integers carefully - this is a common source of lost marks

Graphical Inequalities

To represent inequalities graphically: draw the boundary line (solid for <= or >=, dashed for < or >), then shade the correct region. Test a point like (0,0) to determine which side to shade. The feasible region satisfies all inequalities simultaneously.

Exam Tips

Use a solid line for <= or >= and a dashed line for < or >
Test (0,0) unless the line passes through the origin
Label the required region clearly - shade the region you want or the region you do not want, as instructed

Quadratic Inequalities

To solve a quadratic inequality: (1) solve the corresponding equation to find the critical values, (2) sketch the parabola, (3) read off the solution from the graph. For ax squared + bx + c <= 0 with a > 0, the solution is between the roots. For >= 0, the solution is outside the roots.

Key Formulas

Solve ax squared + bx + c = 0 first to find critical values

Exam Tips

Always sketch the parabola to see which region satisfies the inequality
For <= 0 (below x-axis): solution is between the roots
For >= 0 (above x-axis): solution is outside the roots (two separate intervals)

The Power Rule

Differentiation finds the gradient function (rate of change). For y = ax^n, the derivative is dy/dx = nax^(n-1). Multiply by the power, then reduce the power by 1. Constants differentiate to zero. The derivative represents the gradient of the tangent at any point on the curve.

Key Formulas

If y = ax^n, then dy/dx = nax^(n-1)
If y = c (constant), then dy/dx = 0
If y = f(x) + g(x), then dy/dx = f'(x) + g'(x)

Exam Tips

Rewrite fractions and roots as powers before differentiating: 1/x = x^(-1), sqrt(x) = x^(1/2)
Differentiate each term independently
The derivative of a constant is always zero

Finding the Gradient at a Point

The gradient of a curve at a specific point equals the value of the derivative at that point. Differentiate to find dy/dx, then substitute the x-value.

Key Formulas

Gradient at x = a is dy/dx evaluated at x = a

Exam Tips

The gradient of the TANGENT equals the gradient of the CURVE at that point
The gradient of the NORMAL is the negative reciprocal of the tangent gradient
A positive gradient means the function is increasing, negative means decreasing

Stationary Points

Stationary points occur where dy/dx = 0. To classify: find the second derivative d2y/dx2. If d2y/dx2 > 0, it is a minimum. If d2y/dx2 < 0, it is a maximum. If d2y/dx2 = 0, the test is inconclusive and you need to check the gradient on either side.

Key Formulas

Stationary point: dy/dx = 0
Maximum: d2y/dx2 < 0
Minimum: d2y/dx2 > 0

Exam Tips

Setting dy/dx = 0 is worth a mark on its own - always write this step
Find y-coordinates by substituting back into the ORIGINAL equation, not the derivative
Questions often ask you to 'determine the nature' - this means use the second derivative

Tangents and Normals

The tangent to a curve at a point touches the curve and has the same gradient as the curve at that point. The normal is perpendicular to the tangent at that point. Use y - y1 = m(x - x1) to find the equation of either.

Key Formulas

Tangent gradient = dy/dx at the point
Normal gradient = -1 / (dy/dx at the point)
Line equation: y - y1 = m(x - x1)

Exam Tips

Find the y-coordinate first, then the gradient, then use the line equation formula
For the normal, use the NEGATIVE RECIPROCAL of the tangent gradient
These questions often combine with other topics like solving simultaneous equations

Revision Checklist

Worked Examples

Past Papers

Common Mistakes

Frequently Asked Questions

What algebra topics are covered in IGCSE Maths?

The IGCSE 0580 syllabus covers algebra across both Core and Extended tiers. Key areas include expanding and factorising. Key areas include solving linear equations. Key areas include solving quadratic equations.

How important is algebra in the IGCSE exam?

Algebra is a significant part of the IGCSE Mathematics exam, typically appearing in Paper 2 (non-calculator) and Paper 4 (calculator). Questions range from straightforward calculations to multi-step problems that combine algebra with other topics.

What are the most common mistakes in algebra?

Common mistakes include not showing full working, forgetting to state units, misreading the question, and rushing through calculations. For algebra specifically, make sure you understand the underlying concepts rather than just memorising procedures.

What functions topics are covered in IGCSE Maths?

The IGCSE 0580 syllabus covers functions across both Core and Extended tiers. Key areas include function notation. Key areas include composite functions. Key areas include inverse functions.

How important is functions in the IGCSE exam?

Functions is a significant part of the IGCSE Mathematics exam, typically appearing in Paper 2 (non-calculator) and Paper 4 (calculator). Questions range from straightforward calculations to multi-step problems that combine functions with other topics.

What are the most common mistakes in functions?

Common mistakes include not showing full working, forgetting to state units, misreading the question, and rushing through calculations. For functions specifically, make sure you understand the underlying concepts rather than just memorising procedures.

What sequences topics are covered in IGCSE Maths?

The IGCSE 0580 syllabus covers sequences across both Core and Extended tiers. Key areas include linear (arithmetic) sequences. Key areas include quadratic sequences. Key areas include geometric sequences.

How important is sequences in the IGCSE exam?

Sequences is a significant part of the IGCSE Mathematics exam, typically appearing in Paper 2 (non-calculator) and Paper 4 (calculator). Questions range from straightforward calculations to multi-step problems that combine sequences with other topics.

What are the most common mistakes in sequences?

Common mistakes include not showing full working, forgetting to state units, misreading the question, and rushing through calculations. For sequences specifically, make sure you understand the underlying concepts rather than just memorising procedures.

What inequalities topics are covered in IGCSE Maths?

The IGCSE 0580 syllabus covers inequalities across both Core and Extended tiers. Key areas include linear inequalities. Key areas include double inequalities. Key areas include graphical inequalities.

How important is inequalities in the IGCSE exam?

Inequalities is a significant part of the IGCSE Mathematics exam, typically appearing in Paper 2 (non-calculator) and Paper 4 (calculator). Questions range from straightforward calculations to multi-step problems that combine inequalities with other topics.

What are the most common mistakes in inequalities?

Common mistakes include not showing full working, forgetting to state units, misreading the question, and rushing through calculations. For inequalities specifically, make sure you understand the underlying concepts rather than just memorising procedures.

What differentiation topics are covered in IGCSE Maths?

The IGCSE 0580 syllabus covers differentiation across both Core and Extended tiers. Key areas include the power rule. Key areas include finding the gradient at a point. Key areas include stationary points.

How important is differentiation in the IGCSE exam?

Differentiation is a significant part of the IGCSE Mathematics exam, typically appearing in Paper 2 (non-calculator) and Paper 4 (calculator). Questions range from straightforward calculations to multi-step problems that combine differentiation with other topics.

What are the most common mistakes in differentiation?

Common mistakes include not showing full working, forgetting to state units, misreading the question, and rushing through calculations. For differentiation specifically, make sure you understand the underlying concepts rather than just memorising procedures.

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