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Edexcel IGCSE Maths (4MA1) · Revision notes · Graphs & Functional Relationships

IGCSE Maths Graphs, Sequences and Calculus Revision Notes

Short, plain-English notes on linear & quadratics for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Each of the 7 sub-topics has what you need to know and one example with its answer. Cover the answer and try the example first, then practise the topic with worked solutions.

Linear Graphs

Plotting straight lines, identifying gradients / intercepts, parallel lines (y = mx + c).

A linear graph is y = mx + c where m is the gradient (rise/run) and c is the y-intercept.

Example: Find the gradient of the line through (2, −3) and (5, 6).

Answer: m = (6 − (−3)) ÷ (5 − 2) = 9/3 = 3.

Practise it: Straight Line Graphs questions and answers

Real-Life Graphs

Interpreting distance-time travel graphs and rates of physical change.

‘Real-life graphs’ include distance–time and speed–time — the gradient tells you speed or acceleration.

Example: On a distance-time graph a runner covers 300 m in 60 s. What's their speed?

Answer: Speed = gradient = 300/60 = 5 m/s.

Practise it: Real-Life Graphs questions and answers

Sequences

Generating terms and finding the general algebraic rule (nth term) of linear sequences.

A sequence is a list of numbers with a pattern. The nth-term rule generates any term from its position.

Example: Find the nth term of 3, 7, 11, 15, …

Answer: Common difference 4, first term 3 → nth term = 4n − 1.

Practise it: Sequences questions and answers

Regions on Graphs

Graphing and shading regions defined by single linear inequalities.

Shading a region uses inequalities: e.g. y < 2x + 1 shades below the line.

Example: Shade the region satisfying y ≥ x + 2.

Answer: Draw y = x + 2 as a solid line; shade above it.

Practise it: Inequalities on Graphs questions and answers

Quadratic Graphs

Plotting parabolas, identifying axes of symmetry, roots and turning points.

A quadratic graph is a parabola y = ax² + bx + c. Its vertex is the min (a>0) or max (a<0).

Example: Find the y-intercept and roots of y = x² − 2x − 3.

Answer: y-int = (0, −3). Roots: (x − 3)(x + 1) = 0 → x = 3, −1.

Practise it: Quadratic and Cubic Graphs questions and answers

Differentiation

Differentiating polynomials with the power rule; gradient of a curve at a point.

Differentiating gives the gradient function: bring the power down and reduce it by one (xⁿ → nxⁿ⁻¹).

Example: Differentiate y = 4x³ − 5x + 2.

Answer: dy/dx = 12x² − 5.

Practise it: Differentiation questions and answers · Kinematics questions and answers

Turning Points

Finding stationary points with dy/dx = 0 and classifying them with the second derivative.

At a turning point the gradient is zero: solve dy/dx = 0, then use d²y/dx² (positive → minimum, negative → maximum).

Example: Find the turning point of y = x² − 6x + 1.

Answer: dy/dx = 2x − 6 = 0 → x = 3, y = −8. d²y/dx² = 2 > 0 → minimum (3, −8).

Practise it: Quadratic and Cubic Graphs questions and answers · Differentiation questions and answers

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