Cambridge IGCSE·Mathematics 0580·Extended Estimate Gradients Using Tangents

📋Key knowledge — Functions And Graphs

Formulas & equations

Gradient of a straight line: m = (y₂−y₁) ÷ (x₂−x₁)
Equation of a line: y = mx + c
Gradient of tangent ≈ Δy/Δx over a very small interval (numerical differentiation)
Area under a curve ≈ sum of trapezium areas
For f(x): f⁻¹(x) is the inverse (reflect in y = x); fg(x) means f(g(x))

Key facts & rules

y = mx + c: m is gradient, c is y-intercept.
Parallel lines have the same gradient.
Perpendicular lines: m₁ × m₂ = −1.
To find the gradient at a point on a curve: draw a tangent and calculate its gradient.
Transformations of y = f(x): y = f(x) + a shifts up by a; y = f(x+a) shifts left by a; y = af(x) stretches vertically; y = f(ax) stretches horizontally.
Key graphs to know: y = x², y = x³, y = 1/x, y = aˣ, y = sin x, y = cos x, y = tan x.

IGCSE Functions and Graphs — Estimate gradients using tangents Practice Extended

2 Cambridge IGCSE 0580 questions on estimate gradients using tangents at the Extended tier — each with its full mark scheme. For the Core tier on the same skill, click here.

How to approach estimate gradients using tangents — open the study guide