Tangent and Normal Lines

Finding the equations of tangent and normal lines to curves is a fundamental skill at A-Level. This lesson consolidates the techniques for both Cartesian and parametric curves, as required by the Edexcel 9MA0 specification.

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The Tangent Line

The tangent to a curve at a point P is the straight line that just touches the curve at P. Its gradient equals the gradient of the curve at P, which is found by differentiation.

For y = f(x)

1. Differentiate to find dy/dx.
2. Substitute the x-coordinate of P to find the gradient m.
3. Use the point-gradient form: y − y₁ = m(x − x₁).

Example: Find the tangent to y = x³ − 2x + 1 at x = 1.

dy/dx = 3x² − 2. At x = 1: dy/dx = 3 − 2 = 1. Point: y = 1 − 2 + 1 = 0, so P = (1, 0). Tangent: y − 0 = 1(x − 1), so y = x − 1.

For Parametric Curves