GCSE-ready- Home
- Lessons
- IGCSE Further Pure Mathematics
- Tangents and normals
Tangents and normals
🎬 The doodle video for this lesson is coming soon. Subscribe on YouTube to see it first.
A tangent just touches a curve and has the same gradient as the curve there. A normal is at right angles to the tangent.
Tangent
1. Differentiate and substitute x to find the gradient m.
2. Find the y-coordinate if it is not given.
3. Use y − y1 = m(x − x1).
y = x2 + 3x at (1, 4): m = 2x + 3 = 5, so y − 4 = 5(x − 1), y = 5x − 1.
2. Find the y-coordinate if it is not given.
3. Use y − y1 = m(x − x1).
y = x2 + 3x at (1, 4): m = 2x + 3 = 5, so y − 4 = 5(x − 1), y = 5x − 1.
Normal
The normal is perpendicular to the tangent, so its gradient is −1m.
y = x2 at (2, 4): tangent gradient 4, normal gradient −14
y − 4 = −14(x − 2), which gives x + 4y − 18 = 0.
y = x2 at (2, 4): tangent gradient 4, normal gradient −14
y − 4 = −14(x − 2), which gives x + 4y − 18 = 0.
Any function
This works for any curve you can differentiate.
y = e2x at x = 0: point (0, 1), gradient 2e0 = 2, tangent y = 2x + 1.
y = e2x at x = 0: point (0, 1), gradient 2e0 = 2, tangent y = 2x + 1.
Find the equation of the tangent to y = x3 at x = 2.
- Point: (2, 8)
- dydx = 3x2 = 12
- y − 8 = 12(x − 2)
Answer: y = 12x − 16
Tangent gradient = dydx at the point. Normal gradient = −1m. Then y − y1 = m(x − x1).
Check you have got it
Answer 7 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.