Tangents and Normals for IGCSE Maths
Finding equations of tangent and normal lines to curves. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding tangents and norm
What You Need to Know
Finding equations of tangent and normal lines to curves. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding tangents and normals is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Tangents and Normals
A tangent to a curve at point P is the straight line that touches the curve at P with the same gradient as the curve. The normal at P is perpendicular to the tangent. To find the tangent: differentiate, find dy/dx at the point, then use y−y₁ = m(x−x₁). For the normal: use gradient = −1/(dy/dx). Both appear in IGCSE Extended Paper 4 often as multi-part questions following a stationary points question.
Step-by-Step Method
- 1
Find the gradient of the curve at the point
Differentiate y = f(x), then substitute the x-coordinate of the point to get m = dy/dx at that x.
- 2
For the tangent: use m directly
The tangent gradient equals the curve gradient at that point.
- 3
For the normal: find the perpendicular gradient
Normal gradient = −1/m (negative reciprocal).
- 4
Find the y-coordinate
Substitute x into the original equation to find y (if not given).
- 5
Write the line equation
Use y − y₁ = m(x − x₁) and rearrange into y = mx + c or ax + by = d as required.
Worked Example
Question
Find the equation of the tangent and normal to y = x² − 3x + 4 at the point where x = 3.
Solution
Step 1: Find y at x=3. y = 9 − 9 + 4 = 4. Point: (3, 4). Step 2: Differentiate. dy/dx = 2x − 3 At x=3: dy/dx = 6−3 = 3. Tangent gradient = 3. Step 3: Tangent equation. y − 4 = 3(x − 3) y = 3x − 9 + 4 y = 3x − 5 Step 4: Normal gradient = −1/3. y − 4 = −(1/3)(x − 3) y = −(1/3)x + 1 + 4 y = −(1/3)x + 5 Answers: Tangent: y = 3x−5; Normal: y = −x/3 + 5
Exam Tips for Tangents and Normals
- The tangent has the same gradient as dy/dx at the given point; the normal gradient is −1/m.
- Always find the y-coordinate of the given point by substituting x into y = f(x) — don't assume it.
- The equation y − y₁ = m(x − x₁) is the most reliable way to write the line equation without errors.
- A tangent touches the curve; a normal is perpendicular to it — they are completely different lines at the same point.
Practice Questions
Q1: Find the equation of the tangent to y = x³ − 2x at the point (2, 4).
Show hint
dy/dx = 3x²−2. At x=2: gradient=10. Tangent: y−4=10(x−2) → y=10x−16.
Q2: Find the equation of the normal to y = x² + 5x at the point where x = −1.
Show hint
y(−1)=1−5=−4. dy/dx=2x+5. At x=−1: m=3. Normal gradient=−1/3. y+4=−(1/3)(x+1).
Q3: The tangent to y = ax² + b at x = 2 has equation y = 4x + 1. Find a and b.
Show hint
dy/dx=2ax. At x=2: 4a=4 → a=1. The point (2,9) lies on the curve: 4+b=9 → b=5.
Frequently Asked Questions
What is tangents and normals in IGCSE Maths?
Finding equations of tangent and normal lines to curves.
Is tangents and normals in the Core or Extended syllabus?
Tangents and Normals is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise tangents and normals effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on tangents and normals rather than trying to cover everything at once.
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