A-level · Beta

Tangent and normal equations

Move a point on a curve and construct the two perpendicular lines.

Move the point using its x-coordinate. Compare the tangent and the perpendicular normal.

f(x)=x^{2}

The gradients multiply to −1. Both lines pass through P.

Point P: (1, 1)
Tangent gradient: 2
Normal gradient: -0.5

Tangent: $y-(1)=2(x-(1))$

Normal: $y-(1)=-\frac{1}{2}(x-(1))$

Blue: curve. Green dashed: tangent. Red dashed: normal. Axes use different scales, so judge perpendicularity from gradients rather than the apparent angle. Equations use exact fractions; decimal gradient readouts are rounded.

Predict, test and explain

Step 1 of 3

Predict

When the tangent is horizontal, what is the normal?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

When the tangent is horizontal, what is the normal?

$y=x^2,\quad x=2$

Hint

Find the point and derivative first. Only take a negative reciprocal when the derivative is nonzero.

Reveal answer

A normal is perpendicular to the tangent. When the tangent gradient is zero, the normal is vertical and has no finite gradient.

$y-4=-\tfrac14(x-2)$

Build an explanation

Hint 1

Find the point and derivative first. Only take a negative reciprocal when the derivative is nonzero.

Hint 2

A normal is perpendicular to the tangent. When the tangent gradient is zero, the normal is vertical and has no finite gradient.

Hint 3

Use the model to check a prediction, then explain the result.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the normal to y = x² at x = 2.

The point is (2, 4). Differentiate to get dy/dx = 2x.
At x = 2, the tangent gradient is 4, so the normal gradient is −1/4.
Use point-gradient form: y − 4 = −(x − 2)/4.

Answer: y − 4 = −(x − 2)/4.

Connect this idea

Watch out: A normal is perpendicular to the tangent. When the tangent gradient is zero, the normal is vertical and has no finite gradient.

Specification and learning route

AQA G3 · Edexcel Pure 7.3

AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Differentiation; Point-gradient form; Perpendicular gradients.

AQA specification · Edexcel specification