LearnYMA01WMA11 · Pure Mathematics 1DifferentiationTangents and normals

Differentiation · Subtopic 4 of 4

Tangents and Normals

The tangent at a point has the curve's gradient there; the normal is perpendicular to it, with gradient $-\dfrac1m$ . Either way you need a point and a gradient, then $y-y_1=m(x-x_1)$ .

Edexcel YMA01 WMA11≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find the gradient of the tangent at a point.
Find the equation of a tangent and a normal.
Write answers in the forms $y=mx+c$ or $ax+by+c=0$ .
Find where tangents or normals meet the axes or a given line.

Pause and think

A car skids off a bend. In which direction does it slide: along the curve, along the tangent, or along the normal?

Section 2 of 14

Core idea

Start hereTouching the curve

A tangent is the straight line that just kisses a curve, matching its direction at one point. The normal sticks straight out at right angles, like a hair on a balloon.

The big idea

At $x=a$ : tangent gradient $m=f'(a)$ ; normal gradient $-\frac1m$ . Line: $y-y_1=m(x-x_1)$ .

1. Equations

Tangent and normal to y = x² at a moving point

x = -1.60

x=p

the tangent has gradient

2p

and the normal has gradient

-\dfrac{1}{2p}

. They always cross at right angles.

y-y_1=m(x-x_1),\qquad m_{\text{normal}}=-\frac1{m_{\text{tangent}}}

$(x_1,y_1)$: point on the curve.

Worked example $y=x^2-5x+4$ at $x=4$

Tangent and normal.

Goal: Two equations.
Strategy: Point, gradient, line formula.

$y=16-20+4=0:\ (4,0)$
Point.
$\frac{dy}{dx}=2x-5=3$
At $x=4$ .
$y=3(x-4)=3x-12$
Tangent.
$y=-\tfrac13(x-4)\Rightarrow x+3y-4=0$
Normal gradient $-\frac13$ .

AnswerTangent

y=3x-12

; normal

x+3y-4=0

✓ Check

3\times(-\frac13)=-1

✓.

Worked exampleNormal meets the curve again

Where else does the normal meet $y=x^2-5x+4$ ?

Goal: Point.
Strategy: Equate; $x=4$ is one root already.

$x^2-5x+4=-\tfrac13x+\tfrac43$
Equate.
$3x^2-14x+8=0$
Multiply by 3, collect.
$(3x-2)(x-4)=0\Rightarrow x=\tfrac23$
Factor using the known root.
$y=\tfrac49-\tfrac{10}3+4=\tfrac{10}9$
Substitute.

Answer

(\frac23,\frac{10}9)

✓ CheckOn the normal:

\frac23+\frac{10}3-4=0

✓.

Common misconceptionUsing f′(x) as the gradient Substitute the specific

x

-value to get a number before writing the line.

Pause and think

Tangent to $y=x^2$ at $(3,9)$ ?

Summary

$m=f'(a)$ ; normal $-1/m$ .
$y-y_1=m(x-x_1)$ .

Tangent: Line touching the curve.
Normal: Perpendicular to the tangent.

HistoryTouching lines. "Tangent" comes from the Latin tangere, "to touch". Finding tangents was one of the great problems of 17th-century mathematics; Fermat, Descartes and Barrow (Newton's teacher) each had methods before calculus unified them.

Fun factSparks fly off along tangents. Watch a grinding wheel: the sparks leave in straight lines along the tangent to the wheel at the point where they break off. The same happens to water flicked off a spinning bicycle wheel.

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