A curve $C$ has parametric equations $x = \sin t$, $y = \sin 2t$ for $-\tfrac{\pi}{2} \le t \le \tfrac{\pi}{2}$. Find the equation of the tangent to $C$ at the point where $t = \tfrac{\pi}{6}$, giving your answer in the form $y = mx + c$.

$y = \tfrac{2\sqrt{3}}{3}x + \tfrac{\sqrt{3}}{6}$

How do I find dy/dx from parametric equations?

Differentiate x and y with respect to t, then divide: dy/dx = (dy/dt) ÷ (dx/dt). Substitute the value of t for a numerical gradient.

How do I turn parametric equations into a Cartesian equation?

Eliminate t: make t the subject of the simpler equation and substitute into the other, or, with sin and cos, use sin²t + cos²t = 1.

Is a calculator allowed in IAL P4?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P4 included. Method marks are for the working you write down, so show it.

How long is the IAL P4 exam?

Each IAL maths unit, P4 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does parametric equations come up in P4?

In our mapping of the 4 Pearson P4 papers from June 2024 to June 2025, parametric equations was tested in 4 of them (5 questions in all). Any topic can be examined in any series.

Edexcel IAL P4 Parametric Equations | A Level Math Revision

In P4 a curve can be given by parametric equations

x = f (t)

y = g (t)

. You convert to a Cartesian equation by eliminating

t

, differentiate with

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

to find tangents and normals, and find areas with

\int y \frac{d x}{d t} d t

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Start with one question on parametric differentiation. Stuck? Take a hint. No sign-in needed.

A curve has parametric equations

x = t^{2} + 1

y = 6 t

. Find the gradient of the curve at the point where

t = 1

Choose your answer · keys 1–4

\frac{1}{3}

3

6

2

P4 Parametric Equations notes

Parametric to Cartesian

Make $t$ the subject of one equation and substitute into the other: $x = t + 1$ , $y = t^{2}$ gives $y = (x - 1)^{2}$ . With trig parameters use an identity: $x = 3 \cos θ$ , $y = 3 \sin θ$ gives $x^{2} + y^{2} = 9$ . The range of $t$ gives the domain and range of the curve.

Parametric differentiation

$\frac{d y}{d x} = \frac{d y / d t}{d x / d t}$ . For $x = 2 t$ , $y = t^{2} - 1$ : $\frac{d y}{d x} = \frac{2 t}{2} = t$ . At $t = 3$ the point is $(6, 8)$ and the gradient is $3$ , so the tangent is $y = 3 x - 10$ .

Area under a parametric curve

Area $= \int y d x = \int y \frac{d x}{d t} d t$ , with the limits changed to $t$ -values. For $x = t^{2}$ , $y = 3 t$ between $x = 1$ and $x = 4$ ( $t = 1$ to $t = 2$ ): $\int_{1}^{2} 3 t \cdot 2 t d t = 14$ .

Key formulas

Parametric; implicit Learn it: $\frac{d y}{d x} = \frac{d y / d t}{d x / d t}; \frac{d}{d x} y^{2} = 2 y \frac{d y}{d x}$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P4 formulas on one page (free PDF) →

P4 Parametric Equations worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 8 marks · hard

A curve $C$ has parametric equations $x = t^{2}$ , $y = t^{3} - 3 t$ .
(a) Find $\frac{d y}{d x}$ in terms of $t$ . (3)
(b) Find the coordinates of the points on $C$ where the tangent is horizontal. (3)
(c) Find the Cartesian equation of the tangent at $t = 2$ . (2)

Mark scheme

(a) $\frac{d x}{d t} = 2 t$ , $\frac{d y}{d t} = 3 t^{2} - 3$ M1 A1. $\frac{d y}{d x} = \frac{3 t^{2} - 3}{2 t} = \frac{3 (t^{2} - 1)}{2 t}$ A1.
(b) Horizontal iff $\frac{d y}{d t} = 0$ : $3 t^{2} = 3 \Rightarrow t = \pm 1$ M1. $t = 1$ : $(1, - 2)$ ; $t = - 1$ : $(1, 2)$ A1 A1.
(c) At $t = 2$ : $(x, y) = (4, 2)$ ; gradient $= \frac{3 \cdot 3}{4} = \frac{9}{4}$ ; equation: $y - 2 = \frac{9}{4} (x - 4)$ , i.e. $y = \frac{9}{4} x - 7$ M1 A1.

Worked example 2 · 7 marks · hard

A curve has parametric equations $x = t + 1$ , $y = t^{3} - 12 t$ .
(a) Find the values of $t$ at the stationary points on the curve. (3)
(b) Find the coordinates of the stationary points. (2)
(c) Determine the nature of each stationary point. (2)

Mark scheme

(a) $\frac{d y}{d x} = \frac{3 t^{2} - 12}{1} = 3 t^{2} - 12$ M1 A1. Set $= 0$ : $t^{2} = 4 \Rightarrow t = \pm 2$ A1.
(b) $t = 2$ : $(3, - 16)$ ; $t = - 2$ : $(- 1, 16)$ A1 A1.
(c) $\frac{d^{2} y}{d x^{2}} = 6 t \cdot \frac{d t}{d x} = 6 t$ (since $\frac{d t}{d x} = 1$ ) M1. $t = 2$ : $12 > 0$ ⇒ minimum; $t = - 2$ : $- 12 < 0$ ⇒ maximum A1.

Worked example 3 · 6 marks · medium

A curve $C$ has parametric equations $x = \sin t$ , $y = \sin 2 t$ for $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Find the equation of the tangent to $C$ at the point where $t = \frac{π}{6}$ , giving your answer in the form $y = m x + c$ .

Mark scheme

$\frac{d x}{d t} = \cos t$ , $\frac{d y}{d t} = 2 \cos 2 t$ M1 A1. At $t = \frac{π}{6}$ : $\cos t = \frac{\sqrt{3}}{2}$ ; $\cos 2 t = \cos \frac{π}{3} = \frac{1}{2}$ . So $\frac{d y}{d x} = \frac{2 \cdot 1 / 2}{\sqrt{3} / 2} = \frac{2}{\sqrt{3}}$ M1 A1. Point: $(\sin \frac{π}{6}, \sin \frac{π}{3}) = (\frac{1}{2}, \frac{\sqrt{3}}{2})$ B1. Line: $y - \frac{\sqrt{3}}{2} = \frac{2}{\sqrt{3}} (x - \frac{1}{2}) \Rightarrow y = \frac{2}{\sqrt{3}} x + \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{3}}$ ; combine: $y = \frac{2 \sqrt{3}}{3} x + \frac{\sqrt{3}}{6}$ A1.

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Edexcel IAL P4 Parametric Equations: exam questions and answers, notes and worked examples

Try it first

P4 Parametric Equations notes

Parametric to Cartesian

Parametric differentiation

Area under a parametric curve

Key formulas

P4 Parametric Equations worked examples

Worked example 1 · 8 marks · hard

Mark scheme

Worked example 2 · 7 marks · hard

Mark scheme

Worked example 3 · 6 marks · medium

Mark scheme

Common P4 parametric equations mistakes

P4 Parametric Equations FAQ

How do I find dy/dx from parametric equations?

How do I turn parametric equations into a Cartesian equation?

Is a calculator allowed in IAL P4?

How long is the IAL P4 exam?

How often does parametric equations come up in P4?

Next steps

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