Topic summary · Pure advanced (P4)

Parametric Functions summary

Parametric equations define $x$ and $y$ via a parameter. Differentiate with $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ .

Open Learning Sheet — Parametric Functions Download Parametric Functions answers PDF

Files: P4-Parametric Functions.pdf · P4-Parametric Functions answers.pdf · free public downloads · diagrams in the PDFs

Key learning points

Eliminate $t$ when a Cartesian equation helps.
Gradient is the quotient of the parametric derivatives.
Second derivatives need an extra chain-rule step.
Set up tangents and areas parametrically when asked.

Worked example 1 — Evaluate

$f(x)=2x-1$ . Find $f(5)$ .

Working

9

Substitute.

Worked example 2 — Composite

$f(x)=x+1$ , $g(x)=2x$ . Find $f(g(3))$ .

Working

7

Apply $g$ then $f$ .

Worked example 3 — Inverse

$f(x)=3x+2$ . Find $f^{-1}(x)$ .

Working

f^{-1}(x)=\tfrac{x-2}{3}

Swap and rearrange.

Worked example 4 — Gradient

If $x=t^2$ and $y=t^3$ , find $\frac{dy}{dx}$ .

Working

\frac{3t}{2}\ (t\ne0)

Divide $3t^2$ by $2t$ .

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.

Open Learning Sheet Answers PDF

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