Parametric Functions summary

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

Substitute.

Worked example 2 — Composite

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

Working

Apply $g$ then $f$.

Worked example 3 — Inverse

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

Working

Swap and rearrange.

Worked example 4 — Gradient

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

Working

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.