Parametric points
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Problem

Here is a set of equations defined parametrically in terms of \(t\).

\[\begin{align*} x &= 1+ 4 \cos t \\ y &= t+ 3 \sin t \end{align*}\]

\[\begin{align*} x &= 2 \cos t \\ y &= \sin 2t \end{align*}\]

\[\begin{align*} x &= t^2 \\ y &= t^3-4t \end{align*}\]

\[\begin{align*} x &= t^2+t+1 \\ y &= 2t^2-2 \end{align*}\]

\[\begin{align*} x &= t^3-t \\ y &= t^3-3t \end{align*}\]

\[\begin{align*} x &= 2+ 3 \cos t \\ y &= 4+ 5 \sin t \end{align*}\]

Using the interactive below or the printable cards, match up each equation (blue card) with

the location of a stationary point on the curve (green card),
the location of a critical point on the curve where the tangent is vertical (pink card),
its graph (white card).

(Note that some of the curves have other stationary points and critical points that do not appear on cards.)

\[\begin{align*} x &= 2+ 3 \cos t \\ y &= 4+ 5 \sin t \end{align*}\]

\[\begin{align*} x &= t^3-t \\ y &= t^3-3t \end{align*}\]

\[\begin{align*} x &= t^2+t+1 \\ y &= 2t^2-2 \end{align*}\]

\[\begin{align*} x &= t^2 \\ y &= t^3-4t \end{align*}\]

\[\begin{align*} x &= 2 \cos t \\ y &= \sin 2t \end{align*}\]

\[\begin{align*} x &= 1+ 4 \cos t \\ y &= t+ 3 \sin t \end{align*}\]

Stationary point at \[x=1\]

Stationary point at \[x=0\]

Stationary point at \[x=-\frac{1}{3}\]

Stationary point at \[x=\sqrt{2}\]

Stationary point at \[x=2\]

Stationary point at \[x=\frac{4}{3}\]

Critical point at \[(0,0)\]

Critical point at \[(5,4)\]

Critical point at \[\left(\frac{2}{3\sqrt{3}},\frac{8}{3\sqrt{3}}\right)\]

Critical point at \[(2,0)\]

Critical point at \[(5,0)\]

Critical point at \[\left(\frac{3}{4},-\frac{3}{2}\right)\]