Why do this question?
The problem gives experience of investigating curves, considering
properties like symmetry, making substitutions to give the
parametric equation of the curve, and conversion between Cartesian
and parametric forms.
Differentiation of the parametric form and the use of the Chain
Rule is required to find the turning points.
Detailed guidance is given (scaffolding) to support the learner in
working through the problem.
To encourage independent learning, learners could work in pairs to
discuss and follow the steps in the problem. If they are stuck the
teacher could indicate which part of the guidance given in the
question itself might help them.
At some point a class discussion could be used to review the
process and to make sure all the learners can understand and
complete it. In this review the teacher can point out that the
steps are similar for investigating other curves.
See the scaffolding given in the question itself.
Consider how the two Cartesian graphs $x=f(t)$ and $y=g(t)$ can
give insight into the shape of the parametric graph $x=f(t)$,
$y=g(t)$. Can you use this to visualisehow the graph unfolds as $t$
increases from $0$?
Witch of Agnesi
is a little easier and might be tackled
Squareness is another graph sketching problemon the family of
relations: $x^n + y^n = 1$ where the curves have different forms
for odd and even values of $n$.