How Many Solutions?

Find all the solutions to the this equation.

Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Folium of Descartes

Age 16 to 18Challenge Level

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.

Possible approach
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.

Key questions
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 $0$?

Possible support
The question Witch of Agnesi is a little easier and might be tackled first.

Possible extension
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 $n$.