UTM Ideals Varieties and Algorithm - Chapter 1 Section 3 Exercise 2

Problem:

Solution:

The key trigonometry identity to use is the double angle formula:

$\begin{eqnarray*} \cos(2t) &=& 2\cos^2(t) - 1 \\ y &=& 2x^2 - 1 \end{eqnarray*}$

This is a parabola with vertex $(0, -1)$, it cuts the x-axis as $(-\frac{1}{\sqrt{2}}, 0)$ and $(\frac{1}{\sqrt{2}}, 0)$.
As $t \in [-\pi, \pi ]$, $x \in [-1, 1]$, $y \in [-1, 1]$.