Proposition

Use a trigonometric identity to show that

parametrizes a portion of a parabola. Indicate exactly what portion of the parabola is covered.

Solution

Since $\cos(2t) = 2\cos^2(t) - 1$, $y = 2x^2 - 1$. Since $x, y \in [-1, 1]$, this parametrization covers $y = 2x^2 - 1$ where $x \in [-1, 1]$.