Tracing curves and ellipses

We already saw that $x=\cos(t)$, $y=\sin(t)$ gives a circle traced counter-clockwise. In this demonstration, we'll look at something slightly more complicated: $$x(t) = a\cos(t) + h, \qquad y(t)=b\sin(t)+k,$$ where $a$, $b$, $h$ and $k$ are numbers that you specify.

Investigate the interactive figure to the right. (i) Try changing $a$, $b$, $h$ and $k$, one at a time, to understand what each of them does. (ii) How would you make a circle of radius 2, centered at the origin? (iii) How would you make an ellipse centered at the origin that is twice as wide as it is high? Or half as wide as it is high? (iv) How would you make a circle of radius 3, centered at the point (-1,4)?