Sketch the curve described by $(x,y) = (3\cos t+2, 3\sin t -3)$

Question

Sketch the curve described by these parametric equations. $$\begin{align} x &=3\cos t+2 \\ y &=3\sin t -3 \end{align}$$ for $0 \leq t < 2\pi$.

I found the equation to be $$\left(\frac{y+3}{3}\right)^2+\left(\frac{x-2}{3}\right)^2 = 1$$ hence centre to be $(2,-3)$ and can sketch the graph on the cartesian plane but I don't know how to sketch it with this: $0 \leq t < 2\pi$.

Can someone explain and send an image of what it would look like? Thanks!

Answer 1

Very straightforward, plug in values of $t$: just notice the offset:

Here are some points and the $t$ values that led to them:

Clear now??

Answer 2

Your equation can be written as

$$(x-2)^2+(y+3)^2=9=3^2$$

your curve is then a circle with the point $C(2,-3) $ as center and Radius $ R=3$.