Hi!

If $\displaystyle P(t)$ has coordinates $\displaystyle \left(-\frac{\sqrt{5}}{3},\frac23\right)$, find the coordinates of the point.

a. $\displaystyle P(t + \pi) \rightarrow \left(\frac{\sqrt{5}}{3},-\frac23\right)$

Quadrant 2: moving by π goes half a circle which ends up in Quadrant 4.

b. $\displaystyle P(-t) \rightarrow \left( \frac{\sqrt{5}}{3},-\frac23 \right)$

I just switch the signs.

c. $\displaystyle P(t - \pi) \rightarrow \left( \frac{\sqrt{5}}{3},-\frac23 \right)$

Moving clockwise from Quadrant 2, ending up in Quadrant 4.

d. $\displaystyle P(-t - \pi) \rightarrow \left(-\frac{\sqrt{5}}{3},-\frac23\right)$

From b., moving clockwise from Quadrant 4, ending up in Quadrant 2.