Answer
$ a.\quad$ see image
$ b.\quad \mathrm{r}^{\prime}(t) =\langle 1,2t\rangle$
$ c.\quad$ see image
Work Step by Step
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{llll}
x=t-2 & \Rightarrow & t=x+2 & *\\
y=t^{2}+1 & & &
\end{array}\right.$
Substitute $(*)$ into the equation for y:
$y=((x+2)^{2}+1$
a parabola that opens up, with vertex at $(-2,1)$
$b.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{l}
x=1\\
y=2t
\end{array}\right.$
$\mathrm{r}^{\prime}(t) =\langle 1,2t\rangle$
$c. $
position vector : black$\quad \mathrm{r}(-1)$=$\langle-3,2\rangle$
tangent vector: red$\quad \mathrm{r}^{\prime}(-1)$=$\langle 1,-2\rangle$
