Answer
a. $f(x)=3(x-1)^{2}-2$
b. see image below
c. Minimum value $f(1)=-2$
Work Step by Step
a. Complete the square:
$3x^{2}-6x=3(x^{2}-2x)= 3(x^{2}-2\cdot(x)(1)+1^{2}-1^{2})$
$=(x-1)^{2}-3$
$f(x)=3(x-1)^{2}-3+1$
$f(x)=3(x-1)^{2}-2$
b.
To sketch, begin with the parent function $f_{1}(x)=x^{2},$
(blue, dashed line in the image)
and, since $f(x)=f_{1}(x-1)-2,$
shift the graph to the right by $1$ units,
and down $2$ units
(solid red line, see image)
c.
The graph of $f(x)=a(x-h)^{2}+k$
is a parabola, and,
if $a > 0$, then the quadratic function $f$ opens upward and
has the minimum value $k$ at $x=h=-\displaystyle \frac{b}{2a}$.
Minimum value$:\quad f(1)=-2$