Answer
a. $f(x)=(x+1)^{2}-2$
b. see image below
c. Minimum value $f(-1)= -2$
Work Step by Step
a. Complete the square:
$(x^{2}+2x+1-1) -1=(x+1)^{2}-1-1$
$f(x)=(x+1)^{2}-2$
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 left by 1 unit,
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 $f(-1)=-2$
