Work Step by Step
Note that: $x^2+2x+2 = x^2+2x+(1+1) \\x^2+2x+2=(x^2+2x+1)+1 \\x^2+2x+2=(x+1)^2+1$ Thus, the given function can be written as: $f(x) =(x+1)^2+1$ RECALL: (1) The vertex form of a quadratic function whose vertex is at $(h, k)$ is: $f(x) = a(x-h)^2+k$ (2) The graph of the quadratic function $ax^2+bx+c$ is a parabola that opens: (i) upward when $a \gt 0$; (ii) downward when $a\lt 0$. Using the vertex form in (1) above, the given function has its vertex at $(-1,1)$. Using the rule in (2) above, with $a=1$, the given function's graph is a parabola that opens upward. The parabola that opens upward and whose vertex is at $(-1, 1)$ is the one in Option (D).