To solve this quantitatively and construct the graph of the quadratic function $y = x^2 + 2x + 1$, we first need to calculate the values of $y$ for various values of $x$ in the range from $-3$ to $3$.

Step-by-Step Calculation of $y$ Values:

1. When $x = -3$: $y = (-3)^2 + 2(-3) + 1 = 9 - 6 + 1 = 4$
2. When $x = -2$: $y = (-2)^2 + 2(-2) + 1 = 4 - 4 + 1 = 1$
3. When $x = -1$: $y = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$
4. When $x = 0$: $y = (0)^2 + 2(0) + 1 = 1$
5. When $x = 1$: $y = (1)^2 + 2(1) + 1 = 1 + 2 + 1 = 4$ 6