To find the range of the function $y = (x + 2)^2 - 3$, we need to analyze how the function behaves and determine the set of possible output values (i.e., $y$-values).

Step-by-Step Analysis:

1. Identify the form of the function:
   The function is in the form $y = a(x-h)^2 + k$, which represents a parabola. In this case: $y = (x + 2)^2 - 3$ Here, $a = 1$, $h = -2$, and $k = -3$.

2. Determine the vertex of the parabola:
   The vertex form tells us the vertex of the parabola is at $(h, k)$. For this function, the vertex is at: $(-2, -3)$ The vertex is the lowest point on the graph because the parabola opens upwards (since $a > 0$).

3. Determine the minimum value:
   Since the vertex is the lowest point and the parabola opens upwards, the minimum value of $y$ is $-3$.

4. Range of the function:
   The parabola continues infinitely upwards from $y = -3$. Thus, the range of the function is all values of $y$ greater than or equal to $-3$.

Range in Set Notation:

$\text{Range: } y \geq -3$ or equivalently, $\text{Range: } y \in [-3, \infty)$

Would you like more details or have any questions?

Here are some related questions:

1. How would the range change if the function was $y = -(x + 2)^2 - 3$?
2. How do you find the x-intercepts of the function $y = (x + 2)^2 - 3$?
3. What does the vertex tell you about the function's maximum or minimum value?
4. Can you describe the transformation from the parent function $y = x^2$ to $y = (x + 2)^2 - 3$?
5. How would the graph of $y = (x + 2)^2 - 3$ change if the constant term were positive instead of negative?

Tip: The range of a quadratic function $y = a(x-h)^2 + k$ depends on the sign of $a$: if $a > 0$, the range starts from $k$ and goes to infinity; if $a < 0$, it starts from negative infinity and goes up to $k$.