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 \( k \).

Learn how to find the range of the quadratic function (x + 2)^2 - 3 step-by-step. This problem explores the vertex form of quadratic functions and how to determine the range based on the function's vertex. Suitable for high school students studying quadratic functions.

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