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Let's analyze the function given: \( y = 2(x + 2)^2 - 3 \).

### Step 1: Identify the Type of Function
This function is a quadratic function in vertex form: \( y = a(x - h)^2 + k \), where:
- \( a = 2 \) (which means it opens upwards because \( a > 0 \)),
- \( h = -2 \),
- \( k = -3 \).

### Step 2: Find the Vertex
The vertex of a quadratic function in this form is \( (h, k) \). So, the vertex is:
\[
(-2, -3)
\]
Since the parabola opens upwards, this vertex represents the **minimum** value of the function.

### Step 3: Domain and Range
- **Domain**: For any quadratic function, the domain is all real numbers (\( -\infty, \infty \)).
- **Range**: Since the minimum value is at \( y = -3 \), the range is \( y \geq -3 \).

### Step 4: Correct Option
Now, let's match this with the options provided:
- **Minimum value**: \( -3 \)
- **Domain**: All real numbers
- **Range**: \( y \geq -3 \)

Thus, the correct choice is:
- **First Option**: "Minimum value: -3, domain: all real numbers, range: all real numbers \( \geq -3 \)"

Would you like further details or have any questions?

### Related Questions:
1. How do you find the vertex of a quadratic function in standard form?
2. What happens to the range if \( a \) in the quadratic function is negative?
3. Can you explain how to convert a quadratic function from standard form to vertex form?
4. How do you determine whether a quadratic function has a minimum or maximum value?
5. What is the effect of changing the value of \( h \) in the vertex form of a quadratic equation?

### Tip:
The vertex form of a quadratic function \( y = a(x - h)^2 + k \) is particularly useful for quickly identifying the vertex and determining whether the graph opens upwards or downwards.

Identify the maximum or minimum value and the domain and range of the graph of the function y = 2(x + 2)^2 - 3.

Learn how to identify the minimum value, domain, and range of the quadratic function y = 2(x + 2)^2 - 3. This example demonstrates how to find the vertex and use it to determine the properties of the quadratic graph. Ideal for students in Grades 9-12.

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