https://file.aiquickdraw.com/mathbot/file/173120632842033ykckbd.jpg

Let's analyze the function given in the image and extract the necessary information:

The function is:
\[
f(x) = \frac{1}{2}(x - 2)^2 - 3
\]

This is a quadratic function in vertex form, \( f(x) = a(x - h)^2 + k \), where:
- \( h \) and \( k \) represent the vertex of the parabola.
- \( a \) controls the direction and width of the parabola (if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards).

### 1. **Vertex**
Since the function is in vertex form, we can identify the vertex directly from \( (h, k) \):
- Here, \( h = 2 \) and \( k = -3 \).
- Therefore, the vertex is \( (2, -3) \).

### 2. **Axis of Symmetry**
The axis of symmetry of a parabola in vertex form \( f(x) = a(x - h)^2 + k \) is the line \( x = h \).
- For this function, the axis of symmetry is \( x = 2 \).

### 3. **Domain**
Since this is a quadratic function, the domain is all real numbers.
- In interval notation, the domain is \( (-\infty, \infty) \).

### 4. **Range**
Since \( a = \frac{1}{2} > 0 \), the parabola opens upwards, and the minimum value of \( f(x) \) is at the vertex, \( y = -3 \).
- Therefore, the range is \( [-3, \infty) \).

### Summary of Answers
- **Vertex**: \( (2, -3) \)
- **Axis of Symmetry**: \( x = 2 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [-3, \infty) \)

Would you like further details on any part of this explanation?

Here are some additional questions to consider:
1. What would change in the vertex if we modified the value of \( h \) or \( k \)?
2. How does the coefficient \( a \) affect the width of the parabola?
3. What if \( a \) was negative? How would that affect the direction and range?
4. Can the axis of symmetry ever be something other than \( x = h \) in vertex form?
5. How do you find the vertex if the equation is in standard form instead of vertex form?

**Tip**: When graphing parabolas, always identify the vertex and axis of symmetry first; they serve as central guides for sketching the shape accurately.

Graph the parabola given by f(x) = (1/2)(x - 2)^2 - 3. Give the vertex, axis of symmetry, domain, and range.

Learn how to find the vertex, axis of symmetry, domain, and range of the quadratic function f(x) = (1/2)(x - 2)^2 - 3. This example covers key concepts in Algebra, including vertex form and properties of parabolas. Suitable for students in Grades 9-10.

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