### Problem 29:
**Given function**: \( f(x) = \frac{1}{2}x^2 - 2x + 3 \)

**Instructions**: Typically, for these types of problems, you might be asked to determine the vertex, axis of symmetry, direction of opening, and the minimum or maximum value of the quadratic function.

### Solution:

1. **Identify the coefficients**:  
   The function is given in the standard quadratic form \( f(x) = ax^2 + bx + c \).  
   - Here, \( a = \frac{1}{2} \)
   - \( b = -2 \)
   - \( c = 3 \)

2. **Vertex**:  
   The vertex \( (h, k) \) of the parabola can be found using the formula:  
   \[
   h = -\frac{b}{2a}
   \]
   \[
   k = f(h)
   \]
   - Calculate \( h \):
     \[
     h = -\frac{-2}{2 \times \frac{1}{2}} = \frac{2}{1} = 2
     \]
   - Calculate \( k = f(2) \):
     \[
     k = \frac{1}{2}(2)^2 - 2(2) + 3 = \frac{1}{2}(4) - 4 + 3 = 2 - 4 + 3 = 1
     \]
   - Vertex: \( (2, 1) \)

3. **Axis of Symmetry**:  
   The axis of symmetry is the vertical line that passes through the vertex:
   \[
   x = 2
   \]

4. **Direction of Opening**:
   - Since \( a = \frac{1}{2} \) is positive, the parabola opens **upward**.

5. **Minimum Value**:
   - The minimum value of the function occurs at the vertex, so the minimum value is \( f(2) = 1 \).

### Summary:
- **Vertex**: \( (2, 1) \)
- **Axis of Symmetry**: \( x = 2 \)
- **Direction of Opening**: Upward
- **Minimum Value**: \( 1 \) at \( x = 2 \)

Would you like any additional details or explanations?

### Related Questions:
1. How do you derive the formula for the vertex of a quadratic function?
2. Why does the sign of the coefficient \(a\) affect the direction in which the parabola opens?
3. How do you find the y-intercept of a quadratic function?
4. Can you explain the process of completing the square to convert a quadratic function to vertex form?
5. How does the discriminant determine the number and type of roots in a quadratic equation?

### Tip:
When solving quadratic equations, remember that the vertex provides the highest or lowest point on the graph, which is crucial for understanding the function's behavior.

Explore how to analyze quadratic functions using the vertex formula and understand concepts such as the axis of symmetry and direction of opening. Detailed solution to question 29 provided.

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