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To solve the given quadratic function problem, we need to analyze the function:

\[ y = \frac{1}{2}x^2 + 3x - \frac{5}{2} \]

### Step 1: Determine the Vertex
The quadratic function in the form \( ax^2 + bx + c \) can have its vertex determined using the formula \( x = -\frac{b}{2a} \).

Here, \( a = \frac{1}{2} \) and \( b = 3 \).

\[ x = -\frac{3}{2 \cdot \frac{1}{2}} = -\frac{3}{1} = -3 \]

### Step 2: Calculate the Value of the Function at the Vertex
Substitute \( x = -3 \) back into the function to find the value of \( y \):

\[ y = \frac{1}{2}(-3)^2 + 3(-3) - \frac{5}{2} \]
\[ y = \frac{1}{2}(9) - 9 - \frac{5}{2} \]
\[ y = \frac{9}{2} - 9 - \frac{5}{2} \]
\[ y = \frac{9}{2} - \frac{18}{2} - \frac{5}{2} \]
\[ y = \frac{9 - 18 - 5}{2} \]
\[ y = \frac{-14}{2} \]
\[ y = -7 \]

### Step 3: Determine if this is a Maximum or Minimum
Since the coefficient of \( x^2 \) (which is \( \frac{1}{2} \)) is positive, the parabola opens upwards. This means that the vertex represents the minimum value of the function.

Thus, the function has a minimum value of \( -7 \) when \( x = -3 \).

### Final Answer
The function \( y = \frac{1}{2}x^2 + 3x - \frac{5}{2} \) has a minimum value \( -7 \) when \( x = -3 \).

Would you like any further details or have any questions?

### Suggested Questions
1. Can you explain how to find the vertex of a quadratic function?
2. How do you determine if a quadratic function has a maximum or minimum value?
3. What are the steps to convert a quadratic function into vertex form?
4. How do you solve a quadratic equation using the quadratic formula?
5. Can you provide an example of a quadratic function with a maximum value?

### Tip
When dealing with quadratic functions, always check the sign of the coefficient of \( x^2 \) to determine if the parabola opens upwards (minimum value) or downwards (maximum value).

Explore how to find the minimum value of a quadratic function using vertex analysis. This problem involves solving a quadratic equation with detailed steps and understanding the properties of quadratic functions. Suitable for students in Grades 10-12.

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