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### 1. **Function**: \( f(x) = \frac{1}{2}x^2 - 5x - 3 \)
   - This is a quadratic function of the form \( f(x) = ax^2 + bx + c \), where \( a = \frac{1}{2} \), \( b = -5 \), and \( c = -3 \).

### 2. **Domain**:
   - For quadratic functions, the domain is all real numbers.
   - **Domain**: \( (-\infty, \infty) \)

### 3. **Range**:
   - Since \( a = \frac{1}{2} \) is positive, the parabola opens upwards.
   - To find the range, we need to determine the vertex, which gives the minimum value of the function.
   - The x-coordinate of the vertex is given by \( x = \frac{-b}{2a} = \frac{-(-5)}{2 \times \frac{1}{2}} = 5 \).
   - Substituting \( x = 5 \) into the equation for \( f(x) \):
     \[
     f(5) = \frac{1}{2}(5)^2 - 5(5) - 3 = \frac{1}{2}(25) - 25 - 3 = 12.5 - 25 - 3 = -15.5
     \]
   - Thus, the minimum value is \( -15.5 \), and the range is:
   - **Range**: \( [-15.5, \infty) \)

### 4. **Intercepts**:
   - **Y-intercept**: When \( x = 0 \):
     \[
     f(0) = \frac{1}{2}(0)^2 - 5(0) - 3 = -3
     \]
     So, the y-intercept is \( (0, -3) \).

   - **X-intercepts**: We need to solve \( \frac{1}{2}x^2 - 5x - 3 = 0 \) using the quadratic formula:
     \[
     x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
     \]
     Substituting \( a = \frac{1}{2}, b = -5, c = -3 \):
     \[
     x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \times \frac{1}{2} \times (-3)}}{2 \times \frac{1}{2}} = \frac{5 \pm \sqrt{25 + 6}}{1} = 5 \pm \sqrt{31}
     \]
     So, the x-intercepts are:
     \[
     x = 5 + \sqrt{31}, \quad x = 5 - \sqrt{31}
     \]

### 5. **End Behavior**:
   - As \( x \to \infty \), \( f(x) \to \infty \).
   - As \( x \to -\infty \), \( f(x) \to \infty \).
   - Since the parabola opens upwards, both ends of the graph go to \( +\infty \).

### Filling the chart:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [-15.5, \infty) \)
- **Y-intercept**: \( (0, -3) \)
- **X-intercepts**: \( (5 + \sqrt{31}, 0), (5 - \sqrt{31}, 0) \)
- **End behavior**: As \( x \to \infty, f(x

Find the characteristics of the quadratic function f(x) = (1/2)x^2 - 5x - 3. Fill out details such as domain, range, intercepts, and end behavior.

Understand the properties of the quadratic function f(x) = (1/2)x^2 - 5x - 3. Explore domain, range, intercepts, and end behavior in this detailed math problem breakdown. Ideal for high school students learning about quadratic functions and their characteristics.

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