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To solve the given problem and express the function \( s(x) = 0.5x^2 - 4x + 11 \) in vertex form, we need to complete the square. The vertex form of a quadratic function is \( s(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.

Let's go through the process of completing the square:

1. Start with the function: 
   \[
   s(x) = 0.5x^2 - 4x + 11
   \]
   
2. Factor out the coefficient of \( x^2 \) (0.5) from the first two terms:
   \[
   s(x) = 0.5(x^2 - 8x) + 11
   \]
   
3. Complete the square inside the parentheses. Take half of the coefficient of \( x \) (which is -8), square it, and add and subtract this value inside the parentheses:
   - Half of -8 is -4, and \( (-4)^2 = 16 \).

4. Add and subtract 16 inside the parentheses:
   \[
   s(x) = 0.5(x^2 - 8x + 16 - 16) + 11
   \]
   
5. Simplify the expression. The terms \( x^2 - 8x + 16 \) become \( (x - 4)^2 \), and we have:
   \[
   s(x) = 0.5((x - 4)^2 - 16) + 11
   \]

6. Distribute the 0.5 and simplify:
   \[
   s(x) = 0.5(x - 4)^2 - 8 + 11
   \]
   \[
   s(x) = 0.5(x - 4)^2 + 3
   \]

Now we have the function in vertex form:
\[
s(x) = 0.5(x - 4)^2 + 3
\]

### Answers:
- **(A) Intercepts**: To find the x-intercepts, set \( s(x) = 0 \) and solve for \( x \). For the y-intercept, substitute \( x = 0 \) into the original equation.
- **(B) Vertex**: The vertex is \( (4, 3) \).
- **(C) Maximum or Minimum**: Since the coefficient of \( x^2 \) (0.5) is positive, the parabola opens upwards, and the vertex represents a **minimum**.
- **(D) Range**: The range is \( [3, \infty) \) because the vertex is at 3 and the parabola opens upwards.

Would you like a detailed explanation for any of these steps?

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Here are 5 related questions to further explore the topic:
1. How do you find the x-intercepts of a quadratic function?
2. How can you determine if a function has a maximum or minimum value by looking at the equation?
3. How do you complete the square for quadratic functions with different coefficients?
4. How does the value of the leading coefficient affect the shape of the parabola?
5. What is the importance of the vertex in graphing a quadratic function?

**Tip**: Completing the square is a powerful method not only for converting to vertex form but also for solving quadratic equations.

Find the vertex form of the function s(x) = 0.5x^2 - 4x + 11. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or Minimum (D) Range

Learn how to convert the quadratic function s(x) = 0.5x^2 - 4x + 11 into vertex form using the method of completing the square. This problem also covers intercepts, vertex identification, determining maximum or minimum values, and finding the range. Suitable for students in Grades 9-10.

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