https://file.aiquickdraw.com/mathbot/file/1727362642833o4qphntl.jpg

Let's break down and solve the problem in the image step by step.

We are given the function:

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

### Part (a) 
We need to determine \( f(0) \) and \( f(6) \), and explore the theoretical concepts involved.

1. **Finding \( f(0) \):**
   \[
   f(0) = \frac{1}{2}(0)^2 - 3 = 0 - 3 = -3
   \]

2. **Finding \( f(6) \):**
   \[
   f(6) = \frac{1}{2}(6)^2 - 3 = \frac{1}{2}(36) - 3 = 18 - 3 = 15
   \]

So:
- \( f(0) = -3 \)
- \( f(6) = 15 \)

### Theoretical Concepts:
- **Function Evaluation**: Substituting specific values of \( x \) into the function to find corresponding outputs.
- **Quadratic Function**: \( f(x) = \frac{1}{2}x^2 - 3 \) is a quadratic function, which represents a parabola.
- **Vertex and Symmetry**: The vertex form of a quadratic helps analyze the shape and position of the parabola.

### Part (b)
We are asked to find a function \( g(x) \) that is parallel to \( f(x) \) and passes through the point \( (4, 1) \).

1. **Parallel Function**: If \( g(x) \) is parallel to \( f(x) \), it will have the same quadratic term, i.e., the same form \( g(x) = \frac{1}{2}x^2 + b \), where \( b \) is the constant to be determined.

2. **Using the Point (4, 1)**: We substitute \( x = 4 \) and \( g(x) = 1 \) into the equation for \( g(x) \):
   \[
   g(4) = \frac{1}{2}(4)^2 + b = 1
   \]
   \[
   \frac{1}{2}(16) + b = 1
   \]
   \[
   8 + b = 1
   \]
   \[
   b = 1 - 8 = -7
   \]

Thus, the equation for \( g(x) \) is:
\[
g(x) = \frac{1}{2}x^2 - 7
\]

---

### Summary:
- \( f(0) = -3 \)
- \( f(6) = 15 \)
- \( g(x) = \frac{1}{2}x^2 - 7 \), which is parallel to \( f(x) \) and passes through \( (4, 1) \).

Would you like more details or have any specific questions?

Here are 5 related questions for further exploration:
1. What is the vertex of the quadratic function \( f(x) = \frac{1}{2}x^2 - 3 \)?
2. How do we find the axis of symmetry for a quadratic function?
3. What is the geometric interpretation of a parallel quadratic function?
4. How do changes in the constant term \( b \) affect the graph of a quadratic function?
5. What is the general form of a quadratic equation and how does it relate to this problem?

**Tip:** To find parallel functions, always ensure the coefficients of the highest-degree terms are identical!

Dada f(x) = (1/2)x^2 - 3, determinar analíticamente f(0) y f(6) y hallar g(x) que sea paralela a f(x) y que pase por el punto (4,1).

Learn how to evaluate the quadratic function f(x) = (1/2)x^2 - 3 at specific points and determine a parallel function g(x) that passes through a given point. This problem covers function evaluation, quadratic properties, and parallelism concepts, suitable for Grades 9-12.

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