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To write the quadratic function \( f(x) = x^2 - x - 12 \) in vertex form, we need to complete the square.

### Step 1: Start with the original equation.
\[ f(x) = x^2 - x - 12 \]

### Step 2: Complete the square.
- First, focus on the quadratic and linear terms: \( x^2 - x \).
- To complete the square, take the coefficient of \( x \) (which is \(-1\)), divide by 2, and square it:
  \[
  \left(\frac{-1}{2}\right)^2 = \frac{1}{4}
  \]
- Add and subtract this value inside the equation:
  \[
  f(x) = \left(x^2 - x + \frac{1}{4}\right) - \frac{1}{4} - 12
  \]
- Now, factor the perfect square trinomial:
  \[
  f(x) = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} - 12
  \]
- Combine the constant terms:
  \[
  f(x) = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} - \frac{48}{4} = \left(x - \frac{1}{2}\right)^2 - \frac{49}{4}
  \]

### Final Answer in Vertex Form:
\[ f(x) = \left(x - \frac{1}{2}\right)^2 - \frac{49}{4} \]

### Matching to the Options:
The correct answer corresponds to **Option D**:
\[ f(x) = \left(x - \frac{1}{2}\right)^2 - \frac{49}{4} \]

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. How do you find the vertex of a parabola from its vertex form?
2. What is the significance of completing the square in quadratic equations?
3. How would the process differ if the quadratic coefficient were not 1?
4. Can the vertex form be used to determine the direction in which a parabola opens?
5. How can you verify the vertex form by expanding it back to standard form?

**Tip:** Completing the square is not only useful for converting to vertex form but also plays a crucial role in solving quadratic equations and finding maximum or minimum values in calculus.

Write the following equation in vertex form: f(x) = x^2 - x - 12

Learn how to convert the quadratic function f(x) = x^2 - x - 12 to vertex form by completing the square. This process involves key algebraic concepts like the vertex form of a quadratic function and the method of completing the square. Suitable for students in Grades 9-10.

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