To rewrite the quadratic equation \( y = -x^2 + 4x + 1 \) in vertex form, we follow the process of completing the square. The vertex form of a quadratic equation is:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

### Steps:
1. Start with the given equation:

   \[
   y = -x^2 + 4x + 1
   \]

2. Factor out the coefficient of \( -1 \) from the first two terms:

   \[
   y = -\left(x^2 - 4x\right) + 1
   \]

3. Complete the square:
   - Take half of the coefficient of \( x \), which is \( -4 \), divide it by 2 to get \( -2 \), and then square it to get \( 4 \).
   - Add and subtract this number inside the parentheses:

   \[
   y = -\left(x^2 - 4x + 4 - 4\right) + 1
   \]
   \[
   y = -\left((x - 2)^2 - 4\right) + 1
   \]

4. Simplify the equation:

   \[
   y = -(x - 2)^2 + 4 + 1
   \]
   \[
   y = -(x - 2)^2 + 5
   \]

So, the vertex form of the equation is:

\[
y = -(x - 2)^2 + 5
\]

The vertex of the parabola is \( (2, 5) \).

Would you like more details on any step? Here are 5 related questions for deeper understanding:

1. How do you find the vertex of a parabola from its equation in standard form?
2. What is the geometric meaning of completing the square?
3. How does the coefficient of \( a \) in vertex form affect the parabola's shape?
4. How would the vertex form change if the original equation had a different leading coefficient?
5. How can you convert the vertex form back to standard form?

**Tip:** The sign of the coefficient of \( x^2 \) determines whether the parabola opens upwards (positive) or downwards (negative).

Learn how to rewrite the quadratic equation y = -x² + 4x + 1 in vertex form using the completing the square method. This detailed step-by-step solution explores key algebraic concepts and helps understand the transformation from standard form to vertex form. Suitable for students in Grades 9-12.

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