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).