To rewrite the quadratic equation $y = -x^2 + 4x + 1$ in vertex form, we need to complete the square. The vertex form of a quadratic equation is typically:

$y = a(x - h)^2 + k$

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

Step 1: Factor out the coefficient of $x^2$ from the first two terms.

We want to isolate the quadratic and linear terms:

$y = - (x^2 - 4x) + 1$

Step 2: Complete the square.

To complete the square, take half of the coefficient of $x$, square it, and add and subtract that value inside the parentheses. The coefficient of $x$ is $-4$, so:

$\left( \frac{-4}{2} \right)^2 = 4$

Now, add and subtract 4 inside the parentheses:

$y = - (x^2 - 4x + 4 - 4) + 1$

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

Step 3: Simplify the expression.

Distribute the $-1$ and simplify:

$y = - (x - 2)^2 + 4 + 1$

$y = - (x - 2)^2 + 5$

Final Answer:

The equation in vertex form is:

$y = - (x - 2)^2 + 5$

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

Would you like more details on any of the steps?

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Here are 5 questions based on this process:

1. What is the significance of the vertex form of a quadratic equation?
2. How does completing the square help in finding the vertex of a quadratic equation?
3. Can you identify the direction in which the parabola opens from the vertex form?
4. How would the vertex change if the constant term were different in the original equation?
5. How do you convert a quadratic function from vertex form back to standard form?

Tip: The vertex of a parabola is the maximum or minimum point, and it provides a lot of insight into the graph's behavior!