To rewrite the function $f(x) = -2x^2 + 8x + 1$ in the form $f(x) = a(x - h)^2 + k$, we can complete the square.

1. Start with the original function: $f(x) = -2x^2 + 8x + 1$

2. Factor out $-2$ from the terms involving $x$: $f(x) = -2(x^2 - 4x) + 1$

3. Complete the square inside the parentheses. To do this, take half of the coefficient of $x$ (which is $-4$), square it, and add and subtract it inside the parentheses: $f(x) = -2(x^2 - 4x + 4 - 4) + 1$

4. Rewrite it as a perfect square: $f(x) = -2((x - 2)^2 - 4) + 1$

5. Distribute the $-2$: $f(x) = -2(x - 2)^2 + 8 + 1$

6. Simplify the constants: $f(x) = -2(x - 2)^2 + 9$

Now we can identify $a$, $h$, and $k$:

• $a = -2$
• $h = 2$
• $k = 9$

Thus, the answers are:

• $a = -2$
• $h = 2$
• $k = 9$

Would you like more details on the process of completing the square, or do you have any other questions?

Here are some related questions that may help you further understand the topic:

1. How can completing the square help in graphing a quadratic function?
2. What does the value of $a$ indicate about the direction of the parabola?
3. How does the vertex form of a quadratic function relate to its graph?
4. Can you find the vertex of a quadratic function without converting to vertex form?
5. How does changing $h$ and $k$ affect the graph of the function?

Tip: When completing the square, remember that adding and subtracting the same value inside a bracket does not change the expression's value; it only helps restructure it.