Create a table for the reflection about the origin of the function f(x) = -(x+2)^2...

1. Understand Reflection About the Origin: A reflection about the origin means that for every point $(x, y)$ on the graph of the original function, the point $(-x, -y)$ is on the graph of the reflected function. This transformation can be represented by replacing $x$ with $-x$ and $y$ with $-y$ in the function's equation. So, if the original function is $y = f(x)$, the reflected function is $-y = f(-x)$, which simplifies to $y = -f(-x)$.

1. Identify the Original Function: The given function is $f(x) = -(x+2)^2 + 1$.

3. Determine the Reflected Function: To find the reflection about the origin, we need to find $-f(-x)$. First, let's find $f(-x)$ by substituting $-x$ for $x$ in the original function:
$$f(-x) = -((-x)+2)^2 + 1$$
$$f(-x) = -(-x+2)^2 + 1$$
Now, we multiply this by $-1$ to get $-f(-x)$:
$$-f(-x) = -[ -(-x+2)^2 + 1 ]$$
$$-f(-x) = -[ -(x-2)^2 + 1 ]$$
$$-f(-x) = (x-2)^2 - 1$$

1. Create a Table of Values for $-f(-x)$: The problem asks for a table of values for $-f(-x)$ at $x = 0, 1, 2$. We will use the expression we found in Step 3, which is $-f(-x) = (x-2)^2 - 1$.

- For $x = 0$:
$$-f(-0) = (0-2)^2 - 1$$
$$-f(0) = (-2)^2 - 1$$
$$-f(0) = 4 - 1$$
$$-f(0) = 3$$

- For $x = 1$:
$$-f(-1) = (1-2)^2 - 1$$
$$-f(-1) = (-1)^2 - 1$$
$$-f(-1) = 1 - 1$$
$$-f(-1) = 0$$

- For $x = 2$:
$$-f(-2) = (2-2)^2 - 1$$
$$-f(-2) = (0)^2 - 1$$
$$-f(-2) = 0 - 1$$
$$-f(-2) = -1$$

1. Fill in the Table: Based on the calculations above, the table for $-f(-x)$ at the given $x$ values is:

| x | 0 | 1 | 2 |
|---|---|---|---|
| -f(-x) | 3 | 0 | -1 |