Question

Create a table for the reflection about the origin of the function f(x)=(x+2)^2+1
.
x|0|1|2
-f(-x)|__|__|__

Answer 1

To reflect on the origins of the function \( f(x) = (x + 2)^2 + 1 \) regarding its symmetry, let's calculate the values of \( -f(-x) \) for \( x = 0, 1, \) and \( 2 \).

### Step 1: Calculate \( f(-x) \)

1. When \( x = 0 \):
\[
f(-0) = f(0) = (0 + 2)^2 + 1 = 2^2 + 1 = 4 + 1 = 5
\]
Therefore, \( -f(-0) = -5 \).

2. When \( x = 1 \):
\[
f(-1) = (-1 + 2)^2 + 1 = 1^2 + 1 = 1 + 1 = 2
\]
Therefore, \( -f(-1) = -2 \).

3. When \( x = 2 \):
\[
f(-2) = (-2 + 2)^2 + 1 = 0^2 + 1 = 0 + 1 = 1
\]
Therefore, \( -f(-2) = -1 \).

### Step 2: Construct the table

Here is the table filled out with the values we calculated:

| x | 0 | 1 | 2 |
|---|----|----|----|
| -f(-x) | -5 | -2 | -1 |

This table reflects the values of \( -f(-x) \) for the given \( x \) values.