Question

Create a table for the reflection about the origin of the function f(x)=(x+2)2+1
.

Hint: First create a table of values for f(x)
evaluated at x=0
, x=−1
., and x=−2
.

(2 points)
x 0 1 2
−f(−x)
Remaining Attempts : 3

Answer 1

To analyze the reflection of the function \( f(x) = (x + 2)^2 + 1 \), we first need to create a table of values for \( f(x) \) evaluated at \( x = 0 \), \( x = -1 \), and \( x = -2 \).

### Step 1: Calculate \( f(x) \) at the specified points

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

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

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

### Step 2: Create the table for \( -f(-x) \)

To find the values of \( -f(-x) \), we will evaluate the function \( f(x) \) at the corresponding negative values of \( x \):

- For \( x = 0 \), \( -f(-0) = -f(0) = -5 \)
- For \( x = 1 \), \( -f(-1) = -f(-1) = -2 \)
- For \( x = 2 \), \( -f(-2) = -f(-2) = -1 \)

### Final Table

Here’s the completed table with values for \( f(x) \) and \( -f(-x) \):

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

This shows the values of \( f(x) \) at the specified points and the corresponding values of \( -f(-x) \), which can illustrate the reflection property of the function.