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) () () ()

Answer 1

To reflect on the origin of the function \( f(x) = (x + 2)^2 + 1 \), we first need to calculate the values of \( f(x) \) at the specified \( x \) values, then find \( -f(-x) \) for those values of \( x \).

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

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

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

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

### Step 2: Calculate \( -f(-x) \)

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

2. For \( x = -1 \), \( -f(-(-1)) = -f(1) \):
\[
f(1) = (1 + 2)^2 + 1 = 3^2 + 1 = 9 + 1 = 10
\]
Therefore, \( -f(-(-1)) = -10 \)

3. For \( x = -2 \), \( -f(-(-2)) = -f(2) \):
\[
f(2) = (2 + 2)^2 + 1 = 4^2 + 1 = 16 + 1 = 17
\]
Therefore, \( -f(-(-2)) = -17 \)

### Step 3: Create the table

Here’s how the complete table would look:

| \( x \) | \( f(x) \) | \( -f(-x) \) |
|---------|------------|--------------|
| 0 | 5 | -5 |
| -1 | 2 | -10 |
| -2 | 1 | -17 |

This table shows the values of \( f(x) \) and \( -f(-x) \) for the given \( x \) values, helping us reflect on the function and its properties in relation to the origin.