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 will first create a table of values for \( f(x) \) evaluated at \( x = 0, -1, \) and \( -2 \). We will then calculate \( -f(-x) \) for those values of \( x \).

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

- For \( x = 0 \):
\[
f(0) = (0 + 2)^2 + 1 = (2)^2 + 1 = 4 + 1 = 5
\]

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

- For \( x = -2 \):
\[
f(-2) = (-2 + 2)^2 + 1 = (0)^2 + 1 = 0 + 1 = 1
\]

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

- For \( x = 0 \):
\[
-f(-0) = -f(0) = -5
\]

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

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

### Step 3: Create the table

Now, we can organize our findings in a table format:

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

This table summarizes the reflection of the function \( f(x) \) with respect to the origin, showing both \( f(x) \) and its negative reflection \( -f(-x) \).