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.

x 0 1 2
-f(-x)

Answer 1

To create the table of values for the function \( f(x) = (x + 2)^2 + 1 \) evaluated at \( x = 0 \), \( x = -1 \), and \( x = -2 \), we first need to calculate \( f(x) \) for those values. Afterward, we will compute \( -f(-x) \) for the same \( x \) values.

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

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

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

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

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

We also need to evaluate \( f(-x) \) for the same input values:

1. For \( x = 0 \):
\[
f(-0) = f(0) = 5 \quad \Rightarrow \quad -f(-0) = -5
\]

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

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

### Step 3: Create the Table

Now we can summarize our findings in the following table:

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

This table reflects the values of the function \( f(x) \) as well as the negative values of the function evaluated at the negative of those inputs.