Question

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

Finish this table
x 0 1 2
-f(-x) _ _ _

Answer 2

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

### Step 1: Calculate \( f(x) \) for the given \( x \) values

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: Create the table of values for \( f(x) \)

Now we can summarize these results in a table:

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

### Step 3: Calculate \( -f(-x) \) for the same \( x \) values

Next, we will compute \( f(-x) \) for \( x = 0, -1, \) and \( -2 \):

1. **For \( x = 0 \)**:
\[
f(-0) = f(0) = 5 \implies -f(-0) = -5
\]

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

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

### Step 4: Create the table of values for \( -f(-x) \)

Here is the summary of these calculations:

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

### Final Summary

Here are the final tables summarizing \( f(x) \) and \( -f(-x) \):

#### Table of \( f(x) \)

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

#### Table of \( -f(-x) \)

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

Using these tables, you can analyze the symmetry of the function and its reflection properties!

Answer 3

Let's complete the table for \(-f(-x)\) based on the calculations we did earlier. We need to evaluate \(f(-x)\) at \(x = 0\), \(1\), and \(2\).

Recall that the function is defined as:
\[
f(x) = (x + 2)^2 + 1
\]

Now we will compute \(f(-x)\) for \(x = 0\), \(1\), and \(2\) and then take the negative of those values to find \(-f(-x)\).

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

1. **For \(x = 0\)**:
\[
f(-0) = f(0) = 5 \implies -f(-0) = -5
\]

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

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

### Step 2: Fill in the values in the table

Now we can complete the table for \( -f(-x) \):

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

### Final Table

Here is the completed table for \(-f(-x)\):

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

If you have any further questions or need additional information, feel free to ask!