Question

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 analyze the function \( f(x) = (x + 2)^2 + 1 \) and create a table of values for its reflection over the y-axis (which gives us the values of \( -f(-x) \)), we will first calculate \( f(x) \) for \( x = 0, -1, -2 \), and then find \( -f(-x) \) for those 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: Create the table for \( -f(-x) \)

Now, let's compute \( -f(-x) \) for \( x = 0, 1, 2 \):

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

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

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

### Final Table

Now we can create the final table that summarizes the calculations:

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

This table displays the reflected values for the specified inputs under the transformation \( -f(-x) \).