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)

Answer 1

To reflect on the function \( f(x) = (x + 2)^2 + 1 \), we'll first create a table of values by evaluating \( f(x) \) at \( x = 0 \), \( x = -1 \), and \( x = -2 \).

### Evaluation of \( f(x) \)

Let's calculate \( f(x) \) for the specified values of \( x \):

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

### Table of Values

Now we can organize these values into a table:

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

### Reflection on the Origin

To reflect about the origin, we transform the values of \( f(x) \) from the table:

- The point \( (0, 5) \) reflects to \( (0, -5) \).
- The point \( (-1, 2) \) reflects to \( (1, -2) \).
- The point \( (-2, 1) \) reflects to \( (2, -1) \).

The resulting table for the reflection about the origin is:

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

This table represents the new coordinates after reflecting the original points about the origin.