Question

Create a table for the reflection over the x-axis of the function f(x) = (x − 1)
2
. (2 points

Answer 1

To create a table for the reflection of the function \( f(x) = (x - 1)^2 \) over the x-axis, we need to understand how reflection over the x-axis works. When reflecting a function \( f(x) \) over the x-axis, the new function becomes \( g(x) = -f(x) \). Therefore, for our function, we have:

\[
g(x) = -(x - 1)^2
\]

Now, let's choose some values of \( x \) to create a table. We will calculate both \( f(x) \) and \( g(x) \) at these points.

### Selected Points:
We'll choose \( x = -1, 0, 1, 2, 3 \).

### Calculating Values:

1. **For \( x = -1 \)**:
- \( f(-1) = (-1 - 1)^2 = (-2)^2 = 4 \)
- \( g(-1) = -f(-1) = -4 \)

2. **For \( x = 0 \)**:
- \( f(0) = (0 - 1)^2 = (-1)^2 = 1 \)
- \( g(0) = -f(0) = -1 \)

3. **For \( x = 1 \)**:
- \( f(1) = (1 - 1)^2 = 0^2 = 0 \)
- \( g(1) = -f(1) = 0 \)

4. **For \( x = 2 \)**:
- \( f(2) = (2 - 1)^2 = (1)^2 = 1 \)
- \( g(2) = -f(2) = -1 \)

5. **For \( x = 3 \)**:
- \( f(3) = (3 - 1)^2 = (2)^2 = 4 \)
- \( g(3) = -f(3) = -4 \)

### Reflection Table:

| \( x \) | \( f(x) = (x - 1)^2 \) | \( g(x) = -(x - 1)^2 \) |
|---------|-------------------------|--------------------------|
| -1 | 4 | -4 |
| 0 | 1 | -1 |
| 1 | 0 | 0 |
| 2 | 1 | -1 |
| 3 | 4 | -4 |

This table reflects the values of the function \( f(x) \) and its reflection \( g(x) \) over the x-axis.