Question

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

x -f(x)
0 ?
1 ?
2 ?

Answer 1

To find the reflection of the function \( f(x) = (x - 1)^2 \) over the x-axis, we need to negate the output of the function. That is, the reflected function \( g(x) \) will be \( g(x) = -f(x) \).

Let's calculate \( f(x) \) for the given x-values and then determine the reflected values \( g(x) \):

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

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

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

Now we can fill in the table:

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

Thus, the completed table for the reflection over the x-axis is:

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