Question

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

Answer 1

Table
x 0 1 2
-f(x)

Answer 2

To create a table for the reflection over the x-axis of the function \( f(x) = (x-1)^2 \), we first need to understand that reflecting a function over the x-axis involves changing the sign of the output value \( f(x) \). Therefore, the reflected function can be written as \( g(x) = -f(x) \), which means:

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

Next, we'll create a table of values for both the original function \( f(x) \) and its reflection \( g(x) \).

### Table of Values

| \( x \) | \( f(x) = (x-1)^2 \) | \( g(x) = -(x-1)^2 \) |
|---------|-----------------------|------------------------|
| -2 | 9 | -9 |
| -1 | 4 | -4 |
| 0 | 1 | -1 |
| 1 | 0 | 0 |
| 2 | 1 | -1 |
| 3 | 4 | -4 |
| 4 | 9 | -9 |

### Explanation:
- **For \( x = -2 \)**:
- \( f(-2) = (-2 - 1)^2 = 9 \)
- \( g(-2) = -9 \)

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

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

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

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

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

- **For \( x = 4 \)**:
- \( f(4) = (4 - 1)^2 = 9 \)
- \( g(4) = -9 \)

This completes the table showing the values for the function and its reflection over the x-axis.

Answer 3

Sure! Let's fill in the values of the table for the reflection \( g(x) = -f(x) \) where \( f(x) = (x-1)^2 \). The table will cover \( x = 0 \), \( 1 \), and \( 2 \).

### Function Calculation

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

2. ***Calculate \( -f(x) \):***
- For \( x = 0 \):
\[
-f(0) = -1
\]
- For \( x = 1 \):
\[
-f(1) = 0
\]
- For \( x = 2 \):
\[
-f(2) = -1
\]

### Table of Values

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

This is the completed table for \( -f(x) \) at \( x = 0, 1, 2 \).