Question

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

Answer 1

To create a table for the reflection over the x-axis of the function \( f(x) = (x - 1)^2 \), we need to determine the reflected function. The reflection of \( f(x) \) over the x-axis results in \( -f(x) \).

Thus, the reflected function is:
\[
g(x) = -f(x) = -((x - 1)^2)
\]

Now, let's create a table of values for both \( f(x) \) and \( g(x) \). We'll choose a set of \( x \) values and calculate both functions.

### Table of Values

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

### Summary of the Table

- The function \( f(x) = (x - 1)^2 \) represents a parabola that opens upwards with its vertex at \( (1, 0) \).
- The reflected function \( g(x) = -((x - 1)^2) \) will be a parabola that opens downwards with the same vertex at \( (1, 0) \).

Each value in the \( g(x) \) column is the negative of the corresponding value in the \( f(x) \) column, confirming that \( g(x) \) is a reflection of \( f(x) \) over the x-axis.