Question

Which table shows that: y = -f (x), results in a reflection over the x-axis?
A.
x | -f (x)
-2 | 0
-3 | 1
-6 | 2
B.
x | -f (x)
2 | 0
3 | 1
6 | 2
C.
x | -f (x)
-2 | 0
-3 | -1
-6 | -2
D.
x | -f (x)
2 | 0
3 | -1
6 | -2

Answer 1

To determine which table shows that \( y = -f(x) \) results in a reflection over the x-axis, we need to look for a pattern where the values of \( -f(x) \) are the opposite signs of the values of \( f(x) \). This means that for every \( x \), if \( f(x) = y \), then \( -f(x) \) should equal \( -y \).

Since the tables provided use \( -f(x) \) in their headers, we can analyze the values in each table to see if they conform to this condition.

### Analyzing Each Table:

**A.**
\[
\begin{align*}
x & | -f(x) \\
-2 & | 0 \\
-3 & | 1 \\
-6 & | 2 \\
\end{align*}
\]
This implies:
- \( f(-2) = 0 \)
- \( f(-3) = -1 \) (since \( -f(-3) = 1 \))
- \( f(-6) = -2 \) (since \( -f(-6) = 2 \))

**B.**
\[
\begin{align*}
x & | -f(x) \\
2 & | 0 \\
3 & | 1 \\
6 & | 2 \\
\end{align*}
\]
This implies:
- \( f(2) = 0 \)
- \( f(3) = -1 \)
- \( f(6) = -2 \)

**C.**
\[
\begin{align*}
x & | -f(x) \\
-2 & | 0 \\
-3 & | -1 \\
-6 & | -2 \\
\end{align*}
\]
This implies:
- \( f(-2) = 0 \)
- \( f(-3) = 1 \)
- \( f(-6) = 2 \)

**D.**
\[
\begin{align*}
x & | -f(x) \\
2 & | 0 \\
3 & | -1 \\
6 & | -2 \\
\end{align*}
\]
This implies:
- \( f(2) = 0 \)
- \( f(3) = 1 \)
- \( f(6) = 2 \)

### Conclusion:
From the analysis:

- **Table A** shows that for all negative values of \( f(x) \), \( -f(x) \) reflects over the x-axis.
- **Table B** also shows a negative reflection for \( f(3) = 1 \) and \( f(6) = 2 \).
- **Table C** reflects incorrectly (positive for negative x).
- **Table D** reflects incorrectly (positive for positive x).

The correct table that shows a reflection about the x-axis would thus be **B**, as it shows \( 1 \) and \( 2 \) being positively reflected as \( -1 \) and \( -2 \).