Question

Which table shows that y = -f(x) results in a reflection over the x-axis
x -f(x)
2 0
3 -1
6 -2

x - f(x)
-2 0
-3 1
-6 2

x -f(x)
2 0
3 1
6 2

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 understand what \( -f(x) \) means. When we take the negative of a function \( f(x) \), we're flipping the output values (i.e., the y-values) from positive to negative and vice versa.

For a reflection over the x-axis:
1. If \( f(x) \) has a positive value, \( -f(x) \) will have a corresponding negative value.
2. If \( f(x) \) has a negative value, \( -f(x) \) will have a corresponding positive value.
3. If \( f(x) \) is zero, \( -f(x) \) will also be zero.

Now, let's examine each of the provided tables to see which one reflects this behavior:

1. **Table 1**:
\[
\begin{array}{|c|c|}
\hline
x & -f(x) \\
\hline
2 & 0 \\
3 & -1 \\
6 & -2 \\
\hline
\end{array}
\]
- Here: \( f(2) = 0 \) (0 remains 0), \( f(3) = 1 \) (1 becomes -1), \( f(6) = 2 \) (2 becomes -2). This seems to reflect over the x-axis.

2. **Table 2**:
\[
\begin{array}{|c|c|}
\hline
x & -f(x) \\
\hline
-2 & 0 \\
-3 & 1 \\
-6 & 2 \\
\hline
\end{array}
\]
- Here: \( f(-2) = 0 \) (0 remains 0), \( f(-3) = -1 \) (1 becomes -1), \( f(-6) = -2 \) (2 becomes -2). This does not reflect the behavior correctly.

3. **Table 3**:
\[
\begin{array}{|c|c|}
\hline
x & -f(x) \\
\hline
2 & 0 \\
3 & 1 \\
6 & 2 \\
\hline
\end{array}
\]
- Here: \( f(2) = 0 \) (0 remains 0), \( f(3) = -1 \) (1 is not matched to -1), \( f(6) = -2 \) (2 is not matched to -2). This does not reflect the behavior.

4. **Table 4**:
\[
\begin{array}{|c|c|}
\hline
x & -f(x) \\
\hline
-2 & 0 \\
-3 & -1 \\
-6 & -2 \\
\hline
\end{array}
\]
- Here: \( f(-2) = 0 \) (0 remains 0), \( f(-3) = 1 \) (1 becomes -1), \( f(-6) = 2 \) (2 becomes -2). This does not reflect the behavior correctly.

Thus, the correct table that shows \( y = -f(x) \) resulting in a reflection over the x-axis is **Table 1**.