Which table shows that y = -f(x) results in a reflection over the x-axis

Question

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.