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Question

A graphed function starting on the x axis has an arrow at the other end passes through three points on a coordinate plane. The x-axis ranges from negative 9 to 9 in unit increments and the y-axis ranges from negative 11 to 11 in unit increments.

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

(1 point)
Responses

x −f(x)
2 0
3 1
6 2 x f left parenthesis x right parenthesis 2 0 3 1 6 2

x −f(x)
2 0
3 −1
6 −2x f left parenthesis x right parenthesis 2 0 3 negative 1 6 negative 2

x −f(x)
−2 0
−3 −1
−6 −2 x f left parenthesis x right parenthesis negative 2 0 negative 3 negative 1 negative 6 negative 2

x −f(x)
−2 0
−3 1
−6 2 x f left parenthesis x right parenthesis negative 2 0 negative 3 1 negative 6 2
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Answer 1

To determine which table shows that \( y = -f(x) \) results in a reflection over the x-axis, we need to note how values in the table transform when reflected. In a reflection over the x-axis, the negative of the original function values is taken.

This means:

• If \( f(x) = 0 \), then \( -f(x) = 0 \)
• If \( f(x) = 1 \), then \( -f(x) = -1 \)
• If \( f(x) = 2 \), then \( -f(x) = -2 \)
• If \( f(x) = -1 \), then \( -f(x) = 1 \)
• If \( f(x) = -2 \), then \( -f(x) = 2 \)

Now let’s examine the tables provided to see which one reflects these transformations correctly:

1. Table 1:

   x	−f(x)
   2	0
   3	1
   6	2
   

   Here, the values of \( -f(x) \) do not match any negative counterparts of \( f(x) \).

2. Table 2:

   x	−f(x)
   2	0
   3	−1
   6	−2
   

   Here, if we assume \( f(2) = 0, f(3) = 1, f(6) = 2 \), the transformations match \( -f(x) \) correctly (0, -1, -2).

3. Table 3:

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

   Similarly, this table does not exhibit needed transformations in positive range.

4. Table 4:

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

   It reflects the transformation needed, but values don't exhibit \( -f(x) \) for the needed coordinate range.

The clear choice upon reviewing is Table 2, as the values demonstrate the transformation of \( y = -f(x) \) operating correctly through reflection over the x-axis.

Therefore, Table 2 correctly represents what happens when a function \( y = -f(x) \) is reflected over the x-axis.