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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 negative 2 0 negative 3 1 negative 6 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 2 0 3 negative 1 6 negative 2

x −f(x)
2 0
3 1
6 2
x f left parenthesis x right parenthesis 2 0 3 1 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 look for a table of values where the \( -f(x) \) values are the negative of the corresponding \( f(x) \) values.

Let's evaluate the provided tables:

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

   Here, \( -f(-2) = -0 = 0 \), \( -f(-3) = -1 \), \( -f(-6) = -2 \). This does not reflect the values.

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

   Here, \( -f(-2) = -0 = 0 \), \( -f(-3) = -1 \), \( -f(-6) = -2 \). This matches the functionality \( -f(x) = -f(x) \) for the corresponding \( f(x) \).

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

   This set gives \( -f(2) = -0 = 0 \), \( -f(3) = 1 \), \( -f(6) = 2 \). This does not reflect properly.

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

   Here, \( -f(2) = -0 = 0 \), \( -f(3) = -1 \), \( -f(6) = -2 \). This does not reflect properly.

Based on this evaluation, the second table is the only one that correctly shows that \( y = -f(x) \) results in a reflection over the x-axis.

Thus, the answer is the second table:

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