Use the image to answer the question.

Question

A V shaped function with arrows at both ends passes through 7 plotted points. 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 of the function graphed over the x-axis?

(1 point)
Responses

x −f(x)
4 −3
2 −1
3 −2
x f left parenthesis x right parenthesis 4 negative 3 2 negative 1 3 negative 2

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

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

x −f(x)
1 2
−2
1
−3
2

Answer 1

To determine which table correctly reflects the function over the x-axis (y = -f(x)), we need to look for a table where the values of -f(x) are the negatives of the corresponding values of f(x). This means that for each x value, if f(x) is positive, then -f(x) should be negative, and vice versa.

Let's analyze each of the provided responses:

1. Response A:

   • x: 4, 2, 3
   • -f(x): -3, -1, -2
   • f(x): 4 → -3, 2 → -1, 3 → -2
   • This response is not showing the correct reflection since the f(x) values are not the negatives.

2. Response B:

   • x: -1, 1, -2, 3
   • -f(x): 0, -2, 1
   • f(x): -1 → 0, 1 → -2, -2 → 1
   • This response does not match the reflection condition either.

3. Response C:

   • x: 1, 0, -3
   • -f(x): -2, -1, -2
   • f(x): 1 → -2, 0 → -1, -3 → -2
   • This response doesn't show correct reflection.

4. Response D:

   • x: 1, -2, 3
   • -f(x): 2, 1, -3
   • This response shows:
     • f(1) must be -2 (because -f(1) = 2)
     • f(-2) must be -1 (because -f(-2) = 1)
     • f(3) must be 3 (because -f(3) = -3)
   • This fulfills the reflection condition as the values are correct negatives.

Based on this analysis, Response D provides values that correctly indicate the reflection over the x-axis.

Therefore, the answer is Response D.