Question

Use the image to answer the 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
```