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 2 0 3 1 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

Answer 1

To find the correct table that reflects the equation \( y = -f(x) \), we need to understand that this transformation results in the y-values being the negative of the original y-values. If \( f(x) \) is the original function, then \( -f(x) \) will be the reflection of that function over the x-axis.

Let's analyze the given options:

1. **Option 1:**
- \( x \) | \( -f(x) \)
- 2 | 0
- 3 | 1
- 6 | 2

It implies that \( f(2) = 0 \), \( f(3) = -1 \), and \( f(6) = -2 \) which is inconsistent because \( -f(3) = 1 \) should imply \( f(3) = -1 \), and similarly for others.

2. **Option 2:**
- \( x \) | \( -f(x) \)
- -2 | 0
- -3 | -1
- -6 | -2

It implies that \( f(-2) = 0 \), \( f(-3) = 1 \), and \( f(-6) = 2\), which is again inconsistent because \( -f(-3) = -1 \) must imply \( f(-3) = 1 \).

3. **Option 3:**
- \( x \) | \( -f(x) \)
- 2 | 0
- 3 | -1
- 6 | -2

Here, it implies \( f(2) = 0 \), \( f(3) = 1 \), and \( f(6) = 2 \). This is consistent because \( -f(3) = -1 \) matches.

4. **Option 4:**
- \( x \) | \( -f(x) \)
- -2 | 0
- -3 | 1
- -6 | 2

This implies that \( f(-2) = 0 \), \( f(-3) = -1 \), and \( f(-6) = -2 \), which is inconsistent for the same reasons as above.

The only table that shows \( y = -f(x) \) correctly reflecting over the x-axis is **Option 3**:
- \( x = 2 \) has \( -f(2) = 0 \)
- \( x = 3 \) has \( -f(3) = -1 \), meaning \( f(3) = 1 \)
- \( x = 6 \) has \( -f(6) = -2 \), meaning \( f(6) = 2 \)

Thus, the answer is **Option 3**.