Question

A parabola is drawn on a coordinate plane. Both axes range from negative 5 to 5 in one-unit increments.
The original graph of the function y=f(x)
is shown. Select the table that represents y=f(−x)
.

(1 point)
Responses

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

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

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

x f(−x)
1 2
2 0
3 2

Answer 1

To determine the correct table that represents \( y = f(-x) \), we need to understand that this transformation reflects the graph of the function \( y = f(x) \) across the y-axis. Specifically, for a given value of \( x \), \( f(-x) \) will evaluate the function at the negative of that value.

Given this reflection, you'll need to find the values of \( f(-x) \) for corresponding values of \( x \).

However, since the function \( f(x) \) itself isn't given, we will analyze the options based on the structure of a general parabola:

1. If \( f(x) \) has a certain value at positive \( x \), \( f(-x) \) will have the same value at the corresponding negative value (and vice versa).

Let’s analyze each response:

### Response 1:
| x | \( f(-x) \) |
|-----|--------------|
| -1 | -2 |
| -2 | 0 |
| -3 | -2 |

### Response 2:
| x | \( f(-x) \) |
|-----|--------------|
| -1 | 2 |
| -2 | 0 |
| -3 | 2 |

### Response 3:
| x | \( f(-x) \) |
|-----|--------------|
| 1 | -2 |
| 2 | 0 |
| 3 | -2 |

### Response 4:
| x | \( f(-x) \) |
|-----|--------------|
| 1 | 2 |
| 2 | 0 |
| 3 | 2 |

In general, for a given \( x \), the corresponding \( f(-x) \) should display symmetrical values around the y-axis.

1. **Response 1** gives negative results, which generally does not suggest symmetry present with \( f(x) \).
2. **Response 2** gives positive values for negative \( x \) and is thus potentially consistent with symmetry.
3. **Response 3** reflects negative outcomes and seems inconsistent with expected symmetry.
4. **Response 4** indicates a positive outcome at positive \( x \), complementing the typical parabolic shape.

### Conclusion:
From the analysis, **Response 4** represents a reflection through the y-axis correctly where \( f(-x) \) displays symmetry with positive outputs for corresponding positive and negative \( x \).

Therefore, the correct table that represents \( y = f(-x) \) would be:

**Response 4:**
| x | \( f(-x) \) |
|-----|--------------|
| 1 | 2 |
| 2 | 0 |
| 3 | 2 |