Question

The original graph of the function y = f(x) is shown select the table that represents y = f(-x)
The original graph is an upward u shape in the 2 quadrant with a vertex of (2,0)
x f(-x)
1 -2
2 0
3 -2

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

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

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

Answer 1

To determine which table corresponds to the function \( y = f(-x) \) based on the description of the original function \( y = f(x) \) being an upward U-shape located in the second quadrant with a vertex at (2, 0), we need to understand how the transformation \( f(-x) \) affects the graph.

The transformation \( f(-x) \) reflects the graph of \( f(x) \) across the y-axis. Given that the original graph is in the second quadrant, after reflection:

1. The vertex (2, 0) will move to (-2, 0).
2. The shape of the graph will remain a U-shape, but the region that was in the second quadrant will now extend into the first quadrant.

Now let's inspect the tables provided:

1. **First Table:**
```
x f(-x)
1 -2
2 0
3 -2
```
This does not align with our reflection since the y-values are negative, and a U-shape should not have negative outputs for positive x.

2. **Second Table:**
```
x f(-x)
1 2
2 0
3 2
```
This table appears to give positive values for positive x, but it doesn't reflect the vertex at (-2, 0) that we expect.

3. **Third Table:**
```
x f(-x)
-1 2
-2 0
-3 2
```
This table aligns perfectly since:
- At \( x = -2 \), we have the vertex at \( (0, 2) \).
- At \( x = -1 \) and \( x = -3 \), we still have returns to positive values, indicating it's a U-shape.

4. **Fourth Table:**
```
x f(-x)
-1 -2
-2 0
-3 -2
```
This result does not match our expectations because it has negative y-values for positive x, failing to represent the reflected version of \( f(x) \).

From this analysis, the correct table that represents \( y = f(-x) \) is the **Third Table**:
```
x f(-x)
-1 2
-2 0
-3 2
```