The original graph of the function y = f(x) is shown select the table that represents y = f(-x)

Question

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