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Use the image to answer the 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 1 negative 2 2 0 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

1 answer

To determine the table that represents \( y = f(-x) \) based on the original graph of \( y = f(x) \), we need to identify the points of the original parabola and find their corresponding values when \( x \) is replaced by \( -x \).

Since the original function is mirrored about the y-axis when we evaluate \( f(-x) \), we need to switch the signs of the x-values in each ordered pair, while retaining the y-values.

If we denote original points as \( (x, f(x)) \), the new points for \( f(-x) \) will be \( (-x, f(x)) \).

Assuming the original graph provided points similar to:

  • \( (1, -2) \)
  • \( (2, 0) \)
  • \( (3, -2) \)

Then the corresponding points for \( f(-x) \) would become:

  • \( (-1, -2) \)
  • \( (-2, 0) \)
  • \( (-3, -2) \)

So from the given responses, we need to find the one that shows:

  • When \( x = -1 \), \( f(-x) = -2 \)
  • When \( x = -2 \), \( f(-x) = 0 \)
  • When \( x = -3 \), \( f(-x) = -2 \)

Thus the correct table option is:

\( x, f(-x) \)

  • \(-1, -2\)
  • \(-2, 0\)
  • \(-3, -2\)

This is the representation of \( y = f(-x) \) based on the reflection of the original function.

From the provided options, look for the one that matches \( (-1, -2), (-2, 0), (-3, -2) \).

Please compare this information with the options you have to select the correct response based on these coordinates.