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 1 2 2 0 3 2

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
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1 answer

To determine which table represents \( y = f(-x) \), we need to understand how the function transforms when we replace \( x \) with \( -x \). This transformation reflects the graph of the original function \( y = f(x) \) across the y-axis.

Given the potential values of \( f(-x) \) from the options, let's analyze what the values should resemble based on the original graph of the parabola. Here's a step-by-step guide to solving this:

  • Reflect x-values: For positive \( x \), find corresponding negative \( x \) values (i.e., if \( f(x) = k \), then \( f(-x) \) at the mirrored x-value should also be \( k \)).
  • Identify values in the graph: You need to check each table option to find whether they reflect within the limits set by the values on the original graph.

Since I cannot see the image or the original function graph, I cannot tell you exactly what the values of \( f(-x) \) would be. However, you can follow this approach:

  1. Identify what \( f(x) \) is for \( x = 1, 2, 3 \) in your drawing (the values of the original parabola).
  2. Use those values to find \( f(-1) \), \( f(-2) \), and \( f(-3) \).
  3. Compare the calculated values to the tables provided to find a match.

In essence, you are looking for a table that swaps the outputs for the positive x-values with the respective negative x-values while keeping symmetry about the y-axis.

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