Which of the following statements correctly describes how the parent function f(x)= |x| was dilated to result in the graphed function

Question

On graph (0,0), (-2,1), and (2,1)

Answer 1

To analyze how the parent function \( f(x) = |x| \) was dilated to result in the function represented by the points (0,0), (-2,1), and (2,1), we need to look closely at the coordinates of the points.

The parent function \( f(x) = |x| \) typically produces the following outputs for the corresponding inputs:

• \( f(0) = |0| = 0 \)
• \( f(-2) = |-2| = 2 \)
• \( f(2) = |2| = 2 \)

The graphed points you provided show:

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

From this, we can deduce that the outputs have been modified. Instead of \( f(-2) = 2 \) and \( f(2) = 2 \), now \( f(-2) = 1 \) and \( f(2) = 1 \).

To find how \( f(x) = |x| \) was transformed, we note that \( 1 \) is half of \( 2 \). This suggests that the function was vertically compressed by a factor of \( 1/2 \).

In general, applying a vertical compression by a factor of \( k \) to \( f(x) = |x| \) results in the new function: \[ g(x) = \frac{1}{k} |x| \] For a vertical compression by \( k = 2 \), the resulting new function would be: \[ g(x) = \frac{1}{2}|x| \]

Thus, the correct statement describing the transformation from the parent function to the graphed function is that \( f(x) = |x| \) was vertically compressed by a factor of \( 1/2 \).

Answer 2

Options
Horizontally stretched
Horizontally compressed
Vertically translated up
Vertically translated down

Answer 3

Given the options you provided, we need to analyze the transformations based on the points (0,0), (-2,1), and (2,1) in comparison to the parent function \( f(x) = |x| \).

Parent Function:

• The output for the parent function is:
  • \( f(0) = |0| = 0 \)
  • \( f(-2) = |-2| = 2 \)
  • \( f(2) = |2| = 2 \)

Points Given:

• For the new function:
  • \( g(0) = 0 \)
  • \( g(-2) = 1 \)
  • \( g(2) = 1 \)

Analyzing the Changes:

• At \( x = 0 \), the value remains the same, \( 0 = 0 \).
• At \( x = -2 \) and \( x = 2 \), the values have changed from \( 2 \) (from the parent function) to \( 1 \).

Based on this information:

• The function has not shifted horizontally; therefore, no horizontal stretching or compression occurred.
• There is no evidence of vertical translation upward or downward since the points are not constants across all values: g(-2) and g(2) are simply halved values from the parent function's outputs, but they remain at the same horizontal x-values.

Conclusion:

The transformation that occurred here is not explicitly mentioned in the options. However, if we're forced to classify based just on the given points showing that the output values at \( x = -2 \) and \( x = 2 \) have halved, it suggests a form of vertical compression or scaling that isn't directly stated in the options. None of the provided options correctly describe the transformation, but the closest relation might relate this to a change in vertical behavior.

Final Answer:

• The function does not fit within the provided options accurately based on your points; it seems to have undergone a vertical compression (not listed). If we are to pick, one could argue a vertical translation down regarding the alteration from 2 to 1; however, the most accurate transformation should reference compression.

If needing to select from the given answers, it may fall to interpretation; however, based purely on the mechanics illustrated, they are optimal at "none of the above" with emphasis on vertical compression as a note.