Use the image to answer the question.

Question

A coordinate plane shows two triangles A B C and A double dash B double dash C double dash. Both the axes range from negative 6 to 6 in unit increments.

Identify the sequence of transformations that will carry ΔABC→ΔA"B"C" . Give your answer in the form of a composition transformation mapping.

(1 point)
Responses choose and solve the correct answer carefully and tell me why

(x,y)→(x+4,−y)
left parenthesis x comma y right parenthesis right arrow left parenthesis x plus 4 comma negative y right parenthesis

(x,y)→(x,−y−4)
open paren x comma y close paren rightarrow open paren x comma negative y minus 4 close paren

(x,y)→(x−4,−y)
left parenthesis x comma y right parenthesis right arrow left parenthesis x minus 4 comma negative y right parenthesis

(x,y)→(x,−y+4)
left parenthesis x comma y right parenthesis right arrow left parenthesis x comma negative y plus 4 right parenthesis
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Answer 1

To determine the sequence of transformations that maps triangle \( \Delta ABC \) to triangle \( \Delta A''B''C'' \), we need to understand the specific transformations that have taken place.

1. Reflecting across the x-axis: This transformation takes a point \( (x, y) \) to \( (x, -y) \). This would change the orientation of the triangle.

2. Translating horizontally: A translation can be represented as \( (x, y) \) to \( (x ± a, y) \) where \( a \) represents the distance moved left or right.

To analyze each option:

• Option 1: \( (x,y) \rightarrow (x+4,−y) \)
  This reflects the triangle across the x-axis and then translates it 4 units to the right (which may not match the map between the two triangles).

• Option 2: \( (x,y) \rightarrow (x,−y−4) \)
  This reflects the triangle across the x-axis and then translates it down 4 units.

• Option 3: \( (x,y) \rightarrow (x−4,−y) \)
  This reflects the triangle across the x-axis and then translates it 4 units to the left.

• Option 4: \( (x,y) \rightarrow (x,−y+4) \)
  This reflects the triangle across the x-axis and then translates it up 4 units.

Without the visual representation, identifying the exact transformations requires observing the changes in coordinates from \( \Delta ABC \) to \( \Delta A''B''C'' \). However, based on general understanding:

If triangle \( \Delta A''B''C'' \) appears to be reflected across the x-axis and moved 4 units vertically up or down, then Option 4 or Option 2 may be the likely candidates. If it is predominantly reflecting across the x-axis and moving down, it corresponds with Option 2; reflecting up corresponds with Option 4.

The correct response would depend on the final positions of the vertices after the transformations. If you have access to the coordinate points, it will help decisively choose the correct transformation.

Answer: Depending on the exact locations post-transform, either Option 2 or Option 4 could be the correct transformation, with a hypothesis leaning towards Option 4 in general cases of such transformations for upward movements. If you're sure about the vertical displacement being negative, then Option 2 is applicable.