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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 ΔA(-5,2)B (-2,4)C (-3,0)→ΔA" (-1,-2)B" (2,-4)C" (1,0). Give your answer in the form of a composition transformation mapping.

(1 point)
Responses choose the correct answer carfeully 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 find the transformation that will carry triangle ΔABC with vertices A(-5,2), B(-2,4), and C(-3,0) to triangle ΔA''B''C'' with vertices A''(-1,-2), B''(2,-4), and C''(1,0), we can analyze the changes in coordinates.

1. Transformation of A:

   • Initial point A is at (-5, 2).
   • Final point A'' is at (-1, -2).
   • The x-coordinate changes from -5 to -1, which is an increase of 4.
   • The y-coordinate changes from 2 to -2, which is a reflection across the x-axis (the y-coordinate becomes negative) and a downward shift by 4 units (2 to -2 is a decrease of 4).

2. Transformation of B:

   • Initial point B is at (-2, 4).
   • Final point B'' is at (2, -4).
   • The x-coordinate changes from -2 to 2, which is an increase of 4.
   • The y-coordinate changes from 4 to -4, indicating reflection across the x-axis and a downward shift of 8 (4 to -4 is a decrease of 8).

3. Transformation of C:

   • Initial point C is at (-3, 0).
   • Final point C'' is at (1, 0).
   • The x-coordinate changes from -3 to 1, which is an increase of 4.
   • The y-coordinate remains the same at 0 (no vertical change).

From the analysis above, we notice the pattern of transformations. The x-coordinates of each vertex are increased by 4, and the y-coordinates are changed as follows:

• They are reflected across the x-axis, which is represented by changing the sign of y (−y).
• The second part of the transformation involves vertically shifting by -4 after reflection.

The suitable transformation that encapsulates these changes can be represented as:

\((x,y) \rightarrow (x+4, -y)\)

Accordingly, the correct answer is:

\((x,y) \rightarrow (x+4, -y)\)

This transformation reflects each point across the x-axis (changing the sign of y) and shifts to the right by 4 units (adding 4 to x).