Question

Question
Use the image to answer the 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

(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 will carry triangle \( \Delta A(-5,2) B(-2,4) C(-3,0) \) to triangle \( \Delta A''(-1,-2) B''(2,-4) C''(1,0) \), we need to analyze the changes in the coordinates.

1. **Identify the transformations from the points**:
- The original points are \( A(-5,2) \), \( B(-2,4) \), \( C(-3,0) \).
- The final points are \( A''(-1,-2) \), \( B''(2,-4) \), \( C''(1,0) \).

2. **Analyzing the changes**:
- For the point \( A \):
- From \( (-5,2) \) to \( (-1,-2) \):
- The x-coordinate changes from \(-5\) to \(-1\): \( -5 + 4 = -1\).
- The y-coordinate changes from \(2\) to \(-2\): \( 2 \to -2 \), which indicates a reflection over the x-axis and a translation downward by 4 units (or \( -y - 4 \)).

- For the point \( B \):
- From \( (-2,4) \) to \( (2,-4) \):
- The x-coordinate changes from \(-2\) to \(2\): \(-2 + 4 = 2\).
- The y-coordinate changes from \(4\) to \(-4\): \( 4 \to -4 \) just confirms that there is a reflection over the x-axis and a translation downward by 4 units.

- For the point \( C \):
- From \( (-3,0) \) to \( (1,0) \):
- The x-coordinate changes from \(-3\) to \(1\): \(-3 + 4 = 1\).
- The y-coordinate remains the same at \(0\), confirming the reflection does not affect it.

3. **Overall transformation**:
- It seems that all points undergo:
- A translation of \( +4 \) in the x-direction.
- A reflection over the x-axis followed by a downward shift of \( 4 \) units in the y-direction (\( -y - 4 \)).

4. **Putting it all together**:
- The transformation can be represented as a sequence:
\[
(x,y) \to (x + 4, -y) \to (x + 4, -y - 4)
\]
which simplifies to:
\[
(x,y) \to (x + 4, -y - 4)
\]

After analyzing the options given, the response that matches our derived transformation is:

\[
\text{(x,y) → (x, -y - 4)}
\]

This is the transformation that carries triangle \( \Delta A \) to triangle \( \Delta A'' \).