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 maps ΔA(-5,2)B(-2,4)C(-3,0) to ΔA''(-1,-2)B''(2,-4)C''(1,0), we need to analyze the transformation.

1. Start with the coordinates of the points in ΔA and ΔA'':
- A(-5, 2) → A''(-1, -2)
- B(-2, 4) → B''(2, -4)
- C(-3, 0) → C''(1, 0)

2. First, let's identify the transformations. We need to look for a transformation that first reflects the points over the x-axis and then translates the points.

3. **Reflection over the x-axis**:
- Reflecting points over the x-axis changes the sign of the y-coordinate.
- For A: (-5, 2) → (-5, -2)
- For B: (-2, 4) → (-2, -4)
- For C: (-3, 0) → (-3, 0)

After this reflection, we have:
- A'(-5, -2)
- B'(-2, -4)
- C'(-3, 0)

4. **Translation**:
- Now we need to translate these reflected points to their final coordinates: A''(-1, -2), B''(2, -4), C''(1, 0).
- From A'(-5, -2) to A''(-1, -2), we translate right by 4 (increase x by 4).
- From B'(-2, -4) to B''(2, -4), we also translate right by 4.
- From C'(-3, 0) to C''(1, 0), we translate right by 4.

Combining these transformations gives us:

1. Reflect over the x-axis: \( (x, y) → (x, -y) \)
2. Translate right by 4: \( (x, y) → (x + 4, y) \)

The entire transformation can be represented as:
- Reflect over x-axis: \( (x, y) → (x, -y) \)
- Then translate: \( (x, -y) → (x + 4, -y) \)

So the composition of transformations can be summarized as:
\( (x, y) → (x + 4, -y) \)

Now let’s look at the provided options:
- \( (x,y) → (x + 4, -y) \)

This matches exactly with the transformation we deduced.

Thus, the correct answer is:
\( \text{(x,y)→(x+4,-y)} \)