Question

Use the image to answer the question.

A coordinate plane shows two triangles A (-5,2) B (-2,4) C (-3,0) and A double dash (-1,-2) B double dash (2,-4) C double dash (1,0). 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

(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 sequence of transformations that maps triangle \( \Delta ABC \) to triangle \( \Delta A"B"C" \), we need to analyze the two triangles' coordinates:

- \( A (-5, 2) \), \( B (-2, 4) \), \( C (-3, 0) \)
- \( A" (-1, -2) \), \( B" (2, -4) \), \( C" (1, 0) \)

### Step 1: Translation
To see the translation, we notice that the x-coordinates of \( A \), \( B \), and \( C \) all increase by 4 units.

- From \( -5 \) to \( -1 \): \( -5 + 4 = -1 \)
- From \( -2 \) to \( 2 \): \( -2 + 4 = 2 \)
- From \( -3 \) to \( 1 \): \( -3 + 4 = 1 \)

So, there is a horizontal translation of 4 units to the right.

### Step 2: Reflection
Next, we notice the changes in the y-coordinates:

- The y-coordinate at point \( A \) goes from \( +2 \) to \( -2 \) (which indicates reflection and shifting downward).
- The y-coordinate at point \( B \) goes from \( +4 \) to \( -4 \).
- The y-coordinate at point \( C \) goes from \( 0 \) to \( 0 \) (no change).

The reflection indicates flipping about the x-axis, which changes \( y \) to \( -y \).

### Step 3: Vertical Translation
After reflecting, we need to shift downwards to align \( y \) values:

- From 2 to -2: After reflecting, we get \( -2 \), and it needs to go down another 4 units.
- From 4 to -4: After reflecting, we have \( -4 \) already.
- From 0 to 0: No further shift is needed.

This means we also translate downward by 4 units.

### Final Transformation Composition
Combining these transformations, we first reflect over the x-axis and then translate right by 4 and down by 4. The complete transformation can be represented in function notation as follows:

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

Therefore, the complete transformation can be expressed as:
\[
(x, y) \rightarrow (x + 4, -y - 4)
\]

Looking at the options, the correct transformation sequence is:

**(x,y)→(x,−y−4)**