Question

Identify the sequence of transformations that will carry A(-5,2)B(-2,4)C(-3,0) -> A"(-1,-2)B"(2,-4) C"(1,0). Give your answer in the form of a composition transformation mapping.

a. (x,y) -> (x, -y + 4)
b. (x,y) -> (x + 4, -y)
c. (x,y) -> (x, -y-4)
d. (x,y) -> (x-4, y)

Answer 1

To identify the sequence of transformations that carry the points A(-5,2), B(-2,4), and C(-3,0) to A"(-1,-2), B"(2,-4), and C"(1,0), we will analyze the transformations step by step.

Let's start with the points:
- A(-5,2) transforms to A"(-1,-2)
- B(-2,4) transforms to B"(2,-4)
- C(-3,0) transforms to C"(1,0)

### Finding the transformation:

1. **Identify the changes in the coordinates:**
- For point A: from (-5, 2) to (-1, -2)
- Change in x: -5 to -1 → +4
- Change in y: 2 to -2 → -4

- For point B: from (-2, 4) to (2, -4)
- Change in x: -2 to 2 → +4
- Change in y: 4 to -4 → -8

- For point C: from (-3, 0) to (1, 0)
- Change in x: -3 to 1 → +4
- Change in y: 0 to 0 → 0

Notice the pattern in the x-coordinates: each x-coordinate increases by 4.

2. **Consider transformations available:**
- Let's check option **(a)**: \((x,y) \to (x, -y + 4)\)
- A(-5,2) → A(-5, -2 + 4) = A(-5, 2) (no change)
- This doesn’t work.

- Option **(b)**: \((x,y) \to (x + 4, -y)\)
- A(-5,2) → A(-5 + 4, -2) = A(-1,-2)
- B(-2,4) → B(-2 + 4, -4) = B(2,-4)
- C(-3,0) → C(-3 + 4, 0) = C(1,0)
- This matches our required transformation.

- Option **(c)**: \((x,y) \to (x, -y - 4)\)
- A(-5,2) → A(-5, -2 - 4) = A(-5, -6) (not matching)

- Option **(d)**: \((x,y) \to (x - 4, y)\)
- A(-5,2) → A(-5 - 4, 2) = A(-9, 2) (not matching)

### Conclusion:
The required transformation to map A(-5,2), B(-2,4), C(-3,0) to A"(-1,-2), B"(2,-4), C"(1,0) is:

**(x,y) \to (x + 4, -y)**

Thus, the composition transformation that carries the points is:
**(x,y) → (x + 4, -y)**.

This matches option **(b)**.