Question

ΔABC is reflected across the x-axis and then dilated by a factor of 2 using the point (−2, 1) as the center of dilation. What is the transformation of A(3, 1)?

A.
A"(3, −1)
B.
A"(−6, 2)
C.
A"(6, −2)
D.
A"(8, −3)

Answer 1

The point A(3, 1) after reflection across the x-axis becomes A'(3, -1). Then, after dilation by a factor of 2 using (-2, 1) as the center, it becomes A"(8, -3). Thus, the correct transformation is D. A"(8, -3).

Step-by-step explanation:

The transformation of point A(3, 1) after reflecting across the x-axis and then dilating by a factor of 2 with (-2, 1) as the center of dilation can be found using two steps:

1. Reflection across the x-axis: The y-coordinate changes sign, while the x-coordinate remains the same. So, after reflection, A becomes A'(3, -1).
2. Dilation by a factor of 2 using (-2, 1) as the center of dilation: To perform this dilation, we have to calculate the new coordinates by finding the vector from the center of dilation to the reflected point A' and then scaling that vector by the dilation factor.
   Vector from the center of dilation to A' = A' - center = (3 - (-2), -1 - 1) = (5, -2)
   New coordinates = center + 2 × (vector to A') = (-2, 1) + 2 × (5, -2) = (-2 + 10, 1 - 4) = (8, -3)
   Hence, after the dilation, the point A'(3, -1) becomes A"(8, -3).

Therefore, the correct answer is D. A"(8, -3).