The points A(4, -11), B(5, 3), C(2, 15) and D(1, 1) are the vertices of a parallelogram. If the parallelogram is reflected in the y-axis and then in the origin, find the coordinates of the final images. Check whether it remains a parallelogram. Write down a single transformation that brings the above change.

The graph for this problem is shown below:

First the parallelogram is reflected in y-axis.

We know that,

Rule to find reflection of a point in y-axis :

1. Change the sign of abscissa i.e. x-coordinate.
2. Retain the ordinate i.e. y-coordinate.

∴ Coordinates of
⇒ A(4, -11) on reflection in y-axis becomes A'(-4, -11).
⇒ B(5, 3) on reflection in y-axis becomes B'(-5, 3).
⇒ C(2, 15) on reflection in y-axis becomes C'(-2, 15).
⇒ D(1, 1) on reflection in y-axis becomes D'(-1, 1).

We know that,

Rules to find the reflection of a point in the origin :

1. Change the sign of abscissa i.e. x-coordinate.
2. Change the sign of ordinate i.e. y-coordinate.

∴ Coordinates of
⇒ A'(-4, -11) on reflection in origin becomes A''(4, 11).
⇒ B'(-5, 3) on reflection in origin becomes B''(5, -3).
⇒ C'(-2, 15) on reflection in origin becomes C''(2, -15).
⇒ D'(-1, 1) on reflection in origin becomes D''(1, -1)

From graph we can see that A"B"C"D" is also a parallelogram.

The single transformation that marks the following changes i.e.
A(4, -11) ⇒ A"(4, 11)
B(5, 3) ⇒ B"(5, -3)
C(2, 15) ⇒ C"(2, -15)
D(1, 1) ⇒ D"(1, -1), is reflection in x-axis.

Hence,

1. The coordinates of the final images of the vertices are (4, 11), (5, -3), (2, -15) and (1, -1) respectively.
2. These new images still form parallelogram.
3. The single transformation from ABCD ⇒ A"B"C"D" can be achieved by reflection in x-axis.