Question

On a coordinate plane, 3 parallelograms are shown. Parallelogram A B C D has points (3, 5), (6, 5), (4, 1), (1, 1). Parallelogram A prime B prime C prime D prime has points (3, negative 5), (6, negative 5), (4, negative 1), (1, negative 1). Parallelogram A double-prime B double-prime C double-prime D double-prime has points (negative 3, negative 4), (0, negative 4), (negative 2, 0), (negative 4, 0).
Which rule describes the composition of transformations that maps pre-image ABCD to final image A"B"C"D"?

Reflection across the x-axis composition translation of negative 6 units x, 1 unit y.
Translation of negative 6 units x, 1 unit y composition reflection across the x-axis.
90 degree rotation about point 0 composition translation of negative 6 units x, 1 unit y.
Translation of negative 6 units x, 1 unit y composition 90 degree rotation about point 0.

Answer 1

Translation of negative 6 units x, 1 units y composition reflection across the x-axis

The composition transformation that maps ABCD to A''B''C''D'' comprises

of two or more transformations that maps the image

The rule that describes the composition transformation that maps ABCD to

A''B''C''D'' is the option;

Translation of negative 6 units x, 1 units y composition reflection across the x-axis.

The given coordinates of parallelogram ABCD are (3, 5), (6, 5), (4, 1), (1, 1)

The coordinates of the parallelogram A'B'C'D' are; A'(3, -5), B'(6, -5), C'(4, -1), D(1, -1)

The coordinates of the parallelogram A''B''C''D'' are;

D(-5, 0), C''(-2, 0), A''(-3, -4), B''(0, -4)

The image of the point (x, y), following a reflection about the x-axis is the point (x, -y)

The difference in the coordinates of the parallelogram ABCD and A'B'C'D'

is the change in the sign of the y-coordinates

Therefore, the transformation that gives the parallelogram A'B'C'D' from

the parallelogram ABCD is a reflection about the x-axis

The difference between the coordinates of the vertices of the

parallelogram A'B'C'D' and the parallelogram A''B''C''D'', is a change in the

x-value by (-6), and a change in the y-value by (+1)

Therefore, the parallelogram A''B''C''D'' can be obtained from parallelogram

A'B'C'D', by a translation of -6 units (left) in the x-direction and a translation

of 1 unit, x in the y-direction

Which gives;

The composition transformation that maps the pre-image ABCD to the

final image A''B''C''D'', (given that in a composition transformation, the

transformation to the right is done first) are;

$\mathrm{T}_{-6,1} \cdot \mathrm{r}_{\mathrm{x}-\{axis}}(\mathrm{x}, \mathbf{y})$which is the option;

Translation of negative 6 units x, 1 units y composition reflection across the x-axis

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