Question

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Explanation:

Identify the coordinates of shape II

Using the Coordinate Transformations knowledge point

$$ \text{Vertices of shape II: } A(-6, -6), B(-8, -2), C(-4, -1), D(-2, -1) $$

Determine the transformation mapping

Using the Reflections in Coordinate Plane and Coordinate Transformations knowledge points

$$
LATEXBLOCK0
$$

Verify the vertices of shape I

Let the original vertices of shape I be:

$$ A'(3, -3), B'(4, -1), C'(2, -0.5), D'(1, -0.5) $$

Applying the transformation \( (x, y) \to (-2x, 2y) \) or similar reflection/dilation combinations:
Let's test a reflection across the \( y \)-axis followed by a dilation of factor \( 2 \):

$$ (x, y) \to (-x, y) \to (-2x, 2y) $$

If the original shape I has vertices at:

$$ A'(3, -3), B'(4, -1), C'(2, -0.5), D'(1, -0.5) $$

Then:

$$
LATEXBLOCK1
$$

This perfectly matches the vertices of shape II.

Match with the given options

The sequence of transformations is:

1. A reflection across the \( y \)-axis.
2. A dilation by a scale factor of \( 2 \).

Answer:

• a 90° clockwise rotation about the origin, and then a dilation by a scale factor of 2
• a reflection across the x-axis, and then a dilation by a scale factor of 1.5
• a 90° counterclockwise rotation about the origin, and then a dilation by a scale factor of 1.5
• a reflection across the x-axis, and then a dilation by a scale factor of 2
• a reflection across the y-axis, and then a dilation by a scale factor of 2 (Correct answer)