A rectangle is graphed on a coordinate plane. The table shows the original vertices and the new vertices after a scale factor with center of dilation at (0, 0) is applied.

Question

What rule is applied to the original rectangle to create the new rectangle?

(x, y) -----> (3 halves "x" , 3 halves "y"
3
2​x,
3
2​y)

(x, y) -----> (2 thirds "x" , 2 thirds "y"
2
3​x,
2
3​y)

(x, y) -----> ("x" negative 4 , "y" negative 3
x − 4, y − 3)

(x, y) -----> ("x" positive 4 , "y" negative 3
x + 4, y − 3)

Answer 1

The original rectangle is transformed using one of the rules outlined in your table to create the new rectangle. Let's analyze each of the options:

1. (x, y) -----> (3/2 x, 3/2 y): This represents a dilation with a scale factor of \( \frac{3}{2} \) and it expands the rectangle away from the origin (0,0).

2. (x, y) -----> (2/3 x, 2/3 y): This represents a dilation with a scale factor of \( \frac{2}{3} \) and it reduces the rectangle toward the origin (0,0).

3. (x, y) -----> (x - 4, y - 3): This represents a translation of the rectangle 4 units left and 3 units down.

4. (x, y) -----> (x + 4, y - 3): This represents a translation of the rectangle 4 units right and 3 units down.

Conclusion:

The rules that apply to the original rectangle to create a new rectangle are the dilation options: either \( \left( \frac{3}{2}x, \frac{3}{2}y \right) \) for expansion or \( \left( \frac{2}{3}x, \frac{2}{3}y \right) \) for reduction. The translation rules do not result in dilation but instead shift the rectangle's position on the coordinate plane.

Thus, you have dilation (expansion or reduction) represented by the first two options, specifically \( \left( \frac{3}{2}x, \frac{3}{2}y \right) \) or \( \left( \frac{2}{3}x, \frac{2}{3}y \right) \).