The curves shown in the diagrams undergo a number of transformations in the Cartesian plane. After two reflections, first across the line x = y and then across the y-axis, which of... The curves shown in the diagrams undergo a number of transformations in the Cartesian plane. After two reflections, first across the line x = y and then across the y-axis, which of the four curves corresponds to its mirror image as reflected across the x-axis? Read more

Steps to Solve

1. First Reflection Across the Line (x = y)

To reflect a point ((x, y)) across the line (x = y), we swap the coordinates. The new point becomes ((y, x)).

2. Second Reflection Across the Y-Axis

To reflect the point ((y, x)) across the y-axis, we change the sign of the x-coordinate. The new point becomes ((-y, x)).

3. Determine the Original Transformations

Starting with a generic point ((x, y)):

• After reflecting across (x = y), the point is ((y, x)).
• After reflecting across the y-axis, the point is ((-y, x)).

Thus, the transformation can be expressed as:

• Original: ((x, y) \rightarrow (y, x) \rightarrow (-y, x))

This means the original curve's new image will have points where (y) is now negative of (x) when reflected.

4. Identifying the Corresponding Curve

Now look at the options provided (A, B), and identify which curve matches these transformations.