Question

Write a series of transformations that includes a translation followed by a reflection to map M(4, −3) onto M'(2, 5).

Answer 1

To map M(4, −3) onto M′(2, 5), a translation of 2 units to the left, followed by a reflection over the x-axis, and then an upward translation of 2 units is required.

Step-by-step explanation:

To map point M(4, −3) onto M′(2, 5), a series of transformations including a translation followed by a reflection must be performed. First, we translate point M by moving it to the left by 2 units and up by 8 units, arriving at point (2, 5). This is achieved by subtracting 2 from the x-coordinate and adding 8 to the y-coordinate of M, resulting in M′. However, since we also need to include a reflection, we can consider reflecting over a line after a different translation.

Let's break it down step-by-step:

1. Translation: Translate M(4, −3) to (2, −3) by shifting it left by 2 units. This can be represented by the translation vector T(-2, 0).
2. Reflection: Reflect the translated point over the x-axis. The point (2, −3) becomes (2, 3) after reflection across the x-axis since reflection changes the sign of the y-coordinate.
3. Now, perform another translation, T(0, 2), moving the point (2, 3) up by 2 units to (2, 5), which is the final position M′.

This series of transformations, including a translation followed by a reflection and then another translation, effectively maps the original point M onto M′.