3. Find equations for a transformation that maps a rectangular region in the -plane
   onto the region consisting of the parallelogram with vertices , and
   .

To find the transformation that maps a rectangular region in the -plane onto the parallelogram defined by the vertices and , we can use linear transformation concepts. We can represent the edges of the parallelogram as vectors, such as from to (which gives the vector ) and from to (which gives the vector ).

We can express the transformation as:

where are parameters that would map the unit square (bounded by 0 and 1) in the -plane into parallelogram .

This transformation takes combinations of these vectors weighted by and , effectively stretching and shearing the rectangle into the desired parallelogram shape.

Using the coefficients, you could easily plug in and values from a unit rectangle to see how each point translates into the corresponding point in the parallelogram .

For further fun, if you visualize how moving points in directly translates into the vertices and sides of the parallelogram, it’s like pulling and stretching a rubber sheet – every point moves along defined pathways, resulting in a beautiful geometric transformation!