Name __________________
Worksheet 11.2
Transformations II
Due Mon March 2nd
1. Shearing Transformation
Consider the unit square with vertices A  (0, 0), B  (1, 0), C  (1, 1), D  (0, 1) mapped to the
parallelogram A'B'C'D' where A '  (0, 0), B'  (1, 0), C '  (3, 1), and D '  (2, 1)
(a) Draw the object and image.
(b) Find the transformation matrix.
(c) Describe the transformation in words.
(d) Does the area change? What is the determinant of the transformation matrix?
2. Two-Way Stretch
The quadrilateral PQRS with coordinates P  (1, 0), Q  (1, 1), R  (0, 1), and S  ( 1,  1) is
mapped to P’Q’R’S’ where P '  (3, 0), Q '  (3, 2), R '  (0, 2), and S'  ( 3,  2)
(a) Sketch this transformation of the quadrilateral PQRS
(b) Describe the main features of the transformation, saying particularly how it differs from an
enlargement by a simple dilation.
(c) Find the transformation matrix.
(d) By considering the areas of the square OPQR (O is the origin) and the rectangle OP’Q’R’,
what should the determinant of the matrix be?
(e) Find the determinant of the transformation matrix.
3. More Transformations
For each of the following matrices, draw a diagram to show their effects on the unit square and
describe (as best you can) the transformation(s) which they represent.
1 1
(a) 

0 1 
1 0 
(b) 

2 1
2 1
(c) 

1 2 
4. Multiple Transformations
Give the transformation matrix for each of the following transformations.
(a) A shear parallel to the x-axis with
(0, 1)  (3, 1)
(b) A reflection over the line y = x
(c) An enlargement scale factor 4
(d) Find the matrix that represents the transformation through all three of the above
transformations (First (a), then (b), then (c)). Make sure you multiply in the correct order
(remember that matrix multiplication is not commutative).
(e) What is the area of the final image of a unit square and how does this relate to the product of
the determinants of each of the three transformation matrices?
5. Existence of Inverse Matrices
Consider the matrix that maps î to the point (2, 2) and ˆj to the point (-2, 2)
(a) Find the matrix which accomplishes this transformation.
(b) Try to find an inverse transformation matrix. In other words, any point that first undergoes
the transformation in part (a) will then be transformed back to its original position if multiplied
by this inverse matrix.
3
(c) Transform the point   by multiplying by the matrix from part (a)
3
(d) Transform your answer to part (c) by multiplying by the inverse matrix from part (b). Do
things work out as planned? (They should!)
6. Inverse Transformations
For the following transformations, give the matrix, describe the inverse transformation in words
and give the matrix for this inverse.
(a) Rotation through 90o counterclockwise
(b) Rotation through 180o counterclockwise
(c) Reflection in the y–axis
(d) Reflection in the line y = x
(e) Shear parallel to y–axis with (1, 0)  (1, 2)
(f) Enlargement scale factor 3
1 0 
(g) What happens when you transform a point by multiplying by the identity matrix 
?
0 1 
(g) What happens if you multiply a transformation matrix by its inverse transformation matrix?
You can use the fact to check your work to each part above.
7. More Complicated Transformation
2 1
Consider the transformation whose matrix is 

2 2
(a) Find the image of the unit square under this transformation?
(b) An isometry is a transformation in which the distance between any two points is preserved.
Is this transformation an isometry?
(c) Describe the transformation as best you can (clearly it is a composition of two simpler
transformations).
(bonus) Find the matrix which would undo the transformation (the inverse).
8. Degenerate Transformation
1 3
(a) Transform the unit square by multiplying each vertex by the matrix 

2 6
(b) What is the determinant of this matrix?
(c) Does this matrix have an inverse? Why or why not?
(d) The range of a mapping is the collection of all possible points that can be mapped to (just
like the range of a function). Most transformations we have looked at have had a range of all
possible pairs of numbers. What is the range of this transformation?