Exploring Rotations
Using Matrices and
Combined Transformations

Explorations #2, #3 and #4

Mr. Jay Yohe


Exploration #2

Examine triangle ABC as it is rotated about the origin as shown in the figure below:
 
 

Rotation About the Origin

Click Here to See the Triangle Rotation Animation

Use the diagram above to complete the following chart:


Pre-Image Points 90 ° Counterclockwise Rotation  180 ° Counterclockwise Rotation  270 ° Counterclockwise Rotation 
A(__,__)  A' (__,__)  A''(__,__)  A'''(__,__) 
B(__,__)  B' (__,__)  B''(__,__)  B'''(__,__) 
C(__,__)  C' (__,__)  C''(__,__)  C'''(__,__) 


Suppose point D is an arbitrary point on the triangle with coordinates D(x, y), write a general rule relating the coordinates of any pre-image point (x, y) to its appropriate rotational image point (use the chart below):


Pre-Image Points 90 ° Counterclockwise Rotation  180 ° Counterclockwise Rotation  270 ° Counterclockwise Rotation 
D(x,y)  D' (__,__)  D''(__,__)  D'''(__,__) 


Activity -- Representing Rotations with Matrices
  1. Create Matrix [B] for each of the three rotations 90 ° , 180 ° and 270 ° using the Matrix models discussed in Exploration #1.
  2. On a new sheet of graph paper, graph polygon A,B,C,D,E,F,G. Use the rotational matrices from the exercise above and form the product [B][A]. Graph the rotational polygon for each of the 90 ° , 180 ° and 270 ° rotational angles.
  3. Suppose you want to form the 90 ° , 180 ° and 270 ° rotational angles by using successive transformed polygons in a counterclockwise direction (starting with Polygon A,B,C,D,E,F,G), which matrix [B] would you use to accomplish this task? Explain your answer!

 
 

Exploration #3

Materials --Students need two copies of the school house graphic worksheet (Figure 1 below), rulers, scissors and miras if available.

Procedures--Students will receive two copies of the school house graphic worksheet. Students will cut the school out of one of the worksheets and superimpose it over the other worksheet. Students will move the school house (via rotations, reflections or translations) to match it with the transformations pictured below. Students must find the required matrix operation(s) necessary to re-create the transformation(s). Students should also enter the coordinates of the school house into matrix form on the TI-83 calculator. Students will be required to submit the matrix operation which transforms the original pre-image school house into the figures pictured below.

School House Graphic (Figure #1)
School House


Find the transformation matrix for each picture below:

  1. Transformation 1

  2. Transformation 2

  3. Transformation 3

  4. Transformation 4



Bonus--find two transformations which would create the x-axis reflection pictured below. List both transformation matrices!

Transformation Bonus


 
 

Exploration #4

Given Matrix Matrix A, write the matrix equation to perform the requested transformations:
  1. rotate [A] 90° clockwise, and rotate 180° counterclockwise
  2. rotate [A] 180° counterclockwise, and enlarge to 3 times its size
  3. reduce [A] to .75 of its size, and rotate 270° clockwise
  4. reflect [A] over y=x and translate all x coordinates 4 units right and all y coordinates 10 units down

Back to Exploration #1