Activity #1

Sets of Equations for Transformations

 

1.      Plot and connect points A(2,1), B(5,1), and C(5,7).  Translate this triangle with a vector that is right 6 and up 4.  (Show work on graph paper.)

2.      How could you show the change to the coordinates algebraically? ______,______)

3.      Now find the image of (3,6) under the translation that is down 7 and right 2.  What are the new coordinates?____________   What did you do to the x-coordinate and the y-coordinate to get the image point? ________________  Express this algebraically.  ______,______)

 

A translation, , can be defined as a transformation with equations of the form .

 

4.      Using this algebraic generalization find the image of 2x +4y – 6 = 0 under the translation with equations  and .  Graph the pre-image and the image on your graph paper.  What is the equation of the image? (Hint: x=x’-7)

 

 

 

 

 

 

 

Once you know what to do to each coordinate to generate the translation you desire, you can put the points in Lists 1 and 2 in your graphing calculator and then generate Lists 3 and 4 by using the equation.

Basic Steps for Using Lists in TI-83+

STAT (button)

EDIT

Clear L1 – L4

Enter x-coordinates in List 1

Enter y-coordinates in List 2

Place cursor over the L3 and enter algebraic expression that    will take x to x’, i.e. L1-7

Place cursor over the L4 and enter algebraic expression that    will take y to y’, i.e. L2-2

 

 

5.      Plot and connect points D(-2,1), E(-2,6), and F(-5,1) on your graph paper.  If you reflect this triangle across the x-axis, what are the coordinates of the image under this reflection?  D’(__,__), E’(__,__), F’(__,__)

6.      Now graph the reflection of DDEF across the x-axis, DD’E’F’.

7.      Can you express this algebraically?  x’ = ____ and y’ = ____

 

8.      Now reflect DDEF over the y-axis, DD”E”F”. What are the coordinates of the image under this reflection?  D”(__,__), E”(__,__), F”(__,__)

 

9.      Can you express this algebraically?  x’ = ____ and y’ = ____

 

After having expressed these translations and reflections as sets of linear equations, let’s think about rotations.  Rather than discover these we will just look at an example for rotations:  The equations for a rotation about the origin are .

 Find the image of P(2,3) under a rotation of 60° about the origin.    

Thus, the coordinates of P’ are

 

Practice Problems:

 

Using the equations x’ = x + 2 and y’ = y – 3 (the vector [2,-3] for translation)

  1. Find the image of point (5,7).
  2. Find the image of point (-1, -8).
  3. What was the original point (pre-image) of the image (5,9)?
  4. Find the image of line y = 5x – 4.
  5. Find the image of the line 2x – 6y + 3 = 0

 

  1. Find the image of y = 2x – 4 after it is reflected across the x-axis?  The y-axis?
  2. What is the image of (4, 6) under a rotation of 45° about the origin?
  3. Find the image of (-3, 0) under a rotation of 30° about the origin.

 

 

Special 2x2 Matrices

(Given to student once they have completed activity #2)

Here are some of the special 2x2 matrices representing particular geometric manipulations that are useful in computer graphics.

 

                        Identity

                       A scale change in the x direction

                     Reflection about the y-axis

                     Reflection about the x-axis

                       Shear in y direction

                     Rotation of -90°

                   Rotation of 180°

                     Rotation of -270°

                       Reflection about line y=x

                       Scaling change in both x and y directions

                       Unequal scaling change

         General rotation about origin

Activity #2

Transformations with Matrices

 

This activity will allow students to explore the effect of matrices on various figures. 

 

1.      Create quadrilateral PROB on graph paper using the coordinates listed below.  P(1,4), R(5,4), O(8,2), B(3,2).  Create Matrix A, a 2 x 4, with these ordered pairs, placing the x-coordinates in row 1 and the y-coordinates in row 2.  Then put this Matrix A in your graphing calculator. (Keep Matrix A in your calculator.  The quadrilateral PROB will be used again.)

 

2.      Multiply Matrix B  by Matrix A .  Graph the resulting quadrilateral by plotting the vertices of  = = [C].  Label the new vertices P’, R’, etc.  Compare the original quadrilateral with the new quadrilateral. 

 

a)     Describe how the quadrilaterals are alike and how they are different. 

 

b)     What do you notice about the distance between points P and P’ and the y-axis?

 

c)      What transformation does multiplying Matrix A by Matrix B accomplish?

 

d)     The two matrices result in a reflection over what axis?

 

3.      Look below at the method used to determine the Matrix B used above. Can you use this model to find a matrix for other transformations?

 

 

Determine y-axis reflection matrix

For y-axis reflection   (x,y)®(-x,y)

 

 

4.      Store the product of  into Matrix C.  Then find the product of Matrix B with Matrix C. Graph the coordinates of Matrix C. Describe the result.  What happened and why?

 

Next:

5.      Can you find Matrix B  that will reflect quadrilateral PROB over

the x-axis?  Try your guess and once again compute  =  = D.  Graph the resulting quadrilateral and label the points P’’, R’’, etc.  Compare the original quadrilateral PROB with the new quadrilateral P”R”O”B”. 

a)     Describe how the quadrilaterals alike and how do they differ.   

 

b)     What do you notice about the distance between points P and P’ and the x-axis?

 

c)      What does product Matrix B accomplish?

 

5.      Again, observe the model used to calculate Matrix B.

Determine X-axis reflection matrix

For x-axis reflection   (x,y)®(x,-y)

 

 

6.      If you did everything correctly, you should now have the quadrilateral in three of the four quadrants of the coordinate plane.  If you rotate PROB 180° about the origin, you would have a quadrilateral in all four quadrants.  Remember what the product of two reflections over intersecting lines yields?

a)     Can you find a series of matrix multiplications to rotate the original quadrilateral to the missing quadrant?

 

 

 

b)     Can you find one matrix multiplication to transform the original polygon to the missing quadrant?

 

 

 

7.      Once you find a series of matrix multiplications to transform the quadrilateral to the missing quadrant (or by a single matrix multiplication), draw the quadrilateral in this quadrant, P”’R”’O”’B”’.  All four reflected quadrilaterals should now form a symmetrical image. 

 

8.      On a new sheet of graph paper, graph PROB again. Now find a new matrix [F] =  that will reflect quadrilateral PROB over the line y = x. Possibly draw this line and use a MIRA or paper folding to study the effect before picking your matrix F.  Graph this new quadrilateral. What happens to the coordinates of PROB when you reflect it over y = x?

 

9.      Now reflect PROB over the line y = -x with matrix G = .  Graph this quadrilateral.

 

Next:

10. Get a new piece of graph paper and graph the original quadrilateral PROB.  How could you translate PROB down 5 units and left 3 units?  What Matrix B could you use?  B = . Would you add or multiply it to the original Matrix A?

11. What kind of transformation results from the addition of Matrix B to Matrix A?

 

12. Can you generalize about transformations involving addition of matrices based on this experiment?

 

Now in this part you will use triangle DOG.

 

10. Put the coordinates of Triangle DOG in a 2 x 3 matrix (call it D): D(0,0), O(1,1), and G(2,1) and graph triangle DOG on a coordinate plane.  Using tracing paper, MIRA or whatever tools will be helpful, complete the chart below.

 

Pre-Image Points

90 ° Counterclockwise Rotation 

180 ° Counterclockwise Rotation 

270 ° Counterclockwise Rotation 

D(__,__) 

D' (__,__) 

D''(__,__) 

D'''(__,__) 

O(__,__) 

O' (__,__) 

O''(__,__) 

O'''(__,__) 

G(__,__) 

G' (__,__) 

G''(__,__) 

G'''(__,__) 

 

11. You did 180  rotation earlier.  Now that you have completed the chart above, can you find a matrix that when multiplied by the matrix D will result in the various rotations in the chart? 

 

90 ° Counterclockwise Rotation              180 ° Counterclockwise Rotation            270 ° Counterclockwise Rotation 

M =                      N =                      P =

 

All of the transformations we have worked with have been isometries.  A transformation is an isometry if it preserves distances. 

 

 

 

 

While PROB is still in Matrix A of your calculator, let’s take a brief look at a similarity transformation or dilation.

 

12. Compute the product  where Matrix A is the original quadrilateral and Matrix B = .   =

13. On a new sheet of graph paper graph PROB once again and then graph the new quadrilateral.  Were distances preserved or was this transformation an isometry?  Describe what happened to the quadrilateral.

 

 

 

14. Find and test matrices which accomplish the following: (Graph the resulting ordered pairs in each case.)

·        Enlarge Matrix A by 125% 

·        Reduce Matrix A by 75%

·        Reduce the matrix 50% and slide it down 10 units

15. In your own words, explain how to enlarge and reduce a figure represented by a matrix. 

 

Practice Problems:

Graph the triangle represented by Matrix A =

Write the matrix equation to perform the requested transformations and graph the resulting triangles:

a)     rotate [A] 90° clockwise and then rotate 180° counterclockwise

b)     rotate [A] 180° counterclockwise  and then enlarge to 3 times its size

c)      reduce [A] to .75 of its size and then rotate 270° clockwise

d)     reflect [A] over the line y = x and translate all x coordinates 4 units right and all y coordinates 10 units down