Calculators | Hewlett Packard | Transformations Investigation - Closed

Transformations Investigation - Closed

By Pat Forster If you use or quote from the following program, please acknowledge the author above.

You are going to discover J how matrices can be used to cause transformations such as reflection and rotation. On …………there will be an in class assessment on this work. This investigation must be handed in on that day.

You may work together for the take home part but will work individually for the in-class part. You may choose to use the graphics calculator aplet TRANS for page 4 and 5 of these worksheets.

First go carefully through these geometric transformations so you can recognise them.

Reflection over the y axis	Reflection over the x axis	Translation 3 units in the negative x direction
Translation of 3 units in positive y direction	Reflection over y = x	Reflection over y = -x
90o Rotation about 0 in a clockwise direction	Dilation in the x direction, factor 3 (the x coordinates multiplied by 3)	Dilation in the direction by a factor of -2
Dilation by a factor of 2 (x and y values doubled)	Dilation factor 3 in x direction, factor -2 in y direction	Shear, factor 3 along the x axis : this will be explained in class!

More information!

Remember that in describing a

• dilation - give the direction (x or y) and the factor (positive or negative, remember fractions make the object smaller)
• translation - give distance and direction
• reflection - give the line of reflection
• rotation - give the centre, the degrees, and direction (clockwise / anticlockwise or -angles for anticlockwise, +angles for clockwise)
• shear - give the factor and shear line (this will be done later)

How to use matrices in this work.

The point (x,y) can be written as a matrix.

Example:  (2,   3) is written as

Several points can be written in one matrix so:

(2,   3)  (5,   -4)  (6,   0)  can be written as

A diagram can be written as a matrix by listing its vertices - go around it, don't jump across it.

The matrix for this is

Answer the following questions.

1a)   Premultiplying a diagram matrix by a 2 x 2 matrix, transforms (reflects, rotates etc) the diagram. Premultiply the matrix for the previous rectangle (the object) by and draw the answer (image) on the same axis as the object. Label the image J'K'L'M'

Transformation matrix

x

Object matrix

=

image matrix

1b)   Describe the transformation completely (see top of previous page).

2.   In this question and you will investigate the affect of the
transformation matrix for different values of p and s.
The method is

• Premultiply the object matrix.
• Graph the object ABCD and image A'B'C'D' on the same axes.
• Describe the transformation.

2a)   Complete this one for p = -1, s = 1

x

=

Label the object vertices A, B ... Draw the image and label it A', B' ...

Description of the transformation

2b)   Follow the procedure above for p = 2 and s = 1, using the same object as in 2a).

3.   The rest of this investigation involves a lot of multiplying and drawing which the graphics calculator will do for you if you wish. Select the aplet TRANS. The aplet carries out exactly the same process as you followed in question 2.

(View object gives you the quadrilateral from question 2. For Transform matrix, p and q form the first row of the transformation matrix and r and s form the second row. Return to VIEWS after each step)

What transformation occurs with the following values for p and s in the transformation matrix ?
Write your results in the table.

p	s	Transformation
-2	1
1	2
1.5	1.5

Predict the next ones in your mind before you graph them.

p	s	Transformation
0.5	1
2	1.5
1	-1
-1	-1
-0.5	-0.5

4.   To answer this try other values for p and s if you need to.
For what does the p value do to the object?

For what does the s value do to the object?

5.   Now investigate the effect of the transformation matrix using the same object as before, and q, r values from the table below. Match the four descriptions to the correct lines of the table.

• +90^o rotation about the origin
• -90^o rotation about the origin
• reflection over y = x
• reflection over y = -x

q	r	Transformation
1	1
-1	1
1	-1
-1	-1

6a)   Another form of transformation matrix is

Complete the following, getting the trig values through HOME (make sure MODES is in Degrees)

transformation matrix

x

=

Description of the transformation

6b)   Using the graphics calculator, investigate the above transformation matrix for angles of 30^o, and -45^o. Enter cos30 for the p etc). Write a description of the effect of the transformation.

7.   Try using the new object or your own object (in VIEWS) with one of each type of transformation from questions 3, 5 and 6.

Transformation matrix

Transformation matrix

Transformation matrix

8.   Two basic types of question are asked on this work as follows.

8a)   Given the transformation matrix describe the transformation.
You probably know from the previous questions what the effect of etc is, but an alternative to memorising them is:

• Draw a unit square as shown.

• Use the two corners A, C for the object matrix and premultiply by the transformation matrix. For example, if the transformation matrix is

  x

  A   C

  =

  A'   C'

• Draw the image of the unit square with A' , C' as 2 of the vertices. Description of the transformation.

The image matrix is the same as the transformation matrix so you can skip the multiplying step . Just draw the image using the transformation matrix for the corners A' and C'. This unit square method works for all transformations - practice it with other matrices from question 3 and 5.

8b)   Given the description find the transformation matrix.

Type 1 involving rotations.
What matrix causes a rotation of -20^o about the origin?

Method: substitute in
You need to remember this matrix.

Answer?

Type 2 involving reflections or dilations.
What matrix causes a reflection over the x axis?

draw a unit square for the object, and label it ABCD the same as before reflect A over the x axis labelling the image point A' reflect C over the x axis labelling it C' A' and C' give you the transformation matrix
	Check you answer with question 3.

Practice some more of these using your results in question 3.