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					      Geometry                                                            Name________________________
      Transformations (3): Transformation Equations                                             3/26/07

      Transformations of the plane
      A transformation of the plane is a function whose domain and range are the xy-plane (or sets of points
      within the plane). You are already familiar with geometric descriptions of certain kinds of transformations,
      such as translations, reflections, rotations, and dilations. A pair of transformation equations defines the
      mapping of the coordinates (x,y) to the coordinates of the image (x’,y’).
      Here is an example of a pair of transformation equations:
          x = x + 3
          y  y – 1
             =

    These equations say that given any point (x, y), the image point ( x  y  is found by adding 3 to the
                                                                              , )
    x-coordinate and subtracting 1 from the y-coordinate. In other words, these equations define a
      horizontal translation by 3 units to the right and a vertical translation by 1 unit down.
                                                                
      Many geometrically interesting transformations have fairly simple equations. In fact, all
                                                                   
      translations, reflections, rotations, and dilations can be described using equations of the form
          x = ax + by + e
          y  cx + dy + f
             =

    An important fact about transformation equations having this form is that the image of a line segment
      is always a line segment. Therefore, if you want to see the effect of a transformation on a polygon, you
    just need to find the images of the vertices.

      Problems
      1. Give a geometric description of the transformation defined by each pair of equations. Be as specific
         as possible. (For example, for a reflection, identify the reflection line.) If you’re not sure about an
         answer, try graphing a few pairs of input and output points.
          a. x = x – 2                                             f.   x = 6 – x
             y = y + 5                                                  y  y
                                                                             =

        b. x = 2x                                              g. x = y
           y = y                                                  y = x

        c.   x = x                                             h. x = –y
             y  1 y
                  = 3                                                 y = x

        d. x = x                                               i.   x = x
           y = –y                                                   y = 0

        e.   x = –x                                            j.   x = kx
               y = –y                                                 y = y + c
     

                                                            
                                                            
 Geometry                                                                           Transformations (3)


 2. Write a pair of transformation equations for each of the following. If you’re not sure,
    write down the coordinates for a few pairs of input and output points, try to spot a
    pattern, then turn the pattern into general rules for finding x and y 
                                                                             .
     a. reflection across the y-axis
     b. the translation that maps the point (3, –1) to an image of (8, –4)
                                                       
                                                             
     c. a horizontal dilation that would shrink any object to half of its original width
     d. reflection across the horizontal line y = 4
     e. reflection across the line y = –x
     f. rotation by 180° centered at the origin
     g. rotation clockwise by 90° centered at the origin
     h. projection onto the y-axis (that is, a transformation that maps every point to the
        point on the y-axis having the same y-coordinate)


 3. The transformation x = 4x + 2y, y = 2x + y maps every point of the plane onto a
    line L. By experimenting with different points and their images, find the equation of
    line L.
               
                           
 4. Consider the transformation: x = ax + by + e      y = cx + dy + f
     a. Calculate the images of the vertices of the unit square with vertices: (0, 0), (0, 1),
        (1, 0), and (1, 1).
                          
                                             
     b. What special type of quadrilateral is the image of the unit square? Prove your
        answer. (For example: if you think it’s a rectangle, prove the angles are right
        angles; if you think it’s a parallelogram, prove the sides are parallel; etc.)


 5. A transformation having the form
         x = ax + by   (note there is no e term)
         y = cx + dy   (note there is no f term)
     is called a linear transformation.

     a. Which of the transformations from problem 2 are linear transformations?

     b. Which of the transformations from problem 3 are linear transformations?
     c. Of the four well-known types of transformations (translations, reflections,
        rotations, dilations), there’s one type that’s almost never a linear transformation.
        Which type?

				
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