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.
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
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.