8th grade Unit 9
Transformations
Name
Date
Period:
A transformation is a function that changes the position, shape, and/or size of a figure. The inputs
for the function are points in the plane; the outputs are other points in the plane. A figure that is used
as the input of a transformation is the pre-image. The output is the image.
For example, the transformation T moves point A to point A′.
Point A is the pre-image, and A′ is the image. You can use
function notation to write T (A) = A′. Note that a transformation
is sometimes called a mapping. Transformation T maps point
A to point A′.
Coordinate notation is one way to write a rule for a
transformation on a coordinate plane. The notation uses an
arrow to show how the transformation changes the
coordinates of a general point, (x, y).
For example, the notation (x, y) → (x + 2, y - 3) means that the transformation adds 2 to the xcoordinate of a point and subtracts 3 from its y-coordinate. Thus, this transformation maps the point
(6, 5) to the point (8, 2).
1.
If I write T(E) = F, identify the pre-image and the image.
2.
Consider the transformation given by the rule (x, y)  (x + 1, y + 1). What is the domain of this
function? What is the range? Describe the transformation.
Investigate the effects of various transformations on the given right triangle.



A
Use coordinate notation to help you find the image of each vertex of the triangle.
Plot the images of the vertices.
Connect the images of the vertices to draw the image of the triangle.
(x, y) → (x - 4, y + 3)
B (x, y) → (-x, y)
Adapted from OnCore Mathematics: Geometry
C (x, y) → (-y, x)
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D (x, y) → (2x, 2y)
E (x, y) → (2x, y)
F (x, y) →(x, ½ y)
3.
A transformation preserves distance if the distance between any two points of the pre-image
equals the distance between the corresponding points of the image. Which of the transformations
A-F (above and on the previous page) preserve distance?
4.
A transformation preserves angle measure if the measure of any angle of the pre-image equals
the measure of the corresponding angle of the image. Which of the above transformations
preserve angle measure?
A rigid motion (or isometry) is a transformation that changes the position of a figure without
changing the size or shape of the figure.
The figures below show the pre-image (ΔABC) and image (ΔA′B′C′ ) under a transformation.
Determine whether the transformation appears to be a rigid motion.
5. The transformation does not change the size or shape of the figure. Therefore ________________
6. The transformation changes the shape of the figure. Therefore, ___________________________
Adapted from OnCore Mathematics: Geometry
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Rigid motions have some important properties. These are summarized below.
These properties ensure that if a
figure is determined by certain
points, then its image after a
rigid motion is also determined
by those points.
Point M is the midpoint of
AB . After a rigid motion,
can you conclude that M′
is the midpoint of A' B' ?
Why or why not?
For example, ΔABC is
determined by its vertices, points
A, B, and C. The image of ΔABC
after a rigid motion is the triangle
determined by A′, B′, and C′.
7.
Draw the image of the triangle under the given transformation. Tell whether the transformation
appears to be a rigid motion.
A (x, y) → (x + 3, y)
B (x, y) → (3x, 3y)
D (x, y) → (-x, -y)
E (x, y)→ (x, 3y)
Adapted from OnCore Mathematics: Geometry
C
(x, y) → (x, -y)
F (x, y) → (x - 4, y - 4)
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Translations
A translation slides all points of a figure the same distance in
the same direction. The figure shows a translation of a triangle.
It is convenient to describe
translations using the language of
vectors. A vector is a quantity
that has both direction and magnitude. The initial point of a vector is
the starting point. The terminal point of a vector is the ending point.
The vector at right may be named EF or v .
You can use vectors to give a formal definition of translation.
A translation is a transformation along a vector such that the segment
joining a point and its image has the same length as the vector and is
parallel to the vector.
The notation Tv(P) = P’ says that the image of point P after a translation along vector v is P’.
The figure below shows pre-image ABCDEFGH and translation vector PQ . Trace ABCDEFGH on
patty paper and use the translation vector to draw the image of ABCDEFGH. Once you see the
pattern, draw the translation on the grid below.
8. Write the
coordinates of each
pre-image point and
its corresponding
image on the point.
Also write the
coordinates of the
endpoints of PQ .
Qx – Px = ______
Qy – Py = ______
Another way to
describe PQ is to
describe the change in
x and y coordinates.
For this vector, you
would write <7, -2>
9. Describe how you can use the coordinates of PQ to find the coordinates of the image.
Adapted from OnCore Mathematics: Geometry
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Still another way to describe the translation is to use coordinate notation (x, y) (x + 7, y – 2).
Explain how vector notation <7, -2> and coordinate notation (x, y) (x + 7, y – 2) give you the same
information.
10. You are going to translate this vector using the
translation vector <-3, 2>.
a.
What does <-3, 2> tell you?
b.
Write this in coordinate notation
c.
Now draw the translated triangle on the grid to
the left.
d.
Give a translation that would move the original
triangle completely into Quadrant IV.
11. Is translation a rigid motion? Explain.
12.
a. <3, -2>
b. (x, y) (x – 4, y + 4)
Draw the image of
the figure under the
given translation.
Write the coordinate
notation / vector
notation for each.
13. Use both vector and coordinate notation to name the
translation that maps
ABC to
A'B'C'
What distance does each point move under this translation?
Adapted from OnCore Mathematics: Geometry
Page 5
Reflections
A reflection is a transformation that moves points by
flipping them over a line called the line of reflection.
The figure shows the reflection of quadrilateral ABCD
across line l. Notice that the pre-image and image
are mirror images of each other.
14. Use a mira to draw the reflection of
ABC over line l. Place the mira on
line l and trace the reflection.
a. Use a straightedge to trace both the
pre-image and the image on patty
paper. Fold along line l. Is the image
actually a reflection of
ABC ?
b. Use your straightedge to connect
corresponding points of the image
and pre-image.
15. Write three things you notice about the segments AA', BB' and CC '
Read the formal definition of a reflection below.
A reflection across line l maps a point P to its image P' as follows.


If P is not on line l, then l is the perpendicular bisector of PP'.
P is on line l, then P = P'.
The notation rl (P) = P' says that the image of point P after a reflection across line l is P'.
16. What is a perpendicular bisector? Look this up in your book if you don’t have the definition in your
notebook.
17. Describe how your notices from problem 15 relates to the definition above.
Adapted from OnCore Mathematics: Geometry
Page 6
18. The three figures below show the pre-image reflected over the x-axis, the y-axis and the line
y = x respectively. Write the coordinates of two points and their corresponding reflections on each
figure. Make a conjecture as to how reflection over each line affects the coordinates of the preimage points.
19. Use coordinate geometry to
prove that the mirror line y = x
is the perpendicular bisector of
DD' (Hint: think about what a
perpendicular bisector would
do to DD')
Reflections in a Coordinate Plane
Reflection across the x-axis
(x, y)  (
,
)
Reflection across the y-axis
(x, y)  (
,
)
Reflection across the line y = x
(x, y)  (
,
)
Is reflection a rigid motion? Explain.
Adapted from OnCore Mathematics: Geometry
Page 7
20. You are designing a logo for a bank. The left half of the logo is shown. You will complete the logo
by reflecting this figure across the y-axis. Visualize in your mind what this logo should look like.
Use the table to list the vertices of the left half and the reflected vertices for the right half. Then
graph the vertices and complete the logo. Did your completed logo look like the one you
visualized before you started?
Left Half
(x, y)
Right Half
(___, ___)
(0, 4)
(0, 4)
21. As the first step in designing a logo, you draw the figure shown in the first quadrant of the
coordinate plane. Then you reflect the figure across the x-axis. You complete the design by
reflecting the original figure and its image across the y-axis.
Original
(x, y)
Reflect on
x-axis
(___, ___)
Reflect
original on
y-axis
(___, ___)
Reflect 1st
reflection
on y-axis
(___, ___)
22. When point P is reflected across
the y-axis, its image lies in
Quadrant IV. When point P is
reflected across the line y = x, its
position does not change. What
can you say about the
coordinates of point P?
Adapted from OnCore Mathematics: Geometry
Page 8
Rotation
A rotation turns all points of the plane around a point called the
center of rotation. The angle of rotation tells you the number of
degrees through which points rotate around the center of rotation.
The figure shows a 120° counterclockwise rotation around point P.
When no direction is specified, you can assume the rotation is in the
counterclockwise direction.
While rotations can be any degree and around any point, we will focus on rotations about the origin,
particularly rotations of 90, 180, and 270 counterclockwise about the origin.
For each rotation below, focus on the pre-image points A, B, C and D and their images.
 Find the coordinates of these 4 pre-images and the coordinates of C’ and D’.
 Identify how the rotation affects the coordinates of the image.
 Predict the coordinates of A’ and B’ and verify your prediction.
Rotation 90 about the origin
23. (x, y) (
,
)
Predicted A’ ______
Predicted B’ ______
Adapted from OnCore Mathematics: Geometry
Rotation 180 about the origin
24. (x, y) (
,
)
Predicted A’ ______
Predicted B’ ______
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Rotation 270 about the origin
25. (x, y) (
,
)
Predicted A’ ______
Predicted B’ ______
Summary:
Rotations in a Coordinate Plane
Rotation of 90
(x, y)  (
,
)
Rotation of 180
(x, y)  (
,
)
Rotation of 270
(x, y)  (
,
)
Is rotation a rigid motion? Explain
Draw the image of the figure after the given rotation. Make a table showing the coordinates to help
26. 180
27. 90
28. 270
29. Reflect JKL across the x-axis. Then reflect the image
across the y-axis. Draw the final image of the triangle and
label it J '' K '' L''.
Describe a single rotation that maps
JKL to
J '' K '' L''
Prove this using coordinate notation for any point (x, y)
Adapted from OnCore Mathematics: Geometry
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