Transformations on the Coordinate Plane
Transformations are movements of geometric
figures.
The preimage is the position of the figure
before the transformation, and the image is the
position of the figure after the transformation.
Transformations on the Coordinate Plane
Isometries – transformations that maintain size and
shape (which transformations below are isometries?)
Identify Transformations
Identify the transformation as a reflection, translation,
dilation, or rotation.
Answer: The figure has been increased in size.
This is a dilation.
Identify Transformations
Identify the transformation as a reflection, translation,
dilation, or rotation.
Answer: The figure has been shifted horizontally to the
right. This is a translation.
Identify Transformations
Identify the transformation as a reflection, translation,
dilation, or rotation.
Answer: The figure has been turned around a point.
This is a rotation.
Identify Transformations
Identify the transformation as a reflection, translation,
dilation, or rotation.
Answer: The figure has been flipped over a line.
This is a reflection.
Transformations on the Coordinate Plane
Translations
• Ex.
• Translate the image by the transformation.
• (x + 2, y – 3).
• A (-3, 4) 
• B ( 0, 7) 
• C (2, -5) 
A’ (
B’ (
C’ (
)
)
)
Translations
• Ex.
• State the preimage for the transformation.
• (x - 5, y + 1).
• A( , ) 
• B ( , )
• C( , )
A’ (12, -5)
B’ ( -3, 4 )
C’ ( 0, 1 )
Translation
Graph triangle ABC and its image from the translation
(x, y) >> (x + 5, y – 3).
Answer:
The preimage is
The translated image
is
B
.
A
C
Translation
Triangle JKL has vertices J(2, –3), K(4, 0), and L(6, –3).
State the transformation rule that creates the image
of J(–3, –1), K(–1, 2), L(1, –1).
Answer:
2) Reflections
Reflections (flips) involve a mirror image
created through a line of reflection.
The image and preimage are equal
distance from the line of reflection.
Reflections are isometries:
All preimage and image measurements
(lengths, angles, area, etc.) are equal.
Lines of Reflection
• You can determine algebraic rules for
different lines of reflections in the
coordinate plane.
Algebra Rules for Reflections
Examples
Reflect the point (-3, 4) in the following lines
of reflection. Then, state the algebraic rule for
each:
a) x-axis
b) y-axis
c) y = x
d) y = -x
e) y = 3
Reflection
A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1)
and Z(–3, 1).
Trapezoid WXYZ is reflected over the y-axis. Find the
coordinates of the vertices of the image.
To reflect the figure over the y-axis, multiply each
x-coordinate by –1.
(x, y)
W(–1, 4)
X(4, 4)
Y(4, 1)
Z(–3, 1)
(–x, y)
(1, 4)
(–4, 4)
(–4, 1)
(3, 1)
Answer: The coordinates of the
vertices of the image
are W(1, 4), X(–4, 4),
Y(–4, 1), and Z(3, 1).
Reflection
A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1),
and Z(–3, 1).
Graph trapezoid WXYZ and its image W X Y Z.
Answer:
Graph each vertex of the
trapezoid WXYZ.
Connect the points.
X
Graph each vertex of the
reflected image W X Y Z.
Connect the points.
Y
W
Z
W
X
Z
Y
Reflection
b. Graph parallelogram ABCD and its image A B C D.
Answer:
Reflection
A parallelogram has vertices A(–4, 7), B(2, 7), C(0, 4)
and D(–2, 4).
a. Parallelogram ABCD is reflected over the x-axis.
Find the coordinates of the vertices of the image.
Answer: A(–4, –7), B(2, –7), C(0, –4), D(–2, –4)
Reflection
A parallelogram has vertices A(–4, 7), B(2, 7), C(0, 4)
and D(–2, 4).
a. Parallelogram ABCD is reflected in the line y = -x.
Find the coordinates of the vertices of the image.
Answer: A(-7, 4), B(-7, -2), C(-4, 0), D(-4, 2)
True or False?
1) If N (3, 5) reflects in the line y =3, then
N’ is (3, 1).
2) If M (4, 3) reflects in the line x = 1, then
M’ is (4, -1).
Reflection:
Across x-axis:(x, y)  (x, -y)
(x, y)  (-x, y)
Across y-axis:
Across y = x:(x, y)  (y, x)
(x, y)  (-y, -x)
Across y = -x:
3) Rotations
• Rotations (turns) are directed by an
imaginary arm attached to a center of
rotation. Clocks continually rotate the end
of the arm from the center of the clock.
• Rotations are considered by direction
(clockwise or counterclockwise),
degree turn and center of rotation.
3) Rotations
• All preimage and image measurements
(lengths, angles, area, etc.) are equal.
Clockwise Rotations at the
origin
90
180
270
Algebraic Rule
Examples
• Rotate the point (-3, 4) through the origin
for the following turns:
a) 90 clockwise
180

b)
clockwise
c)
270 clockwise
Examples
• Rotate the point (-3, 4) through the origin
for the following turns:
a) 90 counterclockwise
180

b)
counterclockwise
c)
270 counterclockwise
Rotation
Triangle ABC has vertices A(1, –3), B(3, 1),
and C(5, –2).
Find the coordinates of the image of ABC after it is
rotated 180° about the origin.
To find the coordinates of the image of ABC after a 180°
rotation, multiply both coordinates of each point by –1.
Answer: The coordinates of the vertices of the image are
A(–1, 3), B(–3, –1), and C(–5, 2).
Rotation
Graph the preimage and its image.
Answer:
The preimage is
The translated image
is
.
A
C
B
B
C
A
Rotation
Triangle RST has vertices R(4, 0), S(2, –3), and T(6, –3).
a. Find the coordinates of the image of RST after it is
rotated 90° counterclockwise about the origin.
Answer: R(0, 4), S(3, 2), T(3, 6)
b. Graph the preimage
and the image.
Answer:
Transformations on the Coordinate Plane
Dilation
A trapezoid has vertices E(–1, 2), F(2, 1), G(2, –1),
and H(–1, –2).
Find the coordinates of the dilated trapezoid
E F G H if the scale factor is 2.
To dilate the figure, multiply the coordinates of each
vertex by 2.
Answer: The coordinates of the vertices of the image
are E(–2, 4), F(4, 2), G(4, –2), and
H(–2, –4).
Dilation
Graph the preimage and its image.
Answer:
E
The preimage is
trapezoid EFGH.
F
The image is trapezoid
E F G H .
Notice that the image
has sides that are twice
the length of the sides of
the original figure.
F
E
G
H
H
G
Dilation
A trapezoid has vertices E(–4, 7), F(2, 7), G(0, 4),
and H(–2, 4).
a. Find the coordinates of the dilated trapezoid E F G H
if the scale factor is
Answer:
Dilation
b. Graph the preimage and its image.
Answer: