The document describes reflections and translations, which are types of transformations in geometry. It defines reflections as transformations across a line where corresponding points are an equal distance from the line. Translations are defined as transformations where all points of a figure are moved the same distance in the same direction, making them an isometry. Examples of reflecting and translating points and figures are provided to illustrate these concepts.

1. Obj. 30 Reflections and Translations
The student is able to (I can):
• Identify and draw reflections
• Identify and draw translations

2. transformation A change in the position, size, or shape of a
figure.
A
A´
B
C
B´
C´
preimage
The original figure.
image
The figure after the transformation.
isometry
A transformation that only changes the
position of the figure.

3. A´
A
mapping
B
B´
C
C´
Note: We use “arrow notation” to describe
a transformation and primes (´) to label
the image. This process is called mapping
mapping.
A is mapped to A´ (A → A′)
B is mapped to B´ (B → B′)
C is mapped to C´ (C → C′)
∆ABC is mapped to ∆A´B´C´
(∆ABC → ∆A′B′C′)

4. reflection
A transformation across a line; each point
and its image are the same distance from
the line.
P(x, y) → P′(x, − y)
P(x, y) → P′(− x, y)
P(x, y) → P′(y, x)
Across the x-axis
Across the y-axis
Across the line y=x
P´(—x, y)
y
•
P(x, y)
•
P´(y, x)
x
0
•
P´(x, —y)

5. Examples
Reflect the given vertices across the line.
1. L(-2, 0), H(-1, 4), S(3, 2); y-axis
y
H•
•
•
S
x
L
2. M(-3, 3), A(2, 3), T(2, -1), H(-3, -1); y=x
M
•
•
•
•
H
y=x
A
T

6. Examples
Reflect the given vertices across the line.
1. L(-2, 0), H(-1, 4), S(3, 2); y-axis
y
H•
• H´
S´ •
•
•
•
L
L´(2, 0)
H´(1, 4)
S´(-3, 2)
S
L´
x
2. M(-3, 3), A(2, 3), T(2, -1), H(-3, -1); y=x
M
•
•
A
T´ •
•
•
H
y=x
• A´
•
H´
T
• M´
M´(3, -3)
A´(3, 2)
T´(-1, 2)
H´(-1, -3)

7. 3. Reflect the points
G(-1, 5), E(0, 3), O(2, -4)
a. Across the y-axis: (x, y) → (− x, y)
G´(1, 5), E´(0, 3), O´(-2, -4)
b. Across the x-axis: (x, y) → (x, − y)
G´(-1, -5), E´(0, -3), O´(2, 4)
c. Across the line y=x: (x, y) → (y, x)
G´(5, -1), E´(3, 0), O´(-4, 2)

8. translation
A transformation where all the points of a
figure are moved the same distance in the
same direction. It is an isometry.

9. Examples
What are the coordinates of the translated
points?
1. L(-1, 6)
5 units to the right and 4
units down.
L´(4, 2)
2. R(0, 8)
2 units to the left and 5
units up.
R´(-2, 13)
3. Y(7, -3) 4 units to the left and 3
units down.
Y´(3, -6)

10. vector
A quantity that has both length and
direction.
The vector x, y lists the horizontal and
vertical change from the initial point to the
final point. (Notice the angle brackets
instead of parentheses.)
Example
Translate U(7, 2) along −2, 4
U´(7 — 2, 2 + 4)
U´(5, 6)

11. Examples
Translate the figure with the given vertices
along the given vector.
1. U(-3, -1), T(1, 5), A(6, -3); 4, 4
T´
A´
U´(1, 3), T´(5, 9), A´(10, 1)
2. T(-2, -4), A(-3, 0), M(1, 0), U(2, -4);
−2, 4
A´
M´
U´
T´(-4, 0), A´(-5, 4), M´(-1, 4), U´(0, 0)
3. M(-3, -1), A(5, -3), V(-2, -2); 1, −3
A´
V´
M´(-2, -4), A´(6, -6), V´(-1, -5)