Name:___________________________ Date:______ Period:_____ Transformations – Easter Break Packet Name:___________________________
Date:______ Period:_____
Transformations – Easter Break Packet
Ms. Anderle
Transformations
Line Reflection: A line reflection is a transformation in which a figure is reflected over a given line
as if in a mirror. Each point of the reflection image is the same distance from a line of reflection as
the corresponding point in the original figure. Basically, a line reflection “flips” an object over the
line so that the image appears to be “backwards” much like how a reflected image would appear in a
mirror.
Under a line reflection, distance, angle measure, collinearity, and betweenness are preserved.
Line Reflection Mapping Rules
Reflections in the x-axis
Reflections in the y-axis
Reflections in the y = x line
Reflection in the y = -x line
Reflection in the line y = k
(where k is a constant)
Reflection in the line x = k
(where k is a constant)
rx-axis(x, y) = (x, -y)
ry-axis(x, y) = (-x, y)
ry = x(x, y) = (y, x)
ry=-x(x, y) = (-y, -x)
ry = k(x, y) = (x, 2k – y)
rx = k(x, y) = (2k – x, y)
These rules are applied in the following model problems.
Example 1: Find a reflection in the x-axis of figure ABCD with coordinates A(-4,3), B(-3,5), C(4,6),
and D(0,2).
Solution: Using our mapping rule, the image points are A’(-4,-3), B’(-3,-5), C’(4,-6), and D’(0,-2).
Example 2: Find the reflection across the y = x line of ΔABC with coordinates A(2,1), B(9,6), and
C(5,-4).
Solution: Using our mapping rule, the image points are A’(1,2), B’(6,9), and C’(-4,5).
Homework Problems:
1. Find the reflection in the y = -x line of ΔPQR with the coordinates P(2,2), Q(5,3), and R(3,-1).
2. Find the reflection of the figure whose coordinates are A(-1,3), B(4,6), C(7,2) across the line
represented by x = 2.
3. Find the reflection of the figure whose coordinates are A(-3,-7), B(3,-4), and C(9,-6) across the
line represented by y = -3.
4. State the coordinates of the point (-2,5) under the reflection
a. in the line y = x
b. in the line y = 4
c. in the line x = -2
5. Find the image of the points below in reflected in the line y = -2
a. (1, 4)
b. (-3, 3)
c. (4, -5)
Line Symmetry
If a figure has line symmetry, a line can be drawn through the figure such that both sides of the
figure are “mirror images” of each other. Such a line is also called the axis of symmetry. There can
be more than one line of symmetry in a figure, or there may be none.
One line of symmetry
H
Two lines of symmetry
Four lines of symmetry
Infinite number of lines of symmetry
F
No lines of symmetry
Homework Problems
1) Which letter has vertical line symmetry?
a. S
b. N
c. A
d. B
2) Which letter has vertical but not horizontal line symmetry?
a. X
b. O
c. V
d. E
Draw in the line symmetry to the problems below.
3)
X
4)
M
5)
U
Point Reflection and Symmetry
If a figure is reflected in or through point P, then P is the midpoint of the line segement joining
each point to its corresponding image.
Point Reflection Mapping Rules:
A reflection through the origin is written r origin(x, y) = (-x, -y) or rO(x, y) = (-x, -y), where point O is
the origin.
In coordinate geometry, the usual point of reflection is the origin. If point B(2, 1) is reflected
through the origin, then B’(-2, -1).
A reflection through a given point is written as r (h, k)(x, y) = (2h – x, 2k – y).
Example: Find the image of (-4, 2) under the reflection in the point (2, 2)
r(2, 2)(-4, 2) = (2&middot;2 - -4, 2&middot;2 – 2) = (8, 2)
A figure is said to have point symmetry if the figure coincides with itself when reflected through a
point or when rotated 180&deg; about a point. The point that is the center of reflection or rotation is
called the point of symmetry.
F G
Z SN
Shapes without point symmetry
,
Shapes with point symmetry
,
,
,
Homework Questions
1. What letter has both point symmetry and line symmetry?
a. A
b. H
c. E
d. S
2. What is the image of (k, 2k) after a reflection through the origin.
3. Find the image of (-4, 2) under the reflection in the point (2, 2).
4. Find the image of (-2, -6) under a point reflection in the origin.
5. Find the image of (3, 2) under the reflection in the point (-1, -1)
,
Rotations:
A rotation is a transformation in which a figure is turned around a point called the point of
rotation. In the coordinate plane, this point is typically the origin. Rotations that are
counterclockwise are rotations of a positive degree measure. Rotations that are clockwise are of a
negative degree measure. All rotations are assumed to be clockwise about the origin unless
otherwise stated
Rotation Mapping Rules
R90&deg;(x, y) = (-y, x)
R180&deg;(x, y) = (-x, -y)
R270˚(x, y) = (y, -x)
R360&deg;(x, y) = (x, y)
Rotation of 90&deg; and -270&deg;
Rotation of 180&deg; and -180&deg;
Rotation of 270&deg; and -90&deg;
Rotation of 360&deg; and -360&deg;
Homework Questions
1. What is the image of (-2, 3) under a rotation of:
a) 90&deg;
b) 180&deg;
c) 270&deg;
d) -90&deg;
e) -270&deg;
2. Name the coordinates of each point under a rotation of 90&deg;
a) (5,1)
b) (-3,3)
c) (8, -2)
d) (0,5)
3. Name the coordinates of each point under a rotation of 270&deg;
a) (-3,2)
b) (-2,-2)
c) (4,4)
d) (7,14)
Translation
A translation is a transformation in which each point in a figure slides a certain distance and in the
same direction. On the coordinate plane, the translation is defined by the mapping rule:
Ta, b(x, y) = (x + a, y + b).
Model Problems:
1. What is the image of point P(-3, 2) under the transformation T-2, 6?
Answer: T-2, 6 means add -2 to the x-value (-3) and add 6 to the y-value (2). Therefore, the
image of point P is (-5, 8)
2. A transformation maps P(x,y) onto P’(x – 4, y + 2). Under the same transformation, what are the
coordinates of Q’, the image of Q(2, -3)?
Answer: Since the given transformation is T-4, 2, Q(2, -3) is mapped onto Q’ by adding -4 to
the x-value (2) and 2 to the y-value (-3). Q’(-2, -1).
Homework Problems
1.
What is the image H(-1, 3) under the translation T2, -1?
2. Find the image of (-1,5) if translation T is defined by (x, y)  (x + 3, y – 1)
3. What is the image of P(-2,3) under the translation T4, -2?
4. ***If the transformation Tx, y maps point M(1,-3) onto point M’(-5,8), what is the value of x?
Dilations
A dilation is a transformation in which the size of a figure is changed and the figure is moved.
The dilation mapping rule can be expressed as Dk(x, y) = (kx, ky).
Model Problems:
1. D3(x, y) = (3x, 3y)
2. D1/2(2, 4) = (1, 2)
Note:
-If k = 1, then the image is identical to the original.
-If k = -1, then the image is congruent and is the same as a point reflection.
-If |k| &gt; 1, then the image is similar and larger than the original.
-If |k| &lt; 1, then the image is similar and smaller than the original.
Homework Problems
1) What are the coordinates of point (2, -4) under the dilation D-2?
2) What are the coordinates of point (9, -6) under the dilation D1/3?
3) What are the coordinates of point (12, -4) under the dilation D5?
4) If the dilation Dk(2, -4) = (-1, 2), the scale factor k is equal to:
5) Find the image of (6, -9) under the dilation D2/3.