Translations
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A translation is a transformation that moves a figure by sliding it left, right, up, or down
on the coordinate plane.
For points translated horizontally, the x-coordinates increase or decrease by the amount
of the translation, and the y-coordinates stay the same.
For points translated vertically, the y-coordinates increase or decrease by the amount of
the translation, and the x-coordinates stay the same.
Vertical Transformations
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What do you think will happen to the coordinates if a point or figure is translated
vertically?
Take a look at the vertices of the pre-image (triangle 1) and the image (triangle 2).
Triangle 1 A(2, 4) B(1, 1) C(3, 2)
Triangle 2 A'(2, 0) B'(1, -3) C'(3, -2)
Horizontal Transformations
A translation is a transformation that moves a point or figure in the coordinate plane by sliding it
left, right, up, or down. The image is congruent to the original figure.
Triangle 1 is called the pre-image, and the translated triangle (triangle 2) is called the image. As
with reflections, the vertices of the image are labeled with the same letters as the pre-image, but
there are prime symbols after each letter.
Triangle 2 is the translation of triangle 1. When describing a translation, you need to give the
direction and the distance. So triangle 2 has been translated 3 units to the right of triangle 1.
Every point of the figure has moved right 3 units.
Take a look at the coordinates for each triangle.
Triangle 1 A(2, 4) B(1, 1) C(3, 2)
Triangle 2 A'(5, 4) B'(4, 1) C'(6, 2)
Reflection over the y-axis
Reflect ΔRST over the y-axis. What are the coordinates of the vertices of ΔR'S'T'?
Solution:
You know that the x-coordinates' signs will change and the y-coordinates will stay the same. So
find the coordinates of each vertex of the image and then plot the points.
Think about it!
Each point of the pre-image and each point of the image are the same distance from the line of
reflection, the y-axis:
R is 4 units away from the y-axis in the negative direction.
R' is 4 units away from the y-axis in the positive direction.
Pre-image Image
(x, y)
(-x, y)
R(-4, -1)
R'(4, -1)
S(-2, -1)
S'(2, -1)
T(-2, -4)
T'(2, -4)
So the coordinates of the vertices of ΔRST are R'(4, -1), S'(2, -1), and T'(2, -4). Now you can plot
those points.
Reflections over the x-axis
Example:
Reflect ΔXYZ over the x-axis. What are the coordinates of the vertices of ΔX'Y'Z'?
Solution:
You know that the x-coordinates will not change and the y-coordinates' signs will change. So
find the coordinates of each vertex of the image and then plot the points.
Think about it!
Each point of the pre-image and each point of the image are the same distance from the line of
reflection, the x-axis:
Z is 3 units away from the x-axis in the negative direction.
Z' is 3 units away from the x-axis in the positive direction.
Pre-image Image
(x, y)
(x, -y)
X(-4, -1)
X'(-4, 1)
Y(-1, -1)
Y'(-1, 1)
Z(-3, -3)
Z'(-3, 3)
So the coordinates of the vertices of ΔX'Y'Z' are X'(-4, 1), Y'(-1, 1), and Z'(-3, 3). Now you can
plot those points.