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Lecture 5

MA 108 Lecture 5: Copy of Translations- Geometry- Emily


Department
Mathematics
Course Code
MA 108
Professor
Elyse Suzanne Rogers
Lecture
5

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Transformations:
A geometric transformation is a change (of size, shape, or position) applied to a geometric
figure. The original figure is called the preimage. The figure that results from the
transformation is called the image. In this unit you will learn about the basic geometric
transformations: translation, which slides the figure to a new position without turning, flipping,
or resizing it; reflection in a line, which flips the figure over a given line; rotation, which turns
the figure about a fixed point; dilation, which enlarges or reduces the size of the figure; and
glide reflection, which is a translation followed by a reflection in a line parallel to the direction
of the translation.
All of the above transformations, except dilations, preserve size and shape. The image
and the preimage are congruent. A transformation that preserves size and shape is called an
isometry. Translatlons, rotations, and dilations also preserve the orientation of figures. That
is, if the vertices of the preimage are read in order clockwise or counterclockwise around the
figure, then the corresponding vertices of the image will also be read in order in the same
direction. Reflections and glide reflections reverse the orientation of figures. For these two
types of transformation, a preimage with a clockwise orientation will have an image with a
counterclockwise orientation.
1) Translations: Read about translations online at http://www.mathopenref.com/translate.html
Translations are often performed on figures in the coordinate plane. Every point (x, y) in the
preimage is translated to a corresponding point (x’, y’) in the image.
Translations to the left or right involve changing the x-coordinate of each point in the
preimage, while holding the y-coordinates constant. For example, (x’, y’) = (x + 2, y) indicates a
translation 2 units to the right, while (x’, y’) = (x ─ 3, y) indicates a translation 3 units to the left.
What translations do the following equations represent? (Type in your answers in color.)
1) (x’, y’) = (x ─ 5, y) left 5
2) (x’, y’) = (x + 1, y) right 5
3) (x’, y’) = (x ─ 10, y) left 10
4) (x’, y’) = (x + 6, y) right 6
5) (x’, y’) = (x ─ 7, y) left 7
Translations up or down involve changing the y-coordinate of each point in the preimage,
while holding the x-coordinates constant. For example, (x’, y’) = (x, y + 5) indicates a translation
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5 units up, while (x’, y’) = (x, y ─ 4) indicates a translation 4 units down.
What translations do the following equations represent? (Type in your answers in color.)
1) (x’, y’) = (x, y ─ 5) down 5
2) (x’, y’) = (x, y + 3) up 3
3) (x’, y’) = (x, y ─ 8) down 8
4) (x’, y’) = (x, y + 7) up 7
5) (x’, y’) = (x, y ─ 12) down 12
Any translation in the coordinate plane is a combination of a horizontal translation (left or right)
and a vertical translation (up or down). For example (x’, y’) = (x + 3, y + 5) indicates a
translation 3 units to the right and 5 units up, while (x’, y’) = (x + 1, y ─ 2) indicates a translation
1 unit to the right and 2 units down.
What translations do the following equations represent? (Type in your answers in color.)
1) (x’, y’) = (x + 7, y ─ 4) right 7, down 4
2) (x’, y’) = (x ─ 4, y + 6) left 4, up 6
3) (x’, y’) = (x ─ 2, y ─ 1) left 2, down 1
4) (x’, y’) = (x + 5, y + 2) right 5, up 2
5) (x’, y’) = (x ─ 1, y ─ 3) left 1, down 3
Write a coordinate equation for each translation. (Type in your answers in color.)
1) 4 units left, 3 units up (x’, y’)=(x-4, y+3)
2) 6 units right, 2 units down (x’, y’)=(x+6, y-2)
3) 10 units left, 6 units down (x’, y’)=(x-10, y-6)
4) 3 units right, 7 units up (x’, y’)=(x+3, y+7)
5) 4 units right, 5 units down (x’, y’)=(x+4, y-5)
Translations can also be represented as the sum of two matrices. One of the matrices (the
coordinate matrix) contains the coordinates of the vertices of the preimage. The x-coordinates
of the vertices go in the first row of the matrix and the y-coordinates of the vertices go in the
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