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Lecture 4

# MA 108 Lecture Notes - Lecture 4: Graph Paper, Scalar Multiplication

Department
Mathematics
Course Code
MA 108
Professor
Elyse Suzanne Rogers
Lecture
4

This preview shows half of the first page. to view the full 2 pages of the document. 2) Dilations: A dilation enlarges or reduces the size of a figure. Because a dilation does not
preserve size, it is not an isometry. However, dilations do preserve the shapes of figures. The
preimage and the image will have the same shape, but generally not the same size. Two
figures that have the same shape are said to be similar, and since dilations produce images
that are similar to the preimage, dilations are often called similarity transformations.
A dilation can be completely specified by its scale factor and its center. The scale
factor (usually indicated by the letter k) indicates the factor by which the lengths in the preimage
are multiplied to form the image. For an enlargement, k > 1, and for a reduction, 0 < k < 1.
The center of the dilation is the one fixed point in the plane that does not change when
the dilation is applied. Consider the triangle with vertices at (0, 2), (4, 3), and (1, 0). If this
triangle is enlarged by scale factor of 2, the result will depend on what point we choose for the
center of the dilation. If the origin is the center, the image will be the triangle with vertices at (0,
4), (8, 6), and (2, 0). All of the original coordinates are multiplied by 2. If (0, 2) is the center, the
image will be the triangle with vertices at (0, 2), (8, 4), and (2, -2). If (1, 0) is the center, the
image will be the triangle with vertices at (-1, 4), (7, 6), and (1, 0). All three images are the
same size but their positions depend on where the center of the dilation is. Draw these
triangles, along with the preimage, on a piece of graph paper to get a picture of these dilations.
Most of the time, if you have to describe a dilation using coordinate equations or
matrices, the center of the dilation will be the origin, and your task will be quite simple. In
coordinate notation, a dilation with center at the origin and scale factor k is (x’, y’) = (kx, ky). A
dilation with center at the origin and scale factor 3 is written as (x’, y’) = (3x, 3y). The dilation
(x’, y’) = (0.5x, 0.5y) has its center at the origin and a scale factor of 0.5.
Write a coordinate equation for each dilation. (Type in your answers in color.)
1) Dilation with center (0, 0) and scale factor 0.25. (x, y) = ( 0.25x, 0.25y)
2) Dilation with center (0, 0) and scale factor 1.5. (x, y) = (1.5x, 1.5y)
3) Dilation with center (0, 0) and scale factor 4. (x, y) = (4x, 4y)
4) Dilation with center (0, 0) and scale factor 0.6. (x, y) = (0.6x,0.6y)
5) Dilation with center (0, 0) and scale factor 8. (x, y) = (8x, 8y)
Write a verbal description of each transformation. (Type in your answers in color.)
1) (x, y) = (9x, 9y) Dilation with center (0, 0) and scale factor 9.
2) (x, y) = (0.4x, 0.4y) Dilation with center (0,0) and scale factor 0.4.
3) (x, y) = (5x, 5y) Dilation with center (0,0) and scale factor 5.
4) (x, y) = (1.8x, 1.8y) Dilation with center (0,0) and scale factor 1.8.