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MA 108 Study Guide - Spring 2019, Comprehensive Midterm Notes - Triangle, The Matrix, Positive-Definite Matrix


Department
Mathematics
Course Code
MA 108
Professor
Elyse Suzanne Rogers
Study Guide
Midterm

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MA 108

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5) Compositions: When two or more transformations are combined to produce a single
transformation, the result is called a composition of the transformations. The composition of
two or more isometries is also an isometry. Compositions may preserve or reverse orientation
depending on how many reflections they contain.
An important type of composition is a glide reflection, which is a translation followed by
a reflection in a line parallel to the direction of the translation. A glide reflection is an isometry
that reverses the orientation of figures. For example, suppose a translation 6 units to the right is
followed by a reflection in the x-axis. This glide reflection can be written in coordinate form as
(x’,y’) = ( x + 6, -y ). The glide reflection ( x’, y’ ) = ( -x, y +4 ) is a translation 4 units up followed
by a reflection in the y-axis. The order in which two transformations are performed sometimes
affects the resulting image, but with a glide reflection the image would be the same regardless
of the order of the transformations (translation first or reflection first).
Try these: Write a brief description of each transformation:
1) ( x’, y’ ) = ( x - 3, -y ) a translation 3 units to the left, followed by a reflection in the x-axis.
2) ( x’, y’ ) = ( -x, y - 8 ) a translation 3 units down, followed by a reflection in the y-axis.
3) ( x’, y’ ) = ( y + 2, x + 2 ) a translation 2 units up, and 2 units to the right.
4) ( x’, y’ ) = ( x + 10, -y ) a translation 10 units to the right, followed by a reflection in the x-axis.
5) ( x’, y’ ) = ( -x, y + 6 )a translation 6 units up, followed by a reflection in the y-axis.
Write each transformation in coordinate form:
6) Translation 5 units right and 5 units up, reflection in the line y = x. ( x’, y’ ) = ( x+5, y+5) y,x
7) Translation 8 units left, reflection in the x-axis. ( x’, y’ ) = ( x-8, -y )
8) Translation 5 units down, reflection in the y-axis. ( x’, y’ ) = (- x, y-5 )
9) Translation 4 units right, reflection in the x-axis.( x’, y’ ) = (x+4, -y)
Some transformations can be thought of as a composition of two reflections. A
translation is the composition of two reflections in parallel lines. The direction of the translation
is perpendicular to the lines, and the distance translated is twice the distance between the lines.
A rotation can be expressed as the composition of two reflections in intersecting lines. The
center of the rotation is the point where the lines intersect, and the angle of the rotation is twice
the angle between the lines.
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Compositions of reflections and rotations can be expressed as products of the 2x2
matrices used for the component transformations. For example, if we perform a 90-degree
rotation about the origin, followed by a reflection in the x-axis, then we can determine the overall
effect of this composition by multiplying the matrix that reflects in the x-axis by the matrix that
rotates a figure 90 degrees:
[ 1 0 [ 0 -1 = [ 0 -1
0 -1 ] 1 0 ] -1 0 ]
The product is a reflection in the line y = -x. If we reverse the order of the composition and do
the reflection first, we get:
[ 0 -1 [ 1 0 = [ 0 1
1 0 ] 0 -1 ] 1 0 ]
The product is a reflection in the line y = x. These examples show that the order in which the
transformations are performed can affect the result of the composition. When composing
rotations or reflections, make sure that the matrix for the first transformation is on the right, and
the matrix for the second transformation is on the left.
Try these: Use matrix multiplication to determine the net result of each composition:
1) Reflection in the line y = x, followed by a 270-degree rotation about the origin:
[ 0 1 [ 0 1 = [ 1 0
-1 0 ] 1 0 ] 0 -1 ] , which is the matrix for a reflection in the x-axis.
2) Rotation 270 degrees about the origin, followed by a reflection in the line y = x.
[ 0 1 [ 0 1 = [ -1 0
1 0 ] -1 0 ] 0 1 ]
3) Reflection in the y-axis, followed by a reflection in the line y = -x.
[ -1 0 [ 0 -1 = [ 0 1
0 1 ] -1 0 ] -1 0 ]
4) Rotation 180 degrees about the origin, followed by a reflection in the y-axis.
[ -1 0 [ -1 0 = [ 1 0
0 -1 ] 0 1 ] 0 -1 ]
5) Reflection in the y-axis, followed by a rotation of 180 degrees about the origin.
[ -1 0 [ -1 0 = [ 1 0
0 1 ] 0 -1 ] 0 1 ]
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3) Reflection in a Line: A reflection in a line m maps a preimage point P in the plane to an
image point P’ so that the following properties are true:
1) If P is not on line m, then line m is the perpendicular bisector of segment PP’.
2) If P is on line m, then P = P’.
Line m is called the line of reflection. A reflection is an isometry, so reflections preserve size
(and shape). Reflections reverse the orientation of figures.
While reflection is a quite general concept and the line of reflection can be anywhere in
the plane, we will concentrate on four specific lines of reflection in the coordinate plane: the x-
axis, the y-axis, the line y = x, and the line y = −x.
Case 1: Reflection in the x-axis. When a point is reflected in the x-axis, its x-coordinate stays
the same, while its y-coordinate changes sign. In coordinate notation, (x’, y’) = ( x, −y ). For
example, the point ( 3, 5) has the image (3, -5), and the point ( -2, -7) has the image ( -2, 7).
Any point on the x-axis is its own image.
If a figure is represented as a polygon matrix, then it can be reflected in the x-axis by
multiplying the polygon matrix by the reflection matrix [ 1 0 . The reflection matrix goes in
front
0 -1 ]
of the polygon matrix (this is called “premultiplying”). For example, suppose we want to reflect
triangle ABC in the x-axis, where A = ( 1, 6), B = ( 3, 8), and C = ( 5, 4). The matrix
multiplication and image look like this:
[ 1 0 [ 1 3 5 = [ 1 3 5
0 -1 ] 6 8 4 ] -6 -8 -4 ]
Try these: Find the reflection of each point in the x-axis:
1) ( 9, 4) (9, -4)
2) (-4, 7) ( -4,-7)
3) ( 2, -5) ( 2, 5)
4) ( -5, -3) (-5, 3)
Write the matrix equation to reflect triangle ABC [A = ( 0, -2), B = ( 6, 1), C = ( 4, 3)] in the x-
axis:
[ 1 0 [ 0 6 4 = [ 0 6 4
0 -1 ] -2 1 3 ] 2 -1 -3 ]
Case 2: Reflection in the y-axis. When a point is reflected in the y-axis, its y-coordinate stays
the same, while its x-coordinate changes sign. In coordinate notation, (x’, y’) = ( −x, y ). For
example, the point ( 3, 5) has the image (-3, 5), and the point ( -2, -7) has the image ( 2, -7).
Any point on the y-axis is its own image.
If a figure is represented as a polygon matrix, then it can be reflected in the y-axis by
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