Which of the following transformations will result in an image that maps onto itself?
180 degree rotation around the origin
The first thing I found was MR’s, which said, “Translate ABC using vector AA.” I noticed she was mapping the triangle on the left side of the page onto the triangle on the right side of the page when I looked closer, but even if she had been mapping the triangle onto itself, she had come up with a surprisingly simple solution.
At this point, I decided that a class discussion would be beneficial in limiting additional trivial solutions to this challenge. So far, we’ve discussed transformations that can, of course, map the figure into itself, such as rotating the image around one of its vertices 0 or 360, but which are also fairly basic and uninteresting.
Then I gave them some more time to work. The plan was for them to write a transformation or a series of transformations and then make their partner try it out, following their instructions to the letter. If the instructions didn’t work the first time, the partner assisted in revising them as required.
Several students speculated that the rectangle’s diagonals might be reflection points. Max was designated as the Live Presenter, and he demonstrated what happens when a rectangle is reflected about its diagonal.
Identifying transformations that map a quadrilateral onto itself
A rigid transformation (also known as an isometry) is a plane transformation that maintains length. “Rigid transformations” include reflections, translations, rotations, and variations of these three transformations.
A reflection over line m (notation rm ) is a transformation in which each point of the original figure (pre-image) has an image on the opposite side of the line that is the same distance from the reflection line as the original point. Since the image is the same size and shape as the pre-image, a reflection is called a rigid transformation or isometry.
The distances between the pre-image points and the image points differ (and are not necessarily equal) in this reflection that maps ABC to A’B’C’, but the segments representing these distances are parallel.
Orientation (lettering): The points of the pre-image are lettered clockwise A-B-C in this diagram, while the image is lettered counterclockwise A’-B’-C’. The transformation is known as a non-direct or opposite isometry when lettering changes direction in this way.
Mapping shapes example
We can vertically translate the parabola to create a new parabola that is identical to the original. The graph of the function (y=x2+b) is simply that of a regular parabola with the vertex moved (b) units along the (y)-axis. As a result, the vertex is at ((0,b)). When (b) is positive, the parabola moves upwards; when (b) is negative, the parabola moves downwards.
The parabola can also be translated horizontally. The graph of the function (y=(x-a)2) looks like a regular parabola with the vertex moved (a) units along the (x)-axis. After that, the vertex is located at ((a,0)). If (a) is positive, we shift to the right, while if (a) is negative, we shift to the left.
We reviewed the very critical technique of completing the square in the module Algebra analysis. This method can now be used to find the vertex of quadratics of the form (y=x2+qx+r) that are congruent to the basic parabola and sketch them easily.
Of course, we can use calculus to locate the vertex of a parabola since the derivative would be zero at the vertex’s (x)-coordinate. Even though the (y)-coordinate must be sought, completing the square is usually much faster.
Linear transformations and matrices | essence of linear
In a line L, reflectionA is a transformation that maps every point P in the plane to a point P1, resulting in the following properties: 1. If point P is not on L1, L is the perpendicular bisector of PP1; 2. If point P is not on L1, L is the perpendicular bisector of PP1. When P is on L1, P = P1.
7.3 Theorem A rotation around point O is defined as a reflection in l followed by a reflection in m when two lines, l and m, intersect at point O. The angle of rotation is 2x°, where x° is the acute or right angle that exists between l and m.
7.5 Theorem If lines l and m are parallel, a translation is described as a reflection in line l followed by a reflection in line m. If P11 is P after the two reflections, then PP11 is perpendicular to l and PP11 = 2d, where d is the distance between l and m.