Reflection - The image is a mirrored preimage "a flip." Dilation - The image is a larger or smaller version of the preimage "shrinking" or "enlarging." Transformations in the Coordinate PlaneThere are five different transformations in math: The image is the figure after transformation.
RotationUsing the origin, ( 0 , 0 ), as the point around which a two-dimensional shape rotates, you can easily see rotation in all these figures:A figure does not have to depend on the origin for rotation. Which trapezoid image, red or purple, is a reflection of the green preimage?The purple trapezoid image has been reflected along the x-axis, but you do not need to use a coordinate plane's axis for a reflection. A reflection image is a mirror image of the preimage. That is a reflection or a flip.
Which octagon image below, pink or blue, is a translation of the yellow preimage?The blue octagon is a translation, while the pink octagon has rotated. TranslationA translation moves the figure from its original position on the coordinate plane without changing its orientation. A shear does not stretch dimensions it does change interior angles. To shear it, you "skew it," producing an image of a rhombus:When a figure is sheared, its area is unchanged.
A polygon can be reflected and translated, so the image appears apart and mirrored from its preimage. The image resulting from the transformation will change its size, its shape, or both.There are five different types of transformations, and the transformation of shapes can be combined. Non-Rigid TransformationsA non-rigid transformation can change the size or shape, or both size and shape, of the preimage.Two transformations, dilation and shear, are non-rigid. The image from these transformations will not change its size or shape. Three transformations are rigid.The rigid transformations are reflection, rotation, and translation.
To rotate 90 °: ( x , y ) → ( - y , x ) (multiply the y-value times - 1 and switch the x- and y-values) Mathematically, the graphing instructions look like this:This tells us to add 9 to every x value (moving it to the right) and add 9 to every Y value (moving it up):( - 7 , - 1 ) → ( - 7 + 9 , - 1 + 5 ) → ( 2 , 4 )Do the same mathematics for each vertex and then connect the new points in Quadrants I and IV.Rotation using the coordinate grid is similarly easy using the x-axis and y-axis: Focus on the coordinates of the figure's vertices and then connect them to form the image.Here is a tall, blue rectangle drawn in Quadrant III.We are asked to translate it to new coordinates. The lines also help with drawing the polygons and flat figures. Transformations in the Coordinate PlaneOn a coordinate grid, you can use the x-axis and y-axis to measure every move. What two transformations were carried out on it?The preimage has been rotated and dilated (shrunk) to make the image.
( x , y ) → ( x + m y , y ) to shear h o r i z o n t a l l y Italic letters on a computer are examples of shear.Mathematically, a shear looks like this, where m is the shear factor you wish to apply: If the figure has a vertex at ( - 5 , 4 ) and you are using the y-axis as the line of reflection, then the reflected vertex will be at ( 5 , 4 ).Shearing a figure means fixing one line of the polygon and moving all the other points and lines in a particular direction, in proportion to their distance from the given, fixed-line. To rotate 270 °: ( x , y ) → ( y , - x ) (multiply the x-value times - 1 and switch the x- and y-values)Reflecting a polygon across a line of reflection means counting the distance of each vertex to the line, then counting that same distance away from the line in the other direction.
Transformations, and there are rules that transformations follow in coordinate geometry. If you have an isosceles triangle preimage with legs of 9 f e e t, and you apply a scale factor of 2 3, the image will have legs of 6 f e e t.In summary, a geometric transformation is how a shape moves on a plane or grid.
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