Review of Transformations - Notations and Formulas
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Line Reflections

A reflection is a flip.  It is an opposite isometry - the image does not change size but the lettering is reversed.
 

Reflection in the x-axis:   

When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. 
    or    

Reflection in the y-axis:

When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. 
    or    

Reflection in y = x:

When you reflect a point across the line y = x, the x-coordinate and the y-coordinate change places. 
      or     

Reflection in y = -x:

When you reflect a point across the line y = -x, the x-coordinate and the y-coordinate change places and are negated (the signs are changed). 
   or    



 

Point Reflections

A point reflection exists when a figure is built around a single point called the center of the figure.  It is a direct isometry.
 
Reflection in the Origin: While any point in the coordinate plane may be used as a point of reflection, the most commonly used point is the origin.
    or   



 

Rotations

(assuming center of rotation
 to be the origin)
A rotation turns a figure through an angle about a fixed point called the center.  A positive angle of rotation turns the figure counterclockwise, and a negative angle of rotation turns the figure in a clockwise direction.  It is a direct isometry.

Rotation of 90°:

   

Rotation of 180°:

   (same as point reflection in origin)

Rotation of 270°:

    



 

Dilations

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size NOT an isometryForms similar figures.

Dilation of scale factor k:

The center of the dilation is assumed to be the origin unless otherwise specified.



 

Translations

A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.  It is a direct isometry.
 

Translation of  h, k: