A transformation is a function that moves or changes a figure. The original figure can be called a preimage and the produced figure, an image. Sometimes, transformations are expressed as a mapping. That is, it maps the inputs to the outputs.Here, is the transformation, are the coordinates of the points of the preimage, and are the coordinates of the points of the image.
A composite translation can be seen as the combination of two or more consecutive translations. For example, consider the translations of
Combining these translations creates the composite translation that directly maps
Translate the figure below using the rule
To begin, notice that the mapping translates the figure in both the horizontal and vertical directions. We can perform each translation separately. First, the horizontal translation moves the figure units to the right.
The vertical translation, shifts the figure down units.
We can now construct the image by connecting the new points with line segments.
Determine the translation of the quadrilateral:
To begin, it might simplify the solution to see the translation as two separate translations, one in the vertical direction and one in the horizontal. We'll identify the mapping of
It can be seen that is translated in the vertical direction and in the horizontal direction. Thus, the mapping can be expressed as We can verify this rule by checking the movement of the other points.
When the same rule is applied to the other points, it can be seen that Thus, the rule for this transformation is