Write the translation matrix for each figure. Then find the...

Write the translation matrix for each figure. Then find the coordinates of the image after the translation. Graph the preimage and the image on a coordinate plane.
$\triangle M N O$ with $M(-7,6), N(1,7),$ and $O(-3,1),$ translated 2 units right and 6 units down

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Key Concepts

Translation in Coordinate Geometry

Translation is a transformation that shifts every point of a figure or a space by the same distance in a specific direction. In the coordinate plane, this is achieved by adding a constant value to the x-coordinate and a constant value to the y-coordinate of each point, which moves the entire figure without altering its shape, size, or orientation.

Matrix Representation of Translations

Although translations are not linear transformations in the traditional sense (since they do not preserve the origin), they can be represented in matrix form using homogeneous coordinates. By augmenting the coordinate system (typically with an extra coordinate), a translation can be expressed as a 3×3 matrix that enables the use of matrix multiplication to perform the translation, which is particularly useful in computer graphics and other applications involving multiple transformations.

Preimage and Image in Transformations

In transformation geometry, the preimage refers to the original figure before a transformation is applied, while the image is the resulting figure after the transformation. Understanding this distinction is essential for analyzing how figures change under various operations, such as translations, rotations, or reflections, and for tracking the movement of specific points during these transformations.

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