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Geometric Transformations
Translations, Rotations, Reflections & Dilations
Geometric transformations describe how a figure can move or change position on a coordinate plane while keeping or changing certain properties. They are important because they connect algebra, geometry, and visual reasoning in one system. Students use transformations to analyze symmetry, compare shapes, and model motion in math and science. On a graph, transformations let you predict exactly where every point of a figure will go.
The four main transformations are translation, reflection, rotation, and dilation. A translation slides a figure, a reflection flips it across a line, a rotation turns it around a point, and a dilation changes its size by a scale factor. Each transformation follows a rule that maps original coordinates to new coordinates. By learning these rules, students can sketch images quickly and check whether a transformation preserves length, angle measure, orientation, or size.
Key Facts
- Translation by <a, b>: (x, y) -> (x + a, y + b)
- Reflection across the x-axis: (x, y) -> (x, -y)
- Reflection across the y-axis: (x, y) -> (-x, y)
- Rotation 90 degrees counterclockwise about the origin: (x, y) -> (-y, x)
- Rotation 180 degrees about the origin: (x, y) -> (-x, -y)
- Dilation with scale factor k about the origin: (x, y) -> (kx, ky)
Vocabulary
- Transformation
- A rule that moves or changes a figure to create a new image.
- Preimage
- The original figure before a transformation is applied.
- Image
- The new figure produced after a transformation.
- Scale factor
- The number that tells how much a figure is enlarged or reduced in a dilation.
- Orientation
- The order and direction in which the vertices of a figure are arranged.
Common Mistakes to Avoid
- Mixing up translation and dilation, which is wrong because a translation only shifts a figure while a dilation changes its size.
- Changing both coordinates during a reflection across one axis, which is wrong because reflecting across the x-axis changes only y and reflecting across the y-axis changes only x.
- Using the wrong coordinate rule for a rotation, which is wrong because each angle and direction has a specific mapping such as 90 degrees counterclockwise: (x, y) -> (-y, x).
- Assuming all transformations preserve orientation, which is wrong because reflections reverse orientation while translations and rotations preserve it.
Practice Questions
- 1 Triangle A has vertices (1, 2), (4, 2), and (2, 5). Translate it by <3, -1>. What are the new coordinates of the image?
- 2 Point P is at (-2, 5). First reflect P across the y-axis, then rotate the result 180 degrees about the origin. What are the final coordinates?
- 3 A figure is transformed and the image has the same size and shape as the original, but the vertex order is reversed. Which transformation most likely happened, and why?