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Transformations on the Coordinate Plane
Translation, Reflection, Rotation, Dilation
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Transformations on the coordinate plane describe how a figure moves or changes position while keeping or changing its orientation in a predictable way. They are important because they connect algebra and geometry through ordered pairs and rules. Students use transformations to analyze symmetry, congruence, similarity, and patterns in graphs. These ideas also appear in computer graphics, engineering, and map design.
On a coordinate plane, each transformation follows a rule that changes every point of a figure. A translation slides a figure, a reflection flips it across a line, a rotation turns it around a center, and a dilation resizes it from a center point. By applying coordinate rules such as to , students can predict the exact image of any shape. Comparing the original figure and its image helps reveal which lengths, angles, and orientations stay the same and which change.
Key Facts
- Translation by vector : to
- Reflection across the x-axis: (x, y) to (x, -y)
- Reflection across the y-axis: (x, y) to (-x, y)
- Rotation 90 degrees counterclockwise about the origin: (x, y) to (-y, x)
- Rotation 180 degrees about the origin: (x, y) to (-x, -y)
- Dilation with scale factor k about the origin: (x, y) to (kx, ky)
Vocabulary
- Transformation
- A transformation is a rule that changes the position, orientation, or size of a figure on the coordinate plane.
- Translation
- A translation moves every point of a figure the same distance in the same direction.
- Reflection
- A reflection flips a figure across a line so that each point lands the same distance from the line on the opposite side.
- Rotation
- A rotation turns a figure around a fixed point called the center of rotation.
- Dilation
- A dilation changes the size of a figure by multiplying distances from a center by a scale factor.
Common Mistakes to Avoid
- Using the wrong sign in a translation rule, which moves the figure in the opposite direction from the one intended. Always match right and up with positive values and left and down with negative values.
- Confusing reflection rules, which gives the image on the wrong side of the axis or line. Across the x-axis only y changes sign, and across the y-axis only x changes sign.
- Mixing up the coordinates in a rotation, which produces an incorrect image. For a 90 degree counterclockwise rotation about the origin, switch the coordinates and make the new x negative: (x, y) to (-y, x).
- Assuming all transformations preserve size, which is wrong for dilations. A dilation changes side lengths unless the scale factor is 1, even though the shape stays similar.
Practice Questions
- 1 Triangle ABC has A(1, 2), B(4, 2), and C(2, 5). Translate the triangle by the vector (3, -2). What are the coordinates of A', B', and C'?
- 2 Point P(-3, 4) is rotated 180 degrees about the origin and then reflected across the x-axis. What are the final coordinates of the image?
- 3 A figure and its image have the same side lengths and angle measures, but the order of the vertices appears reversed. Which transformation most likely occurred, and how can you tell?