E1.4 Describe and perform translations, reflections, rotations, and dilations on a Cartesian plane, and predict the results of these transformations.
Skill: Describing and Performing Geometric Transformations on a Cartesian Plane
When shapes are transformed on a Cartesian plane, the coordinates of the original vertices are transformed to create corresponding coordinates known as image points. Each of the transformations can be defined using a mapping rule in which each point is transformed using that rule.
- Mapping rule for translations:
- (x, y) → (x + a, y + b). If a is positive, then the x-value of the image point is a units to the right of the original point. If a is negative, then the x-value of the image point is a units to the left of the original point. If b is positive, then the y-value of the image point is b units up from the original point. If b is negative, then the y-value of the image point is b units down from the original point. For example, (x, y) → (x − 5, y − 2); each image point is 5 units left of and 2 units down from the original point.
- Mapping rules for reflections:
- a shape reflected in the x-axis has mapping rule (x, y) → (x, −y). For example, the vertex of the shape originally at (2, 3) is now at (2, −3).
- a shape reflected in the y-axis has a mapping rule (x, y) → (−x, y). For example, the vertex of the shape originally at (2, 3) is now at (−2, 3).
- Mapping rules for rotations about the origin:
- a shape rotated 90° counterclockwise has mapping rule (x, y) → (−y, x);
- a shape rotated 180° counterclockwise has mapping rule (x, y) → (−x, −y);
- a shape rotated 270° counterclockwise has mapping rule (x, y) → (y, −x).
- Mapping rules for dilations:
- (x, y) → (ax, ay). For example, for (x, y) → (2x, 2y), the coordinates of the image points are double those of the original points. For example, the image point for (−3, 4) is (−6, 8).
Notes
- Translations “slide” a point, segment, or shape by a given distance and direction (vector).
- Reflections “flip” a point, segment, or shape across a reflection line to create its opposite.
- Rotations “turn” a point, segment, or shape around a point of rotation by a given angle.
- Dilations (or dilatations) enlarge or shrink a distance by a given scale factor. Scale factors with an absolute value greater than 1 enlarge the distance, and those with an absolute value of less than 1 reduce the distance. Negative scale factors dilate the shape and rotate it 180°.
- Translations, reflections, and rotations all produce congruent images:
- Lines map to lines of the same length.
- Angles map to angles of the same measure.
- Parallel lines map to parallel lines.
- Dilations (or dilatations) produce scaled images that are similar:
- Lines map to line lengths at a constant scale factor.
- Angles map to angles of the same measure.
- Parallel lines map to parallel lines.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Skill: Predicting the Results of Transformations Such as Translations, Reflections, Rotations and Dilations on a Cartesian Plane
In order to predict the results of transformations, students must understand the effect of the transformations on the coordinates of each of the original vertices.