Transformation in Math: Definition and Types

Transformations in Geometry

• A transformation in math is a mapping of a preimage of a shape or function to an image of the same shape or function.
• There are two main categories of transformations: rigid transformations and non-rigid transformations.
• Rigid transformations do not change the size or shape of the preimage, while non-rigid transformations can change the size and shape of the preimage.

Types of Transformations

• Translation: a rigid transformation in which the location of the preimage is changed, but not its size, shape, or orientation.
• Rotation: a rigid transformation in which the location of the preimage is rotated around a fixed point, but its size and shape are not changed.
• Reflection: a rigid transformation in which the preimage is flipped across a line, but its size and shape are not changed.
• Dilation: a non-rigid transformation in which the shape and orientation of a preimage are not changed, but the image will be either larger or smaller than the original.

Reflection

• A reflection in mathematics is a type of geometrical transformation, where an object is flipped to create a mirror or congruent image.
• The bisector of the plane is known as the line of reflection, and it is perpendicular to the preimage and image.
• To find the reflection of an object, focus on the line of reflection:
  • When the line of reflection is the x-axis: (x, y) transforms into (x, -y).
  • When the line of reflection is the y-axis: (x, y) transforms into (-x, y).
  • When the line of reflection is the Y=X: (x, y) switch and transform to (y, x).

Rotation

• Rotation is one of four geometric transformations, which also include reflection, translation, and dilation.
• Rotation is turning an object clockwise (negative) or counterclockwise (positive) about a given point.
• There are general rules for clockwise and counterclockwise rotations of 90, 180, and 270 about the origin.
  • Counterclockwise:
    • 90 degree rotation: (x, y) ----> (-y, x)
    • 180 degree rotation: (x, y) ----> (-x, -y)
    • 270 degree rotation: (x, y) ----> (y, -x)
  • Clockwise:
    • -90 degree rotation: (x, y) ----> (y, -x)
    • -180 degree rotation: (x, y) ----> (-x, -y)
    • -270 degree rotation: (x, y) ----> (-y, x)

Dilation

• A dilation is the transformation that will change the size of a figure using a center of dilation to stretch away from or to shrink towards.
• A scale factor of dilation is the number used to multiply the pre-image, the original image, by to stretch or shrink according to the center of dilation.
• There are 3 ways a scale factor can affect the pre-image:
  • Multiply by a value greater than 1: the image will stretch away from the center of dilation.
  • Multiply by a value less than 1: the image will shrink towards the center of dilation.
  • Multiply by a value equal to 1: the image will stay the same with no stretching or shrinking.

Similarity and Congruence

• Similar figures are polygons with the same shape, angle sizes, and ratios of side lengths.
• Congruent shapes are shapes that have the same shape and the same size.
• Similar shapes have the same angles and proportional sides, but are different sizes.

Tessellations

• A tessellation is a space-filling arrangement of plane figures that do not overlap or leave gaps.
• There are two main types of tessellations: regular and semi-regular.
• Regular tessellations consist of only one repeated polygon.
• Semi-regular tessellations consist of two or more types of repeated regular polygons.

Triangles

• Triangles are shapes in geometry with three vertices, three sides, and three interior angles.
• Interior angles are the angles that lie inside the triangle formed by the sides.
• Every triangle has three interior angles that must sum to exactly 180.
• There are many different types of triangles in the world of geometry, including:
  • Scalene triangles
  • Isosceles triangles
  • Equilateral triangles
  • Acute triangles
  • Obtuse triangles
  • Right triangles

Pythagorean Theorem

• The Pythagorean Theorem is an equation for finding the sides of a right triangle.
• The rule is based on the lengths of the legs, which are the sides forming the right angle, and the hypotenuse, which is the side opposite the right angle.
• The equation is typically written as a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.

Transformations in Geometry

• A transformation in math is a mapping of a preimage of a shape or function to an image of the same shape or function.
• There are two main categories of transformations: rigid transformations and non-rigid transformations.
• Rigid transformations do not change the size or shape of the preimage, while non-rigid transformations can change the size and shape of the preimage.

Types of Transformations

• Translation: a rigid transformation in which the location of the preimage is changed, but not its size, shape, or orientation.
• Rotation: a rigid transformation in which the location of the preimage is rotated around a fixed point, but its size and shape are not changed.
• Reflection: a rigid transformation in which the preimage is flipped across a line, but its size and shape are not changed.
• Dilation: a non-rigid transformation in which the shape and orientation of a preimage are not changed, but the image will be either larger or smaller than the original.

Reflection

• A reflection in mathematics is a type of geometrical transformation, where an object is flipped to create a mirror or congruent image.
• The bisector of the plane is known as the line of reflection, and it is perpendicular to the preimage and image.
• To find the reflection of an object, focus on the line of reflection:
  • When the line of reflection is the x-axis: (x, y) transforms into (x, -y).
  • When the line of reflection is the y-axis: (x, y) transforms into (-x, y).
  • When the line of reflection is the Y=X: (x, y) switch and transform to (y, x).

Rotation

• Rotation is one of four geometric transformations, which also include reflection, translation, and dilation.
• Rotation is turning an object clockwise (negative) or counterclockwise (positive) about a given point.
• There are general rules for clockwise and counterclockwise rotations of 90, 180, and 270 about the origin.
  • Counterclockwise:
    • 90 degree rotation: (x, y) ----> (-y, x)
    • 180 degree rotation: (x, y) ----> (-x, -y)
    • 270 degree rotation: (x, y) ----> (y, -x)
  • Clockwise:
    • -90 degree rotation: (x, y) ----> (y, -x)
    • -180 degree rotation: (x, y) ----> (-x, -y)
    • -270 degree rotation: (x, y) ----> (-y, x)

Dilation

• A dilation is the transformation that will change the size of a figure using a center of dilation to stretch away from or to shrink towards.
• A scale factor of dilation is the number used to multiply the pre-image, the original image, by to stretch or shrink according to the center of dilation.
• There are 3 ways a scale factor can affect the pre-image:
  • Multiply by a value greater than 1: the image will stretch away from the center of dilation.
  • Multiply by a value less than 1: the image will shrink towards the center of dilation.
  • Multiply by a value equal to 1: the image will stay the same with no stretching or shrinking.

Similarity and Congruence

• Similar figures are polygons with the same shape, angle sizes, and ratios of side lengths.
• Congruent shapes are shapes that have the same shape and the same size.
• Similar shapes have the same angles and proportional sides, but are different sizes.

Tessellations

• A tessellation is a space-filling arrangement of plane figures that do not overlap or leave gaps.
• There are two main types of tessellations: regular and semi-regular.
• Regular tessellations consist of only one repeated polygon.
• Semi-regular tessellations consist of two or more types of repeated regular polygons.

Triangles

• Triangles are shapes in geometry with three vertices, three sides, and three interior angles.
• Interior angles are the angles that lie inside the triangle formed by the sides.
• Every triangle has three interior angles that must sum to exactly 180.
• There are many different types of triangles in the world of geometry, including:
  • Scalene triangles
  • Isosceles triangles
  • Equilateral triangles
  • Acute triangles
  • Obtuse triangles
  • Right triangles

Pythagorean Theorem

• The Pythagorean Theorem is an equation for finding the sides of a right triangle.
• The rule is based on the lengths of the legs, which are the sides forming the right angle, and the hypotenuse, which is the side opposite the right angle.
• The equation is typically written as a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.