Mathematics
Grade 6
15 min
Reflection, rotation, and translation
Reflection, rotation, and translation
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1
Introduction & Learning Objectives
Learning Objectives
Identify and describe the three basic types of geometric transformations: reflection, rotation, and translation.
Perform a translation of a given 2D shape on a coordinate plane.
Perform a reflection of a given 2D shape across the x-axis or y-axis on a coordinate plane.
Perform a rotation of a given 2D shape by 90 or 180 degrees about the origin on a coordinate plane.
Distinguish between a pre-image and its image after a transformation.
Explain how transformations preserve the size and shape of figures.
Apply transformations to solve simple real-world problems.
Have you ever looked in a mirror and seen your reflection? Or watched a Ferris wheel spin around? 🎡 These are everyday examples of how shapes and objects can move without changing their size or f...
2
Key Concepts & Vocabulary
TermDefinitionExample
TransformationA transformation is a mathematical operation that moves or changes a geometric figure in some way to produce a new figure. The original figure is called the pre-image, and the new figure is called the image.If you slide a triangle across a page, the slide is a transformation. The original triangle is the pre-image, and the triangle in its new position is the image.
Pre-imageThe original figure or shape before a transformation is applied.If you have a square with vertices A, B, C, D, this square is the pre-image before you move it.
ImageThe new figure or shape that results after a transformation has been applied to the pre-image.After reflecting the square ABCD, the new square A'B'C'D' (read as A-prime, B-prime, C-prime, D-prime) is t...
3
Core Formulas
Translation Rule
$(x, y) \to (x+a, y+b)$
To translate a point, add 'a' to its x-coordinate for horizontal movement (right if 'a' is positive, left if negative) and 'b' to its y-coordinate for vertical movement (up if 'b' is positive, down if negative).
Reflection across X-axis Rule
$(x, y) \to (x, -y)$
To reflect a point across the x-axis, the x-coordinate stays the same, and the y-coordinate changes to its opposite sign.
Reflection across Y-axis Rule
$(x, y) \to (-x, y)$
To reflect a point across the y-axis, the y-coordinate stays the same, and the x-coordinate changes to its opposite sign.
180-degree Rotation Rule (about origin)
$(x, y) \to (-x, -y)$
To rotate a point 180 degrees about the origin (0,0), change both th...
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Challenging
Point K(3, 5) is first translated 2 units left and 4 units down. Then, the resulting point is reflected across the y-axis. What are the final coordinates?
A.(-1, 1)
B.(1, 1)
C.(-1, -1)
D.(1, -1)
Challenging
A point M at (-2, 1) is translated to become its image M' at (3, -1). Which rule describes this translation?
A.(x, y) → (x - 5, y - 2)
B.(x, y) → (x + 1, y - 2)
C.(x, y) → (x - 5, y + 2)
D.(x, y) → (x + 5, y - 2)
Challenging
A shape's vertex is at A(4, 2). After a single transformation, its image is at A'(-4, -2). What transformation must have occurred?
A.Reflection across the y-axis
B.Reflection across the x-axis
C.180-degree rotation about the origin
D.Translation 8 units left
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